helm/www/lambdadelta/xslt/sitemap.xsl
helm/www/lambdadelta/xslt/versions.xsl
helm/www/lambdadelta/xslt/core.xsl
+helm/www/lambdadelta/xslt/changes.xsl
helm/www/lambdadelta/xslt/chc_45.xsl
helm/www/lambdadelta/xslt/xhtbl.xsl
matita/matita/contribs/lambdadelta/2A
matita/matita/contribs/lambdadelta/*/*_probe.txt
matita/matita/contribs/lambdadelta/*/web/*_sum.tbl
+
+matita/matita/contribs/convergence/
--- /dev/null
+sed s?lambdadelta.info?www.cs.unibo.it/~fguidi/lambdadelta?g html/$1.html > html/$1.alt.html
--- /dev/null
+<?xml version="1.0" encoding="UTF-8"?>
+
+<page xmlns="http://lambdadelta.info/"
+ description = "\lambda\delta home page"
+ title = "\lambda\delta home page"
+ logo = "crux"
+ head = "The Formal Systems of the λδ (\lambda\delta) Family"
+>
+ <sitemap name="sitemap"/>
+
+ <section15 name="changes">Changes</section15>
+ <body>
+ </body>
+ <table name="changes"/>
+
+ <footer/>
+</page>
--- /dev/null
+name "changes"
+
+table {
+ class "gray"
+ [ "version" [ "aspect" [ "" "changes" ]]
+ ]
+
+ class "orange"
+ [ { "λδ-2B" + "(unreleased)" * }
+ {
+ [ [{ "equivalences" * }]
+ { "+" "+" "+" "-" }
+ { "equivalence for full rt-reduction (terms)"
+ "equivalence for whd rt-reduction (terms)"
+ "equivalence for extended rt-reduction (terms, referred lenvs, closures)"
+ "syntactic equivalence (closures) removed"
+ }
+ ]
+ [ [{ "weights" * }]
+ { "+" }
+ { "switch in primitive order relations for closures to enable the exclusion binder"
+ }
+ ]
+ [ [{ "relocation" * }]
+ { "" }
+ { ""
+ }
+ ]
+ [ [{ "syntax" * }]
+ { "+" }
+ { "exclusion binder for lenvs"
+ }
+ ]
+ [ [{ "ground" * }]
+ { "+" "*" "+" "-" }
+ { "rt-transition counters"
+ "generic reference transforming maps as streams of non-negative integers"
+ "extensional equality, labelled transitive closures and streams"
+ "non-negative integers with infinity removed"
+ }
+ ]
+ }
+ ]
+
+ class "orange"
+ [ { "λδ-2A" + "(October 2014)" * }
+ {
+ [ [{ "equivalences" * }]
+ { "+" }
+ { "syntactic equivalence (selected lenvs, referred lenvs, closures)"
+ }
+ ]
+ [ [{ "weights" * }]
+ { "*" "-" }
+ { "primitive order relations for closures"
+ "complex weight (terms) removed"
+ }
+ ]
+ [ [{ "relocation" * }]
+ { "-" }
+ { "level update functions removed"
+ }
+ ]
+ [ [{ "syntax" * }]
+ { "+" "+" "+" "-" "-" }
+ { "polarized binders for terms"
+ "non-negative integer global references for terms"
+ "syntactic support for genvs with typed abstraction, abbreviation"
+ "numbered sorts, application, type annotation removed from lenvs"
+ "exclusion binder removed from terms and lenvs"
+ }
+ ]
+ [ [{ "ground" * }]
+ { "+" "+" }
+ { "lists and non-negative integers with infinity"
+ "library extension for transitive closures and booleans"
+ }
+ ]
+ }
+ ]
+
+ class "red"
+ [ { "λδ-1A" + "(November 2006)" * }
+ {
+ [ [{ "equivalences" * }]
+ { "" }
+ { "equivalence for outer reduction (terms)"
+ }
+ ]
+ [ [{ "weights" * }]
+ { "" "" "" }
+ { "order relations (terms, lenvs, closures) based on weights"
+ "simple weights (terms, lenvs, closures)"
+ "complex weight (terms)"
+ }
+ ]
+ [ [{ "relocation" * }]
+ { "" }
+ { "level update functions"
+ }
+ ]
+ [ [{ "syntax" * }]
+ { "" "" }
+ { "lenvs with non-negative integer sorts, application, typed abstraction, abbreviation, exclusion, type annotation"
+ "terms with non-negative integer sorts and local references, application, typed abstraction, abbreviation, exclusion, type annotation" }
+ ]
+ [ [{ "ground" * }]
+ { "" "" }
+ { "finite reference transforming maps as compositions of basic ones"
+ "library extension for logic and non-negative integers"
+ }
+ ]
+ }
+ ]
+
+}
+
+class "center" { 2 }
-module StringSet = Set.Make (String)
+module StringSet = Set.Make (String)
type file = {
- ddeps: string list; (* direct dependences *)
- rdeps: StringSet.t option (* recursive dependences *)
+ ddeps: string list; (* direct dependences *)
+ rdeps: StringSet.t option (* recursive dependences *)
}
let graph = Hashtbl.create 503
let debug = ref 0
let rec purge dname vdeps = match vdeps with
- | [] -> vdeps
- | hd :: tl -> if hd = dname then tl else hd :: purge dname tl
+ | [] -> vdeps
+ | hd :: tl -> if hd = dname then tl else hd :: purge dname tl
let add fname =
- if fname = "" then () else
- if Hashtbl.mem graph fname then () else
- Hashtbl.add graph fname {ddeps = []; rdeps = None}
+ if fname = "" then () else
+ if Hashtbl.mem graph fname then () else
+ Hashtbl.add graph fname {ddeps = []; rdeps = None}
let add_ddep fname dname =
- if dname = "" then () else
- let file = Hashtbl.find graph fname in
- Hashtbl.replace graph fname {file with ddeps = dname :: file.ddeps}
+ if dname = "" then () else
+ let file = Hashtbl.find graph fname in
+ Hashtbl.replace graph fname {file with ddeps = dname :: file.ddeps}
let init fname dname =
- if !debug land 1 > 0 then Printf.eprintf "init: %s: %s.\n" fname dname;
- add fname; add dname; add_ddep fname dname
+ if !debug land 1 > 0 then Printf.eprintf "init: %s: %s.\n" fname dname;
+ add fname; add dname; add_ddep fname dname
(* vdeps: visited dependences for loop detection *)
let rec compute_from_file vdeps fname file = match file.rdeps with
- | Some rdeps -> rdeps
- | None ->
- if !debug land 2 > 0 then Printf.eprintf " compute file: %s\n" fname;
- let vdeps = fname :: vdeps in
- List.iter (redundant vdeps fname file.ddeps) file.ddeps;
- let rdeps = compute_from_ddeps vdeps file.ddeps in
- Hashtbl.replace graph fname {file with rdeps = Some rdeps};
- rdeps
+ | Some rdeps -> rdeps
+ | None ->
+ if !debug land 2 > 0 then Printf.eprintf " compute file: %s\n" fname;
+ let vdeps = fname :: vdeps in
+ List.iter (redundant vdeps fname file.ddeps) file.ddeps;
+ let rdeps = compute_from_ddeps vdeps file.ddeps in
+ Hashtbl.replace graph fname {file with rdeps = Some rdeps};
+ rdeps
and compute_from_dname vdeps rdeps dname =
- if List.mem dname vdeps then begin
- let loop = purge dname (List.rev vdeps) in
- Printf.printf "circular: %s\n" (String.concat " " loop);
- StringSet.add dname rdeps
- end else
- let file = Hashtbl.find graph dname in
- StringSet.add dname (StringSet.union (compute_from_file vdeps dname file) rdeps)
+ if List.mem dname vdeps then begin
+ let loop = purge dname (List.rev vdeps) in
+ Printf.printf "circular: %s\n" (String.concat " " loop);
+ StringSet.add dname rdeps
+ end else
+ let file = Hashtbl.find graph dname in
+ StringSet.add dname (StringSet.union (compute_from_file vdeps dname file) rdeps)
-and compute_from_ddeps vdeps ddeps =
- List.fold_left (compute_from_dname vdeps) StringSet.empty ddeps
+and compute_from_ddeps vdeps ddeps =
+ List.fold_left (compute_from_dname vdeps) StringSet.empty ddeps
and redundant vdeps fname ddeps dname =
- let rdeps = compute_from_ddeps vdeps (purge dname ddeps) in
- if StringSet.mem dname rdeps then
- Printf.printf "%s: redundant %s\n" fname dname
+ let rdeps = compute_from_ddeps vdeps (purge dname ddeps) in
+ if StringSet.mem dname rdeps then
+ Printf.printf "%s: redundant %s\n" fname dname
-let check () =
- let iter fname file = ignore (compute_from_file [] fname file) in
- Hashtbl.iter iter graph
+let check () =
+ let iter fname file = ignore (compute_from_file [] fname file) in
+ Hashtbl.iter iter graph
let get_unions () =
- let map1 ddeps dname = StringSet.add dname ddeps in
- let map2 fname file (fnames, ddeps) =
- StringSet.add fname fnames, List.fold_left map1 ddeps file.ddeps
- in
- Hashtbl.fold map2 graph (StringSet.empty, StringSet.empty)
+ let map1 ddeps dname = StringSet.add dname ddeps in
+ let map2 fname file (fnames, ddeps) =
+ StringSet.add fname fnames, List.fold_left map1 ddeps file.ddeps
+ in
+ Hashtbl.fold map2 graph (StringSet.empty, StringSet.empty)
let get_leafs () =
- let map fname file fnames =
- if file.ddeps = [] then StringSet.add fname fnames else fnames
- in
- Hashtbl.fold map graph StringSet.empty
+ let map fname file fnames =
+ if file.ddeps = [] then StringSet.add fname fnames else fnames
+ in
+ Hashtbl.fold map graph StringSet.empty
let top () =
- let iter fname = Printf.printf "top: %s\n" fname in
- let fnames, ddeps = get_unions () in
- StringSet.iter iter (StringSet.diff fnames ddeps)
+ let iter fname = Printf.printf "top: %s\n" fname in
+ let fnames, ddeps = get_unions () in
+ StringSet.iter iter (StringSet.diff fnames ddeps)
let leaf () =
- let iter fname = Printf.printf "leaf: %s\n" fname in
- let fnames = get_leafs () in
- StringSet.iter iter fnames
-
-let rec read ich =
- let line = input_line ich in
- begin try Scanf.sscanf line "%s@:include \"%s@\"." init
- with Scanf.Scan_failure _ ->
- begin try Scanf.sscanf line "./%s@:include \"%s@\"." init
- with Scanf.Scan_failure _ ->
- begin try Scanf.sscanf line "%s@:(*%s@*)" (fun _ _ -> ())
- with Scanf.Scan_failure _ ->
- Printf.eprintf "unknown line: %s.\n" line
- end
- end
- end;
- read ich
-
+ let iter fname = Printf.printf "leaf: %s\n" fname in
+ let fnames = get_leafs () in
+ StringSet.iter iter fnames
+
+let back fname =
+ Printf.printf "backward: %s\n" fname;
+ try match (Hashtbl.find graph fname).rdeps with
+ | None -> ()
+ | Some rdeps ->
+ let iter fname = Printf.printf "%s\n" fname in
+ StringSet.iter iter rdeps
+ with Not_found -> ()
+
+let rec read ich =
+ let line = input_line ich in
+ begin try Scanf.sscanf line "%s@:include \"%s@\"." init
+ with Scanf.Scan_failure _ ->
+ begin try Scanf.sscanf line "./%s@:include \"%s@\"." init
+ with Scanf.Scan_failure _ ->
+ begin try Scanf.sscanf line "%s@:(*%s@*)" (fun _ _ -> ())
+ with Scanf.Scan_failure _ ->
+ Printf.eprintf "unknown line: %s.\n" line
+ end
+ end
+ end;
+ read ich
+
let _ =
- let show_check = ref false in
- let show_top = ref false in
- let show_leaf = ref false in
- let process_file name = () in
- let show () =
- if !show_check then check ();
- if !show_top then top ();
- if !show_leaf then leaf ()
- in
- let help = "matitadep [-clt | -d <int> ] < <file>" in
- let help_c = " Print the redundant and looping arcs of the dependences graph" in
- let help_d = "<flags> Set these debug options" in
- let help_l = " Print the leaf nodes of the dependences graph" in
- let help_t = " Print the top nodes of the dependences graph" in
- Arg.parse [
- "-c", Arg.Set show_check, help_c;
- "-d", Arg.Int (fun x -> debug := x), help_d;
- "-l", Arg.Set show_leaf, help_l;
- "-t", Arg.Set show_top, help_t;
- ] process_file help;
- try read stdin with End_of_file -> show ()
+ let show_check = ref false in
+ let show_top = ref false in
+ let show_leaf = ref false in
+ let show_back = ref "" in
+ let process_file name = () in
+ let show () =
+ if !show_check then check ();
+ if !show_top then top ();
+ if !show_leaf then leaf ();
+ if !show_back <> "" then back !show_back
+ in
+ let help = "matitadep [-clt | -d <int> | -b <string> ] < <file>" in
+ let help_b = "<string> Print the backward dependences of this node" in
+ let help_c = " Print the redundant and looping arcs of the dependences graph" in
+ let help_d = "<flags> Set these debug options" in
+ let help_l = " Print the leaf nodes of the dependences graph" in
+ let help_t = " Print the top nodes of the dependences graph" in
+ Arg.parse [
+ "-b", Arg.String ((:=) show_back), help_b;
+ "-c", Arg.Set show_check, help_c;
+ "-d", Arg.Int ((:=) debug), help_d;
+ "-l", Arg.Set show_leaf, help_l;
+ "-t", Arg.Set show_top, help_t;
+ ] process_file help;
+ try read stdin with End_of_file -> show ()
EXEC = xoa
VERSION=0.2.0
-REQUIRES = helm-grafite
+REQUIRES = helm-registry
include ../Makefile.common
let separate = ref false
let clear () =
- incremental := true;
- separate := false;
- R.clear ()
+ incremental := true;
+ separate := false;
+ R.clear ()
let unm_ex s =
- Scanf.sscanf s "%u %u" A.mk_exists
+ Scanf.sscanf s "%u %u" A.mk_exists
let unm_or s =
- Scanf.sscanf s "%u" A.mk_or
+ Scanf.sscanf s "%u" A.mk_or
let unm_and s =
- Scanf.sscanf s "%u" A.mk_and
+ Scanf.sscanf s "%u" A.mk_and
let process_centralized conf =
- let preamble = L.get_preamble conf in
- if R.has "xoa.objects" && R.has "xoa.notations" then begin
- let ooch = L.open_out preamble (R.get_string "xoa.objects") in
- let noch = L.open_out preamble (R.get_string "xoa.notations") in
- List.iter (L.out_include ooch) (R.get_list R.string "xoa.include");
- L.out_include ooch (R.get_string "xoa.notations" ^ ".ma");
- List.iter (E.generate ooch noch) (R.get_list unm_ex "xoa.ex");
- List.iter (E.generate ooch noch) (R.get_list unm_or "xoa.or");
- List.iter (E.generate ooch noch) (R.get_list unm_and "xoa.and");
- close_out noch; close_out ooch
- end
+ let preamble = L.get_preamble conf in
+ if R.has "xoa.objects" && R.has "xoa.notations" then begin
+ let ooch = L.open_out preamble (R.get_string "xoa.objects") in
+ let noch = L.open_out preamble (R.get_string "xoa.notations") in
+ List.iter (L.out_include ooch) (R.get_list R.string "xoa.include");
+ L.out_include ooch (R.get_string "xoa.notations" ^ ".ma");
+ List.iter (E.generate ooch noch) (R.get_list unm_ex "xoa.ex");
+ List.iter (E.generate ooch noch) (R.get_list unm_or "xoa.or");
+ List.iter (E.generate ooch noch) (R.get_list unm_and "xoa.and");
+ close_out noch; close_out ooch
+ end
-let generate (p, o, n) = function
- | A.Exists (c, v) as d ->
+let generate (p, o, n) d =
+ let oname, nname = match d with
+ | A.Exists (c, v) ->
let oname = Printf.sprintf "%s/ex_%u_%u" o c v in
let nname = Printf.sprintf "%s/ex_%u_%u" n c v in
- if !incremental && L.exists_out oname && L.exists_out nname then () else
- begin
- let ooch = L.open_out p oname in
- let noch = L.open_out p nname in
- List.iter (L.out_include ooch) (R.get_list R.string "xoa.include");
- L.out_include ooch (nname ^ ".ma");
- E.generate ooch noch d;
- close_out noch; close_out ooch
- end
- | A.Or c -> ()
- | A.And c -> ()
+ oname, nname
+ | A.Or c ->
+ let oname = Printf.sprintf "%s/or_%u" o c in
+ let nname = Printf.sprintf "%s/or_%u" n c in
+ oname, nname
+ | A.And c ->
+ let oname = Printf.sprintf "%s/and_%u" o c in
+ let nname = Printf.sprintf "%s/and_%u" n c in
+ oname, nname
+ in
+ if !incremental && L.exists_out oname && L.exists_out nname then () else
+ begin
+ let ooch = L.open_out p oname in
+ let noch = L.open_out p nname in
+ List.iter (L.out_include ooch) (R.get_list R.string "xoa.include");
+ L.out_include ooch (nname ^ ".ma");
+ E.generate ooch noch d;
+ close_out noch; close_out ooch
+ end
let process_distributed conf =
- let preamble = L.get_preamble conf in
- if R.has "xoa.objects" && R.has "xoa.notations" then begin
- let st = preamble, R.get_string "xoa.objects", R.get_string "xoa.notations" in
- List.iter (generate st) (R.get_list unm_ex "xoa.ex");
- List.iter (generate st) (R.get_list unm_or "xoa.or");
- List.iter (generate st) (R.get_list unm_and "xoa.and");
- end
+ let preamble = L.get_preamble conf in
+ if R.has "xoa.objects" && R.has "xoa.notations" then begin
+ let st = preamble, R.get_string "xoa.objects", R.get_string "xoa.notations" in
+ List.iter (generate st) (R.get_list unm_ex "xoa.ex");
+ List.iter (generate st) (R.get_list unm_or "xoa.or");
+ List.iter (generate st) (R.get_list unm_and "xoa.and");
+ end
let process conf =
- if !separate then process_distributed conf else process_centralized conf
+ if !separate then process_distributed conf else process_centralized conf
let _ =
- let help = "Usage: xoa [ -Xs | <configuration file> ]*\n" in
- let help_X = " Clear configuration" in
- let help_s = " Generate separate objects" in
- let help_u = " Update existing separate files" in
- Arg.parse [
- "-X", Arg.Unit clear, help_X;
- "-s", Arg.Set separate, help_s;
- "-u", Arg.Clear incremental, help_u;
- ] process help
+ let help = "Usage: xoa [ -Xs | <configuration file> ]*\n" in
+ let help_X = " Clear configuration" in
+ let help_s = " Generate separate objects" in
+ let help_u = " Update existing separate files" in
+ Arg.parse [
+ "-X", Arg.Unit clear, help_X;
+ "-s", Arg.Set separate, help_s;
+ "-u", Arg.Clear incremental, help_u;
+ ] process help
type auto_params = nterm list option * (string*string) list
+type just = [`Term of nterm | `Auto of auto_params]
+
type ntactic =
| NApply of loc * nterm
| NSmartApply of loc * nterm
| NAssumption of loc
| NRepeat of loc * ntactic
| NBlock of loc * ntactic list
+ (* Declarative langauge *)
+ (* Not the best idea to use a string directly, an abstract type for identifiers would be better *)
+ | Assume of loc * string * nterm (* loc, identifier, type *)
+ | Suppose of loc * nterm * string (* loc, assumption, identifier *)
+ | By_just_we_proved of loc * just * nterm * string option (* loc, justification, conclusion, identifier *)
+ | We_need_to_prove of loc * nterm * string option (* loc, newconclusion, identifier *)
+ | BetaRewritingStep of loc * nterm
+ | Bydone of loc * just
+ | ExistsElim of loc * just * string * nterm * nterm * string
+ | AndElim of loc * just * nterm * string * nterm * string
+ | RewritingStep of
+ loc * nterm * [ `Term of nterm | `Auto of auto_params | `Proof | `SolveWith of nterm ] * bool (* last step*)
+ | Obtain of
+ loc * string * nterm
+ | Conclude of
+ loc * nterm
+ | Thesisbecomes of loc * nterm
+ | We_proceed_by_induction_on of loc * nterm * nterm
+ | We_proceed_by_cases_on of loc * nterm * nterm
+ | Byinduction of loc * nterm * string
+ | Case of loc * string * (string * nterm) list
+ (* This is a debug tactic to print the stack to stdout, can be safely removed *)
+ | PrintStack of loc
type nmacro =
| NCheck of loc * nterm
(* Copyright (C) 2004, HELM Team.
- *
+ *
* This file is part of HELM, an Hypertextual, Electronic
* Library of Mathematics, developed at the Computer Science
* Department, University of Bologna, Italy.
- *
+ *
* HELM is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public License
* as published by the Free Software Foundation; either version 2
* of the License, or (at your option) any later version.
- *
+ *
* HELM is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* along with HELM; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place - Suite 330, Boston,
* MA 02111-1307, USA.
- *
+ *
* For details, see the HELM World-Wide-Web page,
* http://helm.cs.unibo.it/
*)
let tactic_terminator = tactical_terminator
let command_terminator = tactical_terminator
-let pp_tactic_pattern status ~map_unicode_to_tex (what, hyp, goal) =
- if what = None && hyp = [] && goal = None then "" else
+let pp_tactic_pattern status ~map_unicode_to_tex (what, hyp, goal) =
+ if what = None && hyp = [] && goal = None then "" else
let what_text =
match what with
| None -> ""
in
Printf.sprintf "%sin %s%s" what_text hyp_text goal_text
+let pp_auto_params params status =
+ match params with
+ | (None,flags) -> String.concat " " (List.map (fun a,b -> a ^ "=" ^ b) flags)
+ | (Some l,flags) -> (String.concat "," (List.map (NotationPp.pp_term status) l)) ^
+ String.concat " " (List.map (fun a,b -> a ^ "=" ^ b) flags)
+;;
+
+let pp_just status just =
+ match just with
+ `Term term -> "using (" ^ NotationPp.pp_term status term ^ ") "
+ | `Auto params ->
+ match params with
+ | (None,[]) -> ""
+ | params -> "by " ^ pp_auto_params params status ^ " "
+;;
+
let rec pp_ntactic status ~map_unicode_to_tex =
let pp_tactic_pattern = pp_tactic_pattern ~map_unicode_to_tex in
function
| NPosbyname (_, s) -> s ^ ":"
| NWildcard _ -> "*:"
| NMerge _ -> "]"
- | NFocus (_,l) ->
- Printf.sprintf "focus %s"
+ | NFocus (_,l) ->
+ Printf.sprintf "focus %s"
(String.concat " " (List.map string_of_int l))
| NUnfocus _ -> "unfocus"
| NSkip _ -> "skip"
| NTry (_,tac) -> "ntry " ^ pp_ntactic status ~map_unicode_to_tex tac
| NAssumption _ -> "nassumption"
- | NBlock (_,l) ->
+ | NBlock (_,l) ->
"(" ^ String.concat " " (List.map (pp_ntactic status ~map_unicode_to_tex) l)^ ")"
| NRepeat (_,t) -> "nrepeat " ^ pp_ntactic status ~map_unicode_to_tex t
+ | Assume (_, ident, term) -> "assume " ^ ident ^ ":(" ^ (NotationPp.pp_term status term) ^ ")"
+ | Suppose (_,term,ident) -> "suppose (" ^ (NotationPp.pp_term status term) ^ ") (" ^ ident ^ ") "
+ | By_just_we_proved (_, just, term1, ident) -> pp_just status just ^ " we proved (" ^
+ (NotationPp.pp_term status term1) ^ ")" ^ (match ident with
+ None -> "" | Some ident -> "(" ^ident^ ")")
+ | We_need_to_prove (_,term,ident) -> "we need to prove (" ^ (NotationPp.pp_term status term) ^ ") " ^
+ (match ident with None -> "" | Some id -> "(" ^ id ^ ")")
+ | BetaRewritingStep (_,t) -> "that is equivalent to (" ^ (NotationPp.pp_term status t) ^ ")"
+ | Bydone (_, just) -> pp_just status just ^ "done"
+ | ExistsElim (_, just, ident, term, term1, ident1) -> pp_just status just ^ "let " ^ ident ^ ": ("
+ ^ (NotationPp.pp_term status term) ^ ") such that (" ^ (NotationPp.pp_term status term1) ^ ") (" ^ ident1 ^ ")"
+ | AndElim (_, just, term1, ident1, term2, ident2) -> pp_just status just ^ " we have (" ^
+ (NotationPp.pp_term status term1) ^ ") (" ^ ident1 ^ ") " ^ "and (" ^ (NotationPp.pp_term status
+ term2)
+ ^ ") (" ^ ident2 ^ ")"
+ | Thesisbecomes (_, t) -> "the thesis becomes (" ^ (NotationPp.pp_term status t) ^ ")"
+ | RewritingStep (_, rhs, just, cont) ->
+ "= (" ^
+ (NotationPp.pp_term status rhs) ^ ")" ^
+ (match just with
+ | `Auto params -> let s = pp_auto_params params status in
+ if s <> "" then " by " ^ s
+ else ""
+ | `Term t -> " exact (" ^ (NotationPp.pp_term status t) ^ ")"
+ | `Proof -> " proof"
+ | `SolveWith t -> " using (" ^ (NotationPp.pp_term status t) ^ ")"
+ )
+ ^ (if cont then " done" else "")
+ | Obtain (_,id,t1) -> "obtain (" ^ id ^ ")" ^ " (" ^ (NotationPp.pp_term status t1) ^ ")"
+ | Conclude (_,t1) -> "conclude (" ^ (NotationPp.pp_term status t1) ^ ")"
+ | We_proceed_by_cases_on (_, term, term1) -> "we proceed by cases on (" ^ NotationPp.pp_term
+ status term ^ ") to prove (" ^ NotationPp.pp_term status term1 ^ ")"
+ | We_proceed_by_induction_on (_, term, term1) -> "we proceed by induction on (" ^
+ NotationPp.pp_term status term ^ ") to prove (" ^ NotationPp.pp_term status term1 ^ ")"
+ | Byinduction (_, term, ident) -> "by induction hypothesis we know (" ^ NotationPp.pp_term status
+ term ^ ") (" ^ ident ^ ")"
+ | Case (_, id, args) ->
+ "case " ^ id ^
+ String.concat " "
+ (List.map (function (id,term) -> "(" ^ id ^ ": (" ^ NotationPp.pp_term status term ^ "))")
+ args)
+ | PrintStack _ -> "print_stack"
;;
let pp_nmacro status = function
desc
| Number_alias (instance,desc) ->
sprintf "alias num (instance %d) = \"%s\"." instance desc
-
+
let pp_associativity = function
| Gramext.LeftA -> "left associative"
| Gramext.RightA -> "right associative"
done;
sprintf "%s%s" (Buffer.contents eta_buf) name
-let pp_interpretation dsc symbol arg_patterns cic_appl_pattern =
+let pp_interpretation dsc symbol arg_patterns cic_appl_pattern =
sprintf "interpretation \"%s\" '%s %s = %s."
dsc symbol
(String.concat " " (List.map pp_argument_pattern arg_patterns))
(NotationPp.pp_cic_appl_pattern cic_appl_pattern)
-
+
let pp_dir_opt = function
| None -> ""
| Some `LeftToRight -> "> "
| Some `RightToLeft -> "< "
-let pp_notation status dir_opt l1_pattern assoc prec l2_pattern =
+let pp_notation status dir_opt l1_pattern assoc prec l2_pattern =
sprintf "notation %s\"%s\" %s %s for %s."
(pp_dir_opt dir_opt)
(pp_l1_pattern status l1_pattern)
(pp_l2_pattern status l2_pattern)
let pp_ncommand status = function
- | UnificationHint (_,t, n) ->
+ | UnificationHint (_,t, n) ->
"unification hint " ^ string_of_int n ^ " " ^ NotationPp.pp_term status t
| NDiscriminator (_,_)
| NInverter (_,_,_,_,_)
| NUnivConstraint (_) -> "not supported"
| NCoercion (_) -> "not supported"
- | NObj (_,obj,index) ->
- (if not index then "-" else "") ^
+ | NObj (_,obj,index) ->
+ (if not index then "-" else "") ^
NotationPp.pp_obj (NotationPp.pp_term status) obj
| NQed (_,true) -> "qed"
| NQed (_,false) -> "qed-"
- | NCopy (_,name,uri,map) ->
- "copy " ^ name ^ " from " ^ NUri.string_of_uri uri ^ " with " ^
- String.concat " and "
- (List.map
- (fun (a,b) -> NUri.string_of_uri a ^ " ↦ " ^ NUri.string_of_uri b)
+ | NCopy (_,name,uri,map) ->
+ "copy " ^ name ^ " from " ^ NUri.string_of_uri uri ^ " with " ^
+ String.concat " and "
+ (List.map
+ (fun (a,b) -> NUri.string_of_uri a ^ " ↦ " ^ NUri.string_of_uri b)
map)
| Include (_,mode,path) -> (* not precise, since path is absolute *)
if mode = WithPreferences then
| Notation (_, dir_opt, l1_pattern, assoc, prec, l2_pattern) ->
pp_notation status dir_opt l1_pattern assoc prec l2_pattern
;;
-
+
let pp_executable status ~map_unicode_to_tex =
function
| NMacro (_, macro) -> pp_nmacro status macro ^ "."
| NTactic (_,tacl) ->
String.concat " " (List.map (pp_ntactic status ~map_unicode_to_tex) tacl)
| NCommand (_, cmd) -> pp_ncommand status cmd ^ "."
-
+
let pp_comment status ~map_unicode_to_tex =
function
| Note (_,"") -> Printf.sprintf "\n"
let pp_statement status =
function
- | Executable (_, ex) -> pp_executable status ex
+ | Executable (_, ex) -> pp_executable status ex
| Comment (_, c) -> pp_comment status c
;;
let eval_ng_tac tac =
+ let just_to_tacstatus_just just text prefix_len =
+ match just with
+ | `Term t -> `Term (text,prefix_len,t)
+ | `Auto (l,params) ->
+ (
+ match l with
+ | None -> `Auto (None,params)
+ | Some l -> `Auto (Some (List.map (fun t -> (text,prefix_len,t)) l),params)
+ )
+ | _ -> assert false
+ in
let rec aux f (text, prefix_len, tac) =
match tac with
| GrafiteAst.NApply (_loc, t) -> NTactics.apply_tac (text,prefix_len,t)
NTactics.block_tac (List.map (fun x -> aux f (text,prefix_len,x)) l)
|GrafiteAst.NRepeat (_,tac) ->
NTactics.repeat_tac (f f (text, prefix_len, tac))
+ |GrafiteAst.Assume (_,id,t) -> Declarative.assume id (text,prefix_len,t)
+ |GrafiteAst.Suppose (_,t,id) -> Declarative.suppose (text,prefix_len,t) id
+ |GrafiteAst.By_just_we_proved (_,j,t1,s) -> Declarative.by_just_we_proved
+ (just_to_tacstatus_just j text prefix_len) (text,prefix_len,t1) s
+ |GrafiteAst.We_need_to_prove (_, t, id) -> Declarative.we_need_to_prove (text,prefix_len,t) id
+ |GrafiteAst.BetaRewritingStep (_, t) -> Declarative.beta_rewriting_step (text,prefix_len,t)
+ | GrafiteAst.Bydone (_, j) -> Declarative.bydone (just_to_tacstatus_just j text prefix_len)
+ | GrafiteAst.ExistsElim (_, just, id1, t1, t2, id2) ->
+ Declarative.existselim (just_to_tacstatus_just just text prefix_len) id1 (text,prefix_len,t1)
+ (text,prefix_len,t2) id2
+ | GrafiteAst.AndElim(_,just,t1,id1,t2,id2) -> Declarative.andelim (just_to_tacstatus_just just
+ text prefix_len) (text,prefix_len,t1) id1 (text,prefix_len,t2) id2
+ | GrafiteAst.Thesisbecomes (_, t1) -> Declarative.thesisbecomes (text,prefix_len,t1)
+ | GrafiteAst.RewritingStep (_,rhs,just,cont) ->
+ Declarative.rewritingstep (text,prefix_len,rhs)
+ (match just with
+ `Term _
+ | `Auto _ -> just_to_tacstatus_just just text prefix_len
+ |`Proof -> `Proof
+ |`SolveWith t -> `SolveWith (text,prefix_len,t)
+ )
+ cont
+ | GrafiteAst.Obtain (_,id,t1) ->
+ Declarative.obtain id (text,prefix_len,t1)
+ | GrafiteAst.Conclude (_,t1) ->
+ Declarative.conclude (text,prefix_len,t1)
+ | GrafiteAst.We_proceed_by_cases_on (_, t, t1) ->
+ Declarative.we_proceed_by_cases_on (text,prefix_len,t) (text,prefix_len,t1)
+ | GrafiteAst.We_proceed_by_induction_on (_, t, t1) ->
+ Declarative.we_proceed_by_induction_on (text,prefix_len,t) (text,prefix_len,t1)
+ | GrafiteAst.Byinduction (_, t, id) -> Declarative.byinduction (text,prefix_len,t) id
+ | GrafiteAst.Case (_,id,params) -> Declarative.case id params
+ | GrafiteAst.PrintStack (_) -> Declarative.print_stack
in
aux aux tac (* trick for non uniform recursion call *)
;;
type by_continuation =
BYC_done
- | BYC_weproved of N.term * string option * N.term option
- | BYC_letsuchthat of string * N.term * string * N.term
+ | BYC_weproved of N.term * string option
+ | BYC_letsuchthat of string * N.term * N.term * string
| BYC_wehaveand of string * N.term * string * N.term
let mk_parser statement lstatus =
| SYMBOL "#"; SYMBOL "_" -> G.NTactic(loc,[ G.NIntro (loc,"_")])
| SYMBOL "*" -> G.NTactic(loc,[ G.NCase1 (loc,"_")])
| SYMBOL "*"; "as"; n=IDENT -> G.NTactic(loc,[ G.NCase1 (loc,n)])
+ | IDENT "assume" ; id = IDENT; SYMBOL ":"; t = tactic_term -> G.NTactic (loc,[G.Assume (loc,id,t)])
+ | IDENT "suppose" ; t = tactic_term ; LPAREN ; id = IDENT ; RPAREN -> G.NTactic (loc,[G.Suppose (loc,t,id)])
+ | "let"; name = IDENT ; SYMBOL <:unicode<def>> ; t = tactic_term ->
+ G.NTactic(loc,[G.NLetIn (loc,(None,[],Some N.UserInput),t,name)])
+ | just =
+ [ IDENT "using"; t=tactic_term -> `Term t
+ | params = auto_params ->
+ let just,params = params in
+ `Auto
+ (match just with
+ | None -> (None,params)
+ | Some (`Univ univ) -> (Some univ,params)
+ (* `Trace behaves exaclty like None for the moment being *)
+ | Some (`Trace) -> (None,params)
+ )
+ ];
+ cont=by_continuation -> G.NTactic (loc,[
+ (match cont with
+ BYC_done -> G.Bydone (loc, just)
+ | BYC_weproved (ty,id) ->
+ G.By_just_we_proved(loc, just, ty, id)
+ | BYC_letsuchthat (id1,t1,t2,id2) ->
+ G.ExistsElim (loc, just, id1, t1, t2, id2)
+ | BYC_wehaveand (id1,t1,id2,t2) ->
+ G.AndElim (loc, just, t1, id1, t2, id2))
+ ])
+ | IDENT "we" ; IDENT "need" ; "to" ; IDENT "prove" ; t = tactic_term ; id = OPT [ LPAREN ; id = IDENT ; RPAREN -> id ] ->
+ G.NTactic (loc,[G.We_need_to_prove (loc, t, id)])
+ | IDENT "that" ; IDENT "is" ; IDENT "equivalent" ; "to" ; t = tactic_term -> G.NTactic(loc,[G.BetaRewritingStep (loc,t)])
+ | IDENT "the" ; IDENT "thesis" ; IDENT "becomes" ; t1=tactic_term -> G.NTactic (loc,[G.Thesisbecomes(loc,t1)])
+ | IDENT "we" ; IDENT "proceed" ; IDENT "by" ; IDENT "cases" ; "on" ; t=tactic_term ; "to" ; IDENT "prove" ; t1=tactic_term ->
+ G.NTactic (loc,[G.We_proceed_by_cases_on (loc, t, t1)])
+ | IDENT "we" ; IDENT "proceed" ; IDENT "by" ; IDENT "induction" ; "on" ; t=tactic_term ; "to" ; IDENT "prove" ; t1=tactic_term ->
+ G.NTactic (loc,[G.We_proceed_by_induction_on (loc, t, t1)])
+ | IDENT "by" ; IDENT "induction" ; IDENT "hypothesis" ; IDENT "we" ; IDENT "know" ; t=tactic_term ; LPAREN ; id = IDENT ; RPAREN ->
+ G.NTactic (loc,[G.Byinduction(loc, t, id)])
+ | IDENT "case" ; id = IDENT ; params=LIST0[LPAREN ; i=IDENT ;
+ SYMBOL":" ; t=tactic_term ; RPAREN -> i,t] ->
+ G.NTactic (loc,[G.Case(loc,id,params)])
+ | IDENT "print_stack" -> G.NTactic (loc,[G.PrintStack loc])
+ (* DO NOT FACTORIZE with the two following, camlp5 sucks*)
+(*
+ | IDENT "conclude";
+ termine = tactic_term;
+ SYMBOL "=" ;
+ t1=tactic_term ;
+ t2 =
+ [ IDENT "using"; t=tactic_term -> `Term t
+ | IDENT "using"; IDENT "once"; term=tactic_term -> `SolveWith term
+ | IDENT "proof" -> `Proof
+ | params = auto_params -> `Auto
+ (
+ let just,params = params in
+ match just with
+ | None -> (None,params)
+ | Some (`Univ univ) -> (Some univ,params)
+ (* `Trace behaves exaclty like None for the moment being *)
+ | Some (`Trace) -> (None,params)
+ )
+ ];
+ cont = rewriting_step_continuation ->
+ G.NTactic (loc,[G.RewritingStep(loc, Some (None,termine), t1, t2, cont)])
+ | IDENT "obtain" ; name = IDENT;
+ termine = tactic_term;
+ SYMBOL "=" ;
+ t1=tactic_term ;
+ t2 =
+ [ IDENT "using"; t=tactic_term -> `Term t
+ | IDENT "using"; IDENT "once"; term=tactic_term -> `SolveWith term
+ | IDENT "proof" -> `Proof
+ | params = auto_params -> `Auto
+ (
+ let just,params = params in
+ match just with
+ | None -> (None,params)
+ | Some (`Univ univ) -> (Some univ,params)
+ (* `Trace behaves exaclty like None for the moment being *)
+ | Some (`Trace) -> (None,params)
+ )
+ ];
+ cont = rewriting_step_continuation ->
+ G.NTactic(loc,[G.RewritingStep(loc, Some (Some name,termine), t1, t2, cont)])
+*)
+ | IDENT "obtain" ; name = IDENT;
+ termine = tactic_term ->
+ G.NTactic(loc,[G.Obtain(loc, name, termine)])
+ | IDENT "conclude" ; termine = tactic_term ->
+ G.NTactic(loc,[G.Conclude(loc, termine)])
+ | SYMBOL "=" ;
+ t1=tactic_term ;
+ t2 =
+ [ IDENT "using"; t=tactic_term -> `Term t
+ | IDENT "using"; IDENT "once"; term=tactic_term -> `SolveWith term
+ | IDENT "proof" -> `Proof
+ | params = auto_params -> `Auto
+ (
+ let just,params = params in
+ match just with
+ | None -> (None,params)
+ | Some (`Univ univ) -> (Some univ,params)
+ (* `Trace behaves exaclty like None for the moment being *)
+ | Some (`Trace) -> (None,params)
+ )
+ ];
+ cont = rewriting_step_continuation ->
+ G.NTactic(loc,[G.RewritingStep(loc, t1, t2, cont)])
]
];
auto_fixed_param: [
]
];
-(* MATITA 1.0
by_continuation: [
- [ WEPROVED; ty = tactic_term ; LPAREN ; id = IDENT ; RPAREN ; t1 = OPT [IDENT "that" ; IDENT "is" ; IDENT "equivalent" ; "to" ; t2 = tactic_term -> t2] -> BYC_weproved (ty,Some id,t1)
- | WEPROVED; ty = tactic_term ; t1 = OPT [IDENT "that" ; IDENT "is" ; IDENT "equivalent" ; "to" ; t2 = tactic_term -> t2] ;
- "done" -> BYC_weproved (ty,None,t1)
+ [ WEPROVED; ty = tactic_term ; id = OPT [ LPAREN ; id = IDENT ; RPAREN -> id] -> BYC_weproved (ty,id)
| "done" -> BYC_done
| "let" ; id1 = IDENT ; SYMBOL ":" ; t1 = tactic_term ;
IDENT "such" ; IDENT "that" ; t2=tactic_term ; LPAREN ;
- id2 = IDENT ; RPAREN -> BYC_letsuchthat (id1,t1,id2,t2)
+ id2 = IDENT ; RPAREN -> BYC_letsuchthat (id1,t1,t2,id2)
| WEHAVE; t1=tactic_term ; LPAREN ; id1=IDENT ; RPAREN ;"and" ; t2=tactic_term ; LPAREN ; id2=IDENT ; RPAREN ->
BYC_wehaveand (id1,t1,id2,t2)
]
];
-*)
-(* MATITA 1.0
+
rewriting_step_continuation : [
[ "done" -> true
| -> false
]
];
-*)
+
(* MATITA 1.0
atomic_tactical:
[ "sequence" LEFTA
let n = List.length i in
let cand = lift n cand in
let cand = pop n (non_stricts add cand t) in
- List.merge (compare) cand c) [] v
+ List.merge (-) cand c) [] v
(* [merge] may duplicates some indices, but I don't mind. *)
| MLmagic t ->
non_stricts add cand t
continuationals.cmo : continuationals.cmi
continuationals.cmx : continuationals.cmi
continuationals.cmi :
+declarative.cmo : nnAuto.cmi nTactics.cmi nTacStatus.cmi nCicElim.cmi \
+ continuationals.cmi declarative.cmi
+declarative.cmx : nnAuto.cmx nTactics.cmx nTacStatus.cmx nCicElim.cmx \
+ continuationals.cmx declarative.cmi
+declarative.cmi : nnAuto.cmi nTacStatus.cmi
nCicElim.cmo : nCicElim.cmi
nCicElim.cmx : nCicElim.cmi
nCicElim.cmi :
continuationals.cmx : continuationals.cmi
continuationals.cmi :
+declarative.cmx : nnAuto.cmx nTactics.cmx nTacStatus.cmx nCicElim.cmx \
+ continuationals.cmx declarative.cmi
+declarative.cmi : nnAuto.cmi nTacStatus.cmi
nCicElim.cmx : nCicElim.cmi
nCicElim.cmi :
nCicTacReduction.cmx : nCicTacReduction.cmi
nCicElim.mli \
nTactics.mli \
nnAuto.mli \
+ declarative.mli \
nDestructTac.mli \
nInversion.mli
type goal = int
+type parameters = (string * string) list
+
module Stack =
struct
type switch = Open of goal | Closed of goal
type locator = int * switch
type tag = [ `BranchTag | `FocusTag | `NoTag ]
- type entry = locator list * locator list * locator list * tag
+ type entry = locator list * locator list * locator list * tag * parameters
type t = entry list
- let empty = [ [], [], [], `NoTag ]
+ let empty = [ [], [], [], `NoTag , []]
let fold ~env ~cont ~todo init stack =
let rec aux acc depth =
function
| [] -> acc
- | (locs, todos, conts, tag) :: tl ->
+ | (locs, todos, conts, tag, _p) :: tl ->
let acc = List.fold_left (fun acc -> env acc depth tag) acc locs in
let acc = List.fold_left (fun acc -> cont acc depth tag) acc conts in
let acc = List.fold_left (fun acc -> todo acc depth tag) acc todos in
let map ~env ~cont ~todo =
let depth = ref ~-1 in
List.map
- (fun (s, t, c, tag) ->
+ (fun (s, t, c, tag, p) ->
incr depth;
let d = !depth in
- env d tag s, todo d tag t, cont d tag c, tag)
+ env d tag s, todo d tag t, cont d tag c, tag, p)
let is_open = function _, Open _ -> true | _ -> false
let close = function n, Open g -> n, Closed g | l -> l
let rec find_goal =
function
| [] -> raise (Failure "Continuationals.find_goal")
- | (l :: _, _ , _ , _) :: _ -> goal_of_loc l
- | ( _ , _ , l :: _, _) :: _ -> goal_of_loc l
- | ( _ , l :: _, _ , _) :: _ -> goal_of_loc l
+ | (l :: _, _ , _ , _, _) :: _ -> goal_of_loc l
+ | ( _ , _ , l :: _, _, _) :: _ -> goal_of_loc l
+ | ( _ , l :: _, _ , _, _) :: _ -> goal_of_loc l
| _ :: tl -> find_goal tl
let is_empty =
function
| [] -> assert false
- | [ [], [], [], `NoTag ] -> true
+ | [ [], [], [], `NoTag , _] -> true
| _ -> false
let of_nmetasenv metasenv =
let goals = List.map (fun (g, _) -> g) metasenv in
- [ zero_pos goals, [], [], `NoTag ]
+ [ zero_pos goals, [], [], `NoTag , []]
let head_switches =
function
- | (locs, _, _, _) :: _ -> List.map switch_of_loc locs
+ | (locs, _, _, _, _) :: _ -> List.map switch_of_loc locs
| [] -> assert false
let head_goals =
function
- | (locs, _, _, _) :: _ -> List.map goal_of_loc locs
+ | (locs, _, _, _, _) :: _ -> List.map goal_of_loc locs
| [] -> assert false
let head_tag =
function
- | (_, _, _, tag) :: _ -> tag
+ | (_, _, _, tag, _) :: _ -> tag
| [] -> assert false
let shift_goals =
function
- | _ :: (locs, _, _, _) :: _ -> List.map goal_of_loc locs
+ | _ :: (locs, _, _, _, _) :: _ -> List.map goal_of_loc locs
| [] -> assert false
| _ -> []
let pp_loc (i, s) = string_of_int i ^ pp_switch s in
let pp_env env = sprintf "[%s]" (String.concat ";" (List.map pp_loc env)) in
let pp_tag = function `BranchTag -> "B" | `FocusTag -> "F" | `NoTag -> "N" in
- let pp_stack_entry (env, todo, cont, tag) =
- sprintf "(%s, %s, %s, %s)" (pp_env env) (pp_env todo) (pp_env cont)
- (pp_tag tag)
+ let pp_par = function [] -> "" | _ as l -> List.fold_left (fun acc (k,v) -> acc ^ "K: " ^ k ^ " V: " ^ v ^ "; ") "" l in
+ let pp_stack_entry (env, todo, cont, tag, parameters) =
+ sprintf "(%s, %s, %s, %s, %s)" (pp_env env) (pp_env todo) (pp_env cont)
+ (pp_tag tag) (pp_par parameters)
in
String.concat " :: " (List.map pp_stack_entry stack)
end
let ostatus, stack =
match cmd, stack with
| _, [] -> assert false
- | Tactical tac, (g, t, k, tag) :: s ->
+ | Tactical tac, (g, t, k, tag, p) :: s ->
(* COMMENTED OUT TO ALLOW PARAMODULATION TO DO A
* auto paramodulation.try assumption.
* EVEN IF NO GOALS ARE LEFT OPEN BY AUTO.
debug_print (lazy ("closed: "
^ String.concat " " (List.map string_of_int gcn)));
let stack =
- (zero_pos gon, t @~- gcn, k @~- gcn, tag) :: deep_close gcn s
+ (zero_pos gon, t @~- gcn, k @~- gcn, tag, p) :: deep_close gcn s
in
sn, stack
- | Dot, ([], _, [], _) :: _ ->
+ | Dot, ([], _, [], _, _) :: _ ->
(* backward compatibility: do-nothing-dot *)
new_stack stack
- | Dot, (g, t, k, tag) :: s ->
+ | Dot, (g, t, k, tag, p) :: s ->
(match filter_open g, k with
- | loc :: loc_tl, _ -> new_stack (([ loc ], t, loc_tl @+ k, tag) :: s)
+ | loc :: loc_tl, _ -> new_stack (([ loc ], t, loc_tl @+ k, tag, p) :: s)
| [], loc :: k ->
assert (is_open loc);
- new_stack (([ loc ], t, k, tag) :: s)
+ new_stack (([ loc ], t, k, tag, p) :: s)
| _ -> fail (lazy "can't use \".\" here"))
| Semicolon, _ -> new_stack stack
- | Branch, (g, t, k, tag) :: s ->
+ | Branch, (g, t, k, tag, p) :: s ->
(match init_pos g with
| [] | [ _ ] -> fail (lazy "too few goals to branch");
| loc :: loc_tl ->
new_stack
- (([ loc ], [], [], `BranchTag) :: (loc_tl, t, k, tag) :: s))
- | Shift, (g, t, k, `BranchTag) :: (g', t', k', tag) :: s ->
+ (([ loc ], [], [], `BranchTag, []) :: (loc_tl, t, k, tag,p) :: s))
+ | Shift, (g, t, k, `BranchTag, p) :: (g', t', k', tag, p') :: s ->
(match g' with
| [] -> fail (lazy "no more goals to shift")
| loc :: loc_tl ->
new_stack
- (([ loc ], t @+ filter_open g @+ k, [],`BranchTag)
- :: (loc_tl, t', k', tag) :: s))
+ (([ loc ], t @+ filter_open g @+ k, [],`BranchTag, p)
+ :: (loc_tl, t', k', tag, p') :: s))
| Shift, _ -> fail (lazy "can't shift goals here")
- | Pos i_s, ([ loc ], t, [],`BranchTag) :: (g', t', k', tag) :: s
+ | Pos i_s, ([ loc ], t, [],`BranchTag, p) :: (g', t', k', tag, p') :: s
when is_fresh loc ->
let l_js = List.filter (fun (i, _) -> List.mem i i_s) ([loc] @+ g') in
new_stack
- ((l_js, t , [],`BranchTag)
- :: (([ loc ] @+ g') @- l_js, t', k', tag) :: s)
+ ((l_js, t , [],`BranchTag, p)
+ :: (([ loc ] @+ g') @- l_js, t', k', tag, p') :: s)
| Pos _, _ -> fail (lazy "can't use relative positioning here")
- | Wildcard, ([ loc ] , t, [], `BranchTag) :: (g', t', k', tag) :: s
+ | Wildcard, ([ loc ] , t, [], `BranchTag, p) :: (g', t', k', tag, p') :: s
when is_fresh loc ->
new_stack
- (([loc] @+ g', t, [], `BranchTag)
- :: ([], t', k', tag) :: s)
+ (([loc] @+ g', t, [], `BranchTag, p)
+ :: ([], t', k', tag, p') :: s)
| Wildcard, _ -> fail (lazy "can't use wildcard here")
- | Merge, (g, t, k,`BranchTag) :: (g', t', k', tag) :: s ->
- new_stack ((t @+ filter_open g @+ g' @+ k, t', k', tag) :: s)
+ | Merge, (g, t, k,`BranchTag,_) :: (g', t', k', tag,p') :: s ->
+ new_stack ((t @+ filter_open g @+ g' @+ k, t', k', tag, p') :: s)
| Merge, _ -> fail (lazy "can't merge goals here")
| Focus [], _ -> assert false
| Focus gs, s ->
if not (List.exists (fun l -> goal_of_loc l = g) stack_locs) then
fail (lazy (sprintf "goal %d not found (or closed)" g)))
gs;
- new_stack ((zero_pos gs, [], [], `FocusTag) :: deep_close gs s)
- | Unfocus, ([], [], [], `FocusTag) :: s -> new_stack s
+ new_stack ((zero_pos gs, [], [], `FocusTag, []) :: deep_close gs s)
+ | Unfocus, ([], [], [], `FocusTag, _) :: s -> new_stack s
| Unfocus, _ -> fail (lazy "can't unfocus, some goals are still open")
in
debug_print (lazy (sprintf "EVAL CONT %s -> %s" (pp_t cmd) (pp stack)));
(** {2 Goal stack} *)
+(* Key value pairs *)
+type parameters = (string * string) list
+
module Stack:
sig
type switch = Open of goal | Closed of goal
type locator = int * switch
type tag = [ `BranchTag | `FocusTag | `NoTag ]
- type entry = locator list * locator list * locator list * tag
+ type entry = locator list * locator list * locator list * tag * parameters
type t = entry list
val empty: t
--- /dev/null
+(* Copyright (C) 2019, HELM Team.
+ *
+ * This file is part of HELM, an Hypertextual, Electronic
+ * Library of Mathematics, developed at the Computer Science
+ * Department, University of Bologna, Italy.
+ *
+ * HELM is free software; you can redistribute it and/or
+ * modify it under the terms of the GNU General Public License
+ * as published by the Free Software Foundation; either version 2
+ * of the License, or (at your option) any later version.
+ *
+ * HELM is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with HELM; if not, write to the Free Software
+ * Foundation, Inc., 59 Temple Place - Suite 330, Boston,
+ * MA 02111-1307, USA.
+ *
+ * For details, see the HELM World-Wide-Web page,
+ * http://cs.unibo.it/helm/.
+*)
+
+open Continuationals.Stack
+module Ast = NotationPt
+open NTactics
+open NTacStatus
+
+type just = [ `Term of NTacStatus.tactic_term | `Auto of NnAuto.auto_params ]
+
+let mk_just status goal =
+ function
+ `Auto (l,params) -> NnAuto.auto_lowtac ~params:(l,params) status goal
+ | `Term t -> apply_tac t
+
+exception NotAProduct
+exception FirstTypeWrong
+exception NotEquivalentTypes
+
+let extract_first_goal_from_status status =
+ let s = status#stack in
+ match s with
+ | [] -> fail (lazy "There's nothing to prove")
+ | (g1, _, _k, _tag1, _) :: _tl ->
+ let goals = filter_open g1 in
+ match goals with
+ [] -> fail (lazy "No goals under focus")
+ | loc::_tl ->
+ let goal = goal_of_loc (loc) in
+ goal ;;
+
+let extract_conclusion_type status goal =
+ let gty = get_goalty status goal in
+ let ctx = ctx_of gty in
+ term_of_cic_term status gty ctx
+;;
+
+let alpha_eq_tacterm_kerterm ty t status goal =
+ let gty = get_goalty status goal in
+ let ctx = ctx_of gty in
+ let status,cicterm = disambiguate status ctx ty `XTNone (*(`XTSome (mk_cic_term ctx t))*) in
+ let (_,_,metasenv,subst,_) = status#obj in
+ let status,ty = term_of_cic_term status cicterm ctx in
+ if NCicReduction.alpha_eq status metasenv subst ctx t ty then
+ true
+ else
+ false
+;;
+
+let are_convertible ty1 ty2 status goal =
+ let gty = get_goalty status goal in
+ let ctx = ctx_of gty in
+ let status,cicterm1 = disambiguate status ctx ty1 `XTNone in
+ let status,cicterm2 = disambiguate status ctx ty2 `XTNone in
+ NTacStatus.are_convertible status ctx cicterm1 cicterm2
+
+let clear_volatile_params_tac status =
+ match status#stack with
+ [] -> fail (lazy "Empty stack")
+ | (g,t,k,tag,p)::tl ->
+ let rec remove_volatile = function
+ [] -> []
+ | (k,_v as hd')::tl' ->
+ let re = Str.regexp "volatile_.*" in
+ if Str.string_match re k 0 then
+ remove_volatile tl'
+ else
+ hd'::(remove_volatile tl')
+ in
+ let newp = remove_volatile p in
+ status#set_stack ((g,t,k,tag,newp)::tl)
+;;
+
+let add_parameter_tac key value status =
+ match status#stack with
+ [] -> status
+ | (g,t,k,tag,p) :: tl -> status#set_stack ((g,t,k,tag,(key,value)::p)::tl)
+;;
+
+
+(* LCF-like tactic that checks whether the conclusion of the sequent of the given goal is a product, checks that
+ the type of the conclusion's bound variable is the same as t1 and then uses an exact_tac with
+ \lambda id: t1. ?. If a t2 is given it checks that t1 ~_{\beta} t2 and uses and exact_tac with \lambda id: t2. ?
+*)
+let lambda_abstract_tac id t1 status goal =
+ match extract_conclusion_type status goal with
+ | status,NCic.Prod (_,t,_) ->
+ if alpha_eq_tacterm_kerterm t1 t status goal then
+ let (_,_,t1) = t1 in
+ block_tac [exact_tac ("",0,(Ast.Binder (`Lambda,(Ast.Ident (id,None),Some t1),Ast.Implicit
+ `JustOne))); clear_volatile_params_tac;
+ add_parameter_tac "volatile_newhypo" id] status
+ else
+ raise FirstTypeWrong
+ | _ -> raise NotAProduct
+
+let assume name ty status =
+ let goal = extract_first_goal_from_status status in
+ try lambda_abstract_tac name ty status goal
+ with
+ | NotAProduct -> fail (lazy "You can't assume without an universal quantification")
+ | FirstTypeWrong -> fail (lazy "The assumed type is wrong")
+ | NotEquivalentTypes -> fail (lazy "The two given types are not equivalent")
+;;
+
+let suppose t1 id status =
+ let goal = extract_first_goal_from_status status in
+ try lambda_abstract_tac id t1 status goal
+ with
+ | NotAProduct -> fail (lazy "You can't suppose without a logical implication")
+ | FirstTypeWrong -> fail (lazy "The supposed proposition is different from the premise")
+ | NotEquivalentTypes -> fail (lazy "The two given propositions are not equivalent")
+;;
+
+let assert_tac t1 t2 status goal continuation =
+ let status,t = extract_conclusion_type status goal in
+ if alpha_eq_tacterm_kerterm t1 t status goal then
+ match t2 with
+ | None -> continuation
+ | Some t2 ->
+ let _status,res = are_convertible t1 t2 status goal in
+ if res then continuation
+ else
+ raise NotEquivalentTypes
+ else
+ raise FirstTypeWrong
+
+let branch_dot_tac status =
+ match status#stack with
+ ([],t,k,tag,p) :: tl ->
+ if List.length t > 0 then
+ status#set_stack (([List.hd t],List.tl t,k,tag,p)::tl)
+ else
+ status
+ | _ -> status
+;;
+
+let status_parameter key status =
+ match status#stack with
+ [] -> ""
+ | (_g,_t,_k,_tag,p)::_ -> try List.assoc key p with _ -> ""
+;;
+
+let beta_rewriting_step t status =
+ let ctx = status_parameter "volatile_context" status in
+ if ctx <> "beta_rewrite" then
+ (
+ let newhypo = status_parameter "volatile_newhypo" status in
+ if newhypo = "" then
+ fail (lazy "Invalid use of 'or equivalently'")
+ else
+ change_tac ~where:("",0,(None,[newhypo,Ast.UserInput],None)) ~with_what:t status
+ )
+ else
+ change_tac ~where:("",0,(None,[],Some
+ Ast.UserInput)) ~with_what:t status
+;;
+
+let done_continuation status =
+ let rec continuation l =
+ match l with
+ [] -> []
+ | (_,t,_,tag,p)::tl ->
+ if tag = `BranchTag then
+ if List.length t > 0 then
+ let continue =
+ let ctx =
+ try List.assoc "context" p
+ with Not_found -> ""
+ in
+ ctx <> "induction" && ctx <> "cases"
+ in
+ if continue then [clear_volatile_params_tac;branch_dot_tac] else
+ [clear_volatile_params_tac]
+ else
+ [merge_tac] @ (continuation tl)
+ else
+ []
+ in
+ continuation status#stack
+;;
+
+let bydone just status =
+ let goal = extract_first_goal_from_status status in
+ let continuation = done_continuation status in
+ let l = [mk_just status goal just] @ continuation in
+ block_tac l status
+;;
+
+let push_goals_tac status =
+ match status#stack with
+ [] -> fail (lazy "Error pushing goals")
+ | (g1,t1,k1,tag1,p1) :: (g2,t2,k2,tag2,p2) :: tl ->
+ if List.length g2 > 0 then
+ status#set_stack ((g1,t1 @+ g2,k1,tag1,p1) :: ([],t2,k2,tag2,p2) :: tl)
+ else status (* Nothing to push *)
+ | _ -> status
+
+let we_need_to_prove t id status =
+ let goal = extract_first_goal_from_status status in
+ match id with
+ | None ->
+ (
+ try assert_tac t None status goal (add_parameter_tac "volatile_context" "beta_rewrite" status)
+ with
+ | FirstTypeWrong -> fail (lazy "The given proposition is not the same as the conclusion")
+ )
+ | Some id ->
+ (
+ block_tac [clear_volatile_params_tac; cut_tac t; branch_tac; shift_tac; intro_tac id; merge_tac; branch_tac;
+ push_goals_tac; add_parameter_tac "volatile_context" "beta_rewrite"
+ ] status
+ )
+;;
+
+let by_just_we_proved just ty id status =
+ let goal = extract_first_goal_from_status status in
+ let just = mk_just status goal just in
+ match id with
+ | None ->
+ assert_tac ty None status goal (block_tac [clear_volatile_params_tac; add_parameter_tac
+ "volatile_context" "beta_rewrite"] status)
+ | Some id ->
+ (
+ block_tac [cut_tac ty; branch_tac; just; shift_tac; intro_tac id; merge_tac;
+ clear_volatile_params_tac; add_parameter_tac "volatile_newhypo" id] status
+ )
+;;
+
+let existselim just id1 t1 t2 id2 status =
+ let goal = extract_first_goal_from_status status in
+ let (_,_,t1) = t1 in
+ let (_,_,t2) = t2 in
+ let just = mk_just status goal just in
+ (block_tac [
+ cut_tac ("",0,(Ast.Appl [Ast.Ident ("ex",None); t1; Ast.Binder (`Lambda,(Ast.Ident
+ (id1,None), Some t1),t2)]));
+ branch_tac ~force:false;
+ just;
+ shift_tac;
+ case1_tac "_";
+ intros_tac ~names_ref:(ref []) [id1;id2];
+ merge_tac;
+ clear_volatile_params_tac
+ ]) status
+;;
+
+let andelim just t1 id1 t2 id2 status =
+ let goal = extract_first_goal_from_status status in
+ let (_,_,t1) = t1 in
+ let (_,_,t2) = t2 in
+ let just = mk_just status goal just in
+ (block_tac [
+ cut_tac ("",0,(Ast.Appl [Ast.Ident ("And",None); t1 ; t2]));
+ branch_tac ~force:false;
+ just;
+ shift_tac;
+ case1_tac "_";
+ intros_tac ~names_ref:(ref []) [id1;id2];
+ merge_tac;
+ clear_volatile_params_tac
+ ]) status
+;;
+
+let type_of_tactic_term status ctx t =
+ let status,cicterm = disambiguate status ctx t `XTNone in
+ let (_,cicty) = typeof status ctx cicterm in
+ cicty
+
+let swap_first_two_goals_tac status =
+ let gstatus =
+ match status#stack with
+ | [] -> assert false
+ | (g,t,k,tag,p) :: s ->
+ match g with
+ | (loc1) :: (loc2) :: tl ->
+ ([loc2;loc1] @+ tl,t,k,tag,p) :: s
+ | _ -> assert false
+ in
+ status#set_stack gstatus
+
+let thesisbecomes t1 = we_need_to_prove t1 None
+;;
+
+let obtain id t1 status =
+ let goal = extract_first_goal_from_status status in
+ let cicgty = get_goalty status goal in
+ let ctx = ctx_of cicgty in
+ let cicty = type_of_tactic_term status ctx t1 in
+ let _,ty = term_of_cic_term status cicty ctx in
+ let (_,_,t1) = t1 in
+ block_tac [ cut_tac ("",0,(Ast.Appl [Ast.Ident ("eq",None); Ast.NCic ty; t1; Ast.Implicit
+ `JustOne]));
+ swap_first_two_goals_tac;
+ branch_tac; shift_tac; shift_tac; intro_tac id; merge_tac; branch_tac; push_goals_tac;
+ add_parameter_tac "volatile_context" "rewrite"
+ ]
+ status
+;;
+
+let conclude t1 status =
+ let goal = extract_first_goal_from_status status in
+ let cicgty = get_goalty status goal in
+ let ctx = ctx_of cicgty in
+ let _,gty = term_of_cic_term status cicgty ctx in
+ match gty with
+ (* The first term of this Appl should probably be "eq" *)
+ NCic.Appl [_;_;plhs;_] ->
+ if alpha_eq_tacterm_kerterm t1 plhs status goal then
+ add_parameter_tac "volatile_context" "rewrite" status
+ else
+ fail (lazy "The given conclusion is different from the left-hand side of the current conclusion")
+ | _ -> fail (lazy "Your conclusion needs to be an equality")
+;;
+
+let rewritingstep rhs just last_step status =
+ let ctx = status_parameter "volatile_context" status in
+ if ctx = "rewrite" then
+ (
+ let goal = extract_first_goal_from_status status in
+ let cicgty = get_goalty status goal in
+ let ctx = ctx_of cicgty in
+ let _,gty = term_of_cic_term status cicgty ctx in
+ let cicty = type_of_tactic_term status ctx rhs in
+ let _,ty = term_of_cic_term status cicty ctx in
+ let just' = (* Extraction of the ""justification"" from the ad hoc justification *)
+ match just with
+ `Auto (univ, params) ->
+ let params =
+ if not (List.mem_assoc "timeout" params) then
+ ("timeout","3")::params
+ else params
+ in
+ let params' =
+ if not (List.mem_assoc "paramodulation" params) then
+ ("paramodulation","1")::params
+ else params
+ in
+ if params = params' then NnAuto.auto_lowtac ~params:(univ, params) status goal
+ else
+ first_tac [NnAuto.auto_lowtac ~params:(univ, params) status goal; NnAuto.auto_lowtac
+ ~params:(univ, params') status goal]
+ | `Term just -> apply_tac just
+ | `SolveWith term -> NnAuto.demod_tac ~params:(Some [term], ["all","1";"steps","1"; "use_ctx","false"])
+ | `Proof -> id_tac
+ in
+ let plhs,prhs,prepare =
+ match gty with (* Extracting the lhs and rhs of the previous equality *)
+ NCic.Appl [_;_;plhs;prhs] -> plhs,prhs,(fun continuation -> continuation status)
+ | _ -> fail (lazy "You are not building an equaility chain")
+ in
+ let continuation =
+ if last_step then
+ let todo = [just'] @ (done_continuation status) in
+ block_tac todo
+ else
+ let (_,_,rhs) = rhs in
+ block_tac [apply_tac ("",0,Ast.Appl [Ast.Ident ("trans_eq",None); Ast.NCic ty; Ast.NCic plhs;
+ rhs; Ast.NCic prhs]); branch_tac; just'; merge_tac]
+ in
+ prepare continuation
+ )
+ else
+ fail (lazy "You are not building an equality chain")
+;;
+
+let rec pp_metasenv_names (metasenv:NCic.metasenv) =
+ match metasenv with
+ [] -> ""
+ | hd :: tl ->
+ let n,conj = hd in
+ let meta_attrs,_,_ = conj in
+ let rec find_name_aux meta_attrs = match meta_attrs with
+ [] -> "Anonymous"
+ | hd :: tl -> match hd with
+ `Name n -> n
+ | _ -> find_name_aux tl
+ in
+ let name = find_name_aux meta_attrs
+ in
+ "[Goal: " ^ (string_of_int n) ^ ", Name: " ^ name ^ "]; " ^ (pp_metasenv_names tl)
+;;
+
+let print_goals_names_tac s (status:#NTacStatus.tac_status) =
+ let (_,_,metasenv,_,_) = status#obj in
+ prerr_endline (s ^" -> Metasenv: " ^ (pp_metasenv_names metasenv)); status
+
+(* Useful as it does not change the order in the list *)
+let rec list_change_assoc k v = function
+ [] -> []
+ | (k',_v' as hd) :: tl -> if k' = k then (k',v) :: tl else hd :: (list_change_assoc k v tl)
+;;
+
+let add_names_to_goals_tac (cl:NCic.constructor list ref) (status:#NTacStatus.tac_status) =
+ let add_name_to_goal name goal metasenv =
+ let (mattrs,ctx,t) = try List.assoc goal metasenv with _ -> assert false in
+ let mattrs = (`Name name) :: (List.filter (function `Name _ -> false | _ -> true) mattrs) in
+ let newconj = (mattrs,ctx,t) in
+ list_change_assoc goal newconj metasenv
+ in
+ let new_goals =
+ (* It's important that this tactic is called before branching and right after the creation of
+ * the new goals, when they are still under focus *)
+ match status#stack with
+ [] -> fail (lazy "Can not add names to an empty stack")
+ | (g,_,_,_,_) :: _tl ->
+ let rec sublist n = function
+ [] -> []
+ | hd :: tl -> if n = 0 then [] else hd :: (sublist (n-1) tl)
+ in
+ List.map (fun _,sw -> goal_of_switch sw) (sublist (List.length !cl) g)
+ in
+ let rec add_names_to_goals g cl metasenv =
+ match g,cl with
+ [],[] -> metasenv
+ | hd::tl, (_,consname,_)::tl' ->
+ add_names_to_goals tl tl' (add_name_to_goal consname hd metasenv)
+ | _,_ -> fail (lazy "There are less goals than constructors")
+ in
+ let (olduri,oldint,metasenv,oldsubst,oldkind) = status#obj in
+ let newmetasenv = add_names_to_goals new_goals !cl metasenv
+ in status#set_obj(olduri,oldint,newmetasenv,oldsubst,oldkind)
+;;
+(*
+ let (olduri,oldint,metasenv,oldsubst,oldkind) = status#obj in
+ let remove_name_from_metaattrs =
+ List.filter (function `Name _ -> false | _ -> true) in
+ let rec add_names_to_metasenv cl metasenv =
+ match cl,metasenv with
+ [],_ -> metasenv
+ | hd :: tl, mhd :: mtl ->
+ let _,consname,_ = hd in
+ let gnum,conj = mhd in
+ let mattrs,ctx,t = conj in
+ let mattrs = [`Name consname] @ (remove_name_from_metaattrs mattrs)
+ in
+ let newconj = mattrs,ctx,t in
+ let newmeta = gnum,newconj in
+ newmeta :: (add_names_to_metasenv tl mtl)
+ | _,[] -> assert false
+ in
+ let newmetasenv = add_names_to_metasenv !cl metasenv in
+ status#set_obj (olduri,oldint,newmetasenv,oldsubst,oldkind)
+*)
+
+let unfocus_branch_tac status =
+ match status#stack with
+ [] -> status
+ | (g,t,k,tag,p) :: tl -> status#set_stack (([],g @+ t,k,tag,p)::tl)
+;;
+
+let we_proceed_by_induction_on t1 t2 status =
+ let goal = extract_first_goal_from_status status in
+ let txt,len,t1 = t1 in
+ let t1 = txt, len, Ast.Appl [t1; Ast.Implicit `Vector] in
+ let indtyinfo = ref None in
+ let sort = ref (NCic.Rel 1) in
+ let cl = ref [] in (* this is a ref on purpose, as the block of code after sort_of_goal_tac in
+ block_tac acts as a block of asynchronous code, in which cl gets modified with the info retrieved
+ with analize_indty_tac, and later used to label each new goal with a costructor name. Using a
+ plain list this doesn't seem to work, as add_names_to_goals_tac would immediately act on an empty
+ list, instead of acting on the list of constructors *)
+ try
+ assert_tac t2 None status goal (block_tac [
+ analyze_indty_tac ~what:t1 indtyinfo;
+ sort_of_goal_tac sort;
+ (fun status ->
+ let ity = HExtlib.unopt !indtyinfo in
+ let NReference.Ref (uri, _) = ref_of_indtyinfo ity in
+ let name =
+ NUri.name_of_uri uri ^ "_" ^
+ snd (NCicElim.ast_of_sort
+ (match !sort with NCic.Sort x -> x | _ -> assert false))
+ in
+ let eliminator =
+ let l = [Ast.Ident (name,None)] in
+ (* Generating an implicit for each argument of the inductive type, plus one the
+ * predicate, plus an implicit for each constructor of the inductive type *)
+ let l = l @ HExtlib.mk_list (Ast.Implicit `JustOne) (ity.leftno+1+ity.consno) in
+ let _,_,t1 = t1 in
+ let l = l @ [t1] in
+ Ast.Appl l
+ in
+ cl := ity.cl;
+ exact_tac ("",0,eliminator) status);
+ add_names_to_goals_tac cl;
+ branch_tac;
+ push_goals_tac;
+ unfocus_branch_tac;
+ add_parameter_tac "context" "induction"
+ ] status)
+ with
+ | FirstTypeWrong -> fail (lazy "What you want to prove is different from the conclusion")
+;;
+
+let we_proceed_by_cases_on ((txt,len,ast1) as t1) t2 status =
+ let goal = extract_first_goal_from_status status in
+ let npt1 = txt, len, Ast.Appl [ast1; Ast.Implicit `Vector] in
+ let indtyinfo = ref None in
+ let cl = ref [] in
+ try
+ assert_tac t2 None status goal (block_tac [
+ analyze_indty_tac ~what:npt1 indtyinfo;
+ cases_tac ~what:t1 ~where:("",0,(None,[],Some
+ Ast.UserInput));
+ (
+ fun status ->
+ let ity = HExtlib.unopt !indtyinfo in
+ cl := ity.cl; add_names_to_goals_tac cl status
+ );
+ branch_tac; push_goals_tac;
+ unfocus_branch_tac;
+ add_parameter_tac "context" "cases"
+ ] status)
+ with
+ | FirstTypeWrong -> fail (lazy "What you want to prove is different from the conclusion")
+;;
+
+let byinduction t1 id = suppose t1 id ;;
+
+let name_of_conj conj =
+ let mattrs,_,_ = conj in
+ let rec search_name mattrs =
+ match mattrs with
+ [] -> "Anonymous"
+ | hd::tl ->
+ match hd with
+ `Name n -> n
+ | _ -> search_name tl
+ in
+ search_name mattrs
+
+let rec loc_of_goal goal l =
+ match l with
+ [] -> fail (lazy "Reached the end")
+ | hd :: tl ->
+ let _,sw = hd in
+ let g = goal_of_switch sw in
+ if g = goal then hd
+ else loc_of_goal goal tl
+;;
+
+let has_focused_goal status =
+ match status#stack with
+ [] -> false
+ | ([],_,_,_,_) :: _tl -> false
+ | _ -> true
+;;
+
+let focus_on_case_tac case status =
+ let (_,_,metasenv,_,_) = status#obj in
+ let rec goal_of_case case metasenv =
+ match metasenv with
+ [] -> fail (lazy "The given case does not exist")
+ | (goal,conj) :: tl ->
+ if name_of_conj conj = case then goal
+ else goal_of_case case tl
+ in
+ let goal_to_focus = goal_of_case case metasenv in
+ let gstatus =
+ match status#stack with
+ [] -> fail (lazy "There is nothing to prove")
+ | (g,t,k,tag,p) :: s ->
+ let loc =
+ try
+ loc_of_goal goal_to_focus t
+ with _ -> fail (lazy "The given case is not part of the current induction/cases analysis
+ context")
+ in
+ let curloc = if has_focused_goal status then
+ let goal = extract_first_goal_from_status status in
+ [loc_of_goal goal g]
+ else []
+ in
+ (((g @- curloc) @+ [loc]),(curloc @+ (t @- [loc])),k,tag,p) :: s
+ in
+ status#set_stack gstatus
+;;
+
+let case id l status =
+ let ctx = status_parameter "context" status in
+ if ctx <> "induction" && ctx <> "cases" then fail (lazy "You can't use case outside of an
+ induction/cases analysis context")
+else
+ (
+ if has_focused_goal status then fail (lazy "Finish the current case before switching")
+ else
+ (
+(*
+ let goal = extract_first_goal_from_status status in
+ let (_,_,metasenv,_,_) = status#obj in
+ let conj = NCicUtils.lookup_meta goal metasenv in
+ let name = name_of_conj conj in
+*)
+ let continuation =
+ let rec aux l =
+ match l with
+ [] -> [id_tac]
+ | (id,ty)::tl ->
+ (try_tac (assume id ("",0,ty))) :: (aux tl)
+ in
+ aux l
+ in
+(* if name = id then block_tac continuation status *)
+(* else *)
+ block_tac ([focus_on_case_tac id] @ continuation) status
+ )
+ )
+;;
+
+let print_stack status = prerr_endline ("PRINT STACK: " ^ (pp status#stack)); id_tac status ;;
+
+(* vim: ts=2: sw=0: et:
+ * *)
--- /dev/null
+(* Copyright (C) 2019, HELM Team.
+ *
+ * This file is part of HELM, an Hypertextual, Electronic
+ * Library of Mathematics, developed at the Computer Science
+ * Department, University of Bologna, Italy.
+ *
+ * HELM is free software; you can redistribute it and/or
+ * modify it under the terms of the GNU General Public License
+ * as published by the Free Software Foundation; either version 2
+ * of the License, or (at your option) any later version.
+ *
+ * HELM is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with HELM; if not, write to the Free Software
+ * Foundation, Inc., 59 Temple Place - Suite 330, Boston,
+ * MA 02111-1307, USA.
+ *
+ * For details, see the HELM World-Wide-Web page,
+ * http://cs.unibo.it/helm/.
+ *)
+
+type just = [ `Term of NTacStatus.tactic_term | `Auto of NnAuto.auto_params ]
+
+val assume : string -> NTacStatus.tactic_term -> 's NTacStatus.tactic
+val suppose : NTacStatus.tactic_term -> string -> 's NTacStatus.tactic
+val we_need_to_prove : NTacStatus.tactic_term -> string option -> 's NTacStatus.tactic
+val beta_rewriting_step : NTacStatus.tactic_term -> 's NTacStatus.tactic
+val bydone : just -> 's NTacStatus.tactic
+val by_just_we_proved : just -> NTacStatus.tactic_term -> string option -> 's NTacStatus.tactic
+val andelim : just -> NTacStatus.tactic_term -> string -> NTacStatus.tactic_term -> string -> 's
+NTacStatus.tactic
+val existselim : just -> string -> NTacStatus.tactic_term -> NTacStatus.tactic_term -> string -> 's
+NTacStatus.tactic
+val thesisbecomes : NTacStatus.tactic_term -> 's NTacStatus.tactic
+val rewritingstep : NTacStatus.tactic_term -> [ `Term of NTacStatus.tactic_term | `Auto of NnAuto.auto_params
+ | `Proof | `SolveWith of NTacStatus.tactic_term ] ->
+ bool (* last step *) -> 's NTacStatus.tactic
+val we_proceed_by_cases_on: NTacStatus.tactic_term -> NTacStatus.tactic_term -> 's NTacStatus.tactic
+val we_proceed_by_induction_on: NTacStatus.tactic_term -> NTacStatus.tactic_term -> 's NTacStatus.tactic
+val byinduction: NTacStatus.tactic_term -> string -> 's NTacStatus.tactic
+val case: string -> (string*NotationPt.term) list -> 's NTacStatus.tactic
+val obtain: string -> NTacStatus.tactic_term -> 's NTacStatus.tactic
+val conclude: NTacStatus.tactic_term -> 's NTacStatus.tactic
+val print_stack : 's NTacStatus.tactic
status, (ctx, t)
;;
+let are_convertible status ctx a b =
+ let status, (_,a) = relocate status ctx a in
+ let status, (_,b) = relocate status ctx b in
+ let _n,_h,metasenv,subst,_o = status#obj in
+ let res = NCicReduction.are_convertible status metasenv subst ctx a b in
+ status, res
+;;
+let are_convertible a b c d = wrap "are_convertible" (are_convertible a b c) d;;
+
let unify status ctx a b =
let status, (_,a) = relocate status ctx a in
let status, (_,b) = relocate status ctx b in
let _,_,_,cl = List.nth tl i in
let consno = List.length cl in
let left, right = HExtlib.split_nth lno args in
- status, (ref, consno, left, right)
+ status, (ref, consno, left, right, cl)
;;
let apply_subst status ctx t =
val analyse_indty:
#pstatus as 'status -> cic_term ->
- 'status * (NReference.reference * int * NCic.term list * NCic.term list)
+ 'status * (NReference.reference * int * NCic.term list * NCic.term list * NCic.constructor list)
val ppterm: #pstatus -> cic_term -> string
val ppcontext: #pstatus -> NCic.context -> string
val normalize:
#pstatus as 'status -> ?delta:int -> NCic.context -> cic_term ->
'status * cic_term
+val are_convertible:
+ #pstatus as 'status -> NCic.context -> cic_term -> cic_term -> 'status * bool
val typeof:
#pstatus as 'status -> NCic.context -> cic_term -> 'status * cic_term
val unify:
let gstatus =
match status#stack with
| [] -> assert false
- | ([], _, [], _) :: _ as stack ->
+ | ([], _, [], _, _) :: _ as stack ->
(* backward compatibility: do-nothing-dot *)
stack
- | (g, t, k, tag) :: s ->
+ | (g, t, k, tag, p) :: s ->
match filter_open g, k with
| loc :: loc_tl, _ ->
- (([ loc ], t, loc_tl @+ k, tag) :: s)
+ (([ loc ], t, loc_tl @+ k, tag, p) :: s)
| [], loc :: k ->
assert (is_open loc);
- (([ loc ], t, k, tag) :: s)
+ (([ loc ], t, k, tag, p) :: s)
| _ -> fail (lazy "can't use \".\" here")
in
status#set_stack gstatus
let gstatus =
match status#stack with
| [] -> assert false
- | (g, t, k, tag) :: s ->
+ | (g, t, k, tag, p) :: s ->
match init_pos g with (* TODO *)
| [] -> fail (lazy "empty goals")
- | [_] when (not force) -> fail (lazy "too few goals to branch")
+ | [_] when (not force) -> fail (lazy "too few goals to branch")
| loc :: loc_tl ->
- ([ loc ], [], [], `BranchTag) :: (loc_tl, t, k, tag) :: s
+ ([ loc ], [], [], `BranchTag, []) :: (loc_tl, t, k, tag, p) :: s
in
status#set_stack gstatus
;;
let shift_tac status =
let gstatus =
match status#stack with
- | (g, t, k, `BranchTag) :: (g', t', k', tag) :: s ->
+ | (g, t, k, `BranchTag, p) :: (g', t', k', tag, p') :: s ->
(match g' with
| [] -> fail (lazy "no more goals to shift")
| loc :: loc_tl ->
- (([ loc ], t @+ filter_open g @+ k, [],`BranchTag)
- :: (loc_tl, t', k', tag) :: s))
+ (([ loc ], t @+ filter_open g @+ k, [],`BranchTag, p)
+ :: (loc_tl, t', k', tag, p') :: s))
| _ -> fail (lazy "can't shift goals here")
in
status#set_stack gstatus
let gstatus =
match status#stack with
| [] -> assert false
- | ([ loc ], t, [],`BranchTag) :: (g', t', k', tag) :: s
+ | ([ loc ], t, [],`BranchTag, p) :: (g', t', k', tag, p') :: s
when is_fresh loc ->
let l_js = List.filter (fun (i, _) -> List.mem i i_s) ([loc] @+ g') in
- ((l_js, t , [],`BranchTag)
- :: (([ loc ] @+ g') @- l_js, t', k', tag) :: s)
+ ((l_js, t , [],`BranchTag, p)
+ :: (([ loc ] @+ g') @- l_js, t', k', tag, p') :: s)
| _ -> fail (lazy "can't use relative positioning here")
in
status#set_stack gstatus
let gstatus =
match status#stack with
| [] -> assert false
- | ([ loc ], t, [],`BranchTag) :: (g', t', k', tag) :: s
+ | ([ loc ], t, [],`BranchTag, p) :: (g', t', k', tag, p') :: s
when is_fresh loc ->
let l_js =
List.filter
match NCicUtils.lookup_meta (goal_of_loc curloc) metasenv with
attrs,_,_ when List.mem (`Name lab) attrs -> true
| _ -> false) ([loc] @+ g') in
- ((l_js, t , [],`BranchTag)
- :: (([ loc ] @+ g') @- l_js, t', k', tag) :: s)
+ ((l_js, t , [],`BranchTag, p)
+ :: (([ loc ] @+ g') @- l_js, t', k', tag, p') :: s)
| _ -> fail (lazy "can't use relative positioning here")
in
status#set_stack gstatus
let gstatus =
match status#stack with
| [] -> assert false
- | ([ loc ] , t, [], `BranchTag) :: (g', t', k', tag) :: s
+ | ([ loc ] , t, [], `BranchTag, p) :: (g', t', k', tag, p') :: s
when is_fresh loc ->
- (([loc] @+ g', t, [], `BranchTag) :: ([], t', k', tag) :: s)
+ (([loc] @+ g', t, [], `BranchTag, p) :: ([], t', k', tag, p') :: s)
| _ -> fail (lazy "can't use wildcard here")
in
status#set_stack gstatus
let gstatus =
match status#stack with
| [] -> assert false
- | (g, t, k,`BranchTag) :: (g', t', k', tag) :: s ->
- ((t @+ filter_open g @+ g' @+ k, t', k', tag) :: s)
+ | (g, t, k,`BranchTag, _) :: (g', t', k', tag, p) :: s ->
+ ((t @+ filter_open g @+ g' @+ k, t', k', tag, p) :: s)
| _ -> fail (lazy "can't merge goals here")
in
status#set_stack gstatus
if not (List.exists (fun l -> goal_of_loc l = g) stack_locs) then
fail (lazy (sprintf "goal %d not found (or closed)" g)))
gs;
- (zero_pos gs, [], [], `FocusTag) :: deep_close gs s
+ (zero_pos gs, [], [], `FocusTag, []) :: deep_close gs s
in
status#set_stack gstatus
;;
let gstatus =
match status#stack with
| [] -> assert false
- | (g, [], [], `FocusTag) :: s when filter_open g = [] -> s
+ | (g, [], [], `FocusTag, _) :: s when filter_open g = [] -> s
| _ as s -> fail (lazy ("can't unfocus, some goals are still open:\n"^
Continuationals.Stack.pp s))
in
let gstatus =
match status#stack with
| [] -> assert false
- | (gl, t, k, tag) :: s ->
+ | (gl, t, k, tag, p) :: s ->
let gl = List.map switch_of_loc gl in
if List.exists (function Open _ -> true | Closed _ -> false) gl then
fail (lazy "cannot skip an open goal")
else
- ([],t,k,tag) :: s
+ ([],t,k,tag,p) :: s
in
status#set_stack gstatus
;;
;;
let exec tac (low_status : #lowtac_status) g =
- let stack = [ [0,Open g], [], [], `NoTag ] in
+ let stack = [ [0,Open g], [], [], `NoTag, [] ] in
let status = change_stack_type low_status stack in
let status = tac status in
(low_status#set_pstatus status)#set_obj status#obj
let distribute_tac tac (status : #tac_status) =
match status#stack with
| [] -> assert false
- | (g, t, k, tag) :: s ->
+ | (g, t, k, tag, p) :: s ->
debug_print (lazy ("context length " ^string_of_int (List.length g)));
let rec aux s go gc =
function
debug_print (lazy ("closed: "
^ String.concat " " (List.map string_of_int gcn)));
let stack =
- (zero_pos gon, t @~- gcn, k @~- gcn, tag) :: deep_close gcn s
+ (zero_pos gon, t @~- gcn, k @~- gcn, tag, p) :: deep_close gcn s
in
((status#set_stack stack)#set_obj(sn:>lowtac_status)#obj)#set_pstatus sn
;;
leftno: int;
consno: int;
reference: NReference.reference;
+ cl: NCic.constructor list;
}
;;
let goalty = get_goalty status goal in
let status, what = disambiguate status (ctx_of goalty) what `XTInd in
let status, ty_what = typeof status (ctx_of what) what in
- let _status, (r,consno,lefts,rights) = analyse_indty status ty_what in
+ let _status, (r,consno,lefts,rights,cl) = analyse_indty status ty_what in
let leftno = List.length lefts in
let rightno = List.length rights in
indtyref := Some {
rightno = rightno; leftno = leftno; consno = consno; reference = r;
+ cl = cl;
};
exec id_tac orig_status goal)
;;
status)
;;
+let pp_ref reference =
+ let NReference.Ref (uri,spec) = reference in
+ let nstring = NUri.string_of_uri uri in
+ (*"Shareno: " ^ (string_of_int nuri) ^*) "Uri: " ^ nstring ^
+ (match spec with
+ | NReference.Decl -> "Decl"
+ | NReference.Def n -> "Def " ^ (string_of_int n)
+ | NReference.Fix (n1,n2,n3) -> "Fix " ^ (string_of_int n1) ^ " " ^ (string_of_int n2) ^ " " ^ (string_of_int n3)(* fixno, recparamno, height *)
+ | NReference.CoFix n -> "CoFix " ^ (string_of_int n)
+ | NReference.Ind (b,n1,n2) -> "Ind " ^ (string_of_bool b) ^ " " ^ (string_of_int n1) ^ " " ^ (string_of_int n2)(* inductive, indtyno, leftno *)
+ | NReference.Con (n1,n2,n3) -> "Con " ^ (string_of_int n1) ^ " " ^ (string_of_int n2) ^ " " ^ (string_of_int n3)(* indtyno, constrno, leftno *)
+ ) ;;
+
+let pp_cl cl =
+ let rec pp_aux acc =
+ match acc with
+ | [] -> ""
+ | (_,consname,_) :: tl -> consname ^ ", " ^ pp_aux tl
+ in
+ pp_aux cl
+;;
+
+let pp_indtyinfo ity = "leftno: " ^ (string_of_int ity.leftno) ^ ", consno: " ^ (string_of_int
+ ity.consno) ^ ", rightno: " ^
+ (string_of_int ity.rightno) ^ ", reference: " ^ (pp_ref ity.reference) ^ ",
+ cl: " ^ (pp_cl ity.cl);;
+
let elim_tac ~what:(txt,len,what) ~where =
let what = txt, len, Ast.Appl [what; Ast.Implicit `Vector] in
let indtyinfo = ref None in
let gty = get_goalty status goal in
let status, what = disambiguate status (ctx_of gty) what `XTInd in
let status, ty = typeof status (ctx_of what) what in
- let status, (ref, consno, _, _) = analyse_indty status ty in
+ let status, (ref, consno, _, _,_) = analyse_indty status ty in
let status, what = term_of_cic_term status what (ctx_of gty) in
let t =
NCic.Match (ref,NCic.Implicit `Term, what,
let constructor_tac ?(num=1) ~args = distribute_tac (fun status goal ->
let gty = get_goalty status goal in
- let status, (r,consno,_,_) = analyse_indty status gty in
+ let status, (r,consno,_,_,_) = analyse_indty status gty in
if num < 1 || num > consno then fail (lazy "Non existant constructor");
let ref = NReference.mk_constructor num r in
let t =
val print_tac: bool -> string -> 's NTacStatus.tactic
+val id_tac: 's NTacStatus.tactic
val dot_tac: 's NTacStatus.tactic
val branch_tac: ?force:bool -> 's NTacStatus.tactic
val shift_tac: 's NTacStatus.tactic
(*(NTacStatus.tac_status -> 'c #NTacStatus.status) ->
(#NTacStatus.tac_status as 'f) -> 'f*)
-type indtyinfo
+(* type indtyinfo *)
+type indtyinfo = {
+ rightno: int;
+ leftno: int;
+ consno: int;
+ reference: NReference.reference;
+ cl: NCic.constructor list;
+ }
val ref_of_indtyinfo : indtyinfo -> NReference.reference
val inversion_tac:
what:NTacStatus.tactic_term -> where:NTacStatus.tactic_pattern ->
's NTacStatus.tactic
+
+val exact_tac: NTacStatus.tactic_term -> 's NTacStatus.tactic
+val first_tac: 's NTacStatus.tactic list -> 's NTacStatus.tactic
+val sort_of_goal_tac: NCic.term ref -> 's NTacStatus.tactic
let toref f tbl t =
match t with
| Ast.NRef n ->
- f tbl n
+ f tbl n
| Ast.NCic _ (* local candidate *)
| _ -> ()
"; uses = " ^ (string_of_int !(v.uses)) ^
"; nom = " ^ (string_of_int !(v.nominations)) in
lazy ("\n\nSTATISTICS:\n" ^
- String.concat "\n" (List.map vstring l))
+ String.concat "\n" (List.map vstring l))
(* ======================= utility functions ========================= *)
module IntSet = Set.Make(struct type t = int let compare = compare end)
debug_print (lazy ("refining: "^(status#ppterm ctx subst metasenv pt)));
let stamp = Unix.gettimeofday () in
let metasenv, subst, pt, pty =
- (* NCicRefiner.typeof status
+ (* NCicRefiner.typeof status
(* (status#set_coerc_db NCicCoercion.empty_db) *)
metasenv subst ctx pt None in
debug_print (lazy ("refined: "^(status#ppterm ctx subst metasenv pt)));
NCicRefiner.RefineFailure msg
| NCicRefiner.Uncertain msg ->
debug_print (lazy ("WARNING U: refining in fast_eq_check failed\n" ^
- snd (Lazy.force msg) ^
- "\n in the environment\n" ^
- status#ppmetasenv subst metasenv)); None
+ snd (Lazy.force msg) ^
+ "\n in the environment\n" ^
+ status#ppmetasenv subst metasenv)); None
| NCicRefiner.AssertFailure msg ->
debug_print (lazy ("WARNING F: refining in fast_eq_check failed" ^
Lazy.force msg ^
- "\n in the environment\n" ^
- status#ppmetasenv subst metasenv)); None
+ "\n in the environment\n" ^
+ status#ppmetasenv subst metasenv)); None
| Sys.Break as e -> raise e
| _ -> None
in
| Error _ -> debug_print (lazy ("no paramod proof found"));[]
;;
-let index_local_equations eq_cache status =
+let index_local_equations eq_cache ?(flag=false) status =
+ if flag then
+ NCicParamod.empty_state
+ else begin
noprint (lazy "indexing equations");
let open_goals = head_goals status#stack in
let open_goal = List.hd open_goals in
| NCicTypeChecker.TypeCheckerFailure _
| NCicTypeChecker.AssertFailure _ -> eq_cache)
eq_cache ctx
+ end
;;
-let index_local_equations2 eq_cache status open_goal lemmas nohyps =
+let index_local_equations2 eq_cache status open_goal lemmas ?flag:(_=false) nohyps =
noprint (lazy "indexing equations");
let eq_cache,lemmas =
match lemmas with
NCicMetaSubst.saturate status ~delta:height metasenv subst ctx ty 0 in
match ty with
| NCic.Const(NReference.Ref (_,NReference.Def _) as nre)
- when nre<>nref ->
- let _, _, bo, _, _, _ = NCicEnvironment.get_checked_def status nre in
- aux metasenv bo (args@moreargs)
+ when nre<>nref ->
+ let _, _, bo, _, _, _ = NCicEnvironment.get_checked_def status nre in
+ aux metasenv bo (args@moreargs)
| NCic.Appl(NCic.Const(NReference.Ref (_,NReference.Def _) as nre)::tl)
- when nre<>nref ->
- let _, _, bo, _, _, _ = NCicEnvironment.get_checked_def status nre in
- aux metasenv (NCic.Appl(bo::tl)) (args@moreargs)
+ when nre<>nref ->
+ let _, _, bo, _, _, _ = NCicEnvironment.get_checked_def status nre in
+ aux metasenv (NCic.Appl(bo::tl)) (args@moreargs)
| _ -> ty,metasenv,(args@moreargs)
in
aux metasenv ty []
match gty with
| NCic.Const(nref)
| NCic.Appl(NCic.Const(nref)::_) ->
- saturate_to_ref status metasenv subst ctx nref ty
+ saturate_to_ref status metasenv subst ctx nref ty
| _ ->
- NCicMetaSubst.saturate status metasenv subst ctx ty 0 in
+ NCicMetaSubst.saturate status metasenv subst ctx ty 0 in
let metasenv,j,inst,_ = NCicMetaSubst.mk_meta metasenv ctx `IsTerm in
let status = status#set_obj (n,h,metasenv,subst,o) in
let pterm = if args=[] then t else
do_types : bool; (* solve goals in Type *)
last : bool; (* last goal: take first solution only *)
candidates: Ast.term list option;
+ local_candidates: bool;
maxwidth : int;
maxsize : int;
maxdepth : int;
let add_to_trace status ~depth cache t =
match t with
| Ast.NRef _ ->
- debug_print ~depth (lazy ("Adding to trace: " ^ NotationPp.pp_term status t));
- {cache with trace = t::cache.trace}
+ debug_print ~depth (lazy ("Adding to trace: " ^ NotationPp.pp_term status t));
+ {cache with trace = t::cache.trace}
| Ast.NCic _ (* local candidate *)
| _ -> (*not an application *) cache
let pptrace status tr =
(lazy ("Proof Trace: " ^ (String.concat ";"
- (List.map (NotationPp.pp_term status) tr))))
+ (List.map (NotationPp.pp_term status) tr))))
(* not used
let remove_from_trace cache t =
match t with
| Ast.NRef _ ->
- (match cache.trace with
- | _::tl -> {cache with trace = tl}
+ (match cache.trace with
+ | _::tl -> {cache with trace = tl}
| _ -> assert false)
| Ast.NCic _ (* local candidate *)
| _ -> (*not an application *) cache *)
let ty = NCicTypeChecker.typeof status subst metasenv ctx t in
let res = branch status (mk_cic_term ctx ty) in
noprint (lazy ("branch factor for: " ^ (ppterm status ct) ^ " = "
- ^ (string_of_int res)));
+ ^ (string_of_int res)));
res
in
let candidates = List.map (fun t -> branch t,t) candidates in
List.sort (fun (a,_) (b,_) -> a - b) candidates in
let candidates = List.map snd candidates in
noprint (lazy ("candidates =\n" ^ (String.concat "\n"
- (List.map (NotationPp.pp_term status) candidates))));
+ (List.map (NotationPp.pp_term status) candidates))));
candidates
let sort_new_elems l =
if level = 0 then []
else match gs with
| [] -> assert false
- | (g,_,_,_)::s ->
+ | (g,_,_,_,_)::s ->
let is_open = function
| (_,Continuationals.Stack.Open i) -> Some i
| (_,Continuationals.Stack.Closed _) -> None
in
- HExtlib.filter_map is_open g @ stack_goals (level-1) s
+ HExtlib.filter_map is_open g @ stack_goals (level-1) s
;;
let open_goals level status = stack_goals level status#stack
(* some flexibility *)
if og_no - old_og_no > res then
(debug_print (lazy ("branch factor for: " ^ (ppterm status cict) ^ " = "
- ^ (string_of_int res) ^ " vs. " ^ (string_of_int og_no)));
+ ^ (string_of_int res) ^ " vs. " ^ (string_of_int og_no)));
debug_print ~depth (lazy "strange application"); None)
else
*) (incr candidate_no; Some ((!candidate_no,t),status))
let perforate_small status subst metasenv context t =
let rec aux = function
| NCic.Appl (hd::tl) ->
- let map t =
- let s = sort_of status subst metasenv context t in
- match s with
- | NCic.Sort(NCic.Type [`Type,u])
- when u=type0 -> NCic.Meta (0,(0,NCic.Irl 0))
- | _ -> aux t
- in
- NCic.Appl (hd::List.map map tl)
+ let map t =
+ let s = sort_of status subst metasenv context t in
+ match s with
+ | NCic.Sort(NCic.Type [`Type,u])
+ when u=type0 -> NCic.Meta (0,(0,NCic.Irl 0))
+ | _ -> aux t
+ in
+ NCic.Appl (hd::List.map map tl)
| t -> t
in
aux t
cands, diff more_cands cands
;;
+let is_a_needed_uri s =
+ s = "cic:/matita/basics/logic/eq.ind" ||
+ s = "cic:/matita/basics/logic/sym_eq.con" ||
+ s = "cic:/matita/basics/logic/trans_eq.con" ||
+ s = "cic:/matita/basics/logic/eq_f3.con" ||
+ s = "cic:/matita/basics/logic/eq_f2.con" ||
+ s = "cic:/matita/basics/logic/eq_f.con"
+
let get_candidates ?(smart=true) ~pfailed depth flags status cache _signature gty =
let universe = status#auto_cache in
let _,_,metasenv,subst,_ = status#obj in
let context = ctx_of gty in
let _, raw_gty = term_of_cic_term status gty context in
let is_prod, _is_eq =
- let status, t = term_of_cic_term status gty context in
- let t = NCicReduction.whd status subst context t in
- match t with
- | NCic.Prod _ -> true, false
- | _ -> false, NCicParamod.is_equation status metasenv subst context t
+ let status, t = term_of_cic_term status gty context in
+ let t = NCicReduction.whd status subst context t in
+ match t with
+ | NCic.Prod _ -> true, false
+ | _ -> false, NCicParamod.is_equation status metasenv subst context t
in
debug_print ~depth (lazy ("gty:" ^ NTacStatus.ppterm status gty));
let is_eq =
let raw_weak_gty, weak_gty =
if smart then
match raw_gty with
- | NCic.Appl _
- | NCic.Const _
- | NCic.Rel _ ->
+ | NCic.Appl _
+ | NCic.Const _
+ | NCic.Rel _ ->
let raw_weak =
perforate_small status subst metasenv context raw_gty in
let weak = mk_cic_term context raw_weak in
noprint ~depth (lazy ("weak_gty:" ^ NTacStatus.ppterm status weak));
Some raw_weak, Some (weak)
- | _ -> None,None
+ | _ -> None,None
else None,None
in
(* we now compute global candidates *)
let global_cands, smart_global_cands =
- let mapf s =
- let to_ast = function
- | NCic.Const r when true
- (*is_relevant statistics r*) -> Some (Ast.NRef r)
- (* | NCic.Const _ -> None *)
- | _ -> assert false in
- HExtlib.filter_map
- to_ast (NDiscriminationTree.TermSet.elements s) in
- let g,l =
- get_cands
- (NDiscriminationTree.DiscriminationTree.retrieve_unifiables universe)
- NDiscriminationTree.TermSet.diff
- NDiscriminationTree.TermSet.empty
- raw_gty raw_weak_gty in
- mapf g, mapf l in
+ let mapf s =
+ let to_ast = function
+ | NCic.Const r when true
+ (*is_relevant statistics r*) -> Some (Ast.NRef r)
+ (* | NCic.Const _ -> None *)
+ | _ -> assert false in
+ HExtlib.filter_map
+ to_ast (NDiscriminationTree.TermSet.elements s) in
+ let g,l =
+ get_cands
+ (NDiscriminationTree.DiscriminationTree.retrieve_unifiables universe)
+ NDiscriminationTree.TermSet.diff
+ NDiscriminationTree.TermSet.empty
+ raw_gty raw_weak_gty in
+ mapf g, mapf l
+ in
+ let global_cands,smart_global_cands =
+ if flags.local_candidates then global_cands,smart_global_cands
+ else let filter = List.filter (function Ast.NRef NReference.Ref (uri,_) -> is_a_needed_uri
+ (NUri.string_of_uri
+ uri) | _ -> false)
+ in filter global_cands,filter smart_global_cands
+ in
(* we now compute local candidates *)
let local_cands,smart_local_cands =
let mapf s =
let to_ast t =
- let _status, t = term_of_cic_term status t context
- in Ast.NCic t in
- List.map to_ast (Ncic_termSet.elements s) in
+ let _status, t = term_of_cic_term status t context
+ in Ast.NCic t in
+ List.map to_ast (Ncic_termSet.elements s) in
let g,l =
get_cands
- (fun ty -> search_in_th ty cache)
- Ncic_termSet.diff Ncic_termSet.empty gty weak_gty in
- mapf g, mapf l in
+ (fun ty -> search_in_th ty cache)
+ Ncic_termSet.diff Ncic_termSet.empty gty weak_gty in
+ mapf g, mapf l
+ in
+ let local_cands,smart_local_cands =
+ if flags.local_candidates then local_cands,smart_local_cands
+ else let filter = List.filter (function Ast.NRef NReference.Ref (uri,_) -> is_a_needed_uri
+ (NUri.string_of_uri
+ uri) | _ -> false)
+ in filter local_cands,filter smart_local_cands
+ in
(* we now splits candidates in facts or not facts *)
let test = is_a_fact_ast status subst metasenv context in
let by,given_candidates =
let status, t = term_of_cic_term status gty context in
let t = NCicReduction.whd status subst context t in
match t with
- | NCic.Prod _ -> true, false
- | _ -> false, NCicParamod.is_equation status metasenv subst context t
+ | NCic.Prod _ -> true, false
+ | _ -> false, NCicParamod.is_equation status metasenv subst context t
in
debug_print ~depth (lazy (string_of_bool is_eq));
(* new *)
NCicMetaSubst.mk_meta
metasenv ctx ~with_type:implication `IsType in
let status = status#set_obj (n,h,metasenv,subst,obj) in
- let status = status#set_stack [([1,Open j],[],[],`NoTag)] in
+ let status = status#set_stack [([1,Open j],[],[],`NoTag,[])] in
try
let status = NTactics.intro_tac "foo" status in
let status =
| _ -> status, facts
;;
-let intros ~depth status cache =
+let intros ~depth status ?(use_given_only=false) cache =
match is_prod status with
| `Inductive _
| `Some _ ->
- let trace = cache.trace in
+ let trace = cache.trace in
let status,facts =
intros_facts ~depth status cache.facts
in
[(0,Ast.Ident("__intros",None)),status], cache
else
(* we reindex the equation from scratch *)
- let unit_eq = index_local_equations status#eq_cache status in
+ let unit_eq = index_local_equations status#eq_cache status ~flag:use_given_only in
let status = NTactics.merge_tac status in
[(0,Ast.Ident("__intros",None)),status],
init_cache ~facts ~unit_eq () ~trace
| _ -> false
;;
-let do_something signature flags status g depth gty cache =
+let do_something signature flags status g depth gty ?(use_given_only=false) cache =
(* if the goal is meta we close it with I:True. This should work
thanks to the toplogical sorting of goals. *)
if is_meta status gty then
let s = NTactics.apply_tac ("",0,t) status in
[(0,t),s], cache
else
- let l0, cache = intros ~depth status cache in
+ let l0, cache = intros ~depth status cache ~use_given_only in
if l0 <> [] then l0, cache
else
(* whd *)
let gstatus =
match status#stack with
| [] -> assert false
- | (goals, t, k, tag) :: s ->
+ | (goals, t, k, tag, p) :: s ->
let g = head_goals status#stack in
let sortedg =
(List.rev (MS.topological_sort g (deps status))) in
let sorted_goals =
List.map (fun i -> List.find (is_it i) goals) sortedg
in
- (sorted_goals, t, k, tag) :: s
+ (sorted_goals, t, k, tag, p) :: s
in
status#set_stack gstatus
;;
let gstatus =
match status#stack with
| [] -> assert false
- | (g, t, k, tag) :: s ->
+ | (g, t, k, tag, p) :: s ->
let is_open = function
| (_,Continuationals.Stack.Open _) -> true
| (_,Continuationals.Stack.Closed _) -> false
in
let g' = List.filter is_open g in
- (g', t, k, tag) :: s
+ (g', t, k, tag, p) :: s
in
status#set_stack gstatus
;;
let rec slice level gs =
if level = 0 then [],[],gs else
match gs with
- | [] -> assert false
- | (g, t, k, tag) :: s ->
+ | [] -> assert false
+ | (g, t, k, tag,p) :: s ->
let f,o,gs = slice (level-1) s in
let f1,o1 = List.partition in_focus g
in
- (f1,[],[],`BranchTag)::f, (o1, t, k, tag)::o, gs
+ (f1,[],[],`BranchTag, [])::f, (o1, t, k, tag, p)::o, gs
in
let gstatus =
let f,o,s = slice level status#stack in f@o@s
let move_to_side level status =
match status#stack with
| [] -> assert false
- | (g,_,_,_)::tl ->
+ | (g,_,_,_,_)::tl ->
let is_open = function
| (_,Continuationals.Stack.Open i) -> Some i
| (_,Continuationals.Stack.Closed _) -> None
in
let others = menv_closure status (stack_goals (level-1) tl) in
List.for_all (fun i -> IntSet.mem i others)
- (HExtlib.filter_map is_open g)
+ (HExtlib.filter_map is_open g)
-let top_cache ~depth:_ top status cache =
+let top_cache ~depth:_ top status ?(use_given_only=false) cache =
if top then
- let unit_eq = index_local_equations status#eq_cache status in
+ let unit_eq = index_local_equations status#eq_cache status ~flag:use_given_only in
{cache with unit_eq = unit_eq}
else cache
-let rec auto_clusters ?(top=false)
- flags signature cache depth status : unit =
+let rec auto_clusters ?(top=false) flags signature cache depth ?(use_given_only=false) status : unit =
debug_print ~depth (lazy ("entering auto clusters at depth " ^
- (string_of_int depth)));
+ (string_of_int depth)));
debug_print ~depth (pptrace status cache.trace);
(* ignore(Unix.select [] [] [] 0.01); *)
let status = clean_up_tac status in
let status = NTactics.merge_tac status in
let cache =
let l,tree = cache.under_inspection in
- match l with
- | [] -> cache (* possible because of intros that cleans the cache *)
- | a::tl -> let tree = rm_from_th a tree a in
- {cache with under_inspection = tl,tree}
+ match l with
+ | [] -> cache (* possible because of intros that cleans the cache *)
+ | a::tl -> let tree = rm_from_th a tree a in
+ {cache with under_inspection = tl,tree}
in
- auto_clusters flags signature cache (depth-1) status
+ auto_clusters flags signature cache (depth-1) status ~use_given_only
else if List.length goals < 2 then
- let cache = top_cache ~depth top status cache in
- auto_main flags signature cache depth status
+ let cache = top_cache ~depth top status cache ~use_given_only in
+ auto_main flags signature cache depth status ~use_given_only
else
let all_goals = open_goals (depth+1) status in
debug_print ~depth (lazy ("goals = " ^
(fun gl ->
if List.length gl > flags.maxwidth then
begin
- debug_print ~depth (lazy "FAIL GLOBAL WIDTH");
- HLog.warn (sprintf "global width (%u) exceeded: %u"
- flags.maxwidth (List.length gl));
- raise (Gaveup cache.failures)
- end else ()) classes;
+ debug_print ~depth (lazy "FAIL GLOBAL WIDTH");
+ HLog.warn (sprintf "global width (%u) exceeded: %u"
+ flags.maxwidth (List.length gl));
+ raise (Gaveup cache.failures)
+ end else ()) classes;
if List.length classes = 1 then
let flags =
{flags with last = (List.length all_goals = 1)} in
- (* no need to cluster *)
- let cache = top_cache ~depth top status cache in
- auto_main flags signature cache depth status
+ (* no need to cluster *)
+ let cache = top_cache ~depth top status cache ~use_given_only in
+ auto_main flags signature cache depth status ~use_given_only
else
let classes = if top then List.rev classes else classes in
debug_print ~depth
let flags =
{flags with last = (List.length gl = 1)} in
let lold = List.length status#stack in
- debug_print ~depth (lazy ("stack length = " ^
- (string_of_int lold)));
+ debug_print ~depth (lazy ("stack length = " ^
+ (string_of_int lold)));
let fstatus = deep_focus_tac (depth+1) gl status in
- let cache = top_cache ~depth top fstatus cache in
+ let cache = top_cache ~depth top fstatus cache ~use_given_only in
try
debug_print ~depth (lazy ("focusing on" ^
String.concat "," (List.map string_of_int gl)));
- auto_main flags signature cache depth fstatus; assert false
+ auto_main flags signature cache depth fstatus ~use_given_only; assert false
with
| Proved(status,trace) ->
- let status = NTactics.merge_tac status in
- let cache = {cache with trace = trace} in
- let lnew = List.length status#stack in
- assert (lold = lnew);
- (status,cache,true)
+ let status = NTactics.merge_tac status in
+ let cache = {cache with trace = trace} in
+ let lnew = List.length status#stack in
+ assert (lold = lnew);
+ (status,cache,true)
| Gaveup failures when top ->
let cache = {cache with failures = failures} in
(status,cache,b)
(status,cache,false) classes
in
let rec final_merge n s =
- if n = 0 then s else final_merge (n-1) (NTactics.merge_tac s)
+ if n = 0 then s else final_merge (n-1) (NTactics.merge_tac s)
in let status = final_merge depth status
in if b then raise (Proved(status,cache.trace)) else raise (Gaveup cache.failures)
and
(* BRAND NEW VERSION *)
-auto_main flags signature cache depth status: unit =
+auto_main flags signature cache depth ?(use_given_only=false) status: unit=
debug_print ~depth (lazy "entering auto main");
debug_print ~depth (pptrace status cache.trace);
debug_print ~depth (lazy ("stack length = " ^
- (string_of_int (List.length status#stack))));
+ (string_of_int (List.length status#stack))));
(* ignore(Unix.select [] [] [] 0.01); *)
let status = sort_tac (clean_up_tac status) in
let goals = head_goals status#stack in
| a::tl -> let tree = rm_from_th a tree a in
{cache with under_inspection = tl,tree}
in
- auto_clusters flags signature cache (depth-1) status
+ auto_clusters flags signature cache (depth-1) status ~use_given_only
| _orig::_ ->
if depth > 0 && move_to_side depth status
then
| a::tl -> let tree = rm_from_th a tree a in
{cache with under_inspection = tl,tree}
in
- auto_clusters flags signature cache (depth-1) status
+ auto_clusters flags signature cache (depth-1) status ~use_given_only
else
let ng = List.length goals in
(* moved inside auto_clusters *)
if ng > flags.maxwidth then begin
debug_print ~depth (lazy "FAIL LOCAL WIDTH");
- HLog.warn (sprintf "local width (%u) exceeded: %u"
- flags.maxwidth ng);
- raise (Gaveup cache.failures)
+ HLog.warn (sprintf "local width (%u) exceeded: %u"
+ flags.maxwidth ng);
+ raise (Gaveup cache.failures)
end else if depth = flags.maxdepth then
- raise (Gaveup cache.failures)
+ raise (Gaveup cache.failures)
else
let status = NTactics.branch_tac ~force:true status in
let g,gctx, gty = current_goal status in
(* for efficiency reasons, in this case we severely cripple the
search depth *)
(debug_print ~depth (lazy ("RAISING DEPTH TO " ^ string_of_int (depth+1)));
- auto_main flags signature cache (depth+1) status)
+ auto_main flags signature cache (depth+1) status ~use_given_only)
(* check for loops *)
else if is_subsumed depth false status closegty (snd cache.under_inspection) then
(debug_print ~depth (lazy "SUBSUMED");
(debug_print ~depth (lazy "ALREADY MET");
raise (Gaveup cache.failures))
else
- let new_sig = height_of_goal g status in
+ let new_sig = height_of_goal g status in
if new_sig < signature then
- (debug_print ~depth (lazy ("news = " ^ (string_of_int new_sig)));
- debug_print ~depth (lazy ("olds = " ^ (string_of_int signature))));
+ (debug_print ~depth (lazy ("news = " ^ (string_of_int new_sig)));
+ debug_print ~depth (lazy ("olds = " ^ (string_of_int signature))));
let alternatives, cache =
- do_something signature flags status g depth gty cache in
+ do_something signature flags status g depth gty cache ~use_given_only in
let loop_cache =
if flags.last then
- let l,tree = cache.under_inspection in
- let l,tree = closegty::l, add_to_th closegty tree closegty in
+ let l,tree = cache.under_inspection in
+ let l,tree = closegty::l, add_to_th closegty tree closegty in
{cache with under_inspection = l,tree}
else cache in
let failures =
List.fold_left
(fun allfailures ((_,t),status) ->
debug_print ~depth
- (lazy ("(re)considering goal " ^
- (string_of_int g) ^" : "^ppterm status gty));
+ (lazy ("(re)considering goal " ^
+ (string_of_int g) ^" : "^ppterm status gty));
debug_print (~depth:depth)
(lazy ("Case: " ^ NotationPp.pp_term status t));
let depth,cache =
- if t=Ast.Ident("__whd",None) ||
+ if t=Ast.Ident("__whd",None) ||
t=Ast.Ident("__intros",None)
then depth, cache
- else depth+1,loop_cache in
- let cache = add_to_trace status ~depth cache t in
+ else depth+1,loop_cache in
+ let cache = add_to_trace status ~depth cache t in
let cache = {cache with failures = allfailures} in
- try
- auto_clusters flags signature cache depth status;
+ try
+ auto_clusters flags signature cache depth status ~use_given_only;
assert false;
- with Gaveup fail ->
- debug_print ~depth (lazy "Failed");
- fail)
- cache.failures alternatives in
+ with Gaveup fail ->
+ debug_print ~depth (lazy "Failed");
+ fail)
+ cache.failures alternatives in
let failures =
if flags.last then
let newfail =
add_to_th newfail failures closegty
else failures in
debug_print ~depth (lazy "no more candidates");
- raise (Gaveup failures)
+ raise (Gaveup failures)
;;
let int name l def =
(* filtering facts *)
in List.filter
(fun t ->
- match t with
- | Ast.NRef (NReference.Ref (u,_)) -> not (is_a_fact_obj s u)
- | _ -> false) trace
+ match t with
+ | Ast.NRef (NReference.Ref (u,_)) -> not (is_a_fact_obj s u)
+ | _ -> false) trace
;;
-let auto_tac ~params:(univ,flags) ?(trace_ref=ref []) status =
+(*CSC: TODO
+
+- auto_params e' una high tactic che prende in input i parametri e poi li
+ processa nel contesto vuoto calcolando i candidate
+
+- astrarla su una auto_params' che prende in input gia' i candidate e un
+ nuovo parametro per evitare il calcolo dei candidate locali che invece
+ diventano vuoti (ovvero: non usare automaticamente tutte le ipotesi, bensi'
+ nessuna)
+
+- reimplementi la auto_params chiamando la auto_params' con il flag a
+ false e il vecchio codice per andare da parametri a candiddati
+ OVVERO: usa tutti le ipotesi locali + candidati globali
+
+- crei un nuovo entry point lowtac che calcola i candidati usando il contesto
+ corrente e poi fa exec della auto_params' con i candidati e il flag a true
+ OVVERO: usa solo candidati globali che comprendono ipotesi locali
+*)
+
+type auto_params = NTacStatus.tactic_term list option * (string * string) list
+
+(*let auto_tac ~params:(univ,flags) ?(trace_ref=ref []) status =*)
+let auto_tac' candidates ~local_candidates ?(use_given_only=false) flags ?(trace_ref=ref []) status =
let oldstatus = status in
let status = (status:> NTacStatus.tac_status) in
let goals = head_goals status#stack in
(NDiscriminationTree.TermSet.elements t))
)));
*)
- let candidates =
- match univ with
- | None -> None
- | Some l ->
- let to_Ast t =
-(* FG: `XTSort here? *)
- let status, res = disambiguate status [] t `XTNone in
- let _,res = term_of_cic_term status res (ctx_of res)
- in Ast.NCic res
- in Some (List.map to_Ast l)
- in
let depth = int "depth" flags 3 in
let size = int "size" flags 10 in
let width = int "width" flags 4 (* (3+List.length goals)*) in
let flags = {
last = true;
candidates = candidates;
+ local_candidates = local_candidates;
maxwidth = width;
maxsize = size;
maxdepth = depth;
let _ = debug_print (lazy("\n\nRound "^string_of_int x^"\n")) in
let flags = { flags with maxdepth = x }
in
- try auto_clusters (~top:true) flags signature cache 0 status;assert false
+ try auto_clusters (~top:true) flags signature cache 0 status ~use_given_only;assert false
(*
try auto_main flags signature cache 0 status;assert false
*)
| Gaveup _ -> up_to (x+1) y
| Proved (s,trace) ->
debug_print (lazy ("proved at depth " ^ string_of_int x));
- List.iter (toref incr_uses statistics) trace;
+ List.iter (toref incr_uses statistics) trace;
let trace = cleanup_trace s trace in
- let _ = debug_print (pptrace status trace) in
+ let _ = debug_print (pptrace status trace) in
let stack =
match s#stack with
- | (g,t,k,f) :: rest -> (filter_open g,t,k,f):: rest
+ | (g,t,k,f,p) :: rest -> (filter_open g,t,k,f,p):: rest
| _ -> assert false
in
let s = s#set_stack stack in
s
;;
+let candidates_from_ctx univ ctx status =
+ match univ with
+ | None -> None
+ | Some l ->
+ let to_Ast t =
+ (* FG: `XTSort here? *)
+ let status, res = disambiguate status ctx t `XTNone in
+ let _,res = term_of_cic_term status res (ctx_of res)
+ in Ast.NCic res
+ in Some ((List.map to_Ast l) @ [Ast.Ident("refl",None); Ast.Ident("sym_eq",None);
+ Ast.Ident("trans_eq",None); Ast.Ident("eq_f",None);
+ Ast.Ident("eq_f2",None); Ast.Ident("eq_f3",None);
+ Ast.Ident("rewrite_r",None); Ast.Ident("rewrite_l",None)
+ ])
+
+let auto_lowtac ~params:(univ,flags) status goal =
+ let gty = get_goalty status goal in
+ let ctx = ctx_of gty in
+ let candidates = candidates_from_ctx univ ctx status in
+ auto_tac' candidates ~local_candidates:true ~use_given_only:false flags ~trace_ref:(ref [])
+
+let auto_tac ~params:(univ,flags) ?(trace_ref=ref []) status =
+ let candidates = candidates_from_ctx univ [] status in
+ auto_tac' candidates ~local_candidates:true ~use_given_only:false flags ~trace_ref status
+
let auto_tac ~params:(_,flags as params) ?trace_ref =
if List.mem_assoc "demod" flags then
demod_tac ~params
\ / This software is distributed as is, NO WARRANTY.
V_______________________________________________________________ *)
+type auto_params = NTacStatus.tactic_term list option * (string * string) list
+
val is_a_fact_obj:
#NTacStatus.pstatus -> NUri.uri -> bool
-val fast_eq_check_tac:
- params:(NTacStatus.tactic_term list option * (string * string) list) ->
- 's NTacStatus.tactic
+val fast_eq_check_tac: params:auto_params -> 's NTacStatus.tactic
-val paramod_tac:
- params:(NTacStatus.tactic_term list option * (string * string) list) ->
- 's NTacStatus.tactic
+val paramod_tac: params:auto_params -> 's NTacStatus.tactic
-val demod_tac:
- params:(NTacStatus.tactic_term list option* (string * string) list) ->
- 's NTacStatus.tactic
+val demod_tac: params:auto_params -> 's NTacStatus.tactic
val smart_apply_tac:
NTacStatus.tactic_term -> 's NTacStatus.tactic
val auto_tac:
- params:(NTacStatus.tactic_term list option * (string * string) list) ->
+ params:auto_params ->
?trace_ref:NotationPt.term list ref ->
's NTacStatus.tactic
+val auto_lowtac: params:auto_params -> #NTacStatus.pstatus -> int -> 's NTacStatus.tactic
+
val keys_of_type:
(#NTacStatus.pstatus as 'a) ->
NTacStatus.cic_term -> 'a * NTacStatus.cic_term list
str \
unix \
lablgtk3 \
-lablgtk3.sourceview3 \
+lablgtk3-sourceview3 \
netstring \
ulex-camlp5 \
zip \
"
FINDLIB_REQUIRES="\
$FINDLIB_CREQUIRES \
-lablgtk3.sourceview3 \
+lablgtk3-sourceview3 \
"
for r in $FINDLIB_LIBSREQUIRES $FINDLIB_REQUIRES
method ppobj obj =
snd (ntxt_of_cic_object ~map_unicode_to_tex:false 80 self obj)
end
+
+let notation_pp_term status term =
+ let to_pres = Content2pres.nterm2pres ?prec:None in
+ let content = term in
+ let size = 80 in
+ let ids_to_nrefs = Hashtbl.create 1 in
+ let pres = to_pres status ~ids_to_nrefs content in
+ let pres = CicNotationPres.mpres_of_box pres in
+ BoxPp.render_to_string ~map_unicode_to_tex:(Helm_registry.get_bool "matita.paste_unicode_as_tex")
+ (function x::_ -> x | _ -> assert false) size pres
+
+let _ = NotationPp.set_pp_term (fun status y -> snd (notation_pp_term (Obj.magic status) y))
$(DEP_INPUT): $(MAS) Makefile
@echo " GREP include"
- $(H)grep "include \"" $^ > $(DEP_INPUT)
+ $(H)grep "include \"" $(MAS) > $(DEP_INPUT)
$(H)echo "$(LINE)" | sed -e 's/\"\. /\"\.\n/g' >> $(DEP_INPUT)
# dep ########################################################################
(* EXAMPLES *****************************************************************)
-(* Extended validy (basic_2B) vs. restricted validity (basic_1A) ************)
+(* Extended validy (λδ-2A) vs. restricted validity (λδ-1B) ******************)
(* Note: extended validity of a closure, height of cnv_appl > 1 *)
-lemma cnv_extended (h) (p): ∀G,L,s. ⦃G, L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛ#0⦄ ⊢ ⓐ#2.#0 ![Ⓕ,h].
+lemma cnv_extended (h) (p) (G) (L):
+ ∀s. ⦃G,L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛ#0⦄ ⊢ ⓐ#2.#0 ![h,𝛚].
#h #p #G #L #s
@(cnv_appl … 2 p … (⋆s) … (⋆s))
-[ #H destruct
+[ //
| /4 width=1 by cnv_sort, cnv_zero, cnv_lref/
| /4 width=1 by cnv_bind, cnv_zero/
| /5 width=3 by cpm_cpms, cpm_lref, cpm_ell, lifts_sort/
qed.
(* Note: restricted validity of the η-expanded closure, height of cnv_appl = 1 **)
-lemma vnv_restricted (h) (p): ∀G,L,s. ⦃G, L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛⓛ{p}⋆s.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ![Ⓣ,h].
+lemma cnv_restricted (h) (p) (G) (L):
+ ∀s. ⦃G,L.ⓛ⋆s.ⓛⓛ{p}⋆s.⋆s.ⓛⓛ{p}⋆s.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ![h,𝟐].
#h #p #G #L #s
@(cnv_appl … 1 p … (⋆s) … (ⓐ#0.#2))
-[ /2 width=1 by/
+[ //
| /4 width=1 by cnv_sort, cnv_zero, cnv_lref/
| @cnv_zero
@cnv_bind //
@(cnv_appl … 1 p … (⋆s) … (⋆s))
- [ /2 width=1 by/
+ [ //
| /2 width=1 by cnv_sort, cnv_zero/
| /4 width=1 by cnv_sort, cnv_zero, cnv_lref, cnv_bind/
| /2 width=3 by cpms_ell, lifts_sort/
qed-.
lemma cpr_inv_Delta_sn (h) (G) (L) (s):
- ∀X. ⦃G, L⦄ ⊢ Delta s ➡[h] X → Delta s = X.
+ ∀X. ⦃G,L⦄ ⊢ Delta s ➡[h] X → Delta s = X.
#h #G #L #s #X #H
elim (cpm_inv_abst1 … H) -H #X1 #X2 #H1 #H2 #H destruct
lapply (cpr_inv_sort1 … H1) -H1 #H destruct
(* Main properties **********************************************************)
-theorem cpr_Omega_12 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega1 s ➡[h] Omega2 s.
+theorem cpr_Omega_12 (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega1 s ➡[h] Omega2 s.
/2 width=1 by cpm_beta/ qed.
-theorem cpr_Omega_23 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega2 s ➡[h] Omega3 s.
+theorem cpr_Omega_23 (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega2 s ➡[h] Omega3 s.
/5 width=3 by cpm_eps, cpm_appl, cpm_bind, cpm_delta, Delta_lifts/ qed.
-theorem cpr_Omega_31 (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega3 s ➡[h] Omega1 s.
+theorem cpr_Omega_31 (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega3 s ➡[h] Omega1 s.
/4 width=3 by cpm_zeta, Delta_lifts, lifts_flat/ qed.
(* Main inversion properties ************************************************)
theorem cpr_inv_Omega1_sn (h) (G) (L) (s):
- ∀X. ⦃G, L⦄ ⊢ Omega1 s ➡[h] X →
+ ∀X. ⦃G,L⦄ ⊢ Omega1 s ➡[h] X →
∨∨ Omega1 s = X | Omega2 s = X.
#h #G #L #s #X #H elim (cpm_inv_appl1 … H) -H *
[ #W2 #T2 #HW2 #HT2 #H destruct
]
qed-.
-theorem cpr_Omega_21_false (h) (G) (L) (s): ⦃G, L⦄ ⊢ Omega2 s ➡[h] Omega1 s → ⊥.
+theorem cpr_Omega_21_false (h) (G) (L) (s): ⦃G,L⦄ ⊢ Omega2 s ➡[h] Omega1 s → ⊥.
#h #G #L #s #H elim (cpm_inv_bind1 … H) -H *
[ #W #T #_ #_ whd in ⊢ (??%?→?); #H destruct
| #X #H #_ #_ #_
⬆*[f] (ApplDelta s0 s) ≘ (ApplDelta s0 s).
/5 width=1 by lifts_sort, lifts_lref, lifts_bind, lifts_flat/ qed.
-lemma cpr_ApplOmega_12 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega1 s0 s ➡[h] ApplOmega2 s0 s.
+lemma cpr_ApplOmega_12 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega1 s0 s ➡[h] ApplOmega2 s0 s.
/2 width=1 by cpm_beta/ qed.
-lemma cpr_ApplOmega_23 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega2 s0 s ➡[h] ApplOmega3 s0 s.
+lemma cpr_ApplOmega_23 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega2 s0 s ➡[h] ApplOmega3 s0 s.
/6 width=3 by cpm_eps, cpm_appl, cpm_bind, cpm_delta, ApplDelta_lifts/ qed.
-lemma cpr_ApplOmega_34 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega3 s0 s ➡[h] ApplOmega4 s0 s.
+lemma cpr_ApplOmega_34 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega3 s0 s ➡[h] ApplOmega4 s0 s.
/4 width=3 by cpm_zeta, ApplDelta_lifts, lifts_sort, lifts_flat/ qed.
-lemma cpxs_ApplOmega_14 (h) (G) (L) (s0) (s): ⦃G, L⦄ ⊢ ApplOmega1 s0 s ⬈*[h] ApplOmega4 s0 s.
+lemma cpxs_ApplOmega_14 (h) (G) (L) (s0) (s): ⦃G,L⦄ ⊢ ApplOmega1 s0 s ⬈*[h] ApplOmega4 s0 s.
/5 width=4 by cpxs_strap1, cpm_fwd_cpx/ qed.
-lemma fqup_ApplOmega_41 (G) (L) (s0) (s): â¦\83G,L,ApplOmega4 s0 sâ¦\84 â\8a\90+ ⦃G,L,ApplOmega1 s0 s⦄.
+lemma fqup_ApplOmega_41 (G) (L) (s0) (s): â¦\83G,L,ApplOmega4 s0 sâ¦\84 â¬\82+ ⦃G,L,ApplOmega1 s0 s⦄.
/2 width=1 by/ qed.
(* Main properties **********************************************************)
-theorem fpbg_refl (h) (o) (G) (L) (s0) (s): ⦃G,L,ApplOmega1 s0 s⦄ >[h,o] ⦃G,L,ApplOmega1 s0 s⦄.
+theorem fpbg_refl (h) (G) (L) (s0) (s): ⦃G,L,ApplOmega1 s0 s⦄ >[h] ⦃G,L,ApplOmega1 s0 s⦄.
/3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ qed.
d2 ≤ d1 → ↑[d2,h2]↑[d1,h1]T = ↑[h2+d1,h1]↑[d2,h2]T.
/3 width=1 by flifts_comp, basic_swap/ qed-.
(*
-lemma flift_join: ∀e1,e2,T. ⬆[e1, e2] ↑[0, e1] T ≡ ↑[0, e1 + e2] T.
+lemma flift_join: ∀e1,e2,T. ⬆[e1,e2] ↑[0,e1] T ≡ ↑[0,e1 + e2] T.
#e1 #e2 #T
lapply (flift_lift T 0 (e1+e2)) #H
elim (lift_split … H e1 e1) -H // #U #H
(* Properties with relocation ***********************************************)
-lemma mf_delta_drops (h) (G): ∀K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 →
+lemma mf_delta_drops (h) (G): ∀K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 →
∀T,L,l. ⬇*[l] L ≘ K.ⓓV1 →
- ∀gv,lv. ⦃G, L⦄ ⊢ ●[gv,⇡[l←#l]lv]T ➡[h] ●[gv,⇡[l←↑[↑l]V2]lv]T.
+ ∀gv,lv. ⦃G,L⦄ ⊢ ●[gv,⇡[l←#l]lv]T ➡[h] ●[gv,⇡[l←↑[↑l]V2]lv]T.
#h #G #K #V1 #V2 #HV #T elim T -T * //
[ #i #L #l #HKL #gv #lv
>mf_lref >mf_lref
(* DENOTATIONAL EQUIVALENCE ************************************************)
definition deq (M): relation4 genv lenv term term ≝
- λG,L,T1,T2. ∀gv,lv. lv ϵ ⟦L⟧[gv] → ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, lv].
+ λG,L,T1,T2. ∀gv,lv. lv ϵ ⟦L⟧[gv] → ⟦T1⟧[gv,lv] ≗{M} ⟦T2⟧[gv,lv].
interpretation "denotational equivalence (model)"
'RingEq M G L T1 T2 = (deq M G L T1 T2).
(* Forward lemmas with context-sensitive parallel reduction for terms *******)
lemma cpr_fwd_deq (h) (M): is_model M → is_extensional M →
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ≗{M} T2.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ≗{M} T2.
#h #M #H1M #H2M #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
[ /2 width=2 by deq_refl/
| #G #K #V1 #V2 #W2 #_ #IH #HVW2 #gv #v #H
inductive li (M) (gv): relation2 lenv (evaluation M) ≝
| li_atom: ∀lv. li M gv (⋆) lv
-| li_abbr: ∀lv,d,L,V. li M gv L lv → ⟦V⟧[gv, lv] = d → li M gv (L.ⓓV) (⫯[0←d]lv)
+| li_abbr: ∀lv,d,L,V. li M gv L lv → ⟦V⟧[gv,lv] = d → li M gv (L.ⓓV) (⫯[0←d]lv)
| li_abst: ∀lv,d,L,W. li M gv L lv → li M gv (L.ⓛW) (⫯[0←d]lv)
| li_unit: ∀lv,d,I,L. li M gv L lv → li M gv (L.ⓤ{I}) (⫯[0←d]lv)
| li_veq : ∀lv1,lv2,L. li M gv L lv1 → lv1 ≗ lv2 → li M gv L lv2
(* Note: application: compatibility *)
mp: compatible_3 … (ap M) (sq M) (sq M) (sq M);
(* Note: interpretation: sort *)
- ms: ∀gv,lv,s. ⟦⋆s⟧{M}[gv, lv] ≗ sv M s;
+ ms: ∀gv,lv,s. ⟦⋆s⟧{M}[gv,lv] ≗ sv M s;
(* Note: interpretation: local reference *)
- ml: ∀gv,lv,i. ⟦#i⟧{M}[gv, lv] ≗ lv i;
+ ml: ∀gv,lv,i. ⟦#i⟧{M}[gv,lv] ≗ lv i;
(* Note: interpretation: global reference *)
- mg: ∀gv,lv,l. ⟦§l⟧{M}[gv, lv] ≗ gv l;
+ mg: ∀gv,lv,l. ⟦§l⟧{M}[gv,lv] ≗ gv l;
(* Note: interpretation: intensional binder *)
- mi: ∀p,gv1,gv2,lv1,lv2,W,T. ⟦W⟧{M}[gv1, lv1] ≗ ⟦W⟧{M}[gv2, lv2] →
- (∀d. ⟦T⟧{M}[gv1, ⫯[0←d]lv1] ≗ ⟦T⟧{M}[gv2, ⫯[0←d]lv2]) →
- ⟦ⓛ{p}W.T⟧[gv1, lv1] ≗ ⟦ⓛ{p}W.T⟧[gv2, lv2];
+ mi: ∀p,gv1,gv2,lv1,lv2,W,T. ⟦W⟧{M}[gv1,lv1] ≗ ⟦W⟧{M}[gv2,lv2] →
+ (∀d. ⟦T⟧{M}[gv1,⫯[0←d]lv1] ≗ ⟦T⟧{M}[gv2,⫯[0←d]lv2]) →
+ ⟦ⓛ{p}W.T⟧[gv1,lv1] ≗ ⟦ⓛ{p}W.T⟧[gv2,lv2];
(* Note: interpretation: abbreviation *)
- md: ∀p,gv,lv,V,T. ⟦ⓓ{p}V.T⟧{M}[gv, lv] ≗ ⟦V⟧[gv, lv] ⊕[p] ⟦T⟧[gv, ⫯[0←⟦V⟧[gv, lv]]lv];
+ md: ∀p,gv,lv,V,T. ⟦ⓓ{p}V.T⟧{M}[gv,lv] ≗ ⟦V⟧[gv,lv] ⊕[p] ⟦T⟧[gv,⫯[0←⟦V⟧[gv,lv]]lv];
(* Note: interpretation: application *)
- ma: ∀gv,lv,V,T. ⟦ⓐV.T⟧{M}[gv, lv] ≗ ⟦V⟧[gv, lv] @ ⟦T⟧[gv, lv];
+ ma: ∀gv,lv,V,T. ⟦ⓐV.T⟧{M}[gv,lv] ≗ ⟦V⟧[gv,lv] @ ⟦T⟧[gv,lv];
(* Note: interpretation: ζ-equivalence *)
mz: ∀d1,d2. d1 ⊕{M}[Ⓣ] d2 ≗ d2;
(* Note: interpretation: ϵ-equivalence *)
- me: ∀gv,lv,W,T. ⟦ⓝW.T⟧{M}[gv, lv] ≗ ⟦T⟧[gv, lv];
+ me: ∀gv,lv,W,T. ⟦ⓝW.T⟧{M}[gv,lv] ≗ ⟦T⟧[gv,lv];
(* Note: interpretation: β-requivalence *)
- mb: ∀p,gv,lv,d,W,T. d @ ⟦ⓛ{p}W.T⟧{M}[gv, lv] ≗ d ⊕[p] ⟦T⟧[gv, ⫯[0←d]lv];
+ mb: ∀p,gv,lv,d,W,T. d @ ⟦ⓛ{p}W.T⟧{M}[gv,lv] ≗ d ⊕[p] ⟦T⟧[gv,⫯[0←d]lv];
(* Note: interpretation: θ-requivalence *)
mh: ∀p,d1,d2,d3. d1 @ (d2 ⊕{M}[p] d3) ≗ d2 ⊕[p] (d1 @ d3)
}.
record is_extensional (M): Prop ≝ {
(* Note: interpretation: extensional abstraction *)
- mx: ∀p,gv1,gv2,lv1,lv2,W1,W2,T1,T2. ⟦W1⟧{M}[gv1, lv1] ≗ ⟦W2⟧{M}[gv2, lv2] →
- (∀d. ⟦T1⟧{M}[gv1, ⫯[0←d]lv1] ≗ ⟦T2⟧{M}[gv2, ⫯[0←d]lv2]) →
- ⟦ⓛ{p}W1.T1⟧[gv1, lv1] ≗ ⟦ⓛ{p}W2.T2⟧[gv2, lv2]
+ mx: ∀p,gv1,gv2,lv1,lv2,W1,W2,T1,T2. ⟦W1⟧{M}[gv1,lv1] ≗ ⟦W2⟧{M}[gv2,lv2] →
+ (∀d. ⟦T1⟧{M}[gv1,⫯[0←d]lv1] ≗ ⟦T2⟧{M}[gv2,⫯[0←d]lv2]) →
+ ⟦ⓛ{p}W1.T1⟧[gv1,lv1] ≗ ⟦ⓛ{p}W2.T2⟧[gv2,lv2]
}.
record is_injective (M): Prop ≝ {
definition tm_dd ≝ term.
-definition tm_sq (h) (T1) (T2) ≝ ⦃⋆, ⋆⦄ ⊢ T1 ⬌*[h] T2.
+definition tm_sq (h) (T1) (T2) ≝ ⦃⋆,⋆⦄ ⊢ T1 ⬌*[h] T2.
definition tm_sv (s) ≝ ⋆s.
lemma ti_comp (M): is_model M →
∀T,gv1,gv2. gv1 ≗ gv2 → ∀lv1,lv2. lv1 ≗ lv2 →
- ⟦T⟧[gv1, lv1] ≗{M} ⟦T⟧[gv2, lv2].
+ ⟦T⟧[gv1,lv1] ≗{M} ⟦T⟧[gv2,lv2].
#M #HM #T elim T -T * [||| #p * | * ]
[ /4 width=5 by seq_trans, seq_sym, ms/
| /4 width=5 by seq_sym, ml, mq/
fact lifts_fwd_vpush_aux (M): is_model M → is_extensional M →
∀f,T1,T2. ⬆*[f] T1 ≘ T2 → ∀m. 𝐁❴m,1❵ = f →
- ∀gv,lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[m←d]lv].
+ ∀gv,lv,d. ⟦T1⟧[gv,lv] ≗{M} ⟦T2⟧[gv,⫯[m←d]lv].
#M #H1M #H2M #f #T1 #T2 #H elim H -f -T1 -T2
[ #f #s #m #Hf #gv #lv #d
@(mq … H1M) [4,5: /3 width=2 by seq_sym, ms/ |1,2: skip ]
lemma lifts_SO_fwd_vpush (M) (gv): is_model M → is_extensional M →
∀T1,T2. ⬆*[1] T1 ≘ T2 →
- ∀lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[0←d]lv].
+ ∀lv,d. ⟦T1⟧[gv,lv] ≗{M} ⟦T2⟧[gv,⫯[0←d]lv].
/2 width=3 by lifts_fwd_vpush_aux/ qed-.
inductive vpushs (M) (gv) (lv): relation2 lenv (evaluation M) ≝
| vpushs_atom: vpushs M gv lv (⋆) lv
-| vpushs_abbr: ∀v,d,K,V. vpushs M gv lv K v → ⟦V⟧[gv, v] = d → vpushs M gv lv (K.ⓓV) (⫯[0←d]v)
+| vpushs_abbr: ∀v,d,K,V. vpushs M gv lv K v → ⟦V⟧[gv,v] = d → vpushs M gv lv (K.ⓓV) (⫯[0←d]v)
| vpushs_abst: ∀v,d,K,V. vpushs M gv lv K v → vpushs M gv lv (K.ⓛV) (⫯[0←d]v)
| vpushs_unit: ∀v,d,I,K. vpushs M gv lv K v → vpushs M gv lv (K.ⓤ{I}) (⫯[0←d]v)
| vpushs_repl: ∀v1,v2,L. vpushs M gv lv L v1 → v1 ≗ v2 → vpushs M gv lv L v2
fact vpushs_inv_abbr_aux (M) (gv) (lv): is_model M →
∀y,L. L ⨁{M}[gv] lv ≘ y →
∀K,V. K.ⓓV = L →
- ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv, v]]v ≗ y.
+ ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv,v]]v ≗ y.
#M #gv #lv #HM #y #L #H elim H -y -L
[ #Y #X #H destruct
| #v #d #K #V #Hv #Hd #_ #Y #X #H destruct
lemma vpushs_inv_abbr (M) (gv) (lv): is_model M →
∀y,K,V. K.ⓓV ⨁{M}[gv] lv ≘ y →
- ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv, v]]v ≗ y.
+ ∃∃v. K ⨁[gv] lv ≘ v & ⫯[0←⟦V⟧[gv,v]]v ≗ y.
/2 width=3 by vpushs_inv_abbr_aux/ qed-.
fact vpushs_inv_abst_aux (M) (gv) (lv): is_model M →
lemma vpushs_fold (M): is_model M → is_extensional M →
∀L,T1,T2,gv,lv.
- (∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv, v] ≗ ⟦T2⟧[gv, v]) →
- ⟦L+T1⟧[gv, lv] ≗{M} ⟦L+T2⟧[gv, lv].
+ (∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv,v] ≗ ⟦T2⟧[gv,v]) →
+ ⟦L+T1⟧[gv,lv] ≗{M} ⟦L+T2⟧[gv,lv].
#M #H1M #H2M #L elim L -L [| #K * [| * ]]
[ #T1 #T2 #gv #lv #H12
>fold_atom >fold_atom
(* Inversion lemmas with fold for restricted closures ***********************)
lemma vpushs_inv_fold (M): is_model M → is_injective M →
- ∀L,T1,T2,gv,lv. ⟦L+T1⟧[gv, lv] ≗{M} ⟦L+T2⟧[gv, lv] →
- ∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv, v] ≗ ⟦T2⟧[gv, v].
+ ∀L,T1,T2,gv,lv. ⟦L+T1⟧[gv,lv] ≗{M} ⟦L+T2⟧[gv,lv] →
+ ∀v. L ⨁[gv] lv ≘ v → ⟦T1⟧[gv,v] ≗ ⟦T2⟧[gv,v].
#M #H1M #H2M #L elim L -L [| #K * [| * ]]
[ #T1 #T2 #gv #lv
>fold_atom >fold_atom #H12 #v #H
(* *)
(**************************************************************************)
+include "static_2/syntax/ac.ma".
include "basic_2/notation/relations/exclaim_5.ma".
-include "basic_2/notation/relations/exclaim_4.ma".
-include "basic_2/notation/relations/exclaimstar_4.ma".
include "basic_2/rt_computation/cpms.ma".
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
(* activate genv *)
(* Basic_2A1: uses: snv *)
-inductive cnv (a) (h): relation3 genv lenv term ≝
-| cnv_sort: ∀G,L,s. cnv a h G L (⋆s)
-| cnv_zero: ∀I,G,K,V. cnv a h G K V → cnv a h G (K.ⓑ{I}V) (#0)
-| cnv_lref: ∀I,G,K,i. cnv a h G K (#i) → cnv a h G (K.ⓘ{I}) (#↑i)
-| cnv_bind: ∀p,I,G,L,V,T. cnv a h G L V → cnv a h G (L.ⓑ{I}V) T → cnv a h G L (ⓑ{p,I}V.T)
-| cnv_appl: ∀n,p,G,L,V,W0,T,U0. (a = Ⓣ → n ≤ 1) → cnv a h G L V → cnv a h G L T →
- ⦃G, L⦄ ⊢ V ➡*[1, h] W0 → ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0 → cnv a h G L (ⓐV.T)
-| cnv_cast: ∀G,L,U,T,U0. cnv a h G L U → cnv a h G L T →
- ⦃G, L⦄ ⊢ U ➡*[h] U0 → ⦃G, L⦄ ⊢ T ➡*[1, h] U0 → cnv a h G L (ⓝU.T)
+inductive cnv (h) (a): relation3 genv lenv term ≝
+| cnv_sort: ∀G,L,s. cnv h a G L (⋆s)
+| cnv_zero: ∀I,G,K,V. cnv h a G K V → cnv h a G (K.ⓑ{I}V) (#0)
+| cnv_lref: ∀I,G,K,i. cnv h a G K (#i) → cnv h a G (K.ⓘ{I}) (#↑i)
+| cnv_bind: ∀p,I,G,L,V,T. cnv h a G L V → cnv h a G (L.ⓑ{I}V) T → cnv h a G L (ⓑ{p,I}V.T)
+| cnv_appl: ∀n,p,G,L,V,W0,T,U0. ad a n → cnv h a G L V → cnv h a G L T →
+ ⦃G,L⦄ ⊢ V ➡*[1,h] W0 → ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0 → cnv h a G L (ⓐV.T)
+| cnv_cast: ∀G,L,U,T,U0. cnv h a G L U → cnv h a G L T →
+ ⦃G,L⦄ ⊢ U ➡*[h] U0 → ⦃G,L⦄ ⊢ T ➡*[1,h] U0 → cnv h a G L (ⓝU.T)
.
interpretation "context-sensitive native validity (term)"
- 'Exclaim a h G L T = (cnv a h G L T).
-
-interpretation "context-sensitive restricted native validity (term)"
- 'Exclaim h G L T = (cnv true h G L T).
-
-interpretation "context-sensitive extended native validity (term)"
- 'ExclaimStar h G L T = (cnv false h G L T).
+ 'Exclaim h a G L T = (cnv h a G L T).
(* Basic inversion lemmas ***************************************************)
-fact cnv_inv_zero_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → X = #0 →
- ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
-#a #h #G #L #X * -G -L -X
+fact cnv_inv_zero_aux (h) (a):
+ ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] → X = #0 →
+ ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![h,a] & L = K.ⓑ{I}V.
+#h #a #G #L #X * -G -L -X
[ #G #L #s #H destruct
| #I #G #K #V #HV #_ /2 width=5 by ex2_3_intro/
| #I #G #K #i #_ #H destruct
]
qed-.
-lemma cnv_inv_zero (a) (h): ∀G,L. ⦃G, L⦄ ⊢ #0 ![a, h] →
- ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
+lemma cnv_inv_zero (h) (a):
+ ∀G,L. ⦃G,L⦄ ⊢ #0 ![h,a] →
+ ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![h,a] & L = K.ⓑ{I}V.
/2 width=3 by cnv_inv_zero_aux/ qed-.
-fact cnv_inv_lref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀i. X = #(↑i) →
- ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
-#a #h #G #L #X * -G -L -X
+fact cnv_inv_lref_aux (h) (a):
+ ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] → ∀i. X = #(↑i) →
+ ∃∃I,K. ⦃G,K⦄ ⊢ #i ![h,a] & L = K.ⓘ{I}.
+#h #a #G #L #X * -G -L -X
[ #G #L #s #j #H destruct
| #I #G #K #V #_ #j #H destruct
| #I #G #L #i #Hi #j #H destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma cnv_inv_lref (a) (h): ∀G,L,i. ⦃G, L⦄ ⊢ #↑i ![a, h] →
- ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
+lemma cnv_inv_lref (h) (a):
+ ∀G,L,i. ⦃G,L⦄ ⊢ #↑i ![h,a] →
+ ∃∃I,K. ⦃G,K⦄ ⊢ #i ![h,a] & L = K.ⓘ{I}.
/2 width=3 by cnv_inv_lref_aux/ qed-.
-fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀l. X = §l → ⊥.
-#a #h #G #L #X * -G -L -X
+fact cnv_inv_gref_aux (h) (a): ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] → ∀l. X = §l → ⊥.
+#h #a #G #L #X * -G -L -X
[ #G #L #s #l #H destruct
| #I #G #K #V #_ #l #H destruct
| #I #G #K #i #_ #l #H destruct
qed-.
(* Basic_2A1: uses: snv_inv_gref *)
-lemma cnv_inv_gref (a) (h): ∀G,L,l. ⦃G, L⦄ ⊢ §l ![a, h] → ⊥.
+lemma cnv_inv_gref (h) (a): ∀G,L,l. ⦃G,L⦄ ⊢ §l ![h,a] → ⊥.
/2 width=8 by cnv_inv_gref_aux/ qed-.
-fact cnv_inv_bind_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] →
- ∀p,I,V,T. X = ⓑ{p,I}V.T →
- ∧∧ ⦃G, L⦄ ⊢ V ![a, h]
- & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
-#a #h #G #L #X * -G -L -X
+fact cnv_inv_bind_aux (h) (a):
+ ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] →
+ ∀p,I,V,T. X = ⓑ{p,I}V.T →
+ ∧∧ ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![h,a].
+#h #a #G #L #X * -G -L -X
[ #G #L #s #q #Z #X1 #X2 #H destruct
| #I #G #K #V #_ #q #Z #X1 #X2 #H destruct
| #I #G #K #i #_ #q #Z #X1 #X2 #H destruct
qed-.
(* Basic_2A1: uses: snv_inv_bind *)
-lemma cnv_inv_bind (a) (h): ∀p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T ![a, h] →
- ∧∧ ⦃G, L⦄ ⊢ V ![a, h]
- & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
+lemma cnv_inv_bind (h) (a):
+ ∀p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![h,a] →
+ ∧∧ ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![h,a].
/2 width=4 by cnv_inv_bind_aux/ qed-.
-fact cnv_inv_appl_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀V,T. X = ⓐV.T →
- ∃∃n,p,W0,U0. a = Ⓣ → n ≤ 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
-#a #h #G #L #X * -L -X
+fact cnv_inv_appl_aux (h) (a):
+ ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] → ∀V,T. X = ⓐV.T →
+ ∃∃n,p,W0,U0. ad a n & ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] &
+ ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0.
+#h #a #G #L #X * -L -X
[ #G #L #s #X1 #X2 #H destruct
| #I #G #K #V #_ #X1 #X2 #H destruct
| #I #G #K #i #_ #X1 #X2 #H destruct
qed-.
(* Basic_2A1: uses: snv_inv_appl *)
-lemma cnv_inv_appl (a) (h): ∀G,L,V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
- ∃∃n,p,W0,U0. a = Ⓣ → n ≤ 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
+lemma cnv_inv_appl (h) (a):
+ ∀G,L,V,T. ⦃G,L⦄ ⊢ ⓐV.T ![h,a] →
+ ∃∃n,p,W0,U0. ad a n & ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] &
+ ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0.
/2 width=3 by cnv_inv_appl_aux/ qed-.
-fact cnv_inv_cast_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀U,T. X = ⓝU.T →
- ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
-#a #h #G #L #X * -G -L -X
+fact cnv_inv_cast_aux (h) (a):
+ ∀G,L,X. ⦃G,L⦄ ⊢ X ![h,a] → ∀U,T. X = ⓝU.T →
+ ∃∃U0. ⦃G,L⦄ ⊢ U ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] &
+ ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0.
+#h #a #G #L #X * -G -L -X
[ #G #L #s #X1 #X2 #H destruct
| #I #G #K #V #_ #X1 #X2 #H destruct
| #I #G #K #i #_ #X1 #X2 #H destruct
]
qed-.
-(* Basic_2A1: uses: snv_inv_appl *)
-lemma cnv_inv_cast (a) (h): ∀G,L,U,T. ⦃G, L⦄ ⊢ ⓝU.T ![a, h] →
- ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
+(* Basic_2A1: uses: snv_inv_cast *)
+lemma cnv_inv_cast (h) (a):
+ ∀G,L,U,T. ⦃G,L⦄ ⊢ ⓝU.T ![h,a] →
+ ∃∃U0. ⦃G,L⦄ ⊢ U ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] &
+ ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0.
/2 width=3 by cnv_inv_cast_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma cnv_fwd_flat (a) (h) (I) (G) (L):
- ∀V,T. ⦃G, L⦄ ⊢ ⓕ{I}V.T ![a,h] →
- ∧∧ ⦃G, L⦄ ⊢ V ![a,h] & ⦃G, L⦄ ⊢ T ![a,h].
-#a #h * #G #L #V #T #H
+lemma cnv_fwd_flat (h) (a) (I) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ⓕ{I}V.T ![h,a] →
+ ∧∧ ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ T ![h,a].
+#h #a * #G #L #V #T #H
[ elim (cnv_inv_appl … H) #n #p #W #U #_ #HV #HT #_ #_
| elim (cnv_inv_cast … H) #U #HV #HT #_ #_
] -H /2 width=1 by conj/
qed-.
+lemma cnv_fwd_pair_sn (h) (a) (I) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ②{I}V.T ![h,a] → ⦃G,L⦄ ⊢ V ![h,a].
+#h #a * [ #p ] #I #G #L #V #T #H
+[ elim (cnv_inv_bind … H) -H #HV #_
+| elim (cnv_fwd_flat … H) -H #HV #_
+] //
+qed-.
+
(* Basic_2A1: removed theorems 3:
shnv_cast shnv_inv_cast snv_shnv_cast
*)
include "basic_2/rt_computation/cpms_aaa.ma".
include "basic_2/dynamic/cnv.ma".
-(* CONTEXT_SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
(* Forward lemmas on atomic arity assignment for terms **********************)
(* Basic_2A1: uses: snv_fwd_aaa *)
-lemma cnv_fwd_aaa (a) (h): ∀G,L,T. ⦃G, L⦄ ⊢ T ![a, h] → ∃A. ⦃G, L⦄ ⊢ T ⁝ A.
-#a #h #G #L #T #H elim H -G -L -T
+lemma cnv_fwd_aaa (h) (a): ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] → ∃A. ⦃G,L⦄ ⊢ T ⁝ A.
+#h #a #G #L #T #H elim H -G -L -T
[ /2 width=2 by aaa_sort, ex_intro/
| #I #G #L #V #_ * /3 width=2 by aaa_zero, ex_intro/
| #I #G #L #K #_ * /3 width=2 by aaa_lref, ex_intro/
(* Forward lemmas with t_bound rt_transition for terms **********************)
-lemma cnv_fwd_cpm_SO (a) (h) (G) (L):
- ∀T. ⦃G, L⦄ ⊢ T ![a, h] → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U.
-#a #h #G #L #T #H
+lemma cnv_fwd_cpm_SO (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U.
+#h #a #G #L #T #H
elim (cnv_fwd_aaa … H) -H #A #HA
/2 width=2 by aaa_cpm_SO/
qed-.
(* Forward lemmas with t_bound rt_computation for terms *********************)
-lemma cnv_fwd_cpms_total (a) (h) (n) (G) (L):
- ∀T. ⦃G, L⦄ ⊢ T ![a, h] → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
-#a #h #n #G #L #T #H
+lemma cnv_fwd_cpms_total (h) (a) (n) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
+#h #a #n #G #L #T #H
elim (cnv_fwd_aaa … H) -H #A #HA
-/2 width=2 by aaa_cpms_total/
+/2 width=2 by cpms_total_aaa/
qed-.
+
+lemma cnv_fwd_cpms_abst_dx_le (h) (a) (G) (L) (W) (p):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
+ ∀n1,U1. ⦃G,L⦄ ⊢ T ➡*[n1,h] ⓛ{p}W.U1 → ∀n2. n1 ≤ n2 →
+ ∃∃U2. ⦃G,L⦄ ⊢ T ➡*[n2,h] ⓛ{p}W.U2 & ⦃G,L.ⓛW⦄ ⊢ U1 ➡*[n2-n1,h] U2.
+#h #a #G #L #W #p #T #H
+elim (cnv_fwd_aaa … H) -H #A #HA
+/2 width=2 by cpms_abst_dx_le_aaa/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cnv_appl_ge (h) (a) (n1) (p) (G) (L):
+ ∀n2. n1 ≤ n2 → ad a n2 →
+ ∀V. ⦃G,L⦄ ⊢ V ![h,a] → ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
+ ∀X. ⦃G,L⦄ ⊢ V ➡*[1,h] X → ∀W. ⦃G,L⦄ ⊢ W ➡*[h] X →
+ ∀U. ⦃G,L⦄ ⊢ T ➡*[n1,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T ![h,a].
+#h #a #n1 #p #G #L #n2 #Hn12 #Ha #V #HV #T #HT #X #HVX #W #HW #X #HTX
+elim (cnv_fwd_cpms_abst_dx_le … HT … HTX … Hn12) #U #HTU #_ -n1
+/4 width=11 by cnv_appl, cpms_bind, cpms_cprs_trans/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/acle.ma".
+include "basic_2/dynamic/cnv_aaa.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Properties with preorder for applicability domains ***********************)
+
+lemma cnv_acle_trans (h) (a1) (a2):
+ a1 ⊆ a2 → ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a1] → ⦃G,L⦄ ⊢ T ![h,a2].
+#h #a1 #a2 #Ha12 #G #L #T #H elim H -G -L -T
+[ /1 width=1 by cnv_sort/
+| /3 width=1 by cnv_zero/
+| /3 width=1 by cnv_lref/
+| /3 width=1 by cnv_bind/
+| #n1 #p #G #L #V #W #T #U #Hn1 #_ #_ #HVW #HTU #IHV #IHT
+ elim (Ha12 … Hn1) -a1 #n2 #Hn2 #Hn12
+ /3 width=11 by cnv_appl_ge/
+| /3 width=3 by cnv_cast/
+]
+qed-.
+
+lemma cnv_acle_omega (h) (a):
+ ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] → ⦃G,L⦄ ⊢ T ![h,𝛚].
+/3 width=3 by cnv_acle_trans, acle_omega/ qed-.
+
+lemma cnv_acle_one (h) (a) (n):
+ ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,𝟏] → ad a n → ⦃G,L⦄ ⊢ T ![h,a].
+/3 width=3 by cnv_acle_trans, acle_one/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpmuwe_cpmuwe.ma".
+include "basic_2/rt_conversion/cpce_drops.ma".
+include "basic_2/dynamic/cnv_cpmuwe.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Properties with context-sensitive parallel eta-conversion for terms ******)
+
+lemma cpce_total_cnv (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∃T2. ⦃G,L⦄ ⊢ T1 ⬌η[h] T2.
+#h #a #G #L #T1 #HT1
+lapply (cnv_fwd_csx … HT1) #H
+generalize in match HT1; -HT1
+@(csx_ind_fpbg … H) -G -L -T1
+#G #L * *
+[ #s #_ #_ /2 width=2 by cpce_sort, ex_intro/
+| #i #H1i #IH #H2i
+ elim (drops_ldec_dec L i) [ * #K #W #HLK | -H1i -IH #HnX ]
+ [ lapply (cnv_inv_lref_pair … H2i … HLK) -H2i #H2W
+ lapply (csx_inv_lref_pair_drops … HLK H1i) -H1i #H1W
+ elim (cpmuwe_total_csx … H1W) -H1W #X #n #HWX
+ elim (abst_dec X) [ * | -IH ]
+ [ #p #V1 #U #H destruct
+ lapply (cpmuwe_fwd_cpms … HWX) -HWX #HWX
+ elim (IH G K V1) -IH
+ [ #V2 #HV12
+ elim (lifts_total V2 (𝐔❴↑i❵)) #W2 #HVW2
+ /3 width=12 by cpce_eta_drops, ex_intro/
+ | /3 width=6 by cnv_cpms_trans, cnv_fwd_pair_sn/
+ | /4 width=6 by fqup_cpms_fwd_fpbg, fpbg_fqu_trans, fqup_lref/
+ ]
+ | #HnX
+ @(ex_intro … (#i))
+ @cpce_zero_drops #n0 #p #K0 #W0 #V0 #U0 #HLK0 #HWU0
+ lapply (drops_mono … HLK0 … HLK) -i -L #H destruct
+ lapply (cpmuwe_abst … HWU0) -HWU0 #HWU0
+ elim (cnv_cpmuwe_mono … H2W … HWU0 … HWX) #_ #H -a -n -n0 -W
+ elim (tweq_inv_abst_sn … H) -V0 -U0 #V0 #U0 #H destruct
+ /2 width=4 by/
+ ]
+ | /5 width=3 by cpce_zero_drops, ex1_2_intro, ex_intro/
+ ]
+| #l #_ #_ /2 width=2 by cpce_gref, ex_intro/
+| #p #I #V1 #T1 #_ #IH #H
+ elim (cnv_inv_bind … H) -H #HV1 #HT1
+ elim (IH … HV1) [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_pair_sn/ ] #V2 #HV12
+ elim (IH … HT1) [| /4 width=1 by fpb_fpbg, fpb_fqu, fqu_bind_dx/ ] #T2 #HT12
+ /3 width=2 by cpce_bind, ex_intro/
+| #I #V1 #T1 #_ #IH #H
+ elim (cnv_fwd_flat … H) -H #HV1 #HT1
+ elim (IH … HV1) [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_pair_sn/ ] #V2 #HV12
+ elim (IH … HT1) [| /3 width=1 by fpb_fpbg, fpb_fqu, fqu_flat_dx/ ] #T2 #HT12
+ /3 width=2 by cpce_flat, ex_intro/
+]
+qed-.
(* *)
(**************************************************************************)
-include "basic_2/rt_equivalence/cpcs_cprs.ma".
-include "basic_2/dynamic/cnv_preserve.ma".
+include "basic_2/rt_computation/csx_aaa.ma".
+include "basic_2/rt_equivalence/cpcs_csx.ma".
+include "basic_2/dynamic/cnv_aaa.ma".
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-(* Forward lemmas with r-equivalence ****************************************)
+(* Properties with r-equivalence ********************************************)
-lemma cnv_cpms_conf_eq (a) (h) (n) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] →
- ∀T1. ⦃G,L⦄ ⊢ T ➡*[n,h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
-#a #h #n #G #L #T #HT #T1 #HT1 #T2 #HT2
-elim (cnv_cpms_conf … HT … HT1 … HT2) -T <minus_n_n #T #HT1 #HT2
-/2 width=3 by cprs_div/
-qed-.
-
-lemma cnv_cpms_fwd_appl_sn_decompose (a) (h) (G) (L):
- ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![a,h] → ∀n,X. ⦃G,L⦄ ⊢ ⓐV.T ➡*[n,h] X →
- ∃∃U. ⦃G,L⦄ ⊢ T ![a,h] & ⦃G,L⦄ ⊢ T ➡*[n,h] U & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X.
-#a #h #G #L #V #T #H0 #n #X #HX
-elim (cnv_inv_appl … H0) #m #p #W #U #_ #_ #HT #_ #_ -m -p -W -U
-elim (cnv_fwd_cpms_total … h n … HT) #U #HTU
-lapply (cpms_appl_dx … V V … HTU) [ // ] #H
-/3 width=8 by cnv_cpms_conf_eq, ex3_intro/
+lemma cnv_cpcs_dec (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∀T2. ⦃G,L⦄ ⊢ T2 ![h,a] →
+ Decidable … (⦃G,L⦄ ⊢ T1 ⬌*[h] T2).
+#h #a #G #L #T1 #HT1 #T2 #HT2
+elim (cnv_fwd_aaa … HT1) -HT1 #A1 #HA1
+elim (cnv_fwd_aaa … HT2) -HT2 #A2 #HA2
+/3 width=2 by csx_cpcs_dec, aaa_csx/
qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpms_cpms.ma".
+include "basic_2/rt_equivalence/cpes.ma".
+include "basic_2/dynamic/cnv.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Properties with t-bound rt-equivalence for terms *************************)
+
+lemma cnv_appl_cpes (h) (a) (G) (L):
+ ∀n. ad a n →
+ ∀V. ⦃G,L⦄ ⊢ V ![h,a] → ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
+ ∀W. ⦃G,L⦄ ⊢ V ⬌*[h,1,0] W →
+ ∀p,U. ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T ![h,a].
+#h #a #G #L #n #Hn #V #HV #T #HT #W *
+/4 width=11 by cnv_appl, cpms_cprs_trans, cpms_bind/
+qed.
+
+lemma cnv_cast_cpes (h) (a) (G) (L):
+ ∀U. ⦃G,L⦄ ⊢ U ![h,a] →
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T → ⦃G,L⦄ ⊢ ⓝU.T ![h,a].
+#h #a #G #L #U #HU #T #HT * /2 width=3 by cnv_cast/
+qed.
+
+(* Inversion lemmas with t-bound rt-equivalence for terms *******************)
+
+lemma cnv_inv_appl_cpes (h) (a) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![h,a] →
+ ∃∃n,p,W,U. ad a n & ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] &
+ ⦃G,L⦄ ⊢ V ⬌*[h,1,0] W & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U.
+#h #a #G #L #V #T #H
+elim (cnv_inv_appl … H) -H #n #p #W #U #Hn #HV #HT #HVW #HTU
+/3 width=7 by cpms_div, ex5_4_intro/
+qed-.
+
+lemma cnv_inv_cast_cpes (h) (a) (G) (L):
+ ∀U,T. ⦃G,L⦄ ⊢ ⓝU.T ![h,a] →
+ ∧∧ ⦃G,L⦄ ⊢ U ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] & ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T.
+#h #a #G #L #U #T #H
+elim (cnv_inv_cast … H) -H
+/3 width=3 by cpms_div, and3_intro/
+qed-.
+
+(* Eliminators with t-bound rt-equivalence for terms ************************)
+
+lemma cnv_ind_cpes (h) (a) (Q:relation3 genv lenv term):
+ (∀G,L,s. Q G L (⋆s)) →
+ (∀I,G,K,V. ⦃G,K⦄ ⊢ V![h,a] → Q G K V → Q G (K.ⓑ{I}V) (#O)) →
+ (∀I,G,K,i. ⦃G,K⦄ ⊢ #i![h,a] → Q G K (#i) → Q G (K.ⓘ{I}) (#(↑i))) →
+ (∀p,I,G,L,V,T. ⦃G,L⦄ ⊢ V![h,a] → ⦃G,L.ⓑ{I}V⦄⊢T![h,a] →
+ Q G L V →Q G (L.ⓑ{I}V) T →Q G L (ⓑ{p,I}V.T)
+ ) →
+ (∀n,p,G,L,V,W,T,U. ad a n → ⦃G,L⦄ ⊢ V![h,a] → ⦃G,L⦄ ⊢ T![h,a] →
+ ⦃G,L⦄ ⊢ V ⬌*[h,1,0]W → ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U →
+ Q G L V → Q G L T → Q G L (ⓐV.T)
+ ) →
+ (∀G,L,U,T. ⦃G,L⦄⊢ U![h,a] → ⦃G,L⦄ ⊢ T![h,a] → ⦃G,L⦄ ⊢ U ⬌*[h,0,1] T →
+ Q G L U → Q G L T → Q G L (ⓝU.T)
+ ) →
+ ∀G,L,T. ⦃G,L⦄⊢ T![h,a] → Q G L T.
+#h #a #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #G #L #T #H
+elim H -G -L -T [5,6: /3 width=7 by cpms_div/ |*: /2 width=1 by/ ]
+qed-.
(* Sub diamond propery with t-bound rt-transition for terms *****************)
fact cnv_cpm_conf_lpr_atom_atom_aux (h) (G) (L1) (L2) (I):
- ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡*[0,h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡*[O,h] T.
+ ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡*[0,h] T & ⦃G,L2⦄ ⊢ ⓪{I} ➡*[O,h] T.
/2 width=3 by ex2_intro/ qed-.
fact cnv_cpm_conf_lpr_atom_ess_aux (h) (G) (L1) (L2) (s):
- ∃∃T. ⦃G,L1⦄ ⊢ ⋆s ➡*[1,h] T & ⦃G,L2⦄ ⊢ ⋆(next h s) ➡*[h] T.
+ ∃∃T. ⦃G,L1⦄ ⊢ ⋆s ➡*[1,h] T & ⦃G,L2⦄ ⊢ ⋆(⫯[h]s) ➡*[h] T.
/3 width=3 by cpm_cpms, ex2_intro/ qed-.
-fact cnv_cpm_conf_lpr_atom_delta_aux (a) (h) (o) (G) (L) (i):
- (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄⊢#i![a,h] →
+fact cnv_cpm_conf_lpr_atom_delta_aux (h) (a) (G) (L) (i):
+ (∀G0,L0,T0. ⦃G,L,#i⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄⊢#i![h,a] →
∀K,V. ⬇*[i]L ≘ K.ⓓV →
∀n,XV. ⦃G,K⦄ ⊢ V ➡[n,h] XV →
∀X. ⬆*[↑i]XV ≘ X →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ #i ➡*[n,h] T & ⦃G,L2⦄ ⊢ X ➡*[h] T.
-#a #h #o #G #L #i #IH #HT #K #V #HLK #n #XV #HVX #X #HXV #L1 #HL1 #L2 #HL2
-lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HV
+#h #a #G #L #i #IH #HT #K #V #HLK #n #XV #HVX #X #HXV #L1 #HL1 #L2 #HL2
+lapply (cnv_inv_lref_pair … HT … HLK) -HT #HV
elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK1 #HV1 #H destruct
elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2
/3 width=6 by cpms_delta_drops, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_atom_ell_aux (a) (h) (o) (G) (L) (i):
- (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄⊢#i![a,h] →
+fact cnv_cpm_conf_lpr_atom_ell_aux (h) (a) (G) (L) (i):
+ (∀G0,L0,T0. ⦃G,L,#i⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄⊢#i![h,a] →
∀K,W. ⬇*[i]L ≘ K.ⓛW →
∀n,XW. ⦃G,K⦄ ⊢ W ➡[n,h] XW →
∀X. ⬆*[↑i]XW ≘ X →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ #i ➡*[↑n,h] T & ⦃G,L2⦄ ⊢ X ➡*[h] T.
-#a #h #o #G #L #i #IH #HT #K #W #HLK #n #XW #HWX #X #HXW #L1 #HL1 #L2 #HL2
-lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HW
+#h #a #G #L #i #IH #HT #K #W #HLK #n #XW #HWX #X #HXW #L1 #HL1 #L2 #HL2
+lapply (cnv_inv_lref_pair … HT … HLK) -HT #HW
elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #HK1 #HW1 #H destruct
elim (lpr_drops_conf … HLK … HL2) -HL2 // #Y2 #H2 #HLK2
/3 width=6 by cpms_ell_drops, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_delta_delta_aux (a) (h) (o) (I) (G) (L) (i):
- (∀G0,L0,T0. ⦃G,L,#i⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄⊢#i![a,h] →
+fact cnv_cpm_conf_lpr_delta_delta_aux (h) (a) (I) (G) (L) (i):
+ (∀G0,L0,T0. ⦃G,L,#i⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄⊢#i![h,a] →
∀K1,V1. ⬇*[i]L ≘ K1.ⓑ{I}V1 → ∀K2,V2. ⬇*[i]L ≘ K2.ⓑ{I}V2 →
∀n1,XV1. ⦃G,K1⦄ ⊢ V1 ➡[n1,h] XV1 → ∀n2,XV2. ⦃G,K2⦄ ⊢ V2 ➡[n2,h] XV2 →
∀X1. ⬆*[↑i]XV1 ≘ X1 → ∀X2. ⬆*[↑i]XV2 ≘ X2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ X1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ X2 ➡*[n1-n2,h] T.
-#a #h #o #I #G #L #i #IH #HT
+#h #a #I #G #L #i #IH #HT
#K #V #HLK #Y #X #HLY #n1 #XV1 #HVX1 #n2 #XV2 #HVX2 #X1 #HXV1 #X2 #HXV2
#L1 #HL1 #L2 #HL2
lapply (drops_mono … HLY … HLK) -HLY #H destruct
-lapply (cnv_lref_fwd_drops … HT … HLK) -HT #HV
+lapply (cnv_inv_lref_pair … HT … HLK) -HT #HV
elim (lpr_drops_conf … HLK … HL1) -HL1 // #Y1 #H1 #HLK1
elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK1 #_ #H destruct
lapply (drops_isuni_fwd_drop2 … HLK1) -V1 // #HLK1
lapply (drops_mono … HLK2 … HLK1) -L -i #H destruct
qed-.
-fact cnv_cpm_conf_lpr_bind_bind_aux (a) (h) (o) (p) (I) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓑ{p,I}V.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![a,h] →
+fact cnv_cpm_conf_lpr_bind_bind_aux (h) (a) (p) (I) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓑ{p,I}V.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀n1,T1. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡*[n1-n2,h] T.
-#a #h #o #p #I #G0 #L0 #V0 #T0 #IH #H0
+#h #a #p #I #G0 #L0 #V0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #n1 #T1 #HT01 #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_bind … H0) -H0 #HV0 #HT0
/3 width=5 by cpms_bind_dx, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_bind_zeta_aux (a) (h) (o) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ +ⓓV.T ![a,h] →
+fact cnv_cpm_conf_lpr_bind_zeta_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ +ⓓV.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢V ➡[h] V1 → ∀n1,T1. ⦃G,L.ⓓV⦄ ⊢ T ➡[n1,h] T1 →
∀T2. ⬆*[1]T2 ≘ T → ∀n2,XT2. ⦃G,L⦄ ⊢ T2 ➡[n2,h] XT2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ +ⓓV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡*[n1-n2,h] T.
-#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#V1 #HV01 #n1 #T1 #HT01 #T2 #HT20 #n2 #XT2 #HXT2
#L1 #HL01 #L2 #HL02
elim (cnv_inv_bind … H0) -H0 #_ #HT0
/3 width=3 by cpms_zeta, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_zeta_zeta_aux (a) (h) (o) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ +ⓓV.T ![a,h] →
+fact cnv_cpm_conf_lpr_zeta_zeta_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,+ⓓV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ +ⓓV.T ![h,a] →
∀T1. ⬆*[1]T1 ≘ T → ∀T2. ⬆*[1]T2 ≘ T →
∀n1,XT1. ⦃G,L⦄ ⊢ T1 ➡[n1,h] XT1 → ∀n2,XT2. ⦃G,L⦄ ⊢ T2 ➡[n2,h] XT2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ XT1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ XT2 ➡*[n1-n2,h] T.
-#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#T1 #HT10 #T2 #HT20 #n1 #XT1 #HXT1 #n2 #XT2 #HXT2
#L1 #HL01 #L2 #HL02
elim (cnv_inv_bind … H0) -H0 #_ #HT0
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_appl_appl_aux (a) (h) (o) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓐV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
+fact cnv_cpm_conf_lpr_appl_appl_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓐV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓐV2.T2 ➡*[n1-n2,h] T.
-#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #n1 #T1 #HT01 #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #HT0 #_ #_ -n0 -p0 -X01 -X02
/3 width=5 by cpms_appl_dx, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_appl_beta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
- (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] →
+fact cnv_cpm_conf_lpr_appl_beta_aux (h) (a) (p) (G) (L) (V) (W) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
∀n1,T1. ⦃G,L⦄ ⊢ ⓛ{p}W.T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡*[n1-n2,h] T.
-#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#h #a #p #G0 #L0 #V0 #W0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #W2 #HW02 #n1 #X #HX #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
/4 width=5 by cpms_beta_dx, cpms_bind_dx, cpm_cast, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_appl_theta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
- (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] →
+fact cnv_cpm_conf_lpr_appl_theta_aux (h) (a) (p) (G) (L) (V) (W) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
∀n1,T1. ⦃G,L⦄ ⊢ ⓓ{p}W.T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 →
∀U2. ⬆*[1]V2 ≘ U2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓐV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡*[n1-n2,h] T.
-#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#h #a #p #G0 #L0 #V0 #W0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #W2 #HW02 #n1 #X #HX #n2 #T2 #HT02 #U2 #HVU2
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
]
qed-.
-fact cnv_cpm_conf_lpr_beta_beta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
- (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![a,h] →
+fact cnv_cpm_conf_lpr_beta_beta_aux (h) (a) (p) (G) (L) (V) (W) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓐV.ⓛ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀W1. ⦃G,L⦄ ⊢ W ➡[h] W1 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
∀n1,T1. ⦃G,L.ⓛW⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓛW⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡*[n1-n2,h] T.
-#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#h #a #p #G0 #L0 #V0 #W0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #W1 #HW01 #W2 #HW02 #n1 #T1 #HT01 #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
/4 width=5 by cpms_bind_dx, cpm_eps, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_theta_theta_aux (a) (h) (o) (p) (G) (L) (V) (W) (T):
- (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![a,h] →
+fact cnv_cpm_conf_lpr_theta_theta_aux (h) (a) (p) (G) (L) (V) (W) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓐV.ⓓ{p}W.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓐV.ⓓ{p}W.T ![h,a] →
∀V1. ⦃G,L⦄ ⊢ V ➡[h] V1 → ∀V2. ⦃G,L⦄ ⊢ V ➡[h] V2 →
∀W1. ⦃G,L⦄ ⊢ W ➡[h] W1 → ∀W2. ⦃G,L⦄ ⊢ W ➡[h] W2 →
∀n1,T1. ⦃G,L.ⓓW⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L.ⓓW⦄ ⊢ T ➡[n2,h] T2 →
∀U1. ⬆*[1]V1 ≘ U1 → ∀U2. ⬆*[1]V2 ≘ U2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡*[n1-n2,h] T.
-#a #h #o #p #G0 #L0 #V0 #W0 #T0 #IH #H0
+#h #a #p #G0 #L0 #V0 #W0 #T0 #IH #H0
#V1 #HV01 #V2 #HV02 #W1 #HW01 #W2 #HW02 #n1 #T1 #HT01 #n2 #T2 #HT02 #U1 #HVU1 #U2 #HVU2
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #HV0 #H0 #_ #_ -n0 -p0 -X01 -X02
/4 width=7 by cpms_appl_dx, cpms_bind_dx, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_cast_cast_aux (a) (h) (o) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_cast_cast_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ ⓝV2.T2 ➡*[n1-n2,h] T.
-#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #V1 #HV01 #n2 #V2 #HV02 #T1 #HT01 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
/3 width=5 by cpms_cast, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_cast_epsilon_aux (a) (h) (o) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_cast_epsilon_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 →
∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
-#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #V1 #HV01 #T1 #HT01 #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
/3 width=3 by cpms_eps, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_cast_ee_aux (a) (h) (o) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_cast_ee_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr h a G0 L0 T0) →
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
∀T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ ⓝV1.T1 ➡*[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-↑n2,h] T.
-#a #h #o #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
+#h #a #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
#n1 #V1 #HV01 #n2 #V2 #HV02 #T1 #HT01
#L1 #HL01 #L2 #HL02 -HV01
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0
/3 width=3 by cpms_eps, ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_epsilon_epsilon_aux (a) (h) (o) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_epsilon_epsilon_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
-#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #T1 #HT01 #n2 #T2 #HT02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #_ #HT0 #_ #_ -X0
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_epsilon_ee_aux (a) (h) (o) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_epsilon_ee_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr h a G0 L0 T0) →
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,T1. ⦃G,L⦄ ⊢ T ➡[n1,h] T1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[↑n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-↑n2,h] T.
-#a #h #o #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
+#h #a #G0 #L0 #V0 #T0 #IH2 #IH1 #H0
#n1 #T1 #HT01 #n2 #V2 #HV02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #HVX0 #HTX0
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_ee_ee_aux (a) (h) (o) (G) (L) (V) (T):
- (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- ⦃G,L⦄ ⊢ ⓝV.T ![a,h] →
+fact cnv_cpm_conf_lpr_ee_ee_aux (h) (a) (G) (L) (V) (T):
+ (∀G0,L0,T0. ⦃G,L,ⓝV.T⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ ⦃G,L⦄ ⊢ ⓝV.T ![h,a] →
∀n1,V1. ⦃G,L⦄ ⊢ V ➡[n1,h] V1 → ∀n2,V2. ⦃G,L⦄ ⊢ V ➡[n2,h] V2 →
∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G,L1⦄ ⊢ V1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ V2 ➡*[n1-n2,h] T.
-#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #V1 #HV01 #n2 #V2 #HV02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #_ #_ #_ -X0
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpm_conf_lpr_aux (a) (h) (o):
- ∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
-#a #h #o #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
+fact cnv_cpm_conf_lpr_aux (h) (a):
+ ∀G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr h a G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_conf_lpr h a G1 L1 T1.
+#h #a #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
[ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #HX1 #n2 #X2 #HX2 #L1 #HL1 #L2 #HL2 destruct
elim (cpm_inv_atom1_drops … HX1) -HX1 *
elim (cpm_inv_atom1_drops … HX2) -HX2 *
- [ #H21 #H22 #H11 #H12 destruct -a -o -L
+ [ #H21 #H22 #H11 #H12 destruct -a -L
<minus_O_n
/2 width=1 by cnv_cpm_conf_lpr_atom_atom_aux/
- | #s2 #H21 #H22 #H23 #H11 #H12 destruct -a -o -L
+ | #s2 #H21 #H22 #H23 #H11 #H12 destruct -a -L
<minus_O_n <minus_n_O
/2 width=1 by cnv_cpm_conf_lpr_atom_ess_aux/
| #K2 #V2 #XV2 #i #HLK2 #HVX2 #HXV2 #H21 #H11 #H12 destruct -IH2
| #m2 #K2 #W2 #XW2 #i #HLK2 #HWX2 #HXW2 #H21 #H22 #H11 #H12 destruct -IH2
<minus_O_n <minus_n_O
@(cnv_cpm_conf_lpr_atom_ell_aux … IH1) -IH1 /1 width=6 by/
- | #H21 #H22 #s1 #H11 #H12 #H13 destruct -a -o -L
+ | #H21 #H22 #s1 #H11 #H12 #H13 destruct -a -L
<minus_O_n <minus_n_O
/3 width=1 by cnv_cpm_conf_lpr_atom_ess_aux, ex2_commute/
- | #s2 #H21 #H22 #H23 #s1 #H11 #H12 #H13 destruct -a -o -L
+ | #s2 #H21 #H22 #H23 #s1 #H11 #H12 #H13 destruct -a -L
<minus_n_n
/2 width=1 by cnv_cpm_conf_lpr_atom_atom_aux/
| #K2 #V2 #XV2 #i2 #_ #_ #_ #H21 #s1 #H11 #H12 #H13 destruct
| #s2 #H21 #H22 #H23 #K1 #V1 #XV1 #i1 #_ #_ #_ #H11 destruct
| #K2 #V2 #XV2 #i2 #HLK2 #HVX2 #HXV2 #H21 #K1 #V1 #XV1 #i1 #HLK1 #HVX1 #HXV1 #H11 destruct -IH2
@(cnv_cpm_conf_lpr_delta_delta_aux … IH1) -IH1 /1 width=13 by/
- | #m2 #K2 #W2 #XW2 #i2 #HLK2 #_ #_ #H21 #H22 #K1 #V1 #XV1 #i1 #HLK1 #_ #_ #H11 destruct -a -o -XW2 -XV1 -HL2 -HL1
+ | #m2 #K2 #W2 #XW2 #i2 #HLK2 #_ #_ #H21 #H22 #K1 #V1 #XV1 #i1 #HLK1 #_ #_ #H11 destruct -a -XW2 -XV1 -HL2 -HL1
elim cnv_cpm_conf_lpr_delta_ell_aux /1 width=8 by/
| #H21 #H22 #m1 #K1 #W1 #XW1 #i1 #HLK1 #HWX1 #HXW1 #H11 #H12 destruct -IH2
<minus_O_n <minus_n_O
@ex2_commute @(cnv_cpm_conf_lpr_atom_ell_aux … IH1) -IH1 /1 width=6 by/
| #s2 #H21 #H22 #H23 #m1 #K1 #W1 #XW1 #i1 #_ #_ #_ #H11 #H12 destruct
- | #K2 #V2 #XV2 #i2 #HLK2 #_ #_ #H21 #m1 #K1 #W1 #XW1 #i1 #HLK1 #_ #_ #H11 #H12 destruct -a -o -XV2 -XW1 -HL2 -HL1
+ | #K2 #V2 #XV2 #i2 #HLK2 #_ #_ #H21 #m1 #K1 #W1 #XW1 #i1 #HLK1 #_ #_ #H11 #H12 destruct -a -XV2 -XW1 -HL2 -HL1
elim cnv_cpm_conf_lpr_delta_ell_aux /1 width=8 by/
| #m2 #K2 #W2 #XW2 #i2 #HLK2 #HWX2 #HXW2 #H21 #H22 #m1 #K1 #W1 #XW1 #i1 #HLK1 #HWX1 #HXW1 #H11 #H12 destruct -IH2
>minus_S_S >minus_S_S
include "ground_2/xoa/ex_5_1.ma".
include "ground_2/xoa/ex_9_3.ma".
include "basic_2/rt_transition/cpm_tdeq.ma".
-include "basic_2/rt_transition/cpr.ma".
include "basic_2/rt_computation/fpbg_fqup.ma".
include "basic_2/dynamic/cnv_fsb.ma".
(* Inversion lemmas with restricted rt-transition for terms *****************)
-lemma cnv_cpr_tdeq_fwd_refl (a) (h) (o) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → T1 ≛[h,o] T2 →
- ⦃G, L⦄ ⊢ T1 ![a,h] → T1 = T2.
-#a #h #o #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
+lemma cnv_cpr_tdeq_fwd_refl (h) (a) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → T1 ≛ T2 → ⦃G,L⦄ ⊢ T1 ![h,a] → T1 = T2.
+#h #a #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
[ //
| #G #K #V1 #V2 #X2 #_ #_ #_ #H1 #_ -a -G -K -V1 -V2
lapply (tdeq_inv_lref1 … H1) -H1 #H destruct //
elim (cnv_fwd_flat … H2) -H2 #HV1 #HT1
/3 width=3 by eq_f2/
| #G #K #V #T1 #X1 #X2 #HXT1 #HX12 #_ #H1 #H2
- elim (cnv_fpbg_refl_false … o … H2) -a
+ elim (cnv_fpbg_refl_false … H2) -a
@(fpbg_tdeq_div … H1) -H1
/3 width=9 by cpm_tdneq_cpm_fpbg, cpm_zeta, tdeq_lifts_inv_pair_sn/
| #G #L #U #T1 #T2 #HT12 #_ #H1 #H2
- elim (cnv_fpbg_refl_false … o … H2) -a
+ elim (cnv_fpbg_refl_false … H2) -a
@(fpbg_tdeq_div … H1) -H1
/3 width=6 by cpm_tdneq_cpm_fpbg, cpm_eps, tdeq_inv_pair_xy_y/
| #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H1 #_
]
qed-.
-lemma cpm_tdeq_inv_bind_sn (a) (h) (o) (n) (p) (I) (G) (L):
- ∀V,T1. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
- ∀X. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛[h,o] X →
- ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛[h,o] T2 & X = ⓑ{p,I}V.T2.
-#a #h #o #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
+lemma cpm_tdeq_inv_bind_sn (h) (a) (n) (p) (I) (G) (L):
+ ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![h,a] →
+ ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
+ ∃∃T2. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
+#h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_bind1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
elim (tdeq_inv_pair … H2) -H2 #_ #H2XV #H2T12
lapply (cnv_cpr_tdeq_fwd_refl … HXV H2XV HV) #H destruct -HXV -H2XV
/2 width=4 by ex5_intro/
| #X1 #HXT1 #HX1 #H1 #H destruct
- elim (cnv_fpbg_refl_false … o … H0) -a
+ elim (cnv_fpbg_refl_false … H0) -a
@(fpbg_tdeq_div … H2) -H2
/3 width=9 by cpm_tdneq_cpm_fpbg, cpm_zeta, tdeq_lifts_inv_pair_sn/
]
qed-.
-lemma cpm_tdeq_inv_appl_sn (a) (h) (o) (n) (G) (L):
- ∀V,T1. ⦃G,L⦄ ⊢ ⓐV.T1 ![a,h] →
- ∀X. ⦃G,L⦄ ⊢ ⓐV.T1 ➡[n,h] X → ⓐV.T1 ≛[h,o] X →
- ∃∃m,q,W,U1,T2. a = Ⓣ → m ≤ 1 & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L⦄ ⊢ V ➡*[1,h] W & ⦃G, L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1
- & ⦃G,L⦄⊢ T1 ![a,h] & ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛[h,o] T2 & X = ⓐV.T2.
-#a #h #o #n #G #L #V #T1 #H0 #X #H1 #H2
+lemma cpm_tdeq_inv_appl_sn (h) (a) (n) (G) (L):
+ ∀V,T1. ⦃G,L⦄ ⊢ ⓐV.T1 ![h,a] →
+ ∀X. ⦃G,L⦄ ⊢ ⓐV.T1 ➡[n,h] X → ⓐV.T1 ≛ X →
+ ∃∃m,q,W,U1,T2. ad a m & ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L⦄ ⊢ V ➡*[1,h] W & ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{q}W.U1
+ & ⦃G,L⦄⊢ T1 ![h,a] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓐV.T2.
+#h #a #n #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_inv_appl1 … H1) -H1 *
[ #XV #T2 #HXV #HT12 #H destruct
elim (tdeq_inv_pair … H2) -H2 #_ #H2XV #H2T12
]
qed-.
-lemma cpm_tdeq_inv_cast_sn (a) (h) (o) (n) (G) (L):
- ∀U1,T1. ⦃G, L⦄ ⊢ ⓝU1.T1 ![a,h] →
- ∀X. ⦃G, L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛[h,o] X →
- ∃∃U0,U2,T2. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 & ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0
- & ⦃G, L⦄ ⊢ U1 ![a,h] & ⦃G, L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛[h,o] U2
- & ⦃G, L⦄ ⊢ T1 ![a,h] & ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛[h,o] T2 & X = ⓝU2.T2.
-#a #h #o #n #G #L #U1 #T1 #H0 #X #H1 #H2
+lemma cpm_tdeq_inv_cast_sn (h) (a) (n) (G) (L):
+ ∀U1,T1. ⦃G,L⦄ ⊢ ⓝU1.T1 ![h,a] →
+ ∀X. ⦃G,L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛ X →
+ ∃∃U0,U2,T2. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 & ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0
+ & ⦃G,L⦄ ⊢ U1 ![h,a] & ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛ U2
+ & ⦃G,L⦄ ⊢ T1 ![h,a] & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓝU2.T2.
+#h #a #n #G #L #U1 #T1 #H0 #X #H1 #H2
elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
[ #U2 #T2 #HU12 #HT12 #H destruct
elim (tdeq_inv_pair … H2) -H2 #_ #H2U12 #H2T12
elim (cnv_inv_cast … H0) -H0 #U0 #HU1 #HT1 #HU10 #HT1U0
/2 width=7 by ex9_3_intro/
| #HT1X
- elim (cnv_fpbg_refl_false … o … H0) -a
+ elim (cnv_fpbg_refl_false … H0) -a
@(fpbg_tdeq_div … H2) -H2
/3 width=6 by cpm_tdneq_cpm_fpbg, cpm_eps, tdeq_inv_pair_xy_y/
| #m #HU1X #H destruct
- elim (cnv_fpbg_refl_false … o … H0) -a
+ elim (cnv_fpbg_refl_false … H0) -a
@(fpbg_tdeq_div … H2) -H2
/3 width=6 by cpm_tdneq_cpm_fpbg, cpm_ee, tdeq_inv_pair_xy_x/
]
qed-.
-lemma cpm_tdeq_inv_bind_dx (a) (h) (o) (n) (p) (I) (G) (L):
- ∀X. ⦃G, L⦄ ⊢ X ![a,h] →
- ∀V,T2. ⦃G, L⦄ ⊢ X ➡[n,h] ⓑ{p,I}V.T2 → X ≛[h,o] ⓑ{p,I}V.T2 →
- ∃∃T1. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛[h,o] T2 & X = ⓑ{p,I}V.T1.
-#a #h #o #n #p #I #G #L #X #H0 #V #T2 #H1 #H2
+lemma cpm_tdeq_inv_bind_dx (h) (a) (n) (p) (I) (G) (L):
+ ∀X. ⦃G,L⦄ ⊢ X ![h,a] →
+ ∀V,T2. ⦃G,L⦄ ⊢ X ➡[n,h] ⓑ{p,I}V.T2 → X ≛ ⓑ{p,I}V.T2 →
+ ∃∃T1. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T1.
+#h #a #n #p #I #G #L #X #H0 #V #T2 #H1 #H2
elim (tdeq_inv_pair2 … H2) #V0 #T1 #_ #_ #H destruct
elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T0 #HV #HT1 #H1T12 #H2T12 #H destruct
/2 width=5 by ex5_intro/
(* Eliminators with restricted rt-transition for terms **********************)
-lemma cpm_tdeq_ind (a) (h) (o) (n) (G) (Q:relation3 …):
- (∀I,L. n = 0 → Q L (⓪{I}) (⓪{I})) →
- (∀L,s. n = 1 → deg h o s 0 → Q L (⋆s) (⋆(next h s))) →
- (∀p,I,L,V,T1. ⦃G,L⦄⊢ V![a,h] → ⦃G,L.ⓑ{I}V⦄⊢T1![a,h] →
- ∀T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛[h,o] T2 →
- Q (L.ⓑ{I}V) T1 T2 → Q L (ⓑ{p,I}V.T1) (ⓑ{p,I}V.T2)
- ) →
- (∀m. (a = Ⓣ → m ≤ 1) →
- ∀L,V. ⦃G,L⦄ ⊢ V ![a,h] → ∀W. ⦃G, L⦄ ⊢ V ➡*[1,h] W →
- ∀p,T1,U1. ⦃G, L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![a,h] →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛[h,o] T2 →
- Q L T1 T2 → Q L (ⓐV.T1) (ⓐV.T2)
- ) →
- (∀L,U0,U1,T1. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 → ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0 →
- ∀U2. ⦃G, L⦄ ⊢ U1 ![a,h] → ⦃G, L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛[h,o] U2 →
- ∀T2. ⦃G, L⦄ ⊢ T1 ![a,h] → ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛[h,o] T2 →
- Q L U1 U2 → Q L T1 T2 → Q L (ⓝU1.T1) (ⓝU2.T2)
- ) →
- ∀L,T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛[h,o] T2 → Q L T1 T2.
-#a #h #o #n #G #Q #IH1 #IH2 #IH3 #IH4 #IH5 #L #T1
+lemma cpm_tdeq_ind (h) (a) (n) (G) (Q:relation3 …):
+ (∀I,L. n = 0 → Q L (⓪{I}) (⓪{I})) →
+ (∀L,s. n = 1 → Q L (⋆s) (⋆(⫯[h]s))) →
+ (∀p,I,L,V,T1. ⦃G,L⦄⊢ V![h,a] → ⦃G,L.ⓑ{I}V⦄⊢T1![h,a] →
+ ∀T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ Q (L.ⓑ{I}V) T1 T2 → Q L (ⓑ{p,I}V.T1) (ⓑ{p,I}V.T2)
+ ) →
+ (∀m. ad a m →
+ ∀L,V. ⦃G,L⦄ ⊢ V ![h,a] → ∀W. ⦃G,L⦄ ⊢ V ➡*[1,h] W →
+ ∀p,T1,U1. ⦃G,L⦄ ⊢ T1 ➡*[m,h] ⓛ{p}W.U1 → ⦃G,L⦄⊢ T1 ![h,a] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ Q L T1 T2 → Q L (ⓐV.T1) (ⓐV.T2)
+ ) →
+ (∀L,U0,U1,T1. ⦃G,L⦄ ⊢ U1 ➡*[h] U0 → ⦃G,L⦄ ⊢ T1 ➡*[1,h] U0 →
+ ∀U2. ⦃G,L⦄ ⊢ U1 ![h,a] → ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → U1 ≛ U2 →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ![h,a] → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ Q L U1 U2 → Q L T1 T2 → Q L (ⓝU1.T1) (ⓝU2.T2)
+ ) →
+ ∀L,T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 → Q L T1 T2.
+#h #a #n #G #Q #IH1 #IH2 #IH3 #IH4 #IH5 #L #T1
@(insert_eq_0 … G) #F
@(fqup_wf_ind_eq (Ⓣ) … F L T1) -L -T1 -F
#G0 #L0 #T0 #IH #F #L * [| * [| * ]]
[ #I #_ #_ #_ #_ #HF #X #H1X #H2X destruct -G0 -L0 -T0
elim (cpm_tdeq_inv_atom_sn … H1X H2X) -H1X -H2X *
[ #H1 #H2 destruct /2 width=1 by/
- | #s #H1 #H2 #H3 #Hs destruct /2 width=1 by/
+ | #s #H1 #H2 #H3 destruct /2 width=1 by/
]
| #p #I #V #T1 #HG #HL #HT #H0 #HF #X #H1X #H2X destruct
elim (cpm_tdeq_inv_bind_sn … H0 … H1X H2X) -H0 -H1X -H2X #T2 #HV #HT1 #H1T12 #H2T12 #H destruct
(* Advanced properties with restricted rt-transition for terms **************)
-lemma cpm_tdeq_free (a) (h) (o) (n) (G) (L):
- ∀T1. ⦃G, L⦄ ⊢ T1 ![a,h] →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛[h,o] T2 →
- ∀F,K. ⦃F, K⦄ ⊢ T1 ➡[n,h] T2.
-#a #h #o #n #G #L #T1 #H0 #T2 #H1 #H2
+lemma cpm_tdeq_free (h) (a) (n) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≛ T2 →
+ ∀F,K. ⦃F,K⦄ ⊢ T1 ➡[n,h] T2.
+#h #a #n #G #L #T1 #H0 #T2 #H1 #H2
@(cpm_tdeq_ind … H0 … H1 H2) -L -T1 -T2
[ #I #L #H #F #K destruct //
-| #L #s #H #_ #F #K destruct //
+| #L #s #H #F #K destruct //
| #p #I #L #V #T1 #_ #_ #T2 #_ #_ #IH #F #K
/2 width=1 by cpm_bind/
| #m #_ #L #V #_ #W #_ #q #T1 #U1 #_ #_ #T2 #_ #_ #IH #F #K
(* Advanced inversion lemmas with restricted rt-transition for terms ********)
-lemma cpm_tdeq_inv_bind_sn_void (a) (h) (o) (n) (p) (I) (G) (L):
- ∀V,T1. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
- ∀X. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛[h,o] X →
- ∃∃T2. ⦃G,L⦄ ⊢ V ![a,h] & ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ![a,h] & ⦃G, L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛[h,o] T2 & X = ⓑ{p,I}V.T2.
-#a #h #o #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
+lemma cpm_tdeq_inv_bind_sn_void (h) (a) (n) (p) (I) (G) (L):
+ ∀V,T1. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ![h,a] →
+ ∀X. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛ X →
+ ∃∃T2. ⦃G,L⦄ ⊢ V ![h,a] & ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ![h,a] & ⦃G,L.ⓧ⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛ T2 & X = ⓑ{p,I}V.T2.
+#h #a #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
elim (cpm_tdeq_inv_bind_sn … H0 … H1 H2) -H0 -H1 -H2 #T2 #HV #HT1 #H1T12 #H2T12 #H
/3 width=5 by ex5_intro, cpm_tdeq_free/
qed-.
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-definition IH_cnv_cpm_tdeq_conf_lpr (a) (h) (o): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
- ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 →
- ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛[h,o] T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛[h,o] T.
+definition IH_cnv_cpm_tdeq_conf_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
+ ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
+ ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛ T & ⦃G,L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛ T.
(* Diamond propery with restricted rt-transition for terms ******************)
-fact cnv_cpm_tdeq_conf_lpr_atom_atom_aux (h) (o) (G0) (L1) (L2) (I):
- ∃∃T. ⦃G0,L1⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛[h,o] T & ⦃G0, L2⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛[h,o] T.
-#h #o #G0 #L1 #L2 #I
+fact cnv_cpm_tdeq_conf_lpr_atom_atom_aux (h) (G0) (L1) (L2) (I):
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T & ⦃G0,L2⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T.
+#h #G0 #L1 #L2 #I
/2 width=5 by ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_atom_ess_aux (h) (o) (G0) (L1) (L2) (s):
- deg h o s 0 →
- ∃∃T. ⦃G0,L1⦄ ⊢ ⋆s ➡[1,h] T & ⋆s ≛[h,o] T & ⦃G0,L2⦄ ⊢ ⋆(next h s) ➡[h] T & ⋆(next h s) ≛[h,o] T.
-#h #o #G0 #L1 #L2 #s #Hs
-/4 width=5 by tdeq_sort, deg_next, ex4_intro/
+fact cnv_cpm_tdeq_conf_lpr_atom_ess_aux (h) (G0) (L1) (L2) (s):
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⋆s ➡[1,h] T & ⋆s ≛ T & ⦃G0,L2⦄ ⊢ ⋆(⫯[h]s) ➡[h] T & ⋆(⫯[h]s) ≛ T.
+#h #G0 #L1 #L2 #s
+/3 width=5 by tdeq_sort, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_bind_bind_aux (a) (h) (o) (p) (I) (G0) (L0) (V0) (T0):
- (â\88\80G,L,T. â¦\83G0,L0,â\93\91{p,I}V0.T0â¦\84 â\8a\90+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr a h o G L T) →
- ⦃G0,L0⦄ ⊢ ⓑ{p,I}V0.T0 ![a,h] →
- ∀n1,T1. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 →
- ∀n2,T2. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 →
+fact cnv_cpm_tdeq_conf_lpr_bind_bind_aux (h) (a) (p) (I) (G0) (L0) (V0) (T0):
+ (â\88\80G,L,T. â¦\83G0,L0,â\93\91{p,I}V0.T0â¦\84 â¬\82+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ ⓑ{p,I}V0.T0 ![h,a] →
+ ∀n1,T1. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
+ ∀n2,T2. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0,L1⦄ ⊢ ⓑ{p,I}V0.T1 ➡[n2-n1,h] T & ⓑ{p,I}V0.T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ ⓑ{p,I}V0.T2 ➡[n1-n2,h] T & ⓑ{p,I}V0.T2 ≛[h,o] T.
-#a #h #o #p #I #G0 #L0 #V0 #T0 #IH #H0
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓑ{p,I}V0.T1 ➡[n2-n1,h] T & ⓑ{p,I}V0.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓑ{p,I}V0.T2 ➡[n1-n2,h] T & ⓑ{p,I}V0.T2 ≛ T.
+#h #a #p #I #G0 #L0 #V0 #T0 #IH #H0
#n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_bind … H0) -H0 #_ #HT0
/3 width=7 by cpm_bind, tdeq_pair, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_appl_appl_aux (a) (h) (o) (G0) (L0) (V0) (T0):
- (â\88\80G,L,T. â¦\83G0,L0,â\93\90V0.T0â¦\84 â\8a\90+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr a h o G L T) →
- ⦃G0,L0⦄ ⊢ ⓐV0.T0 ![a,h] →
- ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 →
- ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 →
+fact cnv_cpm_tdeq_conf_lpr_appl_appl_aux (h) (a) (G0) (L0) (V0) (T0):
+ (â\88\80G,L,T. â¦\83G0,L0,â\93\90V0.T0â¦\84 â¬\82+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ ⓐV0.T0 ![h,a] →
+ ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
+ ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0,L1⦄ ⊢ ⓐV0.T1 ➡[n2-n1,h] T & ⓐV0.T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ ⓐV0.T2 ➡[n1-n2,h] T & ⓐV0.T2 ≛[h,o] T.
-#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓐV0.T1 ➡[n2-n1,h] T & ⓐV0.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓐV0.T2 ➡[n1-n2,h] T & ⓐV0.T2 ≛ T.
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #_ #HT0 #_ #_ -n0 -p0 -X01 -X02
/3 width=7 by cpm_appl, tdeq_pair, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_cast_cast_aux (a) (h) (o) (G0) (L0) (V0) (T0):
- (â\88\80G,L,T. â¦\83G0,L0,â\93\9dV0.T0â¦\84 â\8a\90+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr a h o G L T) →
- ⦃G0,L0⦄ ⊢ ⓝV0.T0 ![a,h] →
- ∀n1,V1. ⦃G0,L0⦄ ⊢ V0 ➡[n1,h] V1 → V0 ≛[h,o] V1 →
- ∀n2,V2. ⦃G0,L0⦄ ⊢ V0 ➡[n2,h] V2 → V0 ≛[h,o] V2 →
- ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛[h,o] T1 →
- ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 →
+fact cnv_cpm_tdeq_conf_lpr_cast_cast_aux (h) (a) (G0) (L0) (V0) (T0):
+ (â\88\80G,L,T. â¦\83G0,L0,â\93\9dV0.T0â¦\84 â¬\82+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ ⓝV0.T0 ![h,a] →
+ ∀n1,V1. ⦃G0,L0⦄ ⊢ V0 ➡[n1,h] V1 → V0 ≛ V1 →
+ ∀n2,V2. ⦃G0,L0⦄ ⊢ V0 ➡[n2,h] V2 → V0 ≛ V2 →
+ ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
+ ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0,L1⦄ ⊢ ⓝV1.T1 ➡[n2-n1,h] T & ⓝV1.T1≛[h,o]T & ⦃G0,L2⦄ ⊢ ⓝV2.T2 ➡[n1-n2,h] T & ⓝV2.T2≛[h,o]T.
-#a #h #o #G0 #L0 #V0 #T0 #IH #H0
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓝV1.T1 ➡[n2-n1,h] T & ⓝV1.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓝV2.T2 ➡[n1-n2,h] T & ⓝV2.T2 ≛ T.
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #V1 #H1V01 #H2V01 #n2 #V2 #H1V02 #H2V02 #T1 #H1T01 #H2T01 #T2 #H1T02 #H2T02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
/3 width=7 by cpm_cast, tdeq_pair, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_aux (a) (h) (o) (G0) (L0) (T0):
- (â\88\80G,L,T. â¦\83G0,L0,T0â¦\84 â\8a\90+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr a h o G L T) →
- ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpm_tdeq_conf_lpr a h o G L T.
-#a #h #o #G0 #L0 #T0 #IH1 #G #L * [| * [| * ]]
+fact cnv_cpm_tdeq_conf_lpr_aux (h) (a) (G0) (L0) (T0):
+ (â\88\80G,L,T. â¦\83G0,L0,T0â¦\84 â¬\82+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpm_tdeq_conf_lpr h a G L T.
+#h #a #G0 #L0 #T0 #IH1 #G #L * [| * [| * ]]
[ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #H1X2 #H2X2 #L1 #HL1 #L2 #HL2 destruct
elim (cpm_tdeq_inv_atom_sn … H1X1 H2X1) -H1X1 -H2X1 *
elim (cpm_tdeq_inv_atom_sn … H1X2 H2X2) -H1X2 -H2X2 *
[ #H21 #H22 #H11 #H12 destruct -a -L
<minus_O_n
/2 width=1 by cnv_cpm_tdeq_conf_lpr_atom_atom_aux/
- | #s2 #H21 #H22 #H23 #Hs2 #H11 #H12 destruct -a -L
+ | #s2 #H21 #H22 #H23 #H11 #H12 destruct -a -L
<minus_O_n <minus_n_O
/2 width=1 by cnv_cpm_tdeq_conf_lpr_atom_ess_aux/
- | #H21 #H22 #s1 #H11 #H12 #H13 #Hs1 destruct -a -L
+ | #H21 #H22 #s1 #H11 #H12 #H13 destruct -a -L
<minus_O_n <minus_n_O
@ex4_commute /2 width=1 by cnv_cpm_tdeq_conf_lpr_atom_ess_aux/
- | #s2 #H21 #H22 #H23 #_ #s1 #H11 #H12 #H13 #_ destruct -a -L
+ | #s2 #H21 #H22 #H23 #s1 #H11 #H12 #H13 destruct -a -L
<minus_n_n
/2 width=1 by cnv_cpm_tdeq_conf_lpr_atom_atom_aux/
]
]
qed-.
-lemma cnv_cpm_tdeq_conf_lpr (a) (h) (o) (G0) (L0) (T0):
- IH_cnv_cpm_tdeq_conf_lpr a h o G0 L0 T0.
-#a #h #o #G0 #L0 #T0
+lemma cnv_cpm_tdeq_conf_lpr (h) (a) (G0) (L0) (T0):
+ IH_cnv_cpm_tdeq_conf_lpr h a G0 L0 T0.
+#h #a #G0 #L0 #T0
@(fqup_wf_ind (Ⓣ) … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH
/3 width=17 by cnv_cpm_tdeq_conf_lpr_aux/
qed-.
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-definition IH_cnv_cpm_tdeq_cpm_trans (a) (h) (o): relation3 genv lenv term ≝
- λG,L,T1. ⦃G, L⦄ ⊢ T1 ![a,h] →
- ∀n1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → T1 ≛[h,o] T →
+definition IH_cnv_cpm_tdeq_cpm_trans (h) (a): relation3 genv lenv term ≝
+ λG,L,T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀n1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → T1 ≛ T →
∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 →
- ∃∃T0. ⦃G,L⦄ ⊢ T1 ➡[n2,h] T0 & ⦃G,L⦄ ⊢ T0 ➡[n1,h] T2 & T0 ≛[h,o] T2.
+ ∃∃T0. ⦃G,L⦄ ⊢ T1 ➡[n2,h] T0 & ⦃G,L⦄ ⊢ T0 ➡[n1,h] T2 & T0 ≛ T2.
(* Transitive properties restricted rt-transition for terms *****************)
-fact cnv_cpm_tdeq_cpm_trans_sub (a) (h) (o) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0, L0, T0⦄ >[h,o] ⦃G, L, T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (â\88\80G,L,T. â¦\83G0,L0,T0â¦\84 â\8a\90+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_cpm_trans a h o G L T) →
- ∀G,L,T1. G0 = G → L0 = L → T0 = T1 → IH_cnv_cpm_tdeq_cpm_trans a h o G L T1.
-#a #h #o #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
+fact cnv_cpm_tdeq_cpm_trans_sub (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (â\88\80G,L,T. â¦\83G0,L0,T0â¦\84 â¬\82+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_cpm_trans h a G L T) →
+ ∀G,L,T1. G0 = G → L0 = L → T0 = T1 → IH_cnv_cpm_tdeq_cpm_trans h a G L T1.
+#h #a #G0 #L0 #T0 #IH2 #IH1 #G #L * [| * [| * ]]
[ #I #_ #_ #_ #_ #n1 #X1 #H1X #H2X #n2 #X2 #HX2 destruct -G0 -L0 -T0
elim (cpm_tdeq_inv_atom_sn … H1X H2X) -H1X -H2X *
[ #H1 #H2 destruct /2 width=4 by ex3_intro/
- | #s #H1 #H2 #H3 #Hs destruct
+ | #s #H1 #H2 #H3 destruct
elim (cpm_inv_sort1 … HX2) -HX2 #H #Hn2 destruct >iter_n_Sm
- /5 width=6 by cpm_sort, tdeq_sort, deg_iter, deg_next, ex3_intro/
+ /3 width=4 by cpm_sort, tdeq_sort, ex3_intro/
]
| #p #I #V1 #T1 #HG #HL #HT #H0 #n1 #X1 #H1X #H2X #n2 #X2 #HX2 destruct
elim (cpm_tdeq_inv_bind_sn … H0 … H1X H2X) -H0 -H1X -H2X #T #_ #H0T1 #H1T1 #H2T1 #H destruct
]
qed-.
-fact cnv_cpm_tdeq_cpm_trans_aux (a) (h) (o) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0, L0, T0⦄ >[h, o] ⦃G, L, T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- IH_cnv_cpm_tdeq_cpm_trans a h o G0 L0 T0.
-#a #h #o #G0 #L0 #T0
+fact cnv_cpm_tdeq_cpm_trans_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ IH_cnv_cpm_tdeq_cpm_trans h a G0 L0 T0.
+#h #a #G0 #L0 #T0
@(fqup_wf_ind (Ⓣ) … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH #IH0
/5 width=10 by cnv_cpm_tdeq_cpm_trans_sub, fqup_fpbg_trans/
qed-.
include "ground_2/lib/arith_2b.ma".
include "basic_2/rt_computation/cprs_cprs.ma".
-include "basic_2/rt_computation/lprs_cpms.ma".
include "basic_2/dynamic/cnv_drops.ma".
include "basic_2/dynamic/cnv_preserve_sub.ma".
+include "basic_2/dynamic/cnv_aaa.ma".
include "basic_2/dynamic/lsubv_cnv.ma".
(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
(* Sub preservation propery with t-bound rt-transition for terms ************)
-fact cnv_cpm_trans_lpr_aux (a) (h) (o):
- ∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_trans_lpr a h G1 L1 T1.
-#a #h #o #G0 #L0 #T0 #IH2 #IH1 #G1 #L1 * * [|||| * ]
+fact cnv_cpm_trans_lpr_aux (h) (a):
+ ∀G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_cnv_cpm_trans_lpr h a G1 L1 T1.
+#h #a #G0 #L0 #T0 #IH2 #IH1 #G1 #L1 * * [|||| * ]
[ #s #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #_ destruct -IH2 -IH1 -H1
elim (cpm_inv_sort1 … H2) -H2 #H #_ destruct //
| #i #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #HL12 destruct -IH2
elim (cnv_cpms_strip_lpr_sub … IH2 … HVW1 … HV12 … HL12 … HL12) [|*: /2 width=2 by fqup_fpbg/ ] -HVW1 -HV12
<minus_n_O <minus_O_n #XW1 #HXW1 #HXV2
elim (cnv_cpms_strip_lpr_sub … IH2 … HTU1 … HT12 … HL12 … HL12) [|*: /2 width=2 by fqup_fpbg/ ] -HTU1 -HT12
- #X #H #HTU2 -IH2 -IH1 -L1 -V1 -T1
- elim (cpms_inv_abst_sn … H) -H #W2 #U2 #HW12 #_ #H destruct
+ #X #H #HTX2 -IH2 -IH1 -L1 -V1 -T1
+ elim (cpms_inv_abst_sn … H) -H #W2 #X2 #HW12 #_ #H destruct
elim (cprs_conf … HXW1 … HW12) -W1 #W1 #HXW1 #HW21
lapply (cpms_trans … HXV2 … HXW1) -XW1 <plus_n_O #HV2W1
- lapply (cpms_trans … HTU2 … (ⓛ{p}W1.U2) ?)
- [3:|*: /2 width=2 by cpms_bind/ ] -W2 <plus_n_O #HTU2
- /4 width=7 by cnv_appl, minus_le_trans_sn/
+ lapply (cpms_trans … HTX2 … (ⓛ{p}W1.X2) ?)
+ [3:|*: /2 width=2 by cpms_bind/ ] -W2 <plus_n_O #HTX2
+ elim (cnv_fwd_cpms_abst_dx_le … HT2 … HTX2 n) -HTX2 [| // ] #U2 #HTU2 #_ -X2
+ /2 width=7 by cnv_appl/
| #q #V2 #W10 #W20 #T10 #T20 #HV12 #HW120 #HT120 #H1 #H2 destruct
elim (cnv_inv_bind … HT1) -HT1 #HW10 #HT10
elim (cpms_inv_abst_sn … HTU1) -HTU1 #W30 #T30 #HW130 #_ #H destruct -T30
elim (cnv_cpms_strip_lpr_sub … IH2 … HVW1 … HV10 … HL12 … HL12) [|*: /2 width=2 by fqup_fpbg/ ] -HVW1 -HV10
<minus_n_O <minus_O_n #XW1 #HXW1 #HXV0
elim (cnv_cpms_strip_lpr_sub … IH2 … HTU0 … HT02 … (L2.ⓓW2) … (L2.ⓓW2)) [|*: /2 width=2 by fqup_fpbg, lpr_pair/ ] -HTU0 -HT02 -HW02
- #X #H #HTU2 -IH2 -IH1 -L1 -W0 -T0 -U1
- elim (cpms_inv_abst_sn … H) -H #W #U2 #HW3 #_ #H destruct -U3
+ #X #H #HTX2 -IH2 -IH1 -L1 -W0 -T0 -U1
+ elim (cpms_inv_abst_sn … H) -H #W #X2 #HW3 #_ #H destruct -U3
lapply (cnv_lifts … HV0 (Ⓣ) … (L2.ⓓW2) … HV02) /3 width=1 by drops_refl, drops_drop/ -HV0 #HV2
elim (cpms_lifts_sn … HXV0 (Ⓣ) … (L2.ⓓW2) … HV02) /3 width=1 by drops_refl, drops_drop/ -V0 #XW2 #HXW12 #HXVW2
lapply (cpms_lifts_bi … HXW1 (Ⓣ) … (L2.ⓓW2) … HW13 … HXW12) /3 width=1 by drops_refl, drops_drop/ -W1 -XW1 #HXW32
elim (cprs_conf … HXW32 … HW3) -W3 #W3 #HXW23 #HW3
lapply (cpms_trans … HXVW2 … HXW23) -XW2 <plus_n_O #H1
- lapply (cpms_trans … HTU2 ? (ⓛ{p}W3.U2) ?) [3:|*:/2 width=2 by cpms_bind/ ] -W #H2
- /5 width=7 by cnv_appl, cnv_bind, minus_le_trans_sn/
+ lapply (cpms_trans … HTX2 ? (ⓛ{p}W3.X2) ?) [3:|*:/2 width=2 by cpms_bind/ ] -W #H2
+ elim (cnv_fwd_cpms_abst_dx_le … HT2 … H2 n) -H2 [| // ] #U2 #HTU2 #_ -X2
+ /3 width=7 by cnv_appl, cnv_bind/
]
| #W1 #T1 #HG0 #HL0 #HT0 #H1 #x #X #H2 #L2 #HL12 destruct
elim (cnv_inv_cast … H1) -H1 #U1 #HW1 #HT1 #HWU1 #HTU1
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cprs_cnr.ma".
+include "basic_2/rt_computation/cpre.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Properties with t-bound evaluation on terms ******************************)
+
+lemma cnv_cpme_trans (h) (a) (n) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T2 ![h,a].
+#h #a #n #G #L #T1 #HT1 #T2 * #HT12 #_
+/2 width=4 by cnv_cpms_trans/
+qed-.
+
+lemma cnv_cpme_cpms_conf (h) (a) (n) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀T1. ⦃G,L⦄ ⊢ T ➡*[n,h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡*[h,n] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T1 ➡*[h] 𝐍⦃T2⦄.
+#h #a #n #G #L #T0 #HT0 #T1 #HT01 #T2 * #HT02 #HT2
+elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 <minus_n_n #T0 #HT10 #HT20
+lapply (cprs_inv_cnr_sn … HT20 HT2) -HT20 #H destruct
+/2 width=1 by cpme_intro/
+qed-.
+
+(* Main properties with evaluation for t-bound rt-transition on terms *****)
+
+theorem cnv_cpme_mono (h) (a) (n) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀T1. ⦃G,L⦄ ⊢ T ➡*[h,n] 𝐍⦃T1⦄ →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡*[h,n] 𝐍⦃T2⦄ → T1 = T2.
+#h #a #n #G #L #T0 #HT0 #T1 * #HT01 #HT1 #T2 * #HT02 #HT2
+elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 <minus_n_n #T0 #HT10 #HT20
+/3 width=7 by cprs_inv_cnr_sn, canc_dx_eq/
+qed-.
(* Sub confluence propery with t-bound rt-computation for terms *************)
-fact cnv_cpms_conf_lpr_tdeq_tdeq_aux (a) (h) (o) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ T0 ![a,h] →
- ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → T0 ≛[h,o] T1 →
- ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛[h,o] T2 →
+fact cnv_cpms_conf_lpr_tdeq_tdeq_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![h,a] →
+ ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → T0 ≛ T1 →
+ ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
-#a #h #o #G #L0 #T0 #IH2 #IH1 #HT0
+#h #a #G #L0 #T0 #IH2 #IH1 #HT0
#n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
#L1 #HL01 #L2 #HL02
elim (cnv_cpms_tdeq_conf_lpr_aux … IH2 IH1 … H1T01 … H1T02 … HL01 … HL02) -IH2 -IH1 -H1T01 -H1T02 -HL01 -HL02
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpms_conf_lpr_refl_tdneq_sub (a) (h) (o) (G0) (L0) (T0) (m21) (m22):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ T0 ![a,h] →
- ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
+fact cnv_cpms_conf_lpr_refl_tdneq_sub (h) (a) (G0) (L0) (T0) (m21) (m22):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![h,a] →
+ ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛ X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ T0 ➡*[m21+m22,h] T& ⦃G0,L2⦄ ⊢ T2 ➡*[h] T.
-#a #h #o #G0 #L0 #T0 #m21 #m22 #IH2 #IH1 #H0
+#h #a #G0 #L0 #T0 #m21 #m22 #IH2 #IH1 #H0
#X2 #HX02 #HnX02 #T2 #HXT2
#L1 #HL01 #L2 #HL02
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX02 … L0 ?) // #HX2
elim (cnv_cpms_strip_lpr_sub … IH1 … HXT2 0 X2 … HL02 L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ]
<minus_n_O <minus_O_n #Y2 #HTY2 #HXY2 -HXT2
elim (IH1 … HXY1 … HXY2 … HL01 … HL02) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ]
--a -o -L0 -X2 <minus_n_O <minus_O_n #Y #HY1 #HY2
+-a -L0 -X2 <minus_n_O <minus_O_n #Y #HY1 #HY2
lapply (cpms_trans … HTY1 … HY1) -Y1 #HT0Y
lapply (cpms_trans … HTY2 … HY2) -Y2 #HT2Y
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpms_conf_lpr_step_tdneq_sub (a) (h) (o) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ T0 ![a,h] →
- ∀X1. ⦃G0,L0⦄ ⊢ T0 ➡[m11,h] X1 → T0 ≛[h,o] X1 → ∀T1. ⦃G0,L0⦄ ⊢ X1 ➡*[m12,h] T1 → X1 ≛[h,o] T1 →
- ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
+fact cnv_cpms_conf_lpr_step_tdneq_sub (h) (a) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![h,a] →
+ ∀X1. ⦃G0,L0⦄ ⊢ T0 ➡[m11,h] X1 → T0 ≛ X1 → ∀T1. ⦃G0,L0⦄ ⊢ X1 ➡*[m12,h] T1 → X1 ≛ T1 →
+ ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛ X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
- ((∀G,L,T. ⦃G0,L0,X1⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,X1⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ∀m21,m22.
- ∀X2. ⦃G0,L0⦄ ⊢ X1 ➡[m21,h] X2 → (X1 ≛[h,o] X2 → ⊥) →
- ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
- ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-m12,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m12-(m21+m22),h]T
+ ((∀G,L,T. ⦃G0,L0,X1⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,X1⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ∀m21,m22.
+ ∀X2. ⦃G0,L0⦄ ⊢ X1 ➡[m21,h] X2 → (X1 ≛ X2 → ⊥) →
+ ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-m12,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m12-(m21+m22),h]T
) →
∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-(m11+m12),h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m11+m12-(m21+m22),h] T.
-#a #h #o #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #HT0
+#h #a #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #HT0
#X1 #H1X01 #H2X01 #T1 #H1XT1 #H2XT1 #X2 #H1X02 #H2X02 #T2 #HXT2
#L1 #HL01 #L2 #HL02 #IH
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … H1X01 … L0 ?) // #HX1
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … H1X02 … L0 ?) // #HX2
elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … H1X01 … H1X02 … L0 … L0) // #Z0 #HXZ10 #HXZ20
-cut (⦃G0,L0,T0⦄ >[h,o] ⦃G0,L0,X2⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
-lapply (fpbg_fpbs_trans ??? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg
+cut (⦃G0, L0, T0⦄ >[h] ⦃G0, L0, X2⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
+lapply (fpbg_fpbs_trans ?? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg
lapply (cnv_cpms_trans_lpr_sub … IH2 … HXZ20 … L0 ?) // #HZ0
elim (IH1 … HXT2 … HXZ20 … L2 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT2 -HXZ20 #Z2 #HTZ2 #HZ02
-elim (tdeq_dec h o X1 Z0) #H2XZ
+elim (tdeq_dec X1 Z0) #H2XZ
[ -IH
elim (cnv_cpms_conf_lpr_tdeq_tdeq_aux … HX1 … H1XT1 H2XT1 … HXZ10 H2XZ … L1 … L0) [2,3: // |4,5: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ]
| -H1XT1 -H2XT1
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpms_conf_lpr_tdeq_tdneq_aux (a) (h) (o) (G0) (L0) (T0) (n1) (m21) (m22):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ T0 ![a,h] →
- ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → T0 ≛[h,o] T1 →
- ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
+fact cnv_cpms_conf_lpr_tdeq_tdneq_aux (h) (a) (G0) (L0) (T0) (n1) (m21) (m22):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![h,a] →
+ ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → T0 ≛ T1 →
+ ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛ X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-n1,h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-(m21+m22),h] T.
-#a #h #o #G0 #L0 #T0 #n1 #m21 #m22 #IH2 #IH1 #HT0
+#h #a #G0 #L0 #T0 #n1 #m21 #m22 #IH2 #IH1 #HT0
#T1 #H1T01 #H2T01
generalize in match m22; generalize in match m21; -m21 -m22
generalize in match IH1; generalize in match IH2;
]
qed-.
-fact cnv_cpms_conf_lpr_tdneq_tdneq_aux (a) (h) (o) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ T0 ![a,h] →
- ∀X1. ⦃G0,L0⦄ ⊢ T0 ➡[m11,h] X1 → (T0 ≛[h,o] X1 → ⊥) → ∀T1. ⦃G0,L0⦄ ⊢ X1 ➡*[m12,h] T1 →
- ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛[h,o] X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
+fact cnv_cpms_conf_lpr_tdneq_tdneq_aux (h) (a) (G0) (L0) (T0) (m11) (m12) (m21) (m22):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ T0 ![h,a] →
+ ∀X1. ⦃G0,L0⦄ ⊢ T0 ➡[m11,h] X1 → (T0 ≛ X1 → ⊥) → ∀T1. ⦃G0,L0⦄ ⊢ X1 ➡*[m12,h] T1 →
+ ∀X2. ⦃G0,L0⦄ ⊢ T0 ➡[m21,h] X2 → (T0 ≛ X2 → ⊥) → ∀T2. ⦃G0,L0⦄ ⊢ X2 ➡*[m22,h] T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[m21+m22-(m11+m12),h] T & ⦃G0,L2⦄ ⊢ T2 ➡*[m11+m12-(m21+m22),h] T.
-#a #h #o #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #H0
+#h #a #G0 #L0 #T0 #m11 #m12 #m21 #m22 #IH2 #IH1 #H0
#X1 #HX01 #HnX01 #T1 #HXT1 #X2 #HX02 #HnX02 #T2 #HXT2
#L1 #HL01 #L2 #HL02
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX01 … L0 ?) // #HX1
lapply (cnv_cpm_trans_lpr_aux … IH1 IH2 … HX02 … L0 ?) // #HX2
elim (cnv_cpm_conf_lpr_aux … IH2 IH1 … HX01 … HX02 … L0 … L0) // #Z0 #HXZ10 #HXZ20
-cut (⦃G0,L0,T0⦄ >[h,o] ⦃G0,L0,X1⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
-lapply (fpbg_fpbs_trans ??? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg
+cut (⦃G0, L0, T0⦄ >[h] ⦃G0, L0, X1⦄) [ /4 width=5 by cpms_fwd_fpbs, cpm_fpb, ex2_3_intro/ ] #H1fpbg (**) (* cut *)
+lapply (fpbg_fpbs_trans ?? G0 ? L0 ? Z0 ? … H1fpbg) [ /2 width=2 by cpms_fwd_fpbs/ ] #H2fpbg
lapply (cnv_cpms_trans_lpr_sub … IH2 … HXZ10 … L0 ?) // #HZ0
elim (IH1 … HXT1 … HXZ10 … L1 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT1 -HXZ10 #Z1 #HTZ1 #HZ01
elim (IH1 … HXT2 … HXZ20 … L2 … L0) [|*: /4 width=2 by fpb_fpbg, cpm_fpb/ ] -HXT2 -HXZ20 #Z2 #HTZ2 #HZ02
/2 width=3 by ex2_intro/
qed-.
-fact cnv_cpms_conf_lpr_aux (a) (h) (o) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpms_conf_lpr a h G L T.
-#a #h #o #G #L #T #IH2 #IH1 #G0 #L0 #T0 #HG #HL #HT
+fact cnv_cpms_conf_lpr_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpms_conf_lpr h a G L T.
+#h #a #G #L #T #IH2 #IH1 #G0 #L0 #T0 #HG #HL #HT
#HT0 #n1 #T1 #HT01 #n2 #T2 #HT02 #L1 #HL01 #L2 #HL02 destruct
-elim (tdeq_dec h o T0 T1) #H2T01
-elim (tdeq_dec h o T0 T2) #H2T02
+elim (tdeq_dec T0 T1) #H2T01
+elim (tdeq_dec T0 T2) #H2T02
[ @(cnv_cpms_conf_lpr_tdeq_tdeq_aux … IH2 IH1) -IH2 -IH1 /2 width=1 by/
| elim (cpms_tdneq_fwd_step_sn_aux … HT02 HT0 H2T02 IH1 IH2) -HT02 -H2T02
#m21 #m22 #X2 #HX02 #HnX02 #HXT2 #H2 destruct
(* Properties with restricted rt-computation for terms **********************)
-fact cpms_tdneq_fwd_step_sn_aux (a) (h) (n) (o) (G) (L) (T1):
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L⦄ ⊢ T1 ![a,h] → (T1 ≛[h,o] T2 → ⊥) →
- (∀G0,L0,T0. ⦃G,L,T1⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- (∀G0,L0,T0. ⦃G,L,T1⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
- ∃∃n1,n2,T0. ⦃G, L⦄ ⊢ T1 ➡[n1,h] T0 & T1 ≛[h,o] T0 → ⊥ & ⦃G, L⦄ ⊢ T0 ➡*[n2,h] T2 & n1+n2 = n.
-#a #h #n #o #G #L #T1 #T2 #H
+fact cpms_tdneq_fwd_step_sn_aux (h) (a) (n) (G) (L) (T1):
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ![h,a] → (T1 ≛ T2 → ⊥) →
+ (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr h a G0 L0 T0) →
+ ∃∃n1,n2,T0. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T0 & T1 ≛ T0 → ⊥ & ⦃G,L⦄ ⊢ T0 ➡*[n2,h] T2 & n1+n2 = n.
+#h #a #n #G #L #T1 #T2 #H
@(cpms_ind_sn … H) -n -T1
[ #_ #H2T2 elim H2T2 -H2T2 //
| #n1 #n2 #T1 #T #H1T1 #H1T2 #IH #H0T1 #H2T12 #IH2 #IH1
- elim (tdeq_dec h o T1 T) #H2T1
- [ elim (tdeq_dec h o T T2) #H2T2
+ elim (tdeq_dec T1 T) #H2T1
+ [ elim (tdeq_dec T T2) #H2T2
[ -IH -IH2 -IH1 -H0T1 /4 width=7 by tdeq_trans, ex4_3_intro/
| lapply (cnv_cpm_trans_lpr_aux … IH2 IH1 … H1T1 L ?) [6:|*: // ] -H1T2 -H2T12 #H0T
elim (IH H0T H2T2) [|*: /4 width=5 by cpm_fpbq, fpbq_fpbg_trans/ ] -IH -IH2 -H0T -H2T2 (**)
]
qed-.
-fact cpms_tdeq_ind_sn (a) (h) (o) (G) (L) (T2) (Q:relation2 …):
- (⦃G, L⦄ ⊢ T2 ![a,h] → Q 0 T2) →
- (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G, L⦄ ⊢ T1 ![a,h] → T1 ≛[h,o] T → ⦃G, L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G, L⦄ ⊢ T ![a,h] → T ≛[h,o] T2 → Q n2 T → Q (n1+n2) T1) →
- ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L⦄ ⊢ T1 ![a,h] → T1 ≛[h,o] T2 →
- (∀G0,L0,T0. ⦃G,L,T1⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr a h G0 L0 T0) →
- (∀G0,L0,T0. ⦃G,L,T1⦄ >[h,o] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr a h G0 L0 T0) →
+fact cpms_tdeq_ind_sn (h) (a) (G) (L) (T2) (Q:relation2 …):
+ (⦃G,L⦄ ⊢ T2 ![h,a] → Q 0 T2) →
+ (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G,L⦄ ⊢ T1 ![h,a] → T1 ≛ T → ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T ![h,a] → T ≛ T2 → Q n2 T → Q (n1+n2) T1) →
+ ∀n,T1. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ![h,a] → T1 ≛ T2 →
+ (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpms_conf_lpr h a G0 L0 T0) →
+ (∀G0,L0,T0. ⦃G,L,T1⦄ >[h] ⦃G0,L0,T0⦄ → IH_cnv_cpm_trans_lpr h a G0 L0 T0) →
Q n T1.
-#a #h #o #G #L #T2 #Q #IB1 #IB2 #n #T1 #H
+#h #a #G #L #T2 #Q #IB1 #IB2 #n #T1 #H
@(cpms_ind_sn … H) -n -T1
[ -IB2 #H0T2 #_ #_ #_ /2 width=1 by/
| #n1 #n2 #T1 #T #H1T1 #H1T2 #IH #H0T1 #H2T12 #IH2 #IH1 -IB1
- elim (tdeq_dec h o T1 T) #H2T1
+ elim (tdeq_dec T1 T) #H2T1
[ lapply (cnv_cpm_trans_lpr_aux … IH2 IH1 … H1T1 L ?) [6:|*: // ] #H0T
lapply (tdeq_canc_sn … H2T1 … H2T12) -H2T12 #H2T2
/6 width=7 by cpm_fpbq, fpbq_fpbg_trans/ (**)
| -IB2 -IH -IH2 -IH1
- elim (cnv_fpbg_refl_false … o … H0T1) -a -Q
+ elim (cnv_fpbg_refl_false … H0T1) -a -Q
/3 width=8 by cpm_tdneq_cpm_cpms_tdeq_sym_fwd_fpbg/
]
]
(* Sub confluence propery with restricted rt-transition for terms ***********)
-fact cnv_cpms_tdeq_strip_lpr_aux (a) (h) (o) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛[h,o] T1 →
- ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛[h,o] T2 →
+fact cnv_cpms_tdeq_strip_lpr_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![h,a] → T0 ≛ T1 →
+ ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛[h,o] T.
-#a #h #o #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛ T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛ T.
+#h #a #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
@(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
[ #H0T1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
<minus_O_n <minus_n_O
]
qed-.
-fact cnv_cpms_tdeq_conf_lpr_aux (a) (h) (o) (G0) (L0) (T0):
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr a h G L T) →
- (∀G,L,T. ⦃G0,L0,T0⦄ >[h,o] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr a h G L T) →
- ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![a,h] → T0 ≛[h,o] T1 →
- ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛[h,o] T2 →
+fact cnv_cpms_tdeq_conf_lpr_aux (h) (a) (G0) (L0) (T0):
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpm_trans_lpr h a G L T) →
+ (∀G,L,T. ⦃G0,L0,T0⦄ >[h] ⦃G,L,T⦄ → IH_cnv_cpms_conf_lpr h a G L T) →
+ ∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ⦃G0,L0⦄ ⊢ T0 ![h,a] → T0 ≛ T1 →
+ ∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡*[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & T1 ≛[h,o] T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛[h,o] T.
-#a #h #o #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & T1 ≛ T & ⦃G0,L2⦄ ⊢ T2 ➡*[n1-n2,h] T & T2 ≛ T.
+#h #a #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
generalize in match IH1; generalize in match IH2;
@(cpms_tdeq_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
[ #H0T1 #IH2 #IH1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpmuwe_csx.ma".
+include "basic_2/rt_equivalence/cpes.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Properties with t-unbound whd evaluation on terms ************************)
+
+lemma cnv_cpmuwe_trans (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡*𝐍𝐖*[h,n] T2 → ⦃G,L⦄ ⊢ T2 ![h,a].
+/3 width=4 by cpmuwe_fwd_cpms, cnv_cpms_trans/ qed-.
+
+lemma cnv_R_cpmuwe_total (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∃n. R_cpmuwe h G L T1 n.
+/4 width=2 by cnv_fwd_fsb, fsb_inv_csx, R_cpmuwe_total_csx/ qed-.
+
+(* Main inversions with head evaluation for t-bound rt-transition on terms **)
+
+theorem cnv_cpmuwe_mono (h) (a) (G) (L):
+ ∀T0. ⦃G,L⦄ ⊢ T0 ![h,a] →
+ ∀n1,T1. ⦃G,L⦄ ⊢ T0 ➡*𝐍𝐖*[h,n1] T1 →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T0 ➡*𝐍𝐖*[h,n2] T2 →
+ ∧∧ ⦃G,L⦄ ⊢ T1 ⬌*[h,n2-n1,n1-n2] T2 & T1 ≅ T2.
+#h #a #G #L #T0 #HT0 #n1 #T1 * #HT01 #HT1 #n2 #T2 * #HT02 #HT2
+elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 #T0 #HT10 #HT20
+/4 width=4 by cpms_div, tweq_canc_dx, conj/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpme_aaa.ma".
+include "basic_2/rt_computation/cnuw_cnuw.ma".
+include "basic_2/rt_computation/cpmuwe.ma".
+include "basic_2/dynamic/cnv_cpme.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Advanced Properties with t-unbound whd evaluation on terms ***************)
+
+lemma cnv_R_cpmuwe_dec (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀n. Decidable (R_cpmuwe h G L T n).
+#h #a #G #L #T1 #HT1 #n
+elim (cnv_fwd_aaa … HT1) #A #HA
+elim (cpme_total_aaa h n … HA) -HA #T2 #HT12
+elim (cnuw_dec h G L T2) #HnT1
+[ /5 width=3 by cpme_fwd_cpms, cpmuwe_intro, ex_intro, or_introl/
+| @or_intror * #T3 * #HT13 #HT3
+ lapply (cnv_cpme_cpms_conf … HT1 … HT13 … HT12) -a -T1 #HT32
+ /4 width=9 by cpme_fwd_cpms, cnuw_cpms_trans/
+]
+qed-.
include "basic_2/rt_computation/cpms_drops.ma".
include "basic_2/dynamic/cnv.ma".
-(* CONTEXT_SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
(* Advanced dproperties *****************************************************)
(* Basic_2A1: uses: snv_lref *)
-lemma cnv_lref_drops (a) (h) (G): ∀I,K,V,i,L. ⦃G, K⦄ ⊢ V ![a, h] →
- ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L⦄ ⊢ #i ![a, h].
-#a #h #G #I #K #V #i elim i -i
+lemma cnv_lref_drops (h) (a) (G):
+ ∀I,K,V,i,L. ⦃G,K⦄ ⊢ V ![h,a] →
+ ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,L⦄ ⊢ #i ![h,a].
+#h #a #G #I #K #V #i elim i -i
[ #L #HV #H
lapply (drops_fwd_isid … H ?) -H // #H destruct
/2 width=1 by cnv_zero/
(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: uses: snv_inv_lref *)
-lemma cnv_inv_lref_drops (a) (h) (G):
- ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] →
- ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ![a, h].
-#a #h #G #i elim i -i
+lemma cnv_inv_lref_drops (h) (a) (G):
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![h,a] →
+ ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ![h,a].
+#h #a #G #i elim i -i
[ #L #H
elim (cnv_inv_zero … H) -H #I #K #V #HV #H destruct
/3 width=5 by drops_refl, ex2_3_intro/
]
qed-.
-(* Advanced forward lemmas **************************************************)
-
-lemma cnv_lref_fwd_drops (a) (h) (G):
- ∀i,L. ⦃G, L⦄ ⊢ #i ![a, h] →
- ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ![a, h].
-#a #h #o #i #L #H #I #K #V #HLK
+lemma cnv_inv_lref_pair (h) (a) (G):
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![h,a] →
+ ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ![h,a].
+#h #a #G #i #L #H #I #K #V #HLK
elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #HX
lapply (drops_mono … HLY … HLK) -L #H destruct //
-qed-.
+qed-.
+
+lemma cnv_inv_lref_atom (h) (a) (b) (G):
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![h,a] → ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⊥.
+#h #a #b #G #i #L #H #Hi
+elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_
+lapply (drops_gen b … HLY) -HLY #HLY
+lapply (drops_mono … HLY … Hi) -L #H destruct
+qed-.
+
+lemma cnv_inv_lref_unit (h) (a) (G):
+ ∀i,L. ⦃G,L⦄ ⊢ #i ![h,a] →
+ ∀I,K. ⬇*[i] L ≘ K.ⓤ{I} → ⊥.
+#h #a #G #i #L #H #I #K #HLK
+elim (cnv_inv_lref_drops … H) -H #Z #Y #X #HLY #_
+lapply (drops_mono … HLY … HLK) -L #H destruct
+qed-.
(* Properties with generic slicing for local environments *******************)
(* Basic_2A1: uses: snv_lift *)
-lemma cnv_lifts (a) (h): ∀G. d_liftable1 (cnv a h G).
-#a #h #G #K #T
+lemma cnv_lifts (h) (a): ∀G. d_liftable1 (cnv h a G).
+#h #a #G #K #T
@(fqup_wf_ind_eq (Ⓣ) … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * * [|||| * ]
[ #s #HG #HK #HT #_ #b #f #L #_ #X #H2 destruct
>(lifts_inv_sort1 … H2) -X -K -f //
(* Inversion lemmas with generic slicing for local environments *************)
(* Basic_2A1: uses: snv_inv_lift *)
-lemma cnv_inv_lifts (a) (h): ∀G. d_deliftable1 (cnv a h G).
-#a #h #G #L #U
+lemma cnv_inv_lifts (h) (a): ∀G. d_deliftable1 (cnv h a G).
+#h #a #G #L #U
@(fqup_wf_ind_eq (Ⓣ) … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * * [|||| * ]
[ #s #HG #HL #HU #H1 #b #f #K #HLK #X #H2 destruct
>(lifts_inv_sort2 … H2) -X -L -f //
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpmuwe_cpmuwe.ma".
+include "basic_2/rt_equivalence/cpes_cpes.ma".
+include "basic_2/dynamic/cnv_cpmuwe.ma". (**) (* should be included by the next *)
+include "basic_2/dynamic/cnv_cpmuwe_cpme.ma".
+include "basic_2/dynamic/cnv_cpes.ma".
+include "basic_2/dynamic/cnv_preserve_cpes.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* main properties with evaluations for rt-transition on terms **************)
+
+theorem cnv_dec (h) (a) (G) (L) (T): ac_props a →
+ Decidable (⦃G,L⦄ ⊢ T ![h,a]).
+#h #a #G #L #T #Ha
+@(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T #G0 #L0 #T0 #IH #G #L * * [|||| * ]
+[ #s #HG #HL #HT destruct -Ha -IH
+ /2 width=1 by cnv_sort, or_introl/
+| #i #HG #HL #HT destruct -Ha
+ elim (drops_F_uni L i) [| * * ]
+ [ /3 width=8 by cnv_inv_lref_atom, or_intror/
+ | /3 width=9 by cnv_inv_lref_unit, or_intror/
+ | #I #V #K #HLK
+ elim (IH G K V) -IH [3: /2 width=2 by fqup_lref/ ]
+ [ /3 width=5 by cnv_lref_drops, or_introl/
+ | /4 width=5 by cnv_inv_lref_pair, or_intror/
+ ]
+ ]
+| #l #HG #HL #HT destruct -Ha -IH
+ /3 width=6 by cnv_inv_gref, or_intror/
+| #p #I #V #T #HG #HL #HT destruct -Ha
+ elim (IH G L V) [| -IH | // ] #HV
+ [ elim (IH G (L.ⓑ{I}V) T) -IH [3: // ] #HT
+ [ /3 width=1 by cnv_bind, or_introl/ ]
+ ]
+ @or_intror #H
+ elim (cnv_inv_bind … H) -H /2 width=1 by/
+| #V #T #HG #HL #HT destruct
+ elim (IH G L V) [| -IH #HV | // ]
+ [ elim (IH G L T) -IH [| #HT #HV | // ]
+ [ #HT #HV
+ elim (cnv_R_cpmuwe_total … HT) #n #Hn
+ elim (dec_min (R_cpmuwe h G L T) … Hn) -Hn
+ [| /2 width=2 by cnv_R_cpmuwe_dec/ ] #n0 #_ -n
+ elim (ac_dec … Ha n0) -Ha
+ [ * #n #Ha #Hn * #X0 #HX0 #_
+ elim (abst_dec X0)
+ [ * #p #W #U0 #H destruct
+ elim (cnv_cpes_dec … 1 0 … HV W) [ #HVW | #HnVW ]
+ [ lapply (cpmuwe_fwd_cpms … HX0) -HX0 #HTU0
+ elim (cnv_fwd_cpms_abst_dx_le … HT … HTU0 … Hn) #U #HTU #_ -U0 -n0
+ /3 width=7 by cnv_appl_cpes, or_introl/
+(* Note: argument type mismatch *)
+ | @or_intror #H -n
+ elim (cnv_inv_appl_cpes … H) -H #m0 #q #WX #UX #_ #_ #_ #HVWX #HTUX
+ lapply (cpmuwe_abst … HTUX) -HTUX #HTUX
+ elim (cnv_cpmuwe_mono … HT … HTUX … HX0) -a -T #H #_
+ elim (cpes_fwd_abst_bi … H) -H #_ #HWX -n0 -m0 -p -q -UX -U0
+ /3 width=3 by cpes_cpes_trans/
+ | lapply (cnv_cpmuwe_trans … HT … HX0) -T #H
+ elim (cnv_inv_bind … H) -H #HW #_ //
+ ]
+(* Note: no expected type *)
+ | #HnX0
+ @or_intror #H
+ elim (cnv_inv_appl_cpes … H) -H #m0 #q #W0 #U0 #_ #_ #_ #_ #HTU0
+ lapply (cpmuwe_abst … HTU0) -HTU0 #HTU0
+ elim (cnv_cpmuwe_mono … HT … HTU0 … HX0) -T #_ #H
+ elim (tweq_inv_abst_sn … H) -W0 -U0 #W0 #U0 #H destruct
+ /2 width=4 by/
+ ]
+(* Note: failed applicability *)
+ | #Hge #_ #Hlt
+ @or_intror #H
+ elim (cnv_inv_appl … H) -H #m0 #q #W0 #U0 #Hm0 #_ #_ #_ #HTU0
+ elim (lt_or_ge m0 n0) #H0 [| /3 width=3 by ex2_intro/ ] -Hm0 -Hge
+ /4 width=6 by cpmuwe_abst, ex_intro/
+ ]
+ ]
+ ]
+ @or_intror #H
+ elim (cnv_inv_appl … H) -H /2 width=1 by/
+| #U #T #HG #HL #HT destruct
+ elim (IH G L U) [| -IH | // ] #HU
+ [ elim (IH G L T) -IH [3: // ] #HT
+ [ elim (cnv_cpes_dec … 0 1 … HU … HT) #HUT
+ [ /3 width=1 by cnv_cast_cpes, or_introl/ ]
+ ]
+ ]
+ @or_intror #H
+ elim (cnv_inv_cast_cpes … H) -H /2 width=1 by/
+]
+qed-.
include "static_2/s_computation/fqus_fqup.ma".
include "basic_2/dynamic/cnv_drops.ma".
-(* CONTEXT_SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
(* Properties with supclosure ***********************************************)
(* Basic_2A1: uses: snv_fqu_conf *)
-lemma cnv_fqu_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h].
-#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma cnv_fqu_conf (h) (a):
+ ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂ ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ T1 ![h,a] → ⦃G2,L2⦄ ⊢ T2 ![h,a].
+#h #a #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I1 #G1 #L1 #V1 #H
elim (cnv_inv_zero … H) -H #I2 #L2 #V2 #HV2 #H destruct //
| * [ #p #I1 | * ] #G1 #L1 #V1 #T1 #H
| elim (cnv_inv_appl … H) -H //
| elim (cnv_inv_cast … H) -H //
]
-| #p #I1 #G1 #L1 #V1 #T1 #H
+| #p #I1 #G1 #L1 #V1 #T1 #_ #H
elim (cnv_inv_bind … H) -H //
| #p #I1 #G1 #L1 #V1 #T1 #H destruct
| * #G1 #L1 #V1 #T1 #H
qed-.
(* Basic_2A1: uses: snv_fquq_conf *)
-lemma cnv_fquq_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h].
-#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [|*]
+lemma cnv_fquq_conf (h) (a):
+ ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮ ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ T1 ![h,a] → ⦃G2,L2⦄ ⊢ T2 ![h,a].
+#h #a #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [|*]
/2 width=5 by cnv_fqu_conf/
qed-.
(* Basic_2A1: uses: snv_fqup_conf *)
-lemma cnv_fqup_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h].
-#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+lemma cnv_fqup_conf (h) (a):
+ ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+ ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ T1 ![h,a] → ⦃G2,L2⦄ ⊢ T2 ![h,a].
+#h #a #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
/3 width=5 by fqup_strap1, cnv_fqu_conf/
qed-.
(* Basic_2A1: uses: snv_fqus_conf *)
-lemma cnv_fqus_conf (a) (h): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ T1 ![a, h] → ⦃G2, L2⦄ ⊢ T2 ![a, h].
-#a #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H [|*]
+lemma cnv_fqus_conf (h) (a):
+ ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂* ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ T1 ![h,a] → ⦃G2,L2⦄ ⊢ T2 ![h,a].
+#h #a #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H [|*]
/2 width=5 by cnv_fqup_conf/
qed-.
(* Forward lemmas with strongly rst-normalizing closures ********************)
+(* Note: this is the "big tree" theorem *)
(* Basic_2A1: uses: snv_fwd_fsb *)
-lemma cnv_fwd_fsb (a) (h) (o): ∀G,L,T. ⦃G, L⦄ ⊢ T ![a, h] → ≥[h, o] 𝐒⦃G, L, T⦄.
-#a #h #o #G #L #T #H elim (cnv_fwd_aaa … H) -H /2 width=2 by aaa_fsb/
+lemma cnv_fwd_fsb (h) (a):
+ ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] → ≥[h] 𝐒⦃G,L,T⦄.
+#h #a #G #L #T #H elim (cnv_fwd_aaa … H) -H /2 width=2 by aaa_fsb/
+qed-.
+
+(* Forward lemmas with strongly rt-normalizing terms ************************)
+
+lemma cnv_fwd_csx (h) (a):
+ ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #a #G #L #T #H
+/3 width=2 by cnv_fwd_fsb, fsb_inv_csx/
qed-.
(* Inversion lemmas with proper parallel rst-computation for closures *******)
-lemma cnv_fpbg_refl_false (a) (h) (o) (G) (L) (T):
- ⦃G, L⦄ ⊢ T ![a,h] → ⦃G, L, T⦄ >[h,o] ⦃G, L, T⦄ → ⊥.
+lemma cnv_fpbg_refl_false (h) (a):
+ ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] → ⦃G,L,T⦄ >[h] ⦃G,L,T⦄ → ⊥.
/3 width=7 by cnv_fwd_fsb, fsb_fpbg_refl_false/ qed-.
(* Main preservation properties *********************************************)
(* Basic_2A1: uses: snv_preserve *)
-lemma cnv_preserve (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] →
- ∧∧ IH_cnv_cpms_conf_lpr a h G L T
- & IH_cnv_cpm_trans_lpr a h G L T.
-#a #h #G #L #T #HT
-letin o ≝ (sd_O h)
-lapply (cnv_fwd_fsb … o … HT) -HT #H
+lemma cnv_preserve (h) (a): ∀G,L,T. ⦃G,L⦄ ⊢ T ![h,a] →
+ ∧∧ IH_cnv_cpms_conf_lpr h a G L T
+ & IH_cnv_cpm_trans_lpr h a G L T.
+#h #a #G #L #T #HT
+lapply (cnv_fwd_fsb … HT) -HT #H
@(fsb_ind_fpbg … H) -G -L -T #G #L #T #_ #IH
@conj [ letin aux ≝ cnv_cpms_conf_lpr_aux | letin aux ≝ cnv_cpm_trans_lpr_aux ]
-@(aux … o … G L T) // #G0 #L0 #T0 #H
+@(aux … G L T) // #G0 #L0 #T0 #H
elim (IH … H) -IH -H //
qed-.
-theorem cnv_cpms_conf_lpr (a) (h) (G) (L) (T): IH_cnv_cpms_conf_lpr a h G L T.
-#a #h #G #L #T #HT elim (cnv_preserve … HT) /2 width=1 by/
+theorem cnv_cpms_conf_lpr (h) (a) (G) (L) (T): IH_cnv_cpms_conf_lpr h a G L T.
+#h #a #G #L #T #HT elim (cnv_preserve … HT) /2 width=1 by/
qed-.
(* Basic_2A1: uses: snv_cpr_lpr *)
-theorem cnv_cpm_trans_lpr (a) (h) (G) (L) (T): IH_cnv_cpm_trans_lpr a h G L T.
-#a #h #G #L #T #HT elim (cnv_preserve … HT) /2 width=2 by/
+theorem cnv_cpm_trans_lpr (h) (a) (G) (L) (T): IH_cnv_cpm_trans_lpr h a G L T.
+#h #a #G #L #T #HT elim (cnv_preserve … HT) /2 width=2 by/
qed-.
(* Advanced preservation properties *****************************************)
-lemma cnv_cpms_conf (a) (h) (G) (L):
- ∀T0. ⦃G,L⦄ ⊢ T0 ![a,h] →
+lemma cnv_cpms_conf (h) (a) (G) (L):
+ ∀T0. ⦃G,L⦄ ⊢ T0 ![h,a] →
∀n1,T1. ⦃G,L⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G,L⦄ ⊢ T0 ➡*[n2,h] T2 →
∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L⦄ ⊢ T2 ➡*[n1-n2,h] T.
/2 width=8 by cnv_cpms_conf_lpr/ qed-.
(* Basic_2A1: uses: snv_cprs_lpr *)
-lemma cnv_cpms_trans_lpr (a) (h) (G) (L) (T): IH_cnv_cpms_trans_lpr a h G L T.
-#a #h #G #L1 #T1 #HT1 #n #T2 #H
+lemma cnv_cpms_trans_lpr (h) (a) (G) (L) (T): IH_cnv_cpms_trans_lpr h a G L T.
+#h #a #G #L1 #T1 #HT1 #n #T2 #H
@(cpms_ind_dx … H) -n -T2 /3 width=6 by cnv_cpm_trans_lpr/
qed-.
-lemma cnv_cpm_trans (a) (h) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
- ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ T2 ![a,h].
+lemma cnv_cpm_trans (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ T2 ![h,a].
/2 width=6 by cnv_cpm_trans_lpr/ qed-.
(* Note: this is the preservation property *)
-lemma cnv_cpms_trans (a) (h) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
- ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T2 ![a,h].
+lemma cnv_cpms_trans (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] →
+ ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T2 ![h,a].
/2 width=6 by cnv_cpms_trans_lpr/ qed-.
-lemma cnv_lpr_trans (a) (h) (G):
- ∀L1,T. ⦃G,L1⦄ ⊢ T ![a,h] → ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T ![a,h].
+lemma cnv_lpr_trans (h) (a) (G):
+ ∀L1,T. ⦃G,L1⦄ ⊢ T ![h,a] → ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T ![h,a].
/2 width=6 by cnv_cpm_trans_lpr/ qed-.
-lemma cnv_lprs_trans (a) (h) (G):
- ∀L1,T. ⦃G,L1⦄ ⊢ T ![a,h] → ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L2⦄ ⊢ T ![a,h].
-#a #h #G #L1 #T #HT #L2 #H
+lemma cnv_lprs_trans (h) (a) (G):
+ ∀L1,T. ⦃G,L1⦄ ⊢ T ![h,a] → ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L2⦄ ⊢ T ![h,a].
+#h #a #G #L1 #T #HT #L2 #H
@(lprs_ind_dx … H) -L2 /2 width=3 by cnv_lpr_trans/
qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_equivalence/cpcs_cprs.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Forward lemmas with r-equivalence ****************************************)
+
+lemma cnv_cpms_conf_eq (h) (a) (n) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
+ ∀T1. ⦃G,L⦄ ⊢ T ➡*[n,h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
+#h #a #n #G #L #T #HT #T1 #HT1 #T2 #HT2
+elim (cnv_cpms_conf … HT … HT1 … HT2) -T <minus_n_n #T #HT1 #HT2
+/2 width=3 by cprs_div/
+qed-.
+
+lemma cnv_cpms_fwd_appl_sn_decompose (h) (a) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T ![h,a] → ∀n,X. ⦃G,L⦄ ⊢ ⓐV.T ➡*[n,h] X →
+ ∃∃U. ⦃G,L⦄ ⊢ T ![h,a] & ⦃G,L⦄ ⊢ T ➡*[n,h] U & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X.
+#h #a #G #L #V #T #H0 #n #X #HX
+elim (cnv_inv_appl … H0) #m #p #W #U #_ #_ #HT #_ #_ -m -p -W -U
+elim (cnv_fwd_cpms_total h … n … HT) #U #HTU
+lapply (cpms_appl_dx … V V … HTU) [ // ] #H
+/3 width=8 by cnv_cpms_conf_eq, ex3_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpme_aaa.ma".
+include "basic_2/rt_computation/cpre_cpre.ma".
+include "basic_2/rt_equivalence/cpes.ma".
+include "basic_2/dynamic/cnv_cpme.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Properties with t-bound rt-equivalence for terms *************************)
+
+lemma cnv_cpes_dec (h) (a) (n1) (n2) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∀T2. ⦃G,L⦄ ⊢ T2 ![h,a] →
+ Decidable … (⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2).
+#h #a #n1 #n2 #G #L #T1 #HT1 #T2 #HT2
+elim (cnv_fwd_aaa … HT1) #A1 #HA1
+elim (cnv_fwd_aaa … HT2) #A2 #HA2
+elim (cpme_total_aaa h n1 … HA1) -HA1 #U1 #HTU1
+elim (cpme_total_aaa h n2 … HA2) -HA2 #U2 #HTU2
+elim (eq_term_dec U1 U2) [ #H destruct | #HnU12 ]
+[ cases HTU1 -HTU1 #HTU1 #_
+ cases HTU2 -HTU2 #HTU2 #_
+ /3 width=3 by cpms_div, or_introl/
+| @or_intror * #T0 #HT10 #HT20
+ lapply (cnv_cpme_cpms_conf … HT1 … HT10 … HTU1) -T1 #H1
+ lapply (cnv_cpme_cpms_conf … HT2 … HT20 … HTU2) -T2 #H2
+ /3 width=6 by cpre_mono/
+]
+qed-.
(* Inductive premises for the preservation results **************************)
-definition IH_cnv_cpm_trans_lpr (a) (h): relation3 genv lenv term ≝
- λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
- ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡[n,h] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
+definition IH_cnv_cpm_trans_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![h,a] →
+ ∀n,T2. ⦃G,L1⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![h,a].
-definition IH_cnv_cpms_trans_lpr (a) (h): relation3 genv lenv term ≝
- λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
- ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡*[n,h] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
+definition IH_cnv_cpms_trans_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L1,T1. ⦃G,L1⦄ ⊢ T1 ![h,a] →
+ ∀n,T2. ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 ![h,a].
-definition IH_cnv_cpm_conf_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
- ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+definition IH_cnv_cpm_conf_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
+ ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
-definition IH_cnv_cpms_strip_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
- ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+definition IH_cnv_cpms_strip_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
+ ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
-definition IH_cnv_cpms_conf_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
- ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
+definition IH_cnv_cpms_conf_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
+ ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡*[n2,h] T2 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G,L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
(* Auxiliary properties for preservation ************************************)
-fact cnv_cpms_trans_lpr_sub (a) (h) (o):
- ∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_trans_lpr a h G1 L1 T1.
-#a #h #o #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
+fact cnv_cpms_trans_lpr_sub (h) (a):
+ ∀G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_trans_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_trans_lpr h a G1 L1 T1.
+#h #a #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
@(cpms_ind_dx … H) -n -T2
/3 width=7 by fpbg_cpms_trans/
qed-.
-fact cnv_cpm_conf_lpr_sub (a) (h) (o):
- ∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
+fact cnv_cpm_conf_lpr_sub (h) (a):
+ ∀G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpm_conf_lpr h a G1 L1 T1.
/3 width=8 by cpm_cpms/ qed-.
-fact cnv_cpms_strip_lpr_sub (a) (h) (o):
- ∀G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
- ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, o] ⦃G1, L1, T1⦄ → IH_cnv_cpms_strip_lpr a h G1 L1 T1.
+fact cnv_cpms_strip_lpr_sub (h) (a):
+ ∀G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_conf_lpr h a G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G0,L0,T0⦄ >[h] ⦃G1,L1,T1⦄ → IH_cnv_cpms_strip_lpr h a G1 L1 T1.
/3 width=8 by cpm_cpms/ qed-.
(* LOCAL ENVIRONMENT REFINEMENT FOR NATIVE VALIDITY *************************)
-inductive lsubv (a) (h) (G): relation lenv ≝
-| lsubv_atom: lsubv a h G (⋆) (⋆)
-| lsubv_bind: ∀I,L1,L2. lsubv a h G L1 L2 → lsubv a h G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsubv_beta: ∀L1,L2,W,V. ⦃G, L1⦄ ⊢ ⓝW.V ![a,h] →
- lsubv a h G L1 L2 → lsubv a h G (L1.ⓓⓝW.V) (L2.ⓛW)
+inductive lsubv (h) (a) (G): relation lenv ≝
+| lsubv_atom: lsubv h a G (⋆) (⋆)
+| lsubv_bind: ∀I,L1,L2. lsubv h a G L1 L2 → lsubv h a G (L1.ⓘ{I}) (L2.ⓘ{I})
+| lsubv_beta: ∀L1,L2,W,V. ⦃G,L1⦄ ⊢ ⓝW.V ![h,a] →
+ lsubv h a G L1 L2 → lsubv h a G (L1.ⓓⓝW.V) (L2.ⓛW)
.
interpretation
"local environment refinement (native validity)"
- 'LRSubEqV a h G L1 L2 = (lsubv a h G L1 L2).
+ 'LRSubEqV h a G L1 L2 = (lsubv h a G L1 L2).
(* Basic inversion lemmas ***************************************************)
-fact lsubv_inv_atom_sn_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → L1 = ⋆ → L2 = ⋆.
-#a #h #G #L1 #L2 * -L1 -L2
+fact lsubv_inv_atom_sn_aux (h) (a) (G):
+ ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 → L1 = ⋆ → L2 = ⋆.
+#h #a #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #_ #H destruct
| #L1 #L2 #W #V #_ #_ #H destruct
qed-.
(* Basic_2A1: uses: lsubsv_inv_atom1 *)
-lemma lsubv_inv_atom_sn (a) (h) (G): ∀L2. G ⊢ ⋆ ⫃![a,h] L2 → L2 = ⋆.
+lemma lsubv_inv_atom_sn (h) (a) (G):
+ ∀L2. G ⊢ ⋆ ⫃![h,a] L2 → L2 = ⋆.
/2 width=6 by lsubv_inv_atom_sn_aux/ qed-.
-fact lsubv_inv_bind_sn_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
- ∀I,K1. L1 = K1.ⓘ{I} →
- ∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
- G ⊢ K1 ⫃![a,h] K2 &
- I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
-#a #h #G #L1 #L2 * -L1 -L2
+fact lsubv_inv_bind_sn_aux (h) (a) (G): ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 →
+ ∀I,K1. L1 = K1.ⓘ{I} →
+ ∨∨ ∃∃K2. G ⊢ K1 ⫃![h,a] K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![h,a] & G ⊢ K1 ⫃![h,a] K2
+ & I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
+#h #a #G #L1 #L2 * -L1 -L2
[ #J #K1 #H destruct
| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by ex2_intro, or_introl/
| #L1 #L2 #W #V #HWV #HL12 #J #K1 #H destruct /3 width=7 by ex4_3_intro, or_intror/
qed-.
(* Basic_2A1: uses: lsubsv_inv_pair1 *)
-lemma lsubv_inv_bind_sn (a) (h) (G): ∀I,K1,L2. G ⊢ K1.ⓘ{I} ⫃![a,h] L2 →
- ∨∨ ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
- G ⊢ K1 ⫃![a,h] K2 &
- I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
+lemma lsubv_inv_bind_sn (h) (a) (G):
+ ∀I,K1,L2. G ⊢ K1.ⓘ{I} ⫃![h,a] L2 →
+ ∨∨ ∃∃K2. G ⊢ K1 ⫃![h,a] K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![h,a] & G ⊢ K1 ⫃![h,a] K2
+ & I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
/2 width=3 by lsubv_inv_bind_sn_aux/ qed-.
-fact lsubv_inv_atom_dx_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → L2 = ⋆ → L1 = ⋆.
-#a #h #G #L1 #L2 * -L1 -L2
+fact lsubv_inv_atom_dx_aux (h) (a) (G):
+ ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 → L2 = ⋆ → L1 = ⋆.
+#h #a #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #_ #H destruct
| #L1 #L2 #W #V #_ #_ #H destruct
qed-.
(* Basic_2A1: uses: lsubsv_inv_atom2 *)
-lemma lsubv_inv_atom2 (a) (h) (G): ∀L1. G ⊢ L1 ⫃![a,h] ⋆ → L1 = ⋆.
+lemma lsubv_inv_atom2 (h) (a) (G):
+ ∀L1. G ⊢ L1 ⫃![h,a] ⋆ → L1 = ⋆.
/2 width=6 by lsubv_inv_atom_dx_aux/ qed-.
-fact lsubv_inv_bind_dx_aux (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
- ∀I,K2. L2 = K2.ⓘ{I} →
- ∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
- G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
-#a #h #G #L1 #L2 * -L1 -L2
+fact lsubv_inv_bind_dx_aux (h) (a) (G):
+ ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 →
+ ∀I,K2. L2 = K2.ⓘ{I} →
+ ∨∨ ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![h,a] &
+ G ⊢ K1 ⫃![h,a] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
+#h #a #G #L1 #L2 * -L1 -L2
[ #J #K2 #H destruct
| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or_introl/
| #L1 #L2 #W #V #HWV #HL12 #J #K2 #H destruct /3 width=7 by ex4_3_intro, or_intror/
qed-.
(* Basic_2A1: uses: lsubsv_inv_pair2 *)
-lemma lsubv_inv_bind_dx (a) (h) (G): ∀I,L1,K2. G ⊢ L1 ⫃![a,h] K2.ⓘ{I} →
- ∨∨ ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. ⦃G, K1⦄ ⊢ ⓝW.V ![a,h] &
- G ⊢ K1 ⫃![a,h] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
+lemma lsubv_inv_bind_dx (h) (a) (G):
+ ∀I,L1,K2. G ⊢ L1 ⫃![h,a] K2.ⓘ{I} →
+ ∨∨ ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. ⦃G,K1⦄ ⊢ ⓝW.V ![h,a] &
+ G ⊢ K1 ⫃![h,a] K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
/2 width=3 by lsubv_inv_bind_dx_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lsubv_inv_abst_sn (a) (h) (G): ∀K1,L2,W. G ⊢ K1.ⓛW ⫃![a,h] L2 →
- ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & L2 = K2.ⓛW.
-#a #h #G #K1 #L2 #W #H
+lemma lsubv_inv_abst_sn (h) (a) (G):
+ ∀K1,L2,W. G ⊢ K1.ⓛW ⫃![h,a] L2 →
+ ∃∃K2. G ⊢ K1 ⫃![h,a] K2 & L2 = K2.ⓛW.
+#h #a #G #K1 #L2 #W #H
elim (lsubv_inv_bind_sn … H) -H // *
#K2 #XW #XV #_ #_ #H1 #H2 destruct
qed-.
(* Basic properties *********************************************************)
(* Basic_2A1: uses: lsubsv_refl *)
-lemma lsubv_refl (a) (h) (G): reflexive … (lsubv a h G).
-#a #h #G #L elim L -L /2 width=1 by lsubv_atom, lsubv_bind/
+lemma lsubv_refl (h) (a) (G): reflexive … (lsubv h a G).
+#h #a #G #L elim L -L /2 width=1 by lsubv_atom, lsubv_bind/
qed.
(* Basic_2A1: removed theorems 3:
(* Forward lemmas with native validity **************************************)
(* Basic_2A1: uses: lsubsv_snv_trans *)
-lemma lsubv_cnv_trans (a) (h) (G):
- ∀L2,T. ⦃G, L2⦄ ⊢ T ![a,h] →
- ∀L1. G ⊢ L1 ⫃![a,h] L2 → ⦃G, L1⦄ ⊢ T ![a,h].
-#a #h #G #L2 #T #H elim H -G -L2 -T //
+lemma lsubv_cnv_trans (h) (a) (G):
+ ∀L2,T. ⦃G,L2⦄ ⊢ T ![h,a] →
+ ∀L1. G ⊢ L1 ⫃![h,a] L2 → ⦃G,L1⦄ ⊢ T ![h,a].
+#h #a #G #L2 #T #H elim H -G -L2 -T //
[ #I #G #K2 #V #HV #IH #L1 #H
elim (lsubv_inv_bind_dx … H) -H * /3 width=1 by cnv_zero/
| #I #G #K2 #i #_ #IH #L1 #H
(* Forward lemmas with context-sensitive r-equivalence for terms ************)
(* Basic_2A1: uses: lsubsv_cprs_trans *)
-lemma lsubv_cpcs_trans (a) (h) (G): lsub_trans … (cpcs h G) (lsubv a h G).
+lemma lsubv_cpcs_trans (h) (a) (G): lsub_trans … (cpcs h G) (lsubv h a G).
/3 width=6 by lsubv_fwd_lsubr, lsubr_cpcs_trans/
qed-.
(* Forward lemmas with t-bound contex-sensitive rt-computation for terms ****)
(* Basic_2A1: uses: lsubsv_cprs_trans lsubsv_scpds_trans *)
-lemma lsubv_cpms_trans (a) (n) (h) (G): lsub_trans … (λL. cpms h G L n) (lsubv a h G).
+lemma lsubv_cpms_trans (h) (a) (n) (G):
+ lsub_trans … (λL. cpms h G L n) (lsubv h a G).
/3 width=6 by lsubv_fwd_lsubr, lsubr_cpms_trans/
qed-.
(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
(* Basic_2A1: includes: lsubsv_drop_O1_conf *)
-lemma lsubv_drops_conf_isuni (a) (h) (G):
- ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
- ∀b,f,K1. 𝐔⦃f⦄ → ⬇*[b,f] L1 ≘ K1 →
- ∃∃K2. G ⊢ K1 ⫃![a,h] K2 & ⬇*[b,f] L2 ≘ K2.
-#a #h #G #L1 #L2 #H elim H -L1 -L2
+lemma lsubv_drops_conf_isuni (h) (a) (G):
+ ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 →
+ ∀b,f,K1. 𝐔⦃f⦄ → ⬇*[b,f] L1 ≘ K1 →
+ ∃∃K2. G ⊢ K1 ⫃![h,a] K2 & ⬇*[b,f] L2 ≘ K2.
+#h #a #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #HL12 #IH #b #f #K1 #Hf #H
elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
(* Basic_2A1: includes: lsubsv_drop_O1_trans *)
-lemma lsubv_drops_trans_isuni (a) (h) (G):
- ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 →
- ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b,f] L2 ≘ K2 →
- ∃∃K1. G ⊢ K1 ⫃![a,h] K2 & ⬇*[b,f] L1 ≘ K1.
-#a #h #G #L1 #L2 #H elim H -L1 -L2
+lemma lsubv_drops_trans_isuni (h) (a) (G):
+ ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 →
+ ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b,f] L2 ≘ K2 →
+ ∃∃K1. G ⊢ K1 ⫃![h,a] K2 & ⬇*[b,f] L1 ≘ K1.
+#h #a #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #HL12 #IH #b #f #K2 #Hf #H
elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
(* Forward lemmas with lenv refinement for atomic arity assignment **********)
(* Basic_2A1: uses: lsubsv_fwd_lsuba *)
-lemma lsubsv_fwd_lsuba (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → G ⊢ L1 ⫃⁝ L2.
-#a #h #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsuba_bind/
+lemma lsubsv_fwd_lsuba (h) (a) (G): ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 → G ⊢ L1 ⫃⁝ L2.
+#h #a #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsuba_bind/
#L1 #L2 #W #V #H #_ #IH
elim (cnv_inv_cast … H) -H #W0 #HW #HV #HW0 #HVW0
elim (cnv_fwd_aaa … HW) -HW #B #HW
(* Forward lemmas with restricted refinement for local environments *********)
(* Basic_2A1: uses: lsubsv_fwd_lsubr *)
-lemma lsubv_fwd_lsubr (a) (h) (G): ∀L1,L2. G ⊢ L1 ⫃![a,h] L2 → L1 ⫃ L2.
-#a #h #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_beta/
+lemma lsubv_fwd_lsubr (h) (a) (G): ∀L1,L2. G ⊢ L1 ⫃![h,a] L2 → L1 ⫃ L2.
+#h #a #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_beta/
qed-.
(* Main properties **********************************************************)
(* Note: not valid in Basic_2A1 *)
-theorem lsubv_trans (a) (h) (G): Transitive … (lsubv a h G).
-#a #h #G #L1 #L #H elim H -L1 -L //
+theorem lsubv_trans (h) (a) (G): Transitive … (lsubv h a G).
+#h #a #G #L1 #L #H elim H -L1 -L //
[ #I #K1 #K #HK1 #IH #Y #H
elim (lsubv_inv_bind_sn … H) -H *
[ #K2 #HK2 #H destruct /3 width=1 by lsubv_bind/
(**************************************************************************)
include "basic_2/notation/relations/colon_6.ma".
-include "basic_2/notation/relations/colon_5.ma".
-include "basic_2/notation/relations/colonstar_5.ma".
include "basic_2/dynamic/cnv.ma".
(* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
-definition nta (a) (h): relation4 genv lenv term term ≝
- λG,L,T,U. ⦃G,L⦄ ⊢ ⓝU.T ![a,h].
+definition nta (h) (a): relation4 genv lenv term term ≝
+ λG,L,T,U. ⦃G,L⦄ ⊢ ⓝU.T ![h,a].
interpretation "native type assignment (term)"
- 'Colon a h G L T U = (nta a h G L T U).
-
-interpretation "restricted native type assignment (term)"
- 'Colon h G L T U = (nta true h G L T U).
-
-interpretation "extended native type assignment (term)"
- 'ColonStar h G L T U = (nta false h G L T U).
+ 'Colon h a G L T U = (nta h a G L T U).
(* Basic properties *********************************************************)
(* Basic_1: was by definition: ty3_sort *)
(* Basic_2A1: was by definition: nta_sort ntaa_sort *)
-lemma nta_sort (a) (h) (G) (L) (s): ⦃G,L⦄ ⊢ ⋆s :[a,h] ⋆(next h s).
-#a #h #G #L #s /2 width=3 by cnv_sort, cnv_cast, cpms_sort/
+lemma nta_sort (h) (a) (G) (L): ∀s. ⦃G,L⦄ ⊢ ⋆s :[h,a] ⋆(⫯[h]s).
+#h #a #G #L #s /2 width=3 by cnv_sort, cnv_cast, cpms_sort/
qed.
-lemma nta_bind_cnv (a) (h) (G) (K):
- ∀V. ⦃G,K⦄ ⊢ V ![a,h] →
- ∀I,T,U. ⦃G,K.ⓑ{I}V⦄ ⊢ T :[a,h] U →
- ∀p. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[a,h] ⓑ{p,I}V.U.
-#a #h #G #K #V #HV #I #T #U #H #p
+lemma nta_bind_cnv (h) (a) (G) (K):
+ ∀V. ⦃G,K⦄ ⊢ V ![h,a] →
+ ∀I,T,U. ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h,a] U →
+ ∀p. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[h,a] ⓑ{p,I}V.U.
+#h #a #G #K #V #HV #I #T #U #H #p
elim (cnv_inv_cast … H) -H #X #HU #HT #HUX #HTX
/3 width=5 by cnv_bind, cnv_cast, cpms_bind_dx/
qed.
(* Basic_2A1: was by definition: nta_cast *)
-lemma nta_cast (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ⦃G,L⦄ ⊢ ⓝU.T :[a,h] U.
-#a #h #G #L #T #U #H
+lemma nta_cast (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ⦃G,L⦄ ⊢ ⓝU.T :[h,a] U.
+#h #a #G #L #T #U #H
elim (cnv_inv_cast … H) #X #HU #HT #HUX #HTX
/3 width=3 by cnv_cast, cpms_eps/
qed.
(* Basic_1: was by definition: ty3_cast *)
-lemma nta_cast_old (a) (h) (G) (L):
- ∀T0,T1. ⦃G,L⦄ ⊢ T0 :[a,h] T1 →
- ∀T2. ⦃G,L⦄ ⊢ T1 :[a,h] T2 → ⦃G,L⦄ ⊢ ⓝT1.T0 :[a,h] ⓝT2.T1.
-#a #h #G #L #T0 #T1 #H1 #T2 #H2
+lemma nta_cast_old (h) (a) (G) (L):
+ ∀T0,T1. ⦃G,L⦄ ⊢ T0 :[h,a] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T1 :[h,a] T2 → ⦃G,L⦄ ⊢ ⓝT1.T0 :[h,a] ⓝT2.T1.
+#h #a #G #L #T0 #T1 #H1 #T2 #H2
elim (cnv_inv_cast … H1) #X1 #_ #_ #HTX1 #HTX01
elim (cnv_inv_cast … H2) #X2 #_ #_ #HTX2 #HTX12
/3 width=3 by cnv_cast, cpms_eps/
(* Basic inversion lemmas ***************************************************)
-lemma nta_inv_gref_sn (a) (h) (G) (L):
- ∀X2,l. ⦃G,L⦄ ⊢ §l :[a,h] X2 → ⊥.
-#a #h #G #L #X2 #l #H
+lemma nta_inv_gref_sn (h) (a) (G) (L):
+ ∀X2,l. ⦃G,L⦄ ⊢ §l :[h,a] X2 → ⊥.
+#h #a #G #L #X2 #l #H
elim (cnv_inv_cast … H) -H #X #_ #H #_ #_
elim (cnv_inv_gref … H)
qed-.
(* Basic_forward lemmas *****************************************************)
-lemma nta_fwd_cnv_sn (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ⦃G,L⦄ ⊢ T ![a,h].
-#a #h #G #L #T #U #H
+lemma nta_fwd_cnv_sn (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ⦃G,L⦄ ⊢ T ![h,a].
+#h #a #G #L #T #U #H
elim (cnv_inv_cast … H) -H #X #_ #HT #_ #_ //
qed-.
(* Note: this is nta_fwd_correct_cnv *)
-lemma nta_fwd_cnv_dx (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ⦃G,L⦄ ⊢ U ![a,h].
-#a #h #G #L #T #U #H
+lemma nta_fwd_cnv_dx (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ⦃G,L⦄ ⊢ U ![h,a].
+#h #a #G #L #T #U #H
elim (cnv_inv_cast … H) -H #X #HU #_ #_ #_ //
qed-.
(* Forward lemmas with atomic arity assignment for terms ********************)
(* Note: this means that no type is a universe *)
-lemma nta_fwd_aaa (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ∃∃A. ⦃G,L⦄ ⊢ T ⁝ A & ⦃G,L⦄ ⊢ U ⁝ A.
-#a #h #G #L #T #U #H
+lemma nta_fwd_aaa (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ∃∃A. ⦃G,L⦄ ⊢ T ⁝ A & ⦃G,L⦄ ⊢ U ⁝ A.
+#h #a #G #L #T #U #H
elim (cnv_fwd_aaa … H) -H #A #H
elim (aaa_inv_cast … H) -H #HU #HT
/2 width=3 by ex2_intro/
(* Advanced inversion lemmas ************************************************)
(* Basic_1: uses: ty3_predicative *)
-lemma nta_abst_predicative (a) (h) (p) (G) (L):
- ∀W,T. ⦃G,L⦄ ⊢ ⓛ{p}W.T :[a,h] W → ⊥.
-#a #h #p #G #L #W #T #H
+lemma nta_abst_predicative (h) (a) (p) (G) (L):
+ ∀W,T. ⦃G,L⦄ ⊢ ⓛ{p}W.T :[h,a] W → ⊥.
+#h #a #p #G #L #W #T #H
elim (nta_fwd_aaa … H) -a -h #X #H #H1W
elim (aaa_inv_abst … H) -p #B #A #H2W #_ #H destruct -T
lapply (aaa_mono … H1W … H2W) -G -L -W #H
elim (discr_apair_xy_x … H)
qed-.
-(* Basic_2A1: uses: ty3_repellent *)
-theorem nta_abst_repellent (a) (h) (p) (G) (K):
- ∀W,T,U1. ⦃G,K⦄ ⊢ ⓛ{p}W.T :[a,h] U1 →
- ∀U2. ⦃G,K.ⓛW⦄ ⊢ T :[a,h] U2 → ⬆*[1] U1 ≘ U2 → ⊥.
-#a #h #p #G #K #W #T #U1 #H1 #U2 #H2 #HU12
+(* Basic_1: uses: ty3_repellent *)
+theorem nta_abst_repellent (h) (a) (p) (G) (K):
+ ∀W,T,U1. ⦃G,K⦄ ⊢ ⓛ{p}W.T :[h,a] U1 →
+ ∀U2. ⦃G,K.ⓛW⦄ ⊢ T :[h,a] U2 → ⬆*[1] U1 ≘ U2 → ⊥.
+#h #a #p #G #K #W #T #U1 #H1 #U2 #H2 #HU12
elim (nta_fwd_aaa … H2) -H2 #A2 #H2T #H2U2
elim (nta_fwd_aaa … H1) -H1 #X1 #H1 #HU1
elim (aaa_inv_abst … H1) -a -h -p #B #A1 #_ #H1T #H destruct
(* Properties with r-equivalence for terms **********************************)
-lemma nta_conv_cnv (a) (h) (G) (L) (T):
- ∀U1. ⦃G,L⦄ ⊢ T :[a,h] U1 →
- ∀U2. ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![a,h] → ⦃G,L⦄ ⊢ T :[a,h] U2.
-#a #h #G #L #T #U1 #H1 #U2 #HU12 #HU2
+lemma nta_conv_cnv (h) (a) (G) (L) (T):
+ ∀U1. ⦃G,L⦄ ⊢ T :[h,a] U1 →
+ ∀U2. ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![h,a] → ⦃G,L⦄ ⊢ T :[h,a] U2.
+#h #a #G #L #T #U1 #H1 #U2 #HU12 #HU2
elim (cnv_inv_cast … H1) -H1 #X1 #HU1 #HT #HUX1 #HTX1
lapply (cpcs_cprs_conf … HUX1 … HU12) -U1 #H
elim (cpcs_inv_cprs … H) -H #X2 #HX12 #HU12
(* Basic_1: was by definition: ty3_conv *)
(* Basic_2A1: was by definition: nta_conv ntaa_conv *)
-lemma nta_conv (a) (h) (G) (L) (T):
- ∀U1. ⦃G,L⦄ ⊢ T :[a,h] U1 →
+lemma nta_conv (h) (a) (G) (L) (T):
+ ∀U1. ⦃G,L⦄ ⊢ T :[h,a] U1 →
∀U2. ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 →
- ∀W2. ⦃G,L⦄ ⊢ U2 :[a,h] W2 → ⦃G,L⦄ ⊢ T :[a,h] U2.
-#a #h #G #L #T #U1 #H1 #U2 #HU12 #W2 #H2
+ ∀W2. ⦃G,L⦄ ⊢ U2 :[h,a] W2 → ⦃G,L⦄ ⊢ T :[h,a] U2.
+#h #a #G #L #T #U1 #H1 #U2 #HU12 #W2 #H2
/3 width=3 by nta_conv_cnv, nta_fwd_cnv_sn/
qed-.
(* Basic_1: was: ty3_gen_sort *)
(* Basic_2A1: was: nta_inv_sort1 *)
-lemma nta_inv_sort_sn (a) (h) (G) (L) (X2):
- ∀s. ⦃G,L⦄ ⊢ ⋆s :[a,h] X2 →
- ∧∧ ⦃G,L⦄ ⊢ ⋆(next h s) ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #X2 #s #H
+lemma nta_inv_sort_sn (h) (a) (G) (L) (X2):
+ ∀s. ⦃G,L⦄ ⊢ ⋆s :[h,a] X2 →
+ ∧∧ ⦃G,L⦄ ⊢ ⋆(⫯[h]s) ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #s #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #_ #HX21 #H
lapply (cpms_inv_sort1 … H) -H #H destruct
/3 width=1 by cpcs_cprs_sn, conj/
qed-.
-lemma nta_inv_ldec_sn_cnv (a) (h) (G) (K) (V):
- ∀X2. ⦃G,K.ⓛV⦄ ⊢ #0 :[a,h] X2 →
- ∃∃U. ⦃G,K⦄ ⊢ V ![a,h] & ⬆*[1] V ≘ U & ⦃G,K.ⓛV⦄ ⊢ U ⬌*[h] X2 & ⦃G,K.ⓛV⦄ ⊢ X2 ![a,h].
-#a #h #G #Y #X #X2 #H
+lemma nta_inv_ldec_sn_cnv (h) (a) (G) (K) (V):
+ ∀X2. ⦃G,K.ⓛV⦄ ⊢ #0 :[h,a] X2 →
+ ∃∃U. ⦃G,K⦄ ⊢ V ![h,a] & ⬆*[1] V ≘ U & ⦃G,K.ⓛV⦄ ⊢ U ⬌*[h] X2 & ⦃G,K.ⓛV⦄ ⊢ X2 ![h,a].
+#h #a #G #Y #X #X2 #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_zero … H1) -H1 #Z #K #V #HV #H destruct
elim (cpms_inv_ell_sn … H2) -H2 *
(**************************************************************************)
include "basic_2/rt_computation/cprs_cprs.ma".
-include "basic_2/rt_computation/lprs_cpms.ma".
+include "basic_2/dynamic/cnv_aaa.ma".
include "basic_2/dynamic/nta.ma".
(* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
(* Properties with advanced rt_computation for terms ************************)
-(* Basic_2A1: was by definition nta_appl ntaa_appl *)
-lemma nta_appl_abst (a) (h) (p) (G) (K):
- ∀V,W. ⦃G,K⦄ ⊢ V :[a,h] W →
- ∀T,U. ⦃G,K.ⓛW⦄ ⊢ T :[a,h] U → ⦃G,K⦄ ⊢ ⓐV.ⓛ{p}W.T :[a,h] ⓐV.ⓛ{p}W.U.
-#a #h #p #G #K #V #W #H1 #T #U #H2
+(* Basic_2A1: uses by definition nta_appl ntaa_appl *)
+lemma nta_appl_abst (h) (a) (p) (G) (L):
+ ∀n. ad a n →
+ ∀V,W. ⦃G,L⦄ ⊢ V :[h,a] W →
+ ∀T,U. ⦃G,L.ⓛW⦄ ⊢ T :[h,a] U → ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T :[h,a] ⓐV.ⓛ{p}W.U.
+#h #a #p #G #L #n #Ha #V #W #H1 #T #U #H2
elim (cnv_inv_cast … H1) -H1 #X1 #HW #HV #HWX1 #HVX1
elim (cnv_inv_cast … H2) -H2 #X2 #HU #HT #HUX2 #HTX2
-/4 width=7 by cnv_bind, cnv_appl, cnv_cast, cpms_appl_dx, cpms_bind_dx/
+/4 width=11 by cnv_appl_ge, cnv_cast, cnv_bind, cpms_appl_dx, cpms_bind_dx/
qed.
(* Basic_1: was by definition: ty3_appl *)
(* Basic_2A1: was nta_appl_old *)
-lemma nta_appl (a) (h) (p) (G) (L):
- ∀V,W. ⦃G,L⦄ ⊢ V :[a,h] W →
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T :[a,h] ⓐV.ⓛ{p}W.U.
-#a #h #p #G #L #V #W #H1 #T #U #H2
+lemma nta_appl (h) (a) (p) (G) (L):
+ ∀n. 1 ≤ n → ad a n →
+ ∀V,W. ⦃G,L⦄ ⊢ V :[h,a] W →
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] ⓛ{p}W.U → ⦃G,L⦄ ⊢ ⓐV.T :[h,a] ⓐV.ⓛ{p}W.U.
+#h #a #p #G #L #n #Hn #Ha #V #W #H1 #T #U #H2
elim (cnv_inv_cast … H1) -H1 #X1 #HW #HV #HWX1 #HVX1
elim (cnv_inv_cast … H2) -H2 #X2 #HU #HT #HUX2 #HTX2
elim (cpms_inv_abst_sn … HUX2) #W0 #U0 #HW0 #HU0 #H destruct
elim (cprs_conf … HWX1 … HW0) -HW0 #X0 #HX10 #HWX0
@(cnv_cast … (ⓐV.ⓛ{p}W0.U0)) (**) (* full auto too slow *)
-[ /3 width=7 by cnv_appl, cpms_bind/
-| /4 width=11 by cnv_appl, cpms_cprs_trans, cpms_bind/
+[ /2 width=11 by cnv_appl_ge/
+| /3 width=11 by cnv_appl_ge, cpms_cprs_trans/
| /2 width=1 by cpms_appl_dx/
| /2 width=1 by cpms_appl_dx/
]
(* Inversion lemmas with advanced rt_computation for terms ******************)
-lemma nta_inv_abst_bi_cnv (a) (h) (p) (G) (K) (W):
- ∀T,U. ⦃G,K⦄ ⊢ ⓛ{p}W.T :[a,h] ⓛ{p}W.U →
- ∧∧ ⦃G,K⦄ ⊢ W ![a,h] & ⦃G,K.ⓛW⦄ ⊢ T :[a,h] U.
-#a #h #p #G #K #W #T #U #H
+lemma nta_inv_abst_bi_cnv (h) (a) (p) (G) (K) (W):
+ ∀T,U. ⦃G,K⦄ ⊢ ⓛ{p}W.T :[h,a] ⓛ{p}W.U →
+ ∧∧ ⦃G,K⦄ ⊢ W ![h,a] & ⦃G,K.ⓛW⦄ ⊢ T :[h,a] U.
+#h #a #p #G #K #W #T #U #H
elim (cnv_inv_cast … H) -H #X #HWU #HWT #HUX #HTX
elim (cnv_inv_bind … HWU) -HWU #HW #HU
elim (cnv_inv_bind … HWT) -HWT #_ #HT
elim (cpms_inv_abst_sn … HUX) -HUX #W0 #X0 #_ #HUX0 #H destruct
-elim (cpms_inv_abst_bi … HTX) -HTX #_ #HTX0 -W0
+elim (cpms_inv_abst_bi … HTX) -HTX #_ #_ #HTX0 -W0
/3 width=3 by cnv_cast, conj/
qed-.
(* Advanced properties ******************************************************)
-lemma nta_ldef (a) (h) (G) (K):
- ∀V,W. ⦃G,K⦄ ⊢ V :[a,h] W →
- ∀U. ⬆*[1] W ≘ U → ⦃G,K.ⓓV⦄ ⊢ #0 :[a,h] U.
-#a #h #G #K #V #W #H #U #HWU
+lemma nta_ldef (h) (a) (G) (K):
+ ∀V,W. ⦃G,K⦄ ⊢ V :[h,a] W →
+ ∀U. ⬆*[1] W ≘ U → ⦃G,K.ⓓV⦄ ⊢ #0 :[h,a] U.
+#h #a #G #K #V #W #H #U #HWU
elim (cnv_inv_cast … H) -H #X #HW #HV #HWX #HVX
lapply (cnv_lifts … HW (Ⓣ) … (K.ⓓV) … HWU) -HW
[ /3 width=3 by drops_refl, drops_drop/ ] #HU
/3 width=5 by cnv_zero, cnv_cast, cpms_delta/
qed.
-lemma nta_ldec_cnv (a) (h) (G) (K):
- ∀W. ⦃G,K⦄ ⊢ W ![a,h] →
- ∀U. ⬆*[1] W ≘ U → ⦃G,K.ⓛW⦄ ⊢ #0 :[a,h] U.
-#a #h #G #K #W #HW #U #HWU
+lemma nta_ldec_cnv (h) (a) (G) (K):
+ ∀W. ⦃G,K⦄ ⊢ W ![h,a] →
+ ∀U. ⬆*[1] W ≘ U → ⦃G,K.ⓛW⦄ ⊢ #0 :[h,a] U.
+#h #a #G #K #W #HW #U #HWU
lapply (cnv_lifts … HW (Ⓣ) … (K.ⓛW) … HWU)
/3 width=5 by cnv_zero, cnv_cast, cpms_ell, drops_refl, drops_drop/
qed.
-lemma nta_lref (a) (h) (I) (G) (K):
- ∀T,i. ⦃G,K⦄ ⊢ #i :[a,h] T →
- ∀U. ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #(↑i) :[a,h] U.
-#a #h #I #G #K #T #i #H #U #HTU
+lemma nta_lref (h) (a) (I) (G) (K):
+ ∀T,i. ⦃G,K⦄ ⊢ #i :[h,a] T →
+ ∀U. ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #(↑i) :[h,a] U.
+#h #a #I #G #K #T #i #H #U #HTU
elim (cnv_inv_cast … H) -H #X #HT #Hi #HTX #H2
lapply (cnv_lifts … HT (Ⓣ) … (K.ⓘ{I}) … HTU) -HT
[ /3 width=3 by drops_refl, drops_drop/ ] #HU
(* Properties with generic slicing for local environments *******************)
-lemma nta_lifts_sn (a) (h) (G): d_liftable2_sn … lifts (nta a h G).
-#a #h #G #K #T1 #T2 #H #b #f #L #HLK #U1 #HTU1
+lemma nta_lifts_sn (h) (a) (G): d_liftable2_sn … lifts (nta a h G).
+#h #a #G #K #T1 #T2 #H #b #f #L #HLK #U1 #HTU1
elim (cnv_inv_cast … H) -H #X #HT2 #HT1 #HT2X #HT1X
elim (lifts_total T2 f) #U2 #HTU2
lapply (cnv_lifts … HT2 … HLK … HTU2) -HT2 #HU2
(* Basic_1: was: ty3_lift *)
(* Basic_2A1: was: nta_lift ntaa_lift *)
-lemma nta_lifts_bi (a) (h) (G): d_liftable2_bi … lifts (nta a h G).
+lemma nta_lifts_bi (h) (a) (G): d_liftable2_bi … lifts (nta a h G).
/3 width=12 by nta_lifts_sn, d_liftable2_sn_bi, lifts_mono/ qed-.
(* Basic_1: was by definition: ty3_abbr *)
(* Basic_2A1: was by definition: nta_ldef ntaa_ldef *)
-lemma nta_ldef_drops (a) (h) (G) (K) (L) (i):
- ∀V,W. ⦃G,K⦄ ⊢ V :[a,h] W →
- ∀U. ⬆*[↑i] W ≘ U → ⬇*[i] L ≘ K.ⓓV → ⦃G,L⦄ ⊢ #i :[a,h] U.
-#a #h #G #K #L #i #V #W #HVW #U #HWU #HLK
+lemma nta_ldef_drops (h) (a) (G) (K) (L) (i):
+ ∀V,W. ⦃G,K⦄ ⊢ V :[h,a] W →
+ ∀U. ⬆*[↑i] W ≘ U → ⬇*[i] L ≘ K.ⓓV → ⦃G,L⦄ ⊢ #i :[h,a] U.
+#h #a #G #K #L #i #V #W #HVW #U #HWU #HLK
elim (lifts_split_trans … HWU (𝐔❴1❵) (𝐔❴i❵)) [| // ] #X #HWX #HXU
/3 width=9 by nta_lifts_bi, nta_ldef/
qed.
-lemma nta_ldec_drops_cnv (a) (h) (G) (K) (L) (i):
- ∀W. ⦃G,K⦄ ⊢ W ![a,h] →
- ∀U. ⬆*[↑i] W ≘ U → ⬇*[i] L ≘ K.ⓛW → ⦃G,L⦄ ⊢ #i :[a,h] U.
-#a #h #G #K #L #i #W #HW #U #HWU #HLK
+lemma nta_ldec_drops_cnv (h) (a) (G) (K) (L) (i):
+ ∀W. ⦃G,K⦄ ⊢ W ![h,a] →
+ ∀U. ⬆*[↑i] W ≘ U → ⬇*[i] L ≘ K.ⓛW → ⦃G,L⦄ ⊢ #i :[h,a] U.
+#h #a #G #K #L #i #W #HW #U #HWU #HLK
elim (lifts_split_trans … HWU (𝐔❴1❵) (𝐔❴i❵)) [| // ] #X #HWX #HXU
/3 width=9 by nta_lifts_bi, nta_ldec_cnv/
qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/cnv_eval.ma".
+include "basic_2/dynamic/nta_preserve.ma".
+
+(* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
+
+(* Properties with evaluations for rt-transition on terms *******************)
+
+lemma nta_typecheck_dec (h) (a) (G) (L): ac_props a →
+ ∀T,U. Decidable … (⦃G,L⦄ ⊢ T :[h,a] U).
+/2 width=1 by cnv_dec/ qed-.
+
+(* Basic_1: uses: ty3_inference *)
+lemma nta_inference_dec (h) (a) (G) (L) (T): ac_props a →
+ Decidable (∃U. ⦃G,L⦄ ⊢ T :[h,a] U).
+#h #a #G #L #T #Ha
+elim (cnv_dec h … G L T Ha) -Ha #HT
+[ /3 width=1 by cnv_nta_sn, or_introl/
+| @or_intror * #U #HTU
+ /3 width=2 by nta_fwd_cnv_sn/
+]
+qed-.
(* Note: this might use fsb_inv_cast (still to be proved) *)
(* Basic_1: uses: ty3_sn3 *)
(* Basic_2A1: uses: nta_fwd_csn *)
-theorem nta_fwd_fsb (a) (h) (o) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U →
- ∧∧ ≥[h,o] 𝐒⦃G,L,T⦄ & ≥[h,o] 𝐒⦃G,L,U⦄.
-#a #h #o #G #L #T #U #H
+theorem nta_fwd_fsb (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U →
+ ∧∧ ≥[h] 𝐒⦃G,L,T⦄ & ≥[h] 𝐒⦃G,L,U⦄.
+#h #a #G #L #T #U #H
elim (cnv_inv_cast … H) #X #HU #HT #_ #_ -X
/3 width=2 by cnv_fwd_fsb, conj/
qed-.
(* Advanced eliminators *****************************************************)
lemma nta_ind_rest_cnv (h) (Q:relation4 …):
- (∀G,L,s. Q G L (⋆s) (⋆(next h s))) →
+ (∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
(∀G,K,V,W,U.
- ⦃G,K⦄ ⊢ V :[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ V :[h,𝟐] W → ⬆*[1] W ≘ U →
Q G K V W → Q G (K.ⓓV) (#0) U
) →
- (∀G,K,W,U. ⦃G,K⦄ ⊢ W ![h] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
+ (∀G,K,W,U. ⦃G,K⦄ ⊢ W ![h,𝟐] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
(∀I,G,K,W,U,i.
- ⦃G,K⦄ ⊢ #i :[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ #i :[h,𝟐] W → ⬆*[1] W ≘ U →
Q G K (#i) W → Q G (K.ⓘ{I}) (#↑i) U
) →
(∀p,I,G,K,V,T,U.
- ⦃G,K⦄ ⊢ V ![h] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h] U →
+ ⦃G,K⦄ ⊢ V ![h,𝟐] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h,𝟐] U →
Q G (K.ⓑ{I}V) T U → Q G K (ⓑ{p,I}V.T) (ⓑ{p,I}V.U)
) →
(∀p,G,L,V,W,T,U.
- ⦃G,L⦄ ⊢ V :[h] W → ⦃G,L⦄ ⊢ T :[h] ⓛ{p}W.U →
+ ⦃G,L⦄ ⊢ V :[h,𝟐] W → ⦃G,L⦄ ⊢ T :[h,𝟐] ⓛ{p}W.U →
Q G L V W → Q G L T (ⓛ{p}W.U) → Q G L (ⓐV.T) (ⓐV.ⓛ{p}W.U)
) →
- (∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h] U → Q G L T U → Q G L (ⓝU.T) U
+ (∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝟐] U → Q G L T U → Q G L (ⓝU.T) U
) →
(∀G,L,T,U1,U2.
- ⦃G,L⦄ ⊢ T :[h] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![h] →
+ ⦃G,L⦄ ⊢ T :[h,𝟐] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![h,𝟐] →
Q G L T U1 → Q G L T U2
) →
- ∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h] U → Q G L T U.
+ ∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝟐] U → Q G L T U.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #G #L #T
@(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T #G0 #L0 #T0 #IH #G #L * * [|||| * ]
[ #s #HG #HL #HT #X #H destruct -IH
qed-.
lemma nta_ind_ext_cnv_mixed (h) (Q:relation4 …):
- (∀G,L,s. Q G L (⋆s) (⋆(next h s))) →
+ (∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
(∀G,K,V,W,U.
- ⦃G,K⦄ ⊢ V :*[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ V :[h,𝛚] W → ⬆*[1] W ≘ U →
Q G K V W → Q G (K.ⓓV) (#0) U
) →
- (∀G,K,W,U. ⦃G,K⦄ ⊢ W !*[h] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
+ (∀G,K,W,U. ⦃G,K⦄ ⊢ W ![h,𝛚] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
(∀I,G,K,W,U,i.
- ⦃G,K⦄ ⊢ #i :*[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ #i :[h,𝛚] W → ⬆*[1] W ≘ U →
Q G K (#i) W → Q G (K.ⓘ{I}) (#↑i) U
) →
(∀p,I,G,K,V,T,U.
- ⦃G,K⦄ ⊢ V !*[h] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :*[h] U →
+ ⦃G,K⦄ ⊢ V ![h,𝛚] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h,𝛚] U →
Q G (K.ⓑ{I}V) T U → Q G K (ⓑ{p,I}V.T) (ⓑ{p,I}V.U)
) →
(∀p,G,L,V,W,T,U.
- ⦃G,L⦄ ⊢ V :*[h] W → ⦃G,L⦄ ⊢ T :*[h] ⓛ{p}W.U →
+ ⦃G,L⦄ ⊢ V :[h,𝛚] W → ⦃G,L⦄ ⊢ T :[h,𝛚] ⓛ{p}W.U →
Q G L V W → Q G L T (ⓛ{p}W.U) → Q G L (ⓐV.T) (ⓐV.ⓛ{p}W.U)
) →
(∀G,L,V,T,U.
- ⦃G,L⦄ ⊢ T :*[h] U → ⦃G,L⦄ ⊢ ⓐV.U !*[h] →
+ ⦃G,L⦄ ⊢ T :[h,𝛚] U → ⦃G,L⦄ ⊢ ⓐV.U ![h,𝛚] →
Q G L T U → Q G L (ⓐV.T) (ⓐV.U)
) →
- (∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U → Q G L (ⓝU.T) U
+ (∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝛚] U → Q G L T U → Q G L (ⓝU.T) U
) →
(∀G,L,T,U1,U2.
- ⦃G,L⦄ ⊢ T :*[h] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 !*[h] →
+ ⦃G,L⦄ ⊢ T :[h,𝛚] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![h,𝛚] →
Q G L T U1 → Q G L T U2
) →
- ∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U.
+ ∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝛚] U → Q G L T U.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T
@(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T #G0 #L0 #T0 #IH #G #L * * [|||| * ]
[ #s #HG #HL #HT #X #H destruct -IH
qed-.
lemma nta_ind_ext_cnv (h) (Q:relation4 …):
- (∀G,L,s. Q G L (⋆s) (⋆(next h s))) →
+ (∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
(∀G,K,V,W,U.
- ⦃G,K⦄ ⊢ V :*[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ V :[h,𝛚] W → ⬆*[1] W ≘ U →
Q G K V W → Q G (K.ⓓV) (#0) U
) →
- (∀G,K,W,U. ⦃G,K⦄ ⊢ W !*[h] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
+ (∀G,K,W,U. ⦃G,K⦄ ⊢ W ![h,𝛚] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
(∀I,G,K,W,U,i.
- ⦃G,K⦄ ⊢ #i :*[h] W → ⬆*[1] W ≘ U →
+ ⦃G,K⦄ ⊢ #i :[h,𝛚] W → ⬆*[1] W ≘ U →
Q G K (#i) W → Q G (K.ⓘ{I}) (#↑i) U
) →
(∀p,I,G,K,V,T,U.
- ⦃G,K⦄ ⊢ V !*[h] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :*[h] U →
+ ⦃G,K⦄ ⊢ V ![h,𝛚] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h,𝛚] U →
Q G (K.ⓑ{I}V) T U → Q G K (ⓑ{p,I}V.T) (ⓑ{p,I}V.U)
) →
(∀p,G,K,V,W,T,U.
- ⦃G,K⦄ ⊢ V :*[h] W → ⦃G,K.ⓛW⦄ ⊢ T :*[h] U →
+ ⦃G,K⦄ ⊢ V :[h,𝛚] W → ⦃G,K.ⓛW⦄ ⊢ T :[h,𝛚] U →
Q G K V W → Q G (K.ⓛW) T U → Q G K (ⓐV.ⓛ{p}W.T) (ⓐV.ⓛ{p}W.U)
) →
(∀G,L,V,T,U.
- ⦃G,L⦄ ⊢ T :*[h] U → ⦃G,L⦄ ⊢ ⓐV.U !*[h] →
+ ⦃G,L⦄ ⊢ T :[h,𝛚] U → ⦃G,L⦄ ⊢ ⓐV.U ![h,𝛚] →
Q G L T U → Q G L (ⓐV.T) (ⓐV.U)
) →
- (∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U → Q G L (ⓝU.T) U
+ (∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝛚] U → Q G L T U → Q G L (ⓝU.T) U
) →
(∀G,L,T,U1,U2.
- ⦃G,L⦄ ⊢ T :*[h] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 !*[h] →
+ ⦃G,L⦄ ⊢ T :[h,𝛚] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 ![h,𝛚] →
Q G L T U1 → Q G L T U2
) →
- ∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U.
+ ∀G,L,T,U. ⦃G,L⦄ ⊢ T :[h,𝛚] U → Q G L T U.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T #U #H
@(nta_ind_ext_cnv_mixed … IH1 IH2 IH3 IH4 IH5 … IH7 IH8 IH9 … H) -G -L -T -U -IH1 -IH2 -IH3 -IH4 -IH5 -IH6 -IH8 -IH9
#p #G #L #V #W #T #U #HVW #HTU #_ #IHTU
(**************************************************************************)
include "basic_2/rt_equivalence/cpcs_cpcs.ma".
-include "basic_2/dynamic/cnv_cpcs.ma".
+include "basic_2/dynamic/cnv_preserve_cpcs.ma".
include "basic_2/dynamic/nta.ma".
(* NATIVE TYPE ASSIGNMENT FOR TERMS *****************************************)
(* Properties based on preservation *****************************************)
-lemma cnv_cpms_nta (a) (h) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∀U.⦃G,L⦄ ⊢ T ➡*[1,h] U → ⦃G,L⦄ ⊢ T :[a,h] U.
+lemma cnv_cpms_nta (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀U.⦃G,L⦄ ⊢ T ➡*[1,h] U → ⦃G,L⦄ ⊢ T :[h,a] U.
/3 width=4 by cnv_cast, cnv_cpms_trans/ qed.
-lemma cnv_nta_sn (a) (h) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ∃U. ⦃G,L⦄ ⊢ T :[a,h] U.
-#a #h #G #L #T #HT
+lemma cnv_nta_sn (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∃U. ⦃G,L⦄ ⊢ T :[h,a] U.
+#h #a #G #L #T #HT
elim (cnv_fwd_cpm_SO … HT) #U #HTU
/4 width=2 by cnv_cpms_nta, cpm_cpms, ex_intro/
qed-.
(* Basic_1: was: ty3_typecheck *)
-lemma nta_typecheck (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ∃T0. ⦃G,L⦄ ⊢ ⓝU.T :[a,h] T0.
+lemma nta_typecheck (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ∃T0. ⦃G,L⦄ ⊢ ⓝU.T :[h,a] T0.
/3 width=1 by cnv_cast, cnv_nta_sn/ qed-.
(* Basic_1: was: ty3_correct *)
(* Basic_2A1: was: ntaa_fwd_correct *)
-lemma nta_fwd_correct (a) (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :[a,h] U → ∃T0. ⦃G,L⦄ ⊢ U :[a,h] T0.
+lemma nta_fwd_correct (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ∃T0. ⦃G,L⦄ ⊢ U :[h,a] T0.
/3 width=2 by nta_fwd_cnv_dx, cnv_nta_sn/ qed-.
lemma nta_pure_cnv (h) (G) (L):
- ∀T,U. ⦃G,L⦄ ⊢ T :*[h] U →
- ∀V. ⦃G,L⦄ ⊢ ⓐV.U !*[h] → ⦃G,L⦄ ⊢ ⓐV.T :*[h] ⓐV.U.
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,𝛚] U →
+ ∀V. ⦃G,L⦄ ⊢ ⓐV.U ![h,𝛚] → ⦃G,L⦄ ⊢ ⓐV.T :[h,𝛚] ⓐV.U.
#h #G #L #T #U #H1 #V #H2
elim (cnv_inv_cast … H1) -H1 #X0 #HU #HT #HUX0 #HTX0
elim (cnv_inv_appl … H2) #n #p #X1 #X2 #_ #HV #_ #HVX1 #HUX2
elim (cnv_cpms_conf … HU … HUX0 … HUX2) -HU -HUX2
<minus_O_n <minus_n_O #X #HX0 #H
elim (cpms_inv_abst_sn … H) -H #X3 #X4 #HX13 #HX24 #H destruct
-@(cnv_cast … (ⓐV.X0)) [2:|*: /2 width=1 by cpms_appl_dx/ ]
-@(cnv_appl … X3) [4: |*: /2 width=7 by cpms_trans, cpms_cprs_trans/ ]
-#H destruct
+@(cnv_cast … (ⓐV.X0)) [2:|*: /2 width=1 by cpms_appl_dx/ ] (**) (* full auto a bit slow *)
+/3 width=10 by cnv_appl, cpms_trans, cpms_cprs_trans/
qed.
(* Basic_1: uses: ty3_sred_wcpr0_pr0 *)
-lemma nta_cpr_conf_lpr (a) (h) (G):
- ∀L1,T1,U. ⦃G,L1⦄ ⊢ T1 :[a,h] U → ∀T2. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 →
- ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L1 #T1 #U #H #T2 #HT12 #L2 #HL12
+lemma nta_cpr_conf_lpr (h) (a) (G):
+ ∀L1,T1,U. ⦃G,L1⦄ ⊢ T1 :[h,a] U → ∀T2. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T2 :[h,a] U.
+#h #a #G #L1 #T1 #U #H #T2 #HT12 #L2 #HL12
/3 width=6 by cnv_cpm_trans_lpr, cpm_cast/
qed-.
(* Basic_1: uses: ty3_sred_pr2 ty3_sred_pr0 *)
-lemma nta_cpr_conf (a) (h) (G) (L):
- ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #H #T2 #HT12
+lemma nta_cpr_conf (h) (a) (G) (L):
+ ∀T1,U. ⦃G,L⦄ ⊢ T1 :[h,a] U →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #H #T2 #HT12
/3 width=6 by cnv_cpm_trans, cpm_cast/
qed-.
(* Note: this is the preservation property *)
(* Basic_1: uses: ty3_sred_pr3 ty3_sred_pr1 *)
-lemma nta_cprs_conf (a) (h) (G) (L):
- ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U →
- ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #H #T2 #HT12
+lemma nta_cprs_conf (h) (a) (G) (L):
+ ∀T1,U. ⦃G,L⦄ ⊢ T1 :[h,a] U →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #H #T2 #HT12
/3 width=6 by cnv_cpms_trans, cpms_cast/
qed-.
(* Basic_1: uses: ty3_cred_pr2 *)
-lemma nta_lpr_conf (a) (h) (G):
- ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[a,h] U →
- ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T :[a,h] U.
-#a #h #G #L1 #T #U #HTU #L2 #HL12
+lemma nta_lpr_conf (h) (a) (G):
+ ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[h,a] U →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L2⦄ ⊢ T :[h,a] U.
+#h #a #G #L1 #T #U #HTU #L2 #HL12
/2 width=3 by cnv_lpr_trans/
qed-.
(* Basic_1: uses: ty3_cred_pr3 *)
-lemma nta_lprs_conf (a) (h) (G):
- ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[a,h] U →
- ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L2⦄ ⊢ T :[a,h] U.
-#a #h #G #L1 #T #U #HTU #L2 #HL12
+lemma nta_lprs_conf (h) (a) (G):
+ ∀L1,T,U. ⦃G,L1⦄ ⊢ T :[h,a] U →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L2⦄ ⊢ T :[h,a] U.
+#h #a #G #L1 #T #U #HTU #L2 #HL12
/2 width=3 by cnv_lprs_trans/
qed-.
(* Inversion lemmas based on preservation ***********************************)
-lemma nta_inv_ldef_sn (a) (h) (G) (K) (V):
- ∀X2. ⦃G,K.ⓓV⦄ ⊢ #0 :[a,h] X2 →
- ∃∃W,U. ⦃G,K⦄ ⊢ V :[a,h] W & ⬆*[1] W ≘ U & ⦃G,K.ⓓV⦄ ⊢ U ⬌*[h] X2 & ⦃G,K.ⓓV⦄ ⊢ X2 ![a,h].
-#a #h #G #Y #X #X2 #H
+lemma nta_inv_ldef_sn (h) (a) (G) (K) (V):
+ ∀X2. ⦃G,K.ⓓV⦄ ⊢ #0 :[h,a] X2 →
+ ∃∃W,U. ⦃G,K⦄ ⊢ V :[h,a] W & ⬆*[1] W ≘ U & ⦃G,K.ⓓV⦄ ⊢ U ⬌*[h] X2 & ⦃G,K.ⓓV⦄ ⊢ X2 ![h,a].
+#h #a #G #Y #X #X2 #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_zero … H1) -H1 #Z #K #V #HV #H destruct
elim (cpms_inv_delta_sn … H2) -H2 *
]
qed-.
-lemma nta_inv_lref_sn (a) (h) (G) (L):
- ∀X2,i. ⦃G,L⦄ ⊢ #↑i :[a,h] X2 →
- ∃∃I,K,T2,U2. ⦃G,K⦄ ⊢ #i :[a,h] T2 & ⬆*[1] T2 ≘ U2 & ⦃G,K.ⓘ{I}⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,K.ⓘ{I}⦄ ⊢ X2 ![a,h] & L = K.ⓘ{I}.
-#a #h #G #L #X2 #i #H
+lemma nta_inv_lref_sn (h) (a) (G) (L):
+ ∀X2,i. ⦃G,L⦄ ⊢ #↑i :[h,a] X2 →
+ ∃∃I,K,T2,U2. ⦃G,K⦄ ⊢ #i :[h,a] T2 & ⬆*[1] T2 ≘ U2 & ⦃G,K.ⓘ{I}⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,K.ⓘ{I}⦄ ⊢ X2 ![h,a] & L = K.ⓘ{I}.
+#h #a #G #L #X2 #i #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_lref … H1) -H1 #I #K #Hi #H destruct
elim (cpms_inv_lref_sn … H2) -H2 *
]
qed-.
-lemma nta_inv_lref_sn_drops_cnv (a) (h) (G) (L):
- ∀X2, i. ⦃G,L⦄ ⊢ #i :[a,h] X2 →
- ∨∨ ∃∃K,V,W,U. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V :[a,h] W & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h]
- | ∃∃K,W,U. ⬇*[i] L ≘ K. ⓛW & ⦃G,K⦄ ⊢ W ![a,h] & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #X2 #i #H
+lemma nta_inv_lref_sn_drops_cnv (h) (a) (G) (L):
+ ∀X2,i. ⦃G,L⦄ ⊢ #i :[h,a] X2 →
+ ∨∨ ∃∃K,V,W,U. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V :[h,a] W & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a]
+ | ∃∃K,W,U. ⬇*[i] L ≘ K. ⓛW & ⦃G,K⦄ ⊢ W ![h,a] & ⬆*[↑i] W ≘ U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #i #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_lref_drops … H1) -H1 #I #K #V #HLK #HV
elim (cpms_inv_lref1_drops … H2) -H2 *
]
qed-.
-lemma nta_inv_bind_sn_cnv (a) (h) (p) (I) (G) (K) (X2):
- ∀V,T. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[a,h] X2 →
- ∃∃U. ⦃G,K⦄ ⊢ V ![a,h] & ⦃G,K.ⓑ{I}V⦄ ⊢ T :[a,h] U & ⦃G,K⦄ ⊢ ⓑ{p,I}V.U ⬌*[h] X2 & ⦃G,K⦄ ⊢ X2 ![a,h].
-#a #h #p * #G #K #X2 #V #T #H
+lemma nta_inv_bind_sn_cnv (h) (a) (p) (I) (G) (K) (X2):
+ ∀V,T. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :[h,a] X2 →
+ ∃∃U. ⦃G,K⦄ ⊢ V ![h,a] & ⦃G,K.ⓑ{I}V⦄ ⊢ T :[h,a] U & ⦃G,K⦄ ⊢ ⓑ{p,I}V.U ⬌*[h] X2 & ⦃G,K⦄ ⊢ X2 ![h,a].
+#h #a #p * #G #K #X2 #V #T #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H2
elim (cnv_inv_bind … H1) -H1 #HV #HT
[ elim (cpms_inv_abbr_sn_dx … H2) -H2 *
(* Basic_1: uses: ty3_gen_appl *)
lemma nta_inv_appl_sn (h) (G) (L) (X2):
- ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :[h] X2 →
- ∃∃p,W,U. ⦃G,L⦄ ⊢ V :[h] W & ⦃G,L⦄ ⊢ T :[h] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h].
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :[h,𝟐] X2 →
+ ∃∃p,W,U. ⦃G,L⦄ ⊢ V :[h,𝟐] W & ⦃G,L⦄ ⊢ T :[h,𝟐] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,𝟐].
#h #G #L #X2 #V #T #H
elim (cnv_inv_cast … H) -H #X #HX2 #H1 #HX2 #H2
-elim (cnv_inv_appl … H1) * [ | #n ] #p #W #U #Hn #HV #HT #HVW #HTU
-[ lapply (cnv_cpms_trans … HT … HTU) #H
- elim (cnv_inv_bind … H) -H #_ #HU
- elim (cnv_fwd_cpm_SO … HU) #U0 #HU0 -HU
- lapply (cpms_step_dx … HTU 1 (ⓛ{p}W.U0) ?) -HTU [ /2 width=1 by cpm_bind/ ] #HTU
-| lapply (le_n_O_to_eq n ?) [ /3 width=1 by le_S_S_to_le/ ] -Hn #H destruct
-]
+elim (cnv_inv_appl … H1) #n #p #W #U #H <H -n #HV #HT #HVW #HTU
/5 width=11 by cnv_cpms_nta, cnv_cpms_conf_eq, cpcs_cprs_div, cpms_appl_dx, ex4_3_intro/
qed-.
(* Basic_2A1: uses: nta_fwd_pure1 *)
lemma nta_inv_pure_sn_cnv (h) (G) (L) (X2):
- ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :*[h] X2 →
- ∨∨ ∃∃p,W,U. ⦃G,L⦄ ⊢ V :*[h] W & ⦃G,L⦄ ⊢ T :*[h] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 !*[h]
- | ∃∃U. ⦃G,L⦄ ⊢ T :*[h] U & ⦃G,L⦄ ⊢ ⓐV.U !*[h] & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 !*[h].
+ ∀V,T. ⦃G,L⦄ ⊢ ⓐV.T :[h,𝛚] X2 →
+ ∨∨ ∃∃p,W,U. ⦃G,L⦄ ⊢ V :[h,𝛚] W & ⦃G,L⦄ ⊢ T :[h,𝛚] ⓛ{p}W.U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,𝛚]
+ | ∃∃U. ⦃G,L⦄ ⊢ T :[h,𝛚] U & ⦃G,L⦄ ⊢ ⓐV.U ![h,𝛚] & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,𝛚].
#h #G #L #X2 #V #T #H
elim (cnv_inv_cast … H) -H #X1 #HX2 #H1 #HX21 #H
elim (cnv_inv_appl … H1) * [| #n ] #p #W0 #T0 #Hn #HV #HT #HW0 #HT0
qed-.
(* Basic_2A1: uses: nta_inv_cast1 *)
-lemma nta_inv_cast_sn (a) (h) (G) (L) (X2):
- ∀U,T. ⦃G,L⦄ ⊢ ⓝU.T :[a,h] X2 →
- ∧∧ ⦃G,L⦄ ⊢ T :[a,h] U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #X2 #U #T #H
+lemma nta_inv_cast_sn (h) (a) (G) (L) (X2):
+ ∀U,T. ⦃G,L⦄ ⊢ ⓝU.T :[h,a] X2 →
+ ∧∧ ⦃G,L⦄ ⊢ T :[h,a] U & ⦃G,L⦄ ⊢ U ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #U #T #H
elim (cnv_inv_cast … H) -H #X0 #HX2 #H1 #HX20 #H2
elim (cnv_inv_cast … H1) #X #HU #HT #HUX #HTX
elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
qed-.
(* Basic_1: uses: ty3_gen_cast *)
-lemma nta_inv_cast_sn_old (a) (h) (G) (L) (X2):
- ∀T0,T1. ⦃G,L⦄ ⊢ ⓝT1.T0 :[a,h] X2 →
- ∃∃T2. ⦃G,L⦄ ⊢ T0 :[a,h] T1 & ⦃G,L⦄ ⊢ T1 :[a,h] T2 & ⦃G,L⦄ ⊢ ⓝT2.T1 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #X2 #T0 #T1 #H
+lemma nta_inv_cast_sn_old (h) (a) (G) (L) (X2):
+ ∀T0,T1. ⦃G,L⦄ ⊢ ⓝT1.T0 :[h,a] X2 →
+ ∃∃T2. ⦃G,L⦄ ⊢ T0 :[h,a] T1 & ⦃G,L⦄ ⊢ T1 :[h,a] T2 & ⦃G,L⦄ ⊢ ⓝT2.T1 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a].
+#h #a #G #L #X2 #T0 #T1 #H
elim (cnv_inv_cast … H) -H #X0 #HX2 #H1 #HX20 #H2
elim (cnv_inv_cast … H1) #X #HT1 #HT0 #HT1X #HT0X
elim (cpms_inv_cast1 … H2) -H2 [ * || * ]
qed-.
(* Basic_1: uses: ty3_gen_lift *)
-(* Note: "⦃G,L⦄ ⊢ U2 ⬌*[h] X2" can be "⦃G,L⦄ ⊢ X2 ➡*[h] U2" *)
-lemma nta_inv_lifts_sn (a) (h) (G):
- ∀L,T2,X2. ⦃G,L⦄ ⊢ T2 :[a,h] X2 →
+(* Note: "⦃G, L⦄ ⊢ U2 ⬌*[h] X2" can be "⦃G, L⦄ ⊢ X2 ➡*[h] U2" *)
+lemma nta_inv_lifts_sn (h) (a) (G):
+ ∀L,T2,X2. ⦃G,L⦄ ⊢ T2 :[h,a] X2 →
∀b,f,K. ⬇*[b,f] L ≘ K → ∀T1. ⬆*[f] T1 ≘ T2 →
- ∃∃U1,U2. ⦃G,K⦄ ⊢ T1 :[a,h] U1 & ⬆*[f] U1 ≘ U2 & ⦃G,L⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![a,h].
-#a #h #G #L #T2 #X2 #H #b #f #K #HLK #T1 #HT12
+ ∃∃U1,U2. ⦃G,K⦄ ⊢ T1 :[h,a] U1 & ⬆*[f] U1 ≘ U2 & ⦃G,L⦄ ⊢ U2 ⬌*[h] X2 & ⦃G,L⦄ ⊢ X2 ![h,a].
+#h #a #G #L #T2 #X2 #H #b #f #K #HLK #T1 #HT12
elim (cnv_inv_cast … H) -H #U2 #HX2 #HT2 #HXU2 #HTU2
lapply (cnv_inv_lifts … HT2 … HLK … HT12) -HT2 #HT1
elim (cpms_inv_lifts_sn … HTU2 … HLK … HT12) -T2 -HLK #U1 #HU12 #HTU1
(* Forward lemmas based on preservation *************************************)
(* Basic_1: was: ty3_unique *)
-theorem nta_mono (a) (h) (G) (L) (T):
- ∀U1. ⦃G,L⦄ ⊢ T :[a,h] U1 → ∀U2. ⦃G,L⦄ ⊢ T :[a,h] U2 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2.
-#a #h #G #L #T #U1 #H1 #U2 #H2
+theorem nta_mono (h) (a) (G) (L) (T):
+ ∀U1. ⦃G,L⦄ ⊢ T :[h,a] U1 → ∀U2. ⦃G,L⦄ ⊢ T :[h,a] U2 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2.
+#h #a #G #L #T #U1 #H1 #U2 #H2
elim (cnv_inv_cast … H1) -H1 #X1 #_ #_ #HUX1 #HTX1
elim (cnv_inv_cast … H2) -H2 #X2 #_ #HT #HUX2 #HTX2
lapply (cnv_cpms_conf_eq … HT … HTX1 … HTX2) -T #HX12
(* Advanced properties ******************************************************)
(* Basic_1: uses: ty3_sconv_pc3 *)
-lemma nta_cpcs_bi (a) (h) (G) (L):
- ∀T1,U1. ⦃G,L⦄ ⊢ T1 :[a,h] U1 → ∀T2,U2. ⦃G,L⦄ ⊢ T2 :[a,h] U2 →
+lemma nta_cpcs_bi (h) (a) (G) (L):
+ ∀T1,U1. ⦃G,L⦄ ⊢ T1 :[h,a] U1 → ∀T2,U2. ⦃G,L⦄ ⊢ T2 :[h,a] U2 →
⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2.
-#a #h #G #L #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
+#h #a #G #L #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
elim (cpcs_inv_cprs … HT12) -HT12 #T0 #HT10 #HT02
/3 width=6 by nta_mono, nta_cprs_conf/
qed-.
(* Properties based on type equivalence and preservation *******************)
(* Basic_1: uses: ty3_tred *)
-lemma nta_cprs_trans (a) (h) (G) (L):
- ∀T,U1. ⦃G,L⦄ ⊢ T :[a,h] U1 → ∀U2. ⦃G,L⦄ ⊢ U1 ➡*[h] U2 → ⦃G,L⦄ ⊢ T :[a,h] U2.
-#a #h #G #L #T #U1 #H #U2 #HU12
+lemma nta_cprs_trans (h) (a) (G) (L):
+ ∀T,U1. ⦃G,L⦄ ⊢ T :[h,a] U1 → ∀U2. ⦃G,L⦄ ⊢ U1 ➡*[h] U2 → ⦃G,L⦄ ⊢ T :[h,a] U2.
+#h #a #G #L #T #U1 #H #U2 #HU12
/4 width=4 by nta_conv_cnv, nta_fwd_cnv_dx, cnv_cpms_trans, cpcs_cprs_dx/
qed-.
(* Basic_1: uses: ty3_sred_back *)
-lemma cprs_nta_trans (a) (h) (G) (L):
- ∀T1,U0. ⦃G,L⦄ ⊢ T1 :[a,h] U0 → ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
- ∀U. ⦃G,L⦄ ⊢ T2 :[a,h] U → ⦃G,L⦄ ⊢ T1 :[a,h] U.
-#a #h #G #L #T1 #U0 #HT1 #T2 #HT12 #U #H
+lemma cprs_nta_trans (h) (a) (G) (L):
+ ∀T1,U0. ⦃G,L⦄ ⊢ T1 :[h,a] U0 → ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
+ ∀U. ⦃G,L⦄ ⊢ T2 :[h,a] U → ⦃G,L⦄ ⊢ T1 :[h,a] U.
+#h #a #G #L #T1 #U0 #HT1 #T2 #HT12 #U #H
lapply (nta_cprs_conf … HT1 … HT12) -HT12 #HT2
/4 width=6 by nta_mono, nta_conv_cnv, nta_fwd_cnv_dx/
qed-.
-lemma cprs_nta_trans_cnv (a) (h) (G) (L):
- ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] → ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
- ∀U. ⦃G,L⦄ ⊢ T2 :[a,h] U → ⦃G,L⦄ ⊢ T1 :[a,h] U.
-#a #h #G #L #T1 #HT1 #T2 #HT12 #U #H
+lemma cprs_nta_trans_cnv (h) (a) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![h,a] → ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
+ ∀U. ⦃G,L⦄ ⊢ T2 :[h,a] U → ⦃G,L⦄ ⊢ T1 :[h,a] U.
+#h #a #G #L #T1 #HT1 #T2 #HT12 #U #H
elim (cnv_nta_sn … HT1) -HT1 #U0 #HT1
/2 width=3 by cprs_nta_trans/
qed-.
(* Basic_1: uses: ty3_sconv *)
-lemma nta_cpcs_conf (a) (h) (G) (L):
- ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U → ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
- ∀U0. ⦃G,L⦄ ⊢ T2 :[a,h] U0 → ⦃G,L⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #HT1 #T2 #HT12 #U0 #HT2
+lemma nta_cpcs_conf (h) (a) (G) (L):
+ ∀T1,U. ⦃G,L⦄ ⊢ T1 :[h,a] U → ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀U0. ⦃G,L⦄ ⊢ T2 :[h,a] U0 → ⦃G,L⦄ ⊢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #HT1 #T2 #HT12 #U0 #HT2
elim (cpcs_inv_cprs … HT12) -HT12 #T0 #HT10 #HT02
/3 width=5 by cprs_nta_trans, nta_cprs_conf/
qed-.
(* Note: type preservation by valid r-equivalence *)
-lemma nta_cpcs_conf_cnv (a) (h) (G) (L):
- ∀T1,U. ⦃G,L⦄ ⊢ T1 :[a,h] U →
- ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L⦄ ⊢ T2 ![a,h] → ⦃G,L⦄ ⊢ T2 :[a,h] U.
-#a #h #G #L #T1 #U #HT1 #T2 #HT12 #HT2
+lemma nta_cpcs_conf_cnv (h) (a) (G) (L):
+ ∀T1,U. ⦃G,L⦄ ⊢ T1 :[h,a] U →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L⦄ ⊢ T2 ![h,a] → ⦃G,L⦄ ⊢ T2 :[h,a] U.
+#h #a #G #L #T1 #U #HT1 #T2 #HT12 #HT2
elim (cnv_nta_sn … HT2) -HT2 #U0 #HT2
/2 width=3 by nta_cpcs_conf/
qed-.
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/prednormal_3.ma".
-include "basic_2/reduction/cpr.ma".
-
-(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************)
-
-definition cnr: relation3 genv lenv term ≝ λG,L. NF … (cpr G L) (eq …).
-
-interpretation
- "normality for context-sensitive reduction (term)"
- 'PRedNormal G L T = (cnr G L T).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cnr_inv_delta: ∀G,L,K,V,i. ⬇[i] L ≡ K.ⓓV → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄ → ⊥.
-#G #L #K #V #i #HLK #H
-elim (lift_total V 0 (i+1)) #W #HVW
-lapply (H W ?) -H [ /3 width=6 by cpr_delta/ ] -HLK #H destruct
-elim (lift_inv_lref2_be … HVW) -HVW /2 width=1 by ylt_inj/
-qed-.
-
-lemma cnr_inv_abst: ∀a,G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓛ{a}V.T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ➡ 𝐍⦃T⦄.
-#a #G #L #V1 #T1 #HVT1 @conj
-[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct //
-| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2 by cpr_bind/ -HT2 #H destruct //
-]
-qed-.
-
-lemma cnr_inv_abbr: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃-ⓓV.T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ➡ 𝐍⦃T⦄.
-#G #L #V1 #T1 #HVT1 @conj
-[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct //
-| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpr_bind/ -HT2 #H destruct //
-]
-qed-.
-
lemma cnr_inv_zeta: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃+ⓓV.T⦄ → ⊥.
#G #L #V #T #H elim (is_lift_dec T 0 1)
[ * #U #HTU
]
qed-.
-lemma cnr_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ & 𝐒⦃T⦄.
-#G #L #V1 #T1 #HVT1 @and3_intro
-[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpr_pair_sn/ -HV2 #H destruct //
-| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpr_flat/ -HT2 #H destruct //
-| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
- [ elim (lift_total V1 0 1) #V2 #HV12
- lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /3 width=3 by tpr_cpr, cpr_theta/ -HV12 #H destruct
- | lapply (H (ⓓ{a}ⓝW1.V1.U1) ?) -H /3 width=1 by tpr_cpr, cpr_beta/ #H destruct
-]
-qed-.
-
-lemma cnr_inv_eps: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓝV.T⦄ → ⊥.
-#G #L #V #T #H lapply (H T ?) -H
-/2 width=4 by cpr_eps, discr_tpair_xy_y/
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: nf2_sort *)
-lemma cnr_sort: ∀G,L,s. ⦃G, L⦄ ⊢ ➡ 𝐍⦃⋆s⦄.
-#G #L #s #X #H
->(cpr_inv_sort1 … H) //
-qed.
-
lemma cnr_lref_free: ∀G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄.
#G #L #i #Hi #X #H elim (cpr_inv_lref1 … H) -H // *
#K #V1 #V2 #HLK lapply (drop_fwd_length_lt2 … HLK) -HLK
#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
qed.
-
-(* Basic_1: was only: nf2_csort_lref *)
-lemma cnr_lref_atom: ∀G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄.
-#G #L #i #HL @cnr_lref_free >(drop_fwd_length … HL) -HL //
-qed.
-
-(* Basic_1: was: nf2_abst *)
-lemma cnr_abst: ∀a,G,L,W,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓛ{a}W.T⦄.
-#a #G #L #W #T #HW #HT #X #H
-elim (cpr_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
->(HW … HW0) -W0 >(HT … HT0) -T0 //
-qed.
-
-(* Basic_1: was only: nf2_appl_lref *)
-lemma cnr_appl_simple: ∀G,L,V,T. ⦃G, L⦄ ⊢ ➡ 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃ⓐV.T⦄.
-#G #L #V #T #HV #HT #HS #X #H
-elim (cpr_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
->(HV … HV0) -V0 >(HT … HT0) -T0 //
-qed.
-
-(* Basic_1: was: nf2_dec *)
-axiom cnr_dec: ∀G,L,T1. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T1⦄ ∨
- ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡ T2 & (T1 = T2 → ⊥).
-
-(* Basic_1: removed theorems 1: nf2_abst_shift *)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/cpr_lift.ma".
-include "basic_2/reduction/cnr.ma".
-
-(* NORMAL TERMS FOR CONTEXT-SENSITIVE REDUCTION *****************************)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was: nf2_lref_abst *)
-lemma cnr_lref_abst: ∀G,L,K,V,i. ⬇[i] L ≡ K. ⓛV → ⦃G, L⦄ ⊢ ➡ 𝐍⦃#i⦄.
-#G #L #K #V #i #HLK #X #H
-elim (cpr_inv_lref1 … H) -H // *
-#K0 #V1 #V2 #HLK0 #_ #_
-lapply (drop_mono … HLK … HLK0) -L #H destruct
-qed.
-
-(* Relocation properties ****************************************************)
-
-(* Basic_1: was: nf2_lift *)
-lemma cnr_lift: ∀G,L0,L,T,T0,c,l,k. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ →
- ⬇[c, l, k] L0 ≡ L → ⬆[l, k] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄.
-#G #L0 #L #T #T0 #c #l #k #HLT #HL0 #HT0 #X #H
-elim (cpr_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
-<(HLT … HT1) in HT0; -L #HT0
->(lift_mono … HT10 … HT0) -T1 -X //
-qed.
-
-(* Note: this was missing in basic_1 *)
-lemma cnr_inv_lift: ∀G,L0,L,T,T0,c,l,k. ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄ →
- ⬇[c, l, k] L0 ≡ L → ⬆[l, k] T ≡ T0 → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
-#G #L0 #L #T #T0 #c #l #k #HLT0 #HL0 #HT0 #X #H
-elim (lift_total X l k) #X0 #HX0
-lapply (cpr_lift … H … HL0 … HT0 … HX0) -L #HTX0
->(HLT0 … HTX0) in HX0; -L0 -X0 #H
->(lift_inj … H … HT0) -T0 -X -c -l -k //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ ➡ 𝐍 break ⦃ term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'PRedNormal $G $L $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/predeval_4.ma".
-include "basic_2/computation/cprs.ma".
-include "basic_2/computation/csx.ma".
-
-(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS *************)
-
-definition cpre: relation4 genv lenv term term ≝
- λG,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 ∧ ⦃G, L⦄ ⊢ ➡ 𝐍⦃T2⦄.
-
-interpretation "evaluation for context-sensitive parallel reduction (term)"
- 'PRedEval G L T1 T2 = (cpre G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was just: nf2_sn3 *)
-lemma csx_cpre: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡* 𝐍⦃T2⦄.
-#h #o #G #L #T1 #H @(csx_ind … H) -T1
-#T1 #_ #IHT1 elim (cnr_dec G L T1) /3 width=3 by ex_intro, conj/
-* #T #H1T1 #H2T1 elim (IHT1 … H2T1) -IHT1 -H2T1 /2 width=2 by cpr_cpx/
-#T2 * /4 width=3 by cprs_strap2, ex_intro, conj/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/cprs_cprs.ma".
-include "basic_2/computation/cpre.ma".
-
-(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS *************)
-
-(* Main properties *********************************************************)
-
-(* Basic_1: was: nf2_pr3_confluence *)
-theorem cpre_mono: ∀G,L,T,T1. ⦃G, L⦄ ⊢ T ➡* 𝐍⦃T1⦄ → ∀T2. ⦃G, L⦄ ⊢ T ➡* 𝐍⦃T2⦄ → T1 = T2.
-#G #L #T #T1 * #H1T1 #H2T1 #T2 * #H1T2 #H2T2
-elim (cprs_conf … H1T1 … H1T2) -T #T #HT1
->(cprs_inv_cnr1 … HT1 H2T1) -T1 #HT2
->(cprs_inv_cnr1 … HT2 H2T2) -T2 //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ * 𝐍 ⦃ break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'PRedEval $G $L $T1 $T2 }.
(* Basic_1: was: pr3_pr1 *)
lemma tprs_cprs: ∀G,L,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
/2 width=3 by lsubr_cprs_trans/ qed.
-
-(* Basic_1: was: nf2_pr3_unfold *)
-lemma cprs_inv_cnr1: ∀G,L,T,U. ⦃G, L⦄ ⊢ T ➡* U → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ → T = U.
-#G #L #T #U #H @(cprs_ind_dx … H) -T //
-#T0 #T #H1T0 #_ #IHT #H2T0
-lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
-qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/preditnormal_4.ma".
+include "static_2/syntax/tueq.ma".
+include "basic_2/rt_transition/cpm.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+definition cnu (h) (G) (L): predicate term ≝
+ λT1. ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → T1 ≅ T2.
+
+interpretation
+ "normality for t-unbound context-sensitive parallel rt-transition (term)"
+ 'PRedITNormal h G L T = (cnu h G L T).
+
+(* Basic properties *********************************************************)
+
+lemma cnu_sort (h) (G) (L): ∀s. ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃⋆s⦄.
+#h #G #L #s1 #n #X #H
+elim (cpm_inv_sort1 … H) -H #H #_ destruct //
+qed.
+
+lemma cnu_ctop (h) (G): ∀i. ⦃G,⋆⦄ ⊢ ⥲[h] 𝐍⦃#i⦄.
+#h #G * [| #i ] #n #X #H
+[ elim (cpm_inv_zero1 … H) -H *
+ [ #H #_ destruct //
+ | #Y #X1 #X2 #_ #_ #H destruct
+ | #m #Y #X1 #X2 #_ #_ #H destruct
+ ]
+| elim (cpm_inv_lref1 … H) -H *
+ [ #H #_ destruct //
+ | #Z #Y #X0 #_ #_ #H destruct
+ ]
+]
+qed.
+
+lemma cnu_zero (h) (G) (L): ∀I. ⦃G,L.ⓤ{I}⦄ ⊢ ⥲[h] 𝐍⦃#0⦄.
+#h #G #L #I #n #X #H
+elim (cpm_inv_zero1 … H) -H *
+[ #H #_ destruct //
+| #Y #X1 #X2 #_ #_ #H destruct
+| #m #Y #X1 #X2 #_ #_ #H destruct
+]
+qed.
+
+lemma cnu_gref (h) (G) (L): ∀l. ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃§l⦄.
+#h #G #L #l1 #n #X #H
+elim (cpm_inv_gref1 … H) -H #H #_ destruct //
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cnr.ma".
+include "basic_2/rt_transition/cnu.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Advanced properties with normal terms for r-transition *******************)
+
+lemma cnu_abst (h) (p) (G) (L):
+ ∀W. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃W⦄ → ∀T.⦃G,L.ⓛW⦄ ⊢ ⥲[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃ⓛ{p}W.T⦄.
+#h #p #G #L #W1 #HW1 #T1 #HT1 #n #X #H
+elim (cpm_inv_abst1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
+<(HW1 … HW12) -W2 /3 width=2 by tueq_bind/
+qed.
+
+lemma cnu_abbr_neg (h) (G) (L):
+ ∀V. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ∀T.⦃G,L.ⓓV⦄ ⊢ ⥲[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃-ⓓV.T⦄.
+#h #G #L #V1 #HV1 #T1 #HT1 #n #X #H
+elim (cpm_inv_abbr1 … H) -H *
+[ #V2 #T2 #HV12 #HT12 #H destruct
+ <(HV1 … HV12) -V2 /3 width=2 by tueq_bind/
+| #X1 #_ #_ #H destruct
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cpm_simple.ma".
+include "basic_2/rt_transition/cnr.ma".
+include "basic_2/rt_transition/cnu.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Advanced properties with simple terms and normal terms for r-transition **)
+
+lemma cnu_appl_simple (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃ⓐV.T⦄.
+#h #G #L #V1 #T1 #HV1 #HT1 #HS #n #X #H
+elim (cpm_inv_appl1_simple … H HS) -H -HS #V2 #T2 #HV12 #HT12 #H destruct
+lapply (HV1 … HV12) -HV1 -HV12 #H destruct
+/3 width=2 by tueq_appl/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tueq_tueq.ma".
+include "basic_2/rt_transition/cpm_tueq.ma".
+include "basic_2/rt_transition/cnu.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Advanced properties ******************************************************)
+
+lemma cnu_tueq_trans (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄ → ∀T2.T1 ≅ T2 → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄.
+#h #G #L #T1 #HT1 #T2 #HT12 #n #T0 #HT20
+@(tueq_canc_sn … HT12)
+elim (tueq_cpm_trans … HT12 … HT20) -T2 #T2 #HT13 #HT30
+/3 width=3 by tueq_trans/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/lifts_tueq.ma".
+include "basic_2/rt_transition/cpm_drops.ma".
+include "basic_2/rt_transition/cnu.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Advanced properties ******************************************************)
+
+lemma cnu_atom_drops (h) (b) (G) (L):
+ ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄.
+#h #b #G #L #i #Hi #n #X #H
+elim (cpm_inv_lref1_drops … H) -H * [ // || #m ] #K #V1 #V2 #HLK
+lapply (drops_gen b … HLK) -HLK #HLK
+lapply (drops_mono … Hi … HLK) -L #H destruct
+qed.
+
+lemma cnu_unit_drops (h) (I) (G) (L):
+ ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄.
+#h #I #G #L #K #i #HLK #n #X #H
+elim (cpm_inv_lref1_drops … H) -H * [ // || #m ] #Y #V1 #V2 #HLY
+lapply (drops_mono … HLK … HLY) -L #H destruct
+qed.
+
+(* Properties with generic relocation ***************************************)
+
+lemma cnu_lifts (h) (G): d_liftable1 … (cnu h G).
+#h #G #K #T #HT #b #f #L #HLK #U #HTU #n #U0 #H
+elim (cpm_inv_lifts_sn … H … HLK … HTU) -b -L #T0 #HTU0 #HT0
+lapply (HT … HT0) -G -K /2 width=6 by tueq_lifts_bi/
+qed-.
+
+(* Inversion lemmas with generic relocation *********************************)
+
+lemma cnu_inv_lifts (h) (G): d_deliftable1 … (cnu h G).
+#h #G #L #U #HU #b #f #K #HLK #T #HTU #n #T0 #H
+elim (cpm_lifts_sn … H … HLK … HTU) -b -K #U0 #HTU0 #HU0
+lapply (HU … HU0) -G -L /2 width=6 by tueq_inv_lifts_bi/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/lifts_tueq.ma".
+include "basic_2/rt_transition/cnu.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Advanced properties with uniform relocation for terms ********************)
+
+lemma cnu_lref (h) (I) (G) (L):
+ ∀i. ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃#i⦄ → ⦃G,L.ⓘ{I}⦄ ⊢ ⥲[h] 𝐍⦃#↑i⦄.
+#h #I #G #L #i #Hi #n #X #H
+elim (cpm_inv_lref1 … H) -H *
+[ #H #_ destruct //
+| #J #K #V #HV #HVX #H destruct
+ lapply (Hi … HV) -Hi -HV #HV
+ elim (tueq_lifts_dx … HV … HVX) -V #Xi #Hi #HX
+ lapply (lifts_inv_lref1_uni … Hi) -Hi #H destruct //
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cnr_tdeq.ma".
+include "basic_2/rt_transition/cnu_drops.ma".
+include "basic_2/rt_transition/cnu_cnr.ma".
+include "basic_2/rt_transition/cnu_cnr_simple.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND RT-TRANSITION *******************************)
+
+(* Properties with context-free sort-irrelevant equivalence for terms *******)
+
+lemma cnu_dec_tdeq (h) (G) (L):
+ ∀T1. ∨∨ ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄
+ | ∃∃n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & (T1 ≛ T2 → ⊥).
+#h #G #L #T1
+@(fqup_wf_ind_eq (Ⓣ) … G L T1) -G -L -T1 #G0 #L0 #T0 #IH #G #L * *
+[ #s #HG #HL #HT destruct -IH
+ /3 width=5 by cnu_sort, or_introl/
+| #i #HG #HL #HT destruct -IH
+ elim (drops_F_uni L i)
+ [ /3 width=7 by cnu_atom_drops, or_introl/
+ | * * [ #I | * #V ] #K #HLK
+ [ /3 width=8 by cnu_unit_drops, or_introl/
+ | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
+ @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpm_delta_drops/ ] #H
+ lapply (tdeq_inv_lref1 … H) -H #H destruct
+ /2 width=5 by lifts_inv_lref2_uni_lt/
+ | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
+ @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpm_ell_drops/ ] #H
+ lapply (tdeq_inv_lref1 … H) -H #H destruct
+ /2 width=5 by lifts_inv_lref2_uni_lt/
+ ]
+ ]
+| #l #HG #HL #HT destruct -IH
+ /3 width=5 by cnu_gref, or_introl/
+| #p * [ cases p ] #V1 #T1 #HG #HL #HT destruct
+ [ elim (cpr_subst h G (L.ⓓV1) T1 0 L V1) [| /2 width=1 by drops_refl/ ] #T2 #X2 #HT12 #HXT2 -IH
+ elim (tdeq_dec T1 T2) [ -HT12 #HT12 | #HnT12 ]
+ [ elim (tdeq_inv_lifts_dx … HT12 … HXT2) -T2 #X1 #HXT1 #_ -X2
+ @or_intror @(ex2_2_intro … X1) [1,2: /2 width=4 by cpm_zeta/ ] #H
+ /2 width=7 by tdeq_lifts_inv_pair_sn/
+ | @or_intror @(ex2_2_intro … (+ⓓV1.T2)) [1,2: /2 width=2 by cpm_bind/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ | elim (cnr_dec_tdeq h G L V1) [ elim (IH G (L.ⓓV1) T1) [| * | // ] | * ] -IH
+ [ #HT1 #HV1 /3 width=7 by cnu_abbr_neg, or_introl/
+ | #n #T2 #HT12 #HnT12 #_
+ @or_intror @(ex2_2_intro … (-ⓓV1.T2)) [1,2: /2 width=2 by cpm_bind/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ | #V2 #HV12 #HnV12
+ @or_intror @(ex2_2_intro … (-ⓓV2.T1)) [1,2: /2 width=2 by cpr_pair_sn/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ | elim (cnr_dec_tdeq h G L V1) [ elim (IH G (L.ⓛV1) T1) [| * | // ] | * ] -IH
+ [ #HT1 #HV1 /3 width=7 by cnu_abst, or_introl/
+ | #n #T2 #HT12 #HnT12 #_
+ @or_intror @(ex2_2_intro … (ⓛ{p}V1.T2)) [1,2: /2 width=2 by cpm_bind/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ | #V2 #HV12 #HnV12
+ @or_intror @(ex2_2_intro … (ⓛ{p}V2.T1)) [1,2: /2 width=2 by cpr_pair_sn/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ ]
+| * #V1 #T1 #HG #HL #HT destruct [| -IH ]
+ [ elim (cnr_dec_tdeq h G L V1) [ elim (IH G L T1) [| * | // ] | * ] -IH
+ [ #HT1 #HV1
+ elim (simple_dec_ex T1) [| * #p * #W1 #U1 #H destruct ]
+ [ /3 width=7 by cnu_appl_simple, or_introl/
+ | elim (lifts_total V1 𝐔❴1❵) #X1 #HVX1
+ @or_intror @(ex2_2_intro … (ⓓ{p}W1.ⓐX1.U1)) [1,2: /2 width=3 by cpm_theta/ ] #H
+ elim (tdeq_inv_pair … H) -H #H destruct
+ | @or_intror @(ex2_2_intro … (ⓓ{p}ⓝW1.V1.U1)) [1,2: /2 width=2 by cpm_beta/ ] #H
+ elim (tdeq_inv_pair … H) -H #H destruct
+ ]
+ | #n #T2 #HT12 #HnT12 #_
+ @or_intror @(ex2_2_intro … (ⓐV1.T2)) [1,2: /2 width=2 by cpm_appl/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ | #V2 #HV12 #HnV12
+ @or_intror @(ex2_2_intro … (ⓐV2.T1)) [1,2: /2 width=2 by cpr_pair_sn/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ | @or_intror @(ex2_2_intro … T1) [1,2: /2 width=2 by cpm_eps/ ] #H
+ /2 width=4 by tdeq_inv_pair_xy_y/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpms_cnu.ma".
+include "basic_2/rt_computation/cpmue.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+
+(* T-UNBOUND EVALUATION FOR T-BOUND RT-TRANSITION ON TERMS ******************)
+
+(* Properties with evaluation for t-unbound rt-transition on terms **********)
+
+lemma cnv_cpmue_trans (a) (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
+ ∀T2,n. ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍*⦃T2⦄ → ⦃G,L⦄ ⊢ T2 ![a,h].
+/3 width=4 by cpmue_fwd_cpms, cnv_cpms_trans/ qed-.
+
+lemma cnv_cpmue_cpms_conf (a) (h) (G) (L):
+ ∀T0. ⦃G,L⦄ ⊢ T0 ![a,h] → ∀T1,n1. ⦃G,L⦄ ⊢ T0 ➡*[n1,h] T1 →
+ ∀T2,n2. ⦃G,L⦄ ⊢ T0 ➡*[h,n2] 𝐍*⦃T2⦄ →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[h,n2-n1] 𝐍*⦃T⦄ & T2 ≅ T.
+#a #h #G #L #T0 #HT0 #T1 #n1 #HT01 #T2 #n2 * #HT02 #HT2
+elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 #T0 #HT10 #HT20
+lapply (cpms_inv_cnu_sn … HT20 HT2) -HT20 #HT20
+/4 width=8 by cpmue_intro, cnu_tueq_trans, ex2_intro/
+qed-.
+
+(* Main properties with evaluation for t-unbound rt-transition on terms *****)
+
+theorem cnv_cpmue_mono (a) (h) (G) (L):
+ ∀T0. ⦃G,L⦄ ⊢ T0 ![a,h] → ∀T1,n1. ⦃G,L⦄ ⊢ T0 ➡*[h,n1] 𝐍*⦃T1⦄ →
+ ∀T2,n2. ⦃G,L⦄ ⊢ T0 ➡*[h,n2] 𝐍*⦃T2⦄ → T1 ≅ T2.
+#a #h #G #L #T0 #HT0 #T1 #n1 * #HT01 #HT1 #T2 #n2 * #HT02 #HT2
+elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 #T0 #HT10 #HT20
+/3 width=8 by cpms_inv_cnu_sn, tueq_canc_dx/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpms_cnu.ma".
+include "basic_2/rt_computation/cpue.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+
+(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
+
+(* Properties with evaluation for t-unbound rt-transition on terms **********)
+
+lemma cnv_cpue_trans (a) (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ![a,h] →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⥲*[h] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T2 ![a,h].
+#a #h #G #L #T1 #HT1 #T2 * #n #HT12 #_
+/2 width=4 by cnv_cpms_trans/
+qed-.
+
+lemma cnv_cpue_cpms_conf (a) (h) (n) (G) (L):
+ ∀T0. ⦃G,L⦄ ⊢ T0 ![a,h] → ∀T1. ⦃G,L⦄ ⊢ T0 ➡*[n,h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T0 ⥲*[h] 𝐍⦃T2⦄ →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ⥲*[h] 𝐍⦃T⦄ & T2 ≅ T.
+#a #h #n1 #G #L #T0 #HT0 #T1 #HT01 #T2 * #n2 #HT02 #HT2
+elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 #T0 #HT10 #HT20
+lapply (cpms_inv_cnu_sn … HT20 HT2) -HT20 #HT20
+/4 width=8 by cnu_tueq_trans, ex2_intro/
+qed-.
+
+(* Main properties with evaluation for t-unbound rt-transition on terms *****)
+
+theorem cnv_cpue_mono (a) (h) (G) (L):
+ ∀T0. ⦃G,L⦄ ⊢ T0 ![a,h] → ∀T1. ⦃G,L⦄ ⊢ T0 ⥲*[h] 𝐍⦃T1⦄ →
+ ∀T2. ⦃G,L⦄ ⊢ T0 ⥲*[h] 𝐍⦃T2⦄ → T1 ≅ T2.
+#a #h #G #L #T0 #HT0 #T1 * #n1 #HT01 #HT1 #T2 * #n2 #HT02 #HT2
+elim (cnv_cpms_conf … HT0 … HT01 … HT02) -T0 #T0 #HT10 #HT20
+/3 width=8 by cpms_inv_cnu_sn, tueq_canc_dx/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/lifts_tueq.ma".
+include "basic_2/rt_transition/cpm.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+
+(* Properties with tail sort-irrelevant equivalence on terms ****************)
+
+lemma cpm_tueq_conf (h) (n) (G) (L) (T0):
+ ∀T1. ⦃G,L⦄ ⊢ T0 ➡[n,h] T1 → ∀T2. T0 ≅ T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T2 ➡[n,h] T & T1 ≅ T.
+#h #n #G #L #T0 #T1 #H @(cpm_ind … H) -G -L -T0 -T1 -n
+[ /2 width=3 by ex2_intro/
+| #G #L #s0 #X2 #H2
+ elim (tueq_inv_sort1 … H2) -H2 #s2 #H destruct
+ /3 width=3 by tueq_sort, ex2_intro/
+| #n #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #X2 #H2
+ >(tueq_inv_lref1 … H2) -X2
+ elim (IH V0) [| // ] -IH #V #HV0 #HV1
+ elim (tueq_lifts_sn … HV1 … HVW1) -V1
+ /3 width=3 by cpm_delta, ex2_intro/
+| #n #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #X2 #H2
+ >(tueq_inv_lref1 … H2) -X2
+ elim (IH V0) [| // ] -IH #V #HV0 #HV1
+ elim (tueq_lifts_sn … HV1 … HVW1) -V1
+ /3 width=3 by cpm_ell, ex2_intro/
+| #n #I #G #K0 #V1 #W1 #i #_ #IH #HVW1 #X2 #H2
+ >(tueq_inv_lref1 … H2) -X2
+ elim (IH (#i)) [| // ] -IH #V #HV0 #HV1
+ elim (tueq_lifts_sn … HV1 … HVW1) -V1
+ /3 width=3 by cpm_lref, ex2_intro/
+| #n #p #I #G #L #V0 #V1 #T0 #T1 #HV01 #_ #_ #IHT #X2 #H2
+ elim (tueq_inv_bind1 … H2) -H2 #T2 #HT02 #H destruct
+ elim (IHT … HT02) -T0 #T #HT2 #HT1
+ /3 width=3 by cpm_bind, tueq_bind, ex2_intro/
+| #n #G #L #V0 #V1 #T0 #T1 #HV10 #_ #_ #IHT #X2 #H2
+ elim (tueq_inv_appl1 … H2) -H2 #T2 #HT02 #H destruct
+ elim (IHT … HT02) -T0 #T #HT2 #HT1
+ /3 width=3 by cpm_appl, tueq_appl, ex2_intro/
+| #n #G #L #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X2 #H2
+ elim (tueq_inv_cast1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (IHV … HV02) -V0 #V #HV2 #HV1
+ elim (IHT … HT02) -T0 #T #HT2 #HT1
+ /3 width=5 by cpm_cast, tueq_cast, ex2_intro/
+| #n #G #L #V0 #U0 #T0 #T1 #HTU0 #_ #IH #X2 #H2
+ elim (tueq_inv_bind1 … H2) -H2 #U2 #HU02 #H destruct
+ elim (tueq_inv_lifts_sn … HU02 … HTU0) -U0 #T2 #HTU2 #HT02
+ elim (IH … HT02) -T0 #T #HT2 #HT1
+ /3 width=3 by cpm_zeta, ex2_intro/
+| #n #G #L #V0 #T0 #T1 #_ #IH #X2 #H2
+ elim (tueq_inv_cast1 … H2) -H2 #V2 #T2 #_ #HT02 #H destruct
+ elim (IH … HT02) -V0 -T0
+ /3 width=3 by cpm_eps, ex2_intro/
+| #n #G #L #V0 #T0 #T1 #_ #IH #X2 #H2
+ elim (tueq_inv_cast1 … H2) -H2 #V2 #T2 #HV02 #_ #H destruct
+ elim (IH … HV02) -V0 -T1
+ /3 width=3 by cpm_ee, ex2_intro/
+| #n #p #G #L #V0 #V1 #W0 #W1 #T0 #T1 #HV01 #HW01 #_ #_ #_ #IHT #X2 #H2
+ elim (tueq_inv_appl1 … H2) -H2 #X #H2 #H destruct
+ elim (tueq_inv_bind1 … H2) -H2 #T2 #HT02 #H destruct
+ elim (IHT … HT02) -T0
+ /4 width=3 by cpm_beta, tueq_cast, tueq_bind, ex2_intro/
+| #n #p #G #L #V0 #V1 #U1 #W0 #W1 #T0 #T1 #HV01 #HW01 #_ #_ #_ #IHT #HVU1 #X2 #H2
+ elim (tueq_inv_appl1 … H2) -H2 #X #H2 #H destruct
+ elim (tueq_inv_bind1 … H2) -H2 #T2 #HT02 #H destruct
+ elim (IHT … HT02) -T0 #T #HT2 #HT1
+ /4 width=3 by cpm_theta, tueq_appl, tueq_bind, ex2_intro/
+]
+qed-.
+
+lemma tueq_cpm_trans (h) (n) (G) (L) (T0):
+ ∀T1. T1 ≅ T0 → ∀T2. ⦃G,L⦄ ⊢ T0 ➡[n,h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[n,h] T & T ≅ T2.
+#h #n #G #L #T0 #T1 #HT10 #T2 #HT02
+elim (cpm_tueq_conf … HT02 T1)
+/3 width=3 by tueq_sym, ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cnu_cnu.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+
+(* Inversion lemmas with normal terms for t-unbound rt-transition ***********)
+
+lemma cpms_inv_cnu_sn (h) (n) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄ → T1 ≅ T2.
+#h #n #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1 //
+#n1 #n2 #T1 #T0 #HT10 #_ #IH #HT1
+/5 width=8 by cnu_tueq_trans, tueq_trans/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/predevalstar_6.ma".
+include "basic_2/rt_transition/cnu.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* T-UNBOUND EVALUATION FOR T-BOUND RT-TRANSITION ON TERMS ******************)
+
+definition cpmue (h) (n) (G) (L): relation2 term term ≝
+ λT1,T2. ∧∧ ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄.
+
+interpretation "t-unbound evaluation for t-bound context-sensitive parallel rt-transition (term)"
+ 'PRedEvalStar h n G L T1 T2 = (cpmue h n G L T1 T2).
+
+definition R_cpmue (h) (G) (L) (T): predicate nat ≝
+ λn. ∃U. ⦃G,L⦄ ⊢ T ➡*[h,n] 𝐍*⦃U⦄.
+
+(* Basic properties *********************************************************)
+
+lemma cpmue_intro (h) (n) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍*⦃T2⦄.
+/2 width=1 by conj/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpmue_fwd_cpms (h) (n) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍*⦃T2⦄ → ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2.
+#h #n #G #L #T1 #T2 * #HT12 #_ //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cpm_cpx.ma".
+include "basic_2/rt_transition/cnu_tdeq.ma".
+include "basic_2/rt_computation/csx.ma".
+include "basic_2/rt_computation/cpmue.ma".
+
+(* T-UNBOUND EVALUATION FOR T-BOUND RT-TRANSITION ON TERMS ******************)
+
+(* Properties with strong normalization for unbound rt-transition for terms *)
+
+lemma cpmue_total_csx (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃∃T2,n. ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍*⦃T2⦄.
+#h #G #L #T1 #H
+@(csx_ind … H) -T1 #T1 #_ #IHT1
+elim (cnu_dec_tdeq h G L T1)
+[ -IHT1 #HT1 /3 width=4 by cpmue_intro, ex1_2_intro/
+| * #n1 #T0 #HT10 #HnT10
+ elim (IHT1 … HnT10) -IHT1 -HnT10 [| /2 width=2 by cpm_fwd_cpx/ ]
+ #T2 #n2 * #HT02 #HT2 /4 width=5 by cpms_step_sn, cpmue_intro, ex1_2_intro/
+]
+qed-.
+
+lemma R_cpmue_total_csx (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃n. R_cpmue h G L T1 n.
+#h #G #L #T1 #H
+elim (cpmue_total_csx … H) -H #T2 #n #HT12
+/3 width=3 by ex_intro (* 2x *)/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/prediteval_5.ma".
+include "basic_2/rt_transition/cnu.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* EVALUATION FOR T-UNBOUND RT-TRANSITION ON TERMS **************************)
+
+definition cpue (h) (G) (L): relation2 term term ≝
+ λT1,T2. ∃∃n. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T2⦄.
+
+interpretation "evaluation for t-unbound context-sensitive parallel rt-transition (term)"
+ 'PRedITEval h G L T1 T2 = (cpue h G L T1 T2).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cpm_cpx.ma".
+include "basic_2/rt_transition/cnu_tdeq.ma".
+include "basic_2/rt_computation/csx.ma".
+include "basic_2/rt_computation/cpue.ma".
+
+(* EVALUATION FOR T-UNBOUND RT-TRANSITION ON TERMS **************************)
+
+(* Properties with strong normalization for unbound rt-transition for terms *)
+
+lemma cpue_total_csx (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G,L⦄ ⊢ T1 ⥲*[h] 𝐍⦃T2⦄.
+#h #G #L #T1 #H
+@(csx_ind … H) -T1 #T1 #_ #IHT1
+elim (cnu_dec_tdeq h G L T1) [ /3 width=4 by ex2_intro, ex_intro/ ] *
+#n1 #T0 #HT10 #HnT10
+elim (IHT1 … HnT10) -IHT1 -HnT10 [| /2 width=2 by cpm_fwd_cpx/ ]
+#T2 * #n2 #HT02 #HT2 /4 width=7 by cpms_step_sn, ex2_intro, ex_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ⥲* [ break term 46 h ] 𝐍 ⦃ break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedITEval $h $G $L $T1 $T2 }.
(* Basic_1: uses: ty3_gen_abst_abst *)
lemma nta_inv_abst_bi
+(* Basic_1: uses: pc3_dec *)
+lemma nta_cpcs_dec
+
(* Advanced properties ******************************************************)
| ntaa_cast: ∀L,T,U,W. ntaa h L T U → ntaa h L U W → ntaa h L (ⓝU. T) U
--- /dev/null
+(* FROM BASIC_1
+
+(* NOTE: This can be generalized removing the last premise *)
+ Lemma ty3_gen_cvoid: (g:?; c:?; t1,t2:?) (ty3 g c t1 t2) ->
+ (e:?; u:?; d:?) (getl d c (CHead e (Bind Void) u)) ->
+ (a:?) (drop (1) d c a) ->
+ (EX y1 y2 | t1 = (lift (1) d y1) &
+ t2 = (lift (1) d y2) &
+ (ty3 g a y1 y2)
+ ).
+
+Lemma ty3_gen_appl_nf2: (g:?; c:?; w,v,x:?) (ty3 g c (THead (Flat Appl) w v) x) ->
+ (EX u t | (pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x) &
+ (ty3 g c v (THead (Bind Abst) u t)) &
+ (ty3 g c w u) &
+ (nf2 c (THead (Bind Abst) u t))
+ ).
+
+Lemma ty3_arity: (g:?; c:?; t1,t2:?) (ty3 g c t1 t2) ->
+ (EX a1 | (arity g c t1 a1) &
+ (arity g c t2 (asucc g a1))
+ ).
+
+Lemma ty3_acyclic: (g:?; c:?; t,u:?)
+ (ty3 g c t u) -> (pc3 c u t) -> (P:Prop) P.
+
+Theorem pc3_abst_dec: (g:?; c:?; u1,t1:?) (ty3 g c u1 t1) ->
+ (u2,t2:?) (ty3 g c u2 t2) ->
+ (EX u v2 | (pc3 c u1 (THead (Bind Abst) u2 u)) &
+ (ty3 g c (THead (Bind Abst) v2 u) t1) &
+ (pr3 c u2 v2) & (nf2 c v2)
+ ) \/
+ ((u:?) (pc3 c u1 (THead (Bind Abst) u2 u)) -> False).
+
+file ty3_nf2_gen
+
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/csx_lsubr.ma".
+include "basic_2/rt_computation/csx_cpxs.ma".
+include "basic_2/rt_computation/sdsx_rdsx.ma".
+
+(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+(* Advanced properties ******************************************************)
+
+lemma rdsx_pair_lpxs (h) (I) (G):
+ ∀K1,V1. G ⊢ ⬈*[h,V1] 𝐒⦃K1⦄ →
+ ∀V2. ⦃G,K1⦄ ⊢ ⬈*[h] 𝐒⦃V2⦄ →
+ (∀K. ⦃G,K1⦄ ⊢ ⬈*[h] K → ⦃G,K⦄ ⊢ V1 ⬈*[h] V2) →
+ G ⊢ ⬈*[h,#0] 𝐒⦃K1.ⓑ{I}V2⦄.
+#h #I #G #K1 #V1 #H
+@(rdsx_ind_lpxs … H) -K1 #K1 #_ #IHK #V2 #H
+@(csx_ind_cpxs … H) -V2 #V2 #HV2 #IHV #HK
+@rdsx_intro_lpxs #Y #HY #HnY
+elim (lpxs_inv_pair_sn … HY) -HY #K3 #V3 #HK13 #HV23 #H destruct
+elim (tdeq_dec V2 V3)
+[ -IHV -HV23 #HV23
+ @(rdsx_rdeq_trans … (K3.ⓑ{I}V2)) [| /2 width=1 by rdeq_pair_refl/ ]
+ @(IHK … HK13) -IHK
+ [
+ |
+ | /3 width=3 by lpxs_trans/
+ ]
+| -IHK -HnY #HnV23
+ @(rdsx_lpxs_trans … (K1.ⓑ{I}V3)) [| /2 width=1 by lpxs_bind_refl_dx/ ]
+ @(IHV … HV23 HnV23) -IHV -HnV23
+ #K #HK
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 • * ⬌ * [ break term 46 h, break term 46 o, break term 46 n1, break term 46 n2 ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'DPConvStar $h $o $n1 $n2 $G $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/dpconvstar_8.ma".
-include "basic_2/computation/scpds.ma".
-
-(* STRATIFIED DECOMPOSED PARALLEL EQUIVALENCE FOR TERMS *********************)
-
-definition scpes: ∀h. sd h → nat → nat → relation4 genv lenv term term ≝
- λh,o,d1,d2,G,L,T1,T2.
- ∃∃T. ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d1] T & ⦃G, L⦄ ⊢ T2 •*➡*[h, o, d2] T.
-
-interpretation "stratified decomposed parallel equivalence (term)"
- 'DPConvStar h o d1 d2 G L T1 T2 = (scpes h o d1 d2 G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma scpds_div: ∀h,o,G,L,T1,T2,T,d1,d2.
- ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d1] T → ⦃G, L⦄ ⊢ T2 •*➡*[h, o, d2] T →
- ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2.
-/2 width=3 by ex2_intro/ qed.
-
-lemma scpes_sym: ∀h,o,G,L,T1,T2,d1,d2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2 →
- ⦃G, L⦄ ⊢ T2 •*⬌*[h, o, d2, d1] T1.
-#h #o #G #L #T1 #T2 #L1 #d2 * /2 width=3 by scpds_div/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/computation/scpds_aaa.ma".
-include "basic_2/equivalence/scpes.ma".
-
-(* DECOMPOSED EXTENDED PARALLEL EQUIVALENCE FOR TERMS ***********************)
-
-(* Main inversion lemmas about atomic arity assignment on terms *************)
-
-theorem scpes_aaa_mono: ∀h,o,G,L,T1,T2,d1,d2. ⦃G, L⦄ ⊢ T1 •*⬌*[h, o, d1, d2] T2 →
- ∀A1. ⦃G, L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T2 ⁝ A2 →
- A1 = A2.
-#h #o #G #L #T1 #T2 #d1 #d2 * #T #HT1 #HT2 #A1 #HA1 #A2 #HA2
-lapply (scpds_aaa_conf … HA1 … HT1) -T1 #HA1
-lapply (scpds_aaa_conf … HA2 … HT2) -T2 #HA2
-lapply (aaa_mono … HA1 … HA2) -L -T //
-qed-.
(* Advanced properties ******************************************************)
-lemma scpes_refl: ∀h,o,G,L,T,d1,d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T ▪[h, o] d1 →
- ⦃G, L⦄ ⊢ T •*⬌*[h, o, d2, d2] T.
-#h #o #G #L #T #d1 #d2 #Hd21 #Hd1
-elim (da_lstas … Hd1 … d2) #U #HTU #_
-/3 width=3 by scpds_div, lstas_scpds/
-qed.
-
lemma lstas_scpes_trans: ∀h,o,G,L,T1,d0,d1. ⦃G, L⦄ ⊢ T1 ▪[h, o] d0 → d1 ≤ d0 →
∀T. ⦃G, L⦄ ⊢ T1 •*[h, d1] T →
∀T2,d,d2. ⦃G, L⦄ ⊢ T •*⬌*[h,o,d,d2] T2 → ⦃G, L⦄ ⊢ T1 •*⬌*[h,o,d1+d,d2] T2.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/rdsx.ma".
+
+(* STRONGLY NORMALIZING SELECTED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+(* Basic_2A1: uses: lcosx *)
+inductive sdsx (h) (G): rtmap → predicate lenv ≝
+| sdsx_atom: ∀f. sdsx h G f (⋆)
+| sdsx_push: ∀f,I,K. sdsx h G f K → sdsx h G (⫯f) (K.ⓘ{I})
+| sdsx_unit: ∀f,I,K. sdsx h G f K → sdsx h G (↑f) (K.ⓤ{I})
+| sdsx_pair: ∀f,I,K,V. G ⊢ ⬈*[h,V] 𝐒⦃K⦄ →
+ sdsx h G f K → sdsx h G (↑f) (K.ⓑ{I}V)
+.
+
+interpretation
+ "strong normalization for unbound context-sensitive parallel rt-transition on selected entries (local environment)"
+ 'PRedTySNStrong h f G L = (sdsx h G f L).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact sdsx_inv_push_aux (h) (G):
+ ∀g,L. G ⊢ ⬈*[h,g] 𝐒⦃L⦄ →
+ ∀f,I,K. g = ⫯f → L = K.ⓘ{I} → G ⊢ ⬈*[h,f] 𝐒⦃K⦄.
+#h #G #g #L * -g -L
+[ #f #g #J #L #_ #H destruct
+| #f #I #K #HK #g #J #L #H1 #H2 destruct //
+| #f #I #K #_ #g #J #L #H #_
+ elim (discr_next_push … H)
+| #f #I #K #V #_ #_ #g #J #L #H #_
+ elim (discr_next_push … H)
+]
+qed-.
+
+lemma sdsx_inv_push (h) (G):
+ ∀f,I,K. G ⊢ ⬈*[h,⫯f] 𝐒⦃K.ⓘ{I}⦄ → G ⊢ ⬈*[h,f] 𝐒⦃K⦄.
+/2 width=6 by sdsx_inv_push_aux/ qed-.
+
+fact sdsx_inv_unit_aux (h) (G):
+ ∀g,L. G ⊢ ⬈*[h,g] 𝐒⦃L⦄ →
+ ∀f,I,K. g = ↑f → L = K.ⓤ{I} → G ⊢ ⬈*[h,f] 𝐒⦃K⦄.
+#h #G #g #L * -g -L
+[ #f #g #J #L #_ #H destruct
+| #f #I #K #_ #g #J #L #H #_
+ elim (discr_push_next … H)
+| #f #I #K #HK #g #J #L #H1 #H2 destruct //
+| #f #I #K #V #_ #_ #g #J #L #_ #H destruct
+]
+qed-.
+
+lemma sdsx_inv_unit (h) (G):
+ ∀f,I,K. G ⊢ ⬈*[h,↑f] 𝐒⦃K.ⓤ{I}⦄ → G ⊢ ⬈*[h,f] 𝐒⦃K⦄.
+/2 width=6 by sdsx_inv_unit_aux/ qed-.
+
+fact sdsx_inv_pair_aux (h) (G):
+ ∀g,L. G ⊢ ⬈*[h,g] 𝐒⦃L⦄ →
+ ∀f,I,K,V. g = ↑f → L = K.ⓑ{I}V →
+ ∧∧ G ⊢ ⬈*[h,V] 𝐒⦃K⦄ & G ⊢ ⬈*[h,f] 𝐒⦃K⦄.
+#h #G #g #L * -g -L
+[ #f #g #J #L #W #_ #H destruct
+| #f #I #K #_ #g #J #L #W #H #_
+ elim (discr_push_next … H)
+| #f #I #K #_ #g #J #L #W #_ #H destruct
+| #f #I #K #V #HV #HK #g #J #L #W #H1 #H2 destruct
+ /2 width=1 by conj/
+]
+qed-.
+
+(* Basic_2A1: uses: lcosx_inv_pair *)
+lemma sdsx_inv_pair (h) (G):
+ ∀f,I,K,V. G ⊢ ⬈*[h,↑f] 𝐒⦃K.ⓑ{I}V⦄ →
+ ∧∧ G ⊢ ⬈*[h,V] 𝐒⦃K⦄ & G ⊢ ⬈*[h,f] 𝐒⦃K⦄.
+/2 width=6 by sdsx_inv_pair_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma sdsx_inv_pair_gen (h) (G):
+ ∀g,I,K,V. G ⊢ ⬈*[h,g] 𝐒⦃K.ⓑ{I}V⦄ →
+ ∨∨ ∃∃f. G ⊢ ⬈*[h,f] 𝐒⦃K⦄ & g = ⫯f
+ | ∃∃f. G ⊢ ⬈*[h,V] 𝐒⦃K⦄ & G ⊢ ⬈*[h,f] 𝐒⦃K⦄ & g = ↑f.
+#h #G #g #I #K #V #H
+elim (pn_split g) * #f #Hf destruct
+[ lapply (sdsx_inv_push … H) -H /3 width=3 by ex2_intro, or_introl/
+| elim (sdsx_inv_pair … H) -H /3 width=3 by ex3_intro, or_intror/
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma sdsx_fwd_bind (h) (G):
+ ∀g,I,K. G ⊢ ⬈*[h,g] 𝐒⦃K.ⓘ{I}⦄ → G ⊢ ⬈*[h,⫱g] 𝐒⦃K⦄.
+#h #G #g #I #K
+elim (pn_split g) * #f #Hf destruct
+[ #H lapply (sdsx_inv_push … H) -H //
+| cases I -I #I
+ [ #H lapply (sdsx_inv_unit … H) -H //
+ | #V #H elim (sdsx_inv_pair … H) -H //
+ ]
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma sdsx_eq_repl_back (h) (G):
+ ∀L. eq_repl_back … (λf. G ⊢ ⬈*[h,f] 𝐒⦃L⦄).
+#h #G #L #f1 #H elim H -L -f1
+[ //
+| #f1 #I #L #_ #IH #x2 #H
+ elim (eq_inv_px … H) -H /3 width=3 by sdsx_push/
+| #f1 #I #L #_ #IH #x2 #H
+ elim (eq_inv_nx … H) -H /3 width=3 by sdsx_unit/
+| #f1 #I #L #V #HV #_ #IH #x2 #H
+ elim (eq_inv_nx … H) -H /3 width=3 by sdsx_pair/
+]
+qed-.
+
+lemma sdsx_eq_repl_fwd (h) (G):
+ ∀L. eq_repl_fwd … (λf. G ⊢ ⬈*[h,f] 𝐒⦃L⦄).
+#h #G #L @eq_repl_sym /2 width=3 by sdsx_eq_repl_back/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: lcosx_O *)
+lemma sdsx_isid (h) (G):
+ ∀f. 𝐈⦃f⦄ → ∀L. G ⊢ ⬈*[h,f] 𝐒⦃L⦄.
+#h #G #f #Hf #L elim L -L
+/3 width=3 by sdsx_eq_repl_back, sdsx_push, eq_push_inv_isid/
+qed.
+
+(* Basic_2A1: removed theorems 2:
+ lcosx_drop_trans_lt lcosx_inv_succ
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/scl.ma".
+include "basic_2/rt_computation/rdsx_drops.ma".
+include "basic_2/rt_computation/rdsx_lpxs.ma".
+include "basic_2/rt_computation/sdsx.ma".
+
+axiom pippo (h) (f) (G) (V:term):
+ ∀L1. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀L2. L1 ⊐ⓧ[f] L2 → G ⊢ ⬈*[h,V] 𝐒⦃L2⦄.
+
+
+(* STRONGLY NORMALIZING SELECTED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+(* Properties with strongly normalizing referred local environments *********)
+
+(* Basic_2A1: uses: lsx_cpx_trans_lcosx *)
+lemma rdsx_cpx_trans_sdsx (h):
+ ∀G,L0,T1,T2. ⦃G,L0⦄ ⊢ T1 ⬈[h] T2 →
+ ∀f. G ⊢ ⬈*[h,f] 𝐒⦃L0⦄ → ∀L. L0 ⊐ⓧ[f] L →
+ G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
+#h #G #L0 #T1 #T2 #H @(cpx_ind … H) -G -L0 -T1 -T2
+[ //
+| //
+| #I #G #K0 #V1 #V2 #W2 #_ #IH #HVW2 #g #H1 #Y #H2 #H3
+ elim (sdsx_inv_pair_gen … H1) -H1 *
+ [ #f #HK0 #H destruct
+ elim (scl_inv_push_sn … H2) -H2 #K #HK #H destruct
+ /4 width=8 by rdsx_lifts, rdsx_fwd_pair, drops_refl, drops_drop/
+ | #f #HV1 #HK0 #H destruct
+ elim (scl_inv_next_sn … H2) -H2 #K #HK #H destruct
+ /4 width=8 by pippo, rdsx_lifts, drops_refl, drops_drop/
+ ]
+| #I0 #G #K0 #T #U #i #_ #IH #HTU #g #H1 #Y #H2 #H3
+ lapply (sdsx_fwd_bind … H1) -H1 #HK0
+ elim (scl_fwd_bind_sn … H2) -H2 #I #K #HK #H destruct
+ /6 width=8 by rdsx_inv_lifts, rdsx_lifts, drops_refl, drops_drop/
+| #p #I #G #L0 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #f #H1 #L #H2 #H3
+ elim (rdsx_inv_bind_void … H3) -H3 #HV1 #HT1
+ @rdsx_bind_void
+ [ /2 width=3 by/
+ | @(IHT12 (↑f) … HT1)
+ [ @(sdsx_pair … H1)
+ | /2 width=1 by scl_next/
+
+ /4 width=2 by lsubsx_pair, rdsx_bind_void/
+| #I0 #G #L0 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #f #L #HL0 #HL
+ elim (rdsx_inv_flat … HL) -HL /3 width=2 by rdsx_flat/
+| #G #L0 #V #U1 #T1 #T2 #HTU1 #_ #IHT12 #f #L #HL0 #HL
+ elim (rdsx_inv_bind … HL) -HL #HV #HU1
+ /5 width=8 by rdsx_inv_lifts, drops_refl, drops_drop/
+| #G #L0 #V #T1 #T2 #_ #IHT12 #f #L #HL0 #HL
+ elim (rdsx_inv_flat … HL) -HL /2 width=2 by/
+| #G #L0 #V1 #V2 #T #_ #IHV12 #f #L #HL0 #HL
+ elim (rdsx_inv_flat … HL) -HL /2 width=2 by/
+| #p #G #L0 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #f #L #HL0 #HL
+ elim (rdsx_inv_flat … HL) -HL #HV1 #HL
+ elim (rdsx_inv_bind … HL) -HL #HW1 #HT1
+ /4 width=2 by lsubsx_pair, rdsx_bind_void, rdsx_flat/
+| #p #G #L0 #V1 #V2 #U2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #HVU2 #f #L #HL0 #HL
+ elim (rdsx_inv_flat … HL) -HL #HV1 #HL
+ elim (rdsx_inv_bind … HL) -HL #HW1 #HT1
+ /6 width=8 by lsubsx_pair, rdsx_lifts, rdsx_bind_void, rdsx_flat, drops_refl, drops_drop/
+]
+qed-.
+
+(* Advanced properties of strongly normalizing referred local environments **)
+
+(* Basic_2A1: uses: lsx_cpx_trans_O *)
+lemma rdsx_cpx_trans (h):
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
+ G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
+/3 width=6 by rdsx_cpx_trans_lsubsx, lsubsx_refl/ qed-.
+
+lemma rdsx_cpxs_trans (h):
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
+#h #G #L #T1 #T2 #H
+@(cpxs_ind_dx ???????? H) -T1 //
+/3 width=3 by rdsx_cpx_trans/
+qed-.
+*)
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/colon_7.ma".
-include "basic_2/notation/relations/colon_6.ma".
-include "basic_2/notation/relations/colonstar_6.ma".
+include "basic_2/notation/relations/colonstar_7.ma".
include "basic_2/dynamic/cnv.ma".
(* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
-definition ntas (a) (h) (n) (G) (L): relation term ≝ λT,U.
- ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] & ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[n,h] U0.
+definition ntas (h) (a) (n) (G) (L): relation term ≝ λT,U.
+ ∃∃U0. ⦃G,L⦄ ⊢ U ![h,a] & ⦃G,L⦄ ⊢ T ![h,a] & ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[n,h] U0.
interpretation "iterated native type assignment (term)"
- 'Colon a h n G L T U = (ntas a h n G L T U).
-
-interpretation "restricted iterated native type assignment (term)"
- 'Colon h n G L T U = (ntas true h n G L T U).
-
-interpretation "extended iterated native type assignment (term)"
- 'ColonStar h n G L T U = (ntas false h n G L T U).
+ 'ColonStar h a n G L T U = (ntas h a n G L T U).
(* Basic properties *********************************************************)
-lemma ntas_refl (a) (h) (G) (L):
- ∀T. ⦃G,L⦄ ⊢ T ![a,h] → ⦃G,L⦄ ⊢ T :[a,h,0] T.
+lemma ntas_intro (h) (a) (n) (G) (L):
+ ∀U. ⦃G,L⦄ ⊢ U ![h,a] → ∀T. ⦃G,L⦄ ⊢ T ![h,a] →
+ ∀U0. ⦃G,L⦄ ⊢ U ➡*[h] U0 → ⦃G,L⦄ ⊢ T ➡*[n,h] U0 → ⦃G,L⦄ ⊢ T :*[h,a,n] U.
/2 width=3 by ex4_intro/ qed.
+
+lemma ntas_refl (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ⦃G,L⦄ ⊢ T :*[h,a,0] T.
+/2 width=3 by ntas_intro/ qed.
+
+lemma ntas_sort (h) (a) (n) (G) (L):
+ ∀s. ⦃G,L⦄ ⊢ ⋆s :*[h,a,n] ⋆((next h)^n s).
+#h #a #n #G #L #s
+/2 width=3 by ntas_intro, cnv_sort, cpms_sort/
+qed.
+
+lemma ntas_bind_cnv (h) (a) (n) (G) (K):
+ ∀V. ⦃G,K⦄ ⊢ V ![h,a] →
+ ∀I,T,U. ⦃G,K.ⓑ{I}V⦄ ⊢ T :*[h,a,n] U →
+ ∀p. ⦃G,K⦄ ⊢ ⓑ{p,I}V.T :*[h,a,n] ⓑ{p,I}V.U.
+#h #a #n #G #K #V #HV #I #T #U
+* #X #HU #HT #HUX #HTX #p
+/3 width=5 by ntas_intro, cnv_bind, cpms_bind_dx/
+qed.
+
+(* Basic_forward lemmas *****************************************************)
+
+lemma ntas_fwd_cnv_sn (h) (a) (n) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :*[h,a,n] U → ⦃G,L⦄ ⊢ T ![h,a].
+#h #a #n #G #L #T #U
+* #X #_ #HT #_ #_ //
+qed-.
+
+(* Note: this is ntas_fwd_correct_cnv *)
+lemma ntas_fwd_cnv_dx (h) (a) (n) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :*[h,a,n] U → ⦃G,L⦄ ⊢ U ![h,a].
+#h #a #n #G #L #T #U
+* #X #HU #_ #_ #_ //
+qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-(*
-include "basic_2/dynamic/nta_lift.ma".
-include "basic_2/hod/ntas.ma".
-
-(* HIGHER ORDER NATIVE TYPE ASSIGNMENT ON TERMS *****************************)
-
-(* Advanced properties on native type assignment for terms ******************)
-
-lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h, L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h, L⦄ ⊢ T : U →
- ∀V. ⦃h, L⦄ ⊢ V : W → ⦃h, L⦄ ⊢ ⓐV.T : ⓐV.U.
-#h #L #U #W #Y #H @(ntas_ind_dx … H) -U /2 width=1/ /3 width=2/
-qed.
-
-axiom pippo: ∀h,L,T,W,Y. ⦃h, L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h, L⦄ ⊢ T : U →
- ∃Z. ⦃h, L⦄ ⊢ U :* ⓛW.Z.
-(* REQUIRES SUBJECT CONVERSION
-#h #L #T #W #Y #H @(ntas_ind_dx … H) -T
-[ #U #HYU
- elim (nta_fwd_correct … HYU) #U0 #HU0
- elim (nta_inv_bind1 … HYU) #W0 #Y0 #HW0 #HY0 #HY0U
-*)
-
-(* Advanced inversion lemmas on native type assignment for terms ************)
-
-fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h, L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X →
- ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V &
- L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T.
-#h #L #Z #U #H elim H -L -Z -U
-[ #L #k #X #Y #H destruct
-| #L #K #V #W #U #i #_ #_ #_ #_ #X #Y #H destruct
-| #L #K #W #V #U #i #_ #_ #_ #_ #X #Y #H destruct
-| #I #L #V #W #T #U #_ #_ #_ #_ #X #Y #H destruct
-| #L #V #W #Z #U #HVW #HZU #_ #_ #X #Y #H destruct /2 width=7/
-| #L #V #W #Z #U #HZU #_ #_ #IHUW #X #Y #H destruct
- elim (IHUW U Y ?) -IHUW // /3 width=9/
-| #L #Z #U #_ #_ #X #Y #H destruct
-| #L #Z #U1 #U2 #V2 #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct
- elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #W #V #T #HYW #HXV #HU1 #HVT
- lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=7/
-]
-qed.
-
-(* Basic_1: was only: ty3_gen_appl *)
-lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X : U →
- ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V &
- L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T.
-/2 width=3/ qed-.
-
-axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h, L⦄ ⊢ ⓐZ.ⓛY.X : U →
- ∃∃W. ⦃h, L⦄ ⊢ Z : Y & ⦃h, L⦄ ⊢ ⓛY.X : ⓛY.W &
- L ⊢ ⓐZ.ⓛY.W ⬌* U.
-(* REQUIRES SUBJECT REDUCTION
-#h #L #Z #Y #X #U #H
-elim (nta_inv_pure1 … H) -H #W #V #T #HZW #HXV #HVU #HVT
-elim (nta_inv_bind1 … HXV) -HXV #Y0 #X0 #HY0 #HX0 #HX0V
-lapply (cpcs_trans … (ⓐZ.ⓛY.X0) … HVU) -HVU /2 width=1/ -HX0V #HX0U
-@(ex3_1_intro … HX0U) /2 width=2/
-*)
-*)
(* *)
(**************************************************************************)
-include "basic_2/dynamic/nta.ma".
-include "basic_2/i_dynamic/ntas.ma".
+include "basic_2/dynamic/nta_preserve.ma".
+include "basic_2/i_dynamic/ntas_preserve.ma".
(* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
+(* Properties with native type assignment for terms *************************)
+
+lemma nta_ntas (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :[h,a] U → ⦃G,L⦄ ⊢ T :*[h,a,1] U.
+#h #a #G #L #T #U #H
+elim (cnv_inv_cast … H) -H /2 width=3 by ntas_intro/
+qed-.
+
+(* Inversion lemmas with native type assignment for terms *******************)
+
+lemma ntas_inv_nta (h) (a) (G) (L):
+ ∀T,U. ⦃G,L⦄ ⊢ T :*[h,a,1] U → ⦃G,L⦄ ⊢ T :[h,a] U.
+#h #a #G #L #T #U
+* /2 width=3 by cnv_cast/
+qed-.
+
+(* Note: this follows from ntas_inv_appl_sn *)
+lemma nta_inv_appl_sn_ntas (h) (a) (G) (L) (V) (T):
+ ∀X. ⦃G,L⦄ ⊢ ⓐV.T :[h,a] X →
+ ∨∨ ∃∃p,W,U,U0. ad a 0 & ⦃G,L⦄ ⊢ V :[h,a] W & ⦃G,L⦄ ⊢ T :*[h,a,0] ⓛ{p}W.U0 & ⦃G,L.ⓛW⦄ ⊢ U0 :[h,a] U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X & ⦃G,L⦄ ⊢ X ![h,a]
+ | ∃∃n,p,W,U,U0. ad a (↑n) & ⦃G,L⦄ ⊢ V :[h,a] W & ⦃G,L⦄ ⊢ T :[h,a] U & ⦃G,L⦄ ⊢ U :*[h,a,n] ⓛ{p}W.U0 & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X & ⦃G,L⦄ ⊢ X ![h,a].
+#h #a #G #L #V #T #X #H
+(*
+lapply (nta_ntas … H) -H #H
+elim (ntas_inv_appl_sn … H) -H * #n #p #W #U #U0 #Hn #Ha #HVW #HTU #HU #HUX #HX
+[ elim (eq_or_gt n) #H destruct
+ [ <minus_n_O in HU; #HU -Hn
+ /4 width=8 by ntas_inv_nta, ex6_4_intro, or_introl/
+ | lapply (le_to_le_to_eq … Hn H) -Hn -H #H destruct
+ <minus_n_n in HU; #HU
+ @or_intror
+ @(ex6_5_intro … Ha … HUX HX) -Ha -X
+ [2,4: /2 width=2 by ntas_inv_nta/
+ |6: @ntas_refl
+*)
+elim (cnv_inv_cast … H) -H #X0 #HX #HVT #HX0 #HTX0
+elim (cnv_inv_appl … HVT) #n #p #W #U0 #Ha #HV #HT #HVW #HTU0
+elim (eq_or_gt n) #Hn destruct
+[ elim (cnv_fwd_cpms_abst_dx_le … HT … HTU0 1) [| // ] <minus_n_O #U #H #HU0
+ lapply (cpms_appl_dx … V V … H) [ // ] -H #H
+ elim (cnv_cpms_conf … HVT … HTX0 … H) -HVT -HTX0 -H <minus_n_n #X1 #HX01 #HUX1
+ lapply (cpms_trans … HX0 … HX01) -X0 #HX1
+ lapply (cprs_div … HUX1 … HX1) -X1 #HUX
+ lapply (cnv_cpms_trans … HT … HTU0) #H
+ elim (cnv_inv_bind … H) -H #_ #HU0
+ /4 width=8 by cnv_cpms_ntas, cnv_cpms_nta, ex6_4_intro, or_introl/
+| >(plus_minus_m_m_commutative … Hn) in HTU0; #H
+ elim (cpms_inv_plus … H) -H #U #HTU #HU0
+ lapply (cpms_appl_dx … V V … HTU) [ // ] #H
+ elim (cnv_cpms_conf … HVT … HTX0 … H) -HVT -HTX0 -H <minus_n_n #X1 #HX01 #HUX1
+ lapply (cpms_trans … HX0 … HX01) -X0 #HX1
+ lapply (cprs_div … HUX1 … HX1) -X1 #HUX
+ <(S_pred … Hn) in Ha; -Hn #Ha
+ /5 width=10 by cnv_cpms_ntas, cnv_cpms_nta, cnv_cpms_trans, ex6_5_intro, or_intror/
+]
+qed-.
+
(*
definition ntas: sh → lenv → relation term ≝
(* Basic eliminators ********************************************************)
axiom ntas_ind_dx: ∀h,L,T2. ∀R:predicate term. R T2 →
- (∀T1,T. ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → R T → R T1) →
- ∀T1. ⦃h, L⦄ ⊢ T1 :* T2 → R T1.
+ (∀T1,T. ⦃h,L⦄ ⊢ T1 : T → ⦃h,L⦄ ⊢ T :* T2 → R T → R T1) →
+ ∀T1. ⦃h,L⦄ ⊢ T1 :* T2 → R T1.
(*
#h #L #T2 #R #HT2 #IHT2 #T1 #HT12
@(star_ind_dx … HT2 IHT2 … HT12) //
(* Basic properties *********************************************************)
lemma ntas_strap1: ∀h,L,T1,T,T2.
- ⦃h, L⦄ ⊢ T1 :* T → ⦃h, L⦄ ⊢ T : T2 → ⦃h, L⦄ ⊢ T1 :* T2.
+ ⦃h,L⦄ ⊢ T1 :* T → ⦃h,L⦄ ⊢ T : T2 → ⦃h,L⦄ ⊢ T1 :* T2.
/2 width=3/ qed.
lemma ntas_strap2: ∀h,L,T1,T,T2.
- ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → ⦃h, L⦄ ⊢ T1 :* T2.
+ ⦃h,L⦄ ⊢ T1 : T → ⦃h,L⦄ ⊢ T :* T2 → ⦃h,L⦄ ⊢ T1 :* T2.
/2 width=3/ qed.
*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_equivalence/cpcs_cprs.ma".
+include "basic_2/dynamic/cnv_preserve.ma".
+include "basic_2/i_dynamic/ntas.ma".
+
+(* ITERATED NATIVE TYPE ASSIGNMENT FOR TERMS ********************************)
+
+(* Properties based on preservation *****************************************)
+
+lemma cnv_cpms_ntas (h) (a) (G) (L):
+ ∀T. ⦃G,L⦄ ⊢ T ![h,a] → ∀n,U.⦃G,L⦄ ⊢ T ➡*[n,h] U → ⦃G,L⦄ ⊢ T :*[h,a,n] U.
+/3 width=4 by ntas_intro, cnv_cpms_trans/ qed.
+
+(* Inversion lemmas based on preservation ***********************************)
+
+lemma ntas_inv_appl_sn (h) (a) (m) (G) (L) (V) (T):
+ ∀X. ⦃G,L⦄ ⊢ ⓐV.T :*[h,a,m] X →
+ ∨∨ ∃∃n,p,W,U,U0. n ≤ m & ad a n & ⦃G,L⦄ ⊢ V :*[h,a,1] W & ⦃G,L⦄ ⊢ T :*[h,a,n] ⓛ{p}W.U0 & ⦃G,L.ⓛW⦄ ⊢ U0 :*[h,a,m-n] U & ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.U ⬌*[h] X & ⦃G,L⦄ ⊢ X ![h,a]
+ | ∃∃n,p,W,U,U0. m ≤ n & ad a n & ⦃G,L⦄ ⊢ V :*[h,a,1] W & ⦃G,L⦄ ⊢ T :*[h,a,m] U & ⦃G,L⦄ ⊢ U :*[h,a,n-m] ⓛ{p}W.U0 & ⦃G,L⦄ ⊢ ⓐV.U ⬌*[h] X & ⦃G,L⦄ ⊢ X ![h,a].
+#h #a #m #G #L #V #T #X
+* #X0 #HX #HVT #HX0 #HTX0
+elim (cnv_inv_appl … HVT) #n #p #W #U0 #Ha #HV #HT #HVW #HTU0
+elim (le_or_ge n m) #Hnm
+[ elim (cnv_fwd_cpms_abst_dx_le … HT … HTU0 … Hnm) #U #H #HU0
+ lapply (cpms_appl_dx … V V … H) [ // ] -H #H
+ elim (cnv_cpms_conf … HVT … HTX0 … H) -HVT -HTX0 -H <minus_n_n #X1 #HX01 #HUX1
+ lapply (cpms_trans … HX0 … HX01) -X0 #HX1
+ lapply (cprs_div … HUX1 … HX1) -X1 #HUX
+ lapply (cnv_cpms_trans … HT … HTU0) #H
+ elim (cnv_inv_bind … H) -H #_ #HU0
+ /4 width=11 by cnv_cpms_ntas, ex7_5_intro, or_introl/
+| >(plus_minus_m_m_commutative … Hnm) in HTU0; #H
+ elim (cpms_inv_plus … H) -H #U #HTU #HU0
+ lapply (cpms_appl_dx … V V … HTU) [ // ] #H
+ elim (cnv_cpms_conf … HVT … HTX0 … H) -HVT -HTX0 -H <minus_n_n #X1 #HX01 #HUX1
+ lapply (cpms_trans … HX0 … HX01) -X0 #HX1
+ lapply (cprs_div … HUX1 … HX1) -X1 #HUX
+ /5 width=11 by cnv_cpms_ntas, cnv_cpms_trans, ex7_5_intro, or_intror/
+]
+qed-.
+
+(*
+(* Advanced properties on native type assignment for terms ******************)
+
+lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h,L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h,L⦄ ⊢ T : U →
+ ∀V. ⦃h,L⦄ ⊢ V : W → ⦃h,L⦄ ⊢ ⓐV.T : ⓐV.U.
+#h #L #U #W #Y #H @(ntas_ind_dx … H) -U /2 width=1/ /3 width=2/
+qed.
+
+axiom pippo: ∀h,L,T,W,Y. ⦃h,L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h,L⦄ ⊢ T : U →
+ ∃Z. ⦃h,L⦄ ⊢ U :* ⓛW.Z.
+(* REQUIRES SUBJECT CONVERSION
+#h #L #T #W #Y #H @(ntas_ind_dx … H) -T
+[ #U #HYU
+ elim (nta_fwd_correct … HYU) #U0 #HU0
+ elim (nta_inv_bind1 … HYU) #W0 #Y0 #HW0 #HY0 #HY0U
+*)
+
+(* Advanced inversion lemmas on native type assignment for terms ************)
+
+fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h,L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X →
+ ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V &
+ L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T.
+#h #L #Z #U #H elim H -L -Z -U
+[ #L #k #X #Y #H destruct
+| #L #K #V #W #U #i #_ #_ #_ #_ #X #Y #H destruct
+| #L #K #W #V #U #i #_ #_ #_ #_ #X #Y #H destruct
+| #I #L #V #W #T #U #_ #_ #_ #_ #X #Y #H destruct
+| #L #V #W #Z #U #HVW #HZU #_ #_ #X #Y #H destruct /2 width=7/
+| #L #V #W #Z #U #HZU #_ #_ #IHUW #X #Y #H destruct
+ elim (IHUW U Y ?) -IHUW // /3 width=9/
+| #L #Z #U #_ #_ #X #Y #H destruct
+| #L #Z #U1 #U2 #V2 #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct
+ elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #W #V #T #HYW #HXV #HU1 #HVT
+ lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=7/
+]
+qed.
+
+(* Basic_1: was only: ty3_gen_appl *)
+lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h,L⦄ ⊢ ⓐY.X : U →
+ ∃∃W,V,T. ⦃h,L⦄ ⊢ Y : W & ⦃h,L⦄ ⊢ X : V &
+ L ⊢ ⓐY.V ⬌* U & ⦃h,L⦄ ⊢ V :* ⓛW.T.
+/2 width=3/ qed-.
+
+axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h,L⦄ ⊢ ⓐZ.ⓛY.X : U →
+ ∃∃W. ⦃h,L⦄ ⊢ Z : Y & ⦃h,L⦄ ⊢ ⓛY.X : ⓛY.W &
+ L ⊢ ⓐZ.ⓛY.W ⬌* U.
+(* REQUIRES SUBJECT REDUCTION
+#h #L #Z #Y #X #U #H
+elim (nta_inv_pure1 … H) -H #W #V #T #HZW #HXV #HVU #HVT
+elim (nta_inv_bind1 … HXV) -HXV #Y0 #X0 #HY0 #HX0 #HX0V
+lapply (cpcs_trans … (ⓐZ.ⓛY.X0) … HVU) -HVU /2 width=1/ -HX0V #HX0U
+@(ex3_1_intro … HX0U) /2 width=2/
+*)
+*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :[ break term 46 h ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'Colon $h $G $L $T1 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :[ break term 46 a, break term 46 h ] break term 46 T2 )"
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :[ break term 46 h, break term 46 a ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'Colon $a $h $G $L $T1 $T2 }.
+ for @{ 'Colon $h $a $G $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :[ break term 46 a, break term 46 h, break term 46 n ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'Colon $a $h $n $G $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :*[ break term 46 h ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'ColonStar $h $G $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :*[ break term 46 h, break term 46 n ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'ColonStar $h $n $G $L $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 :*[ break term 46 h, break term 46 a, break term 46 n ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'ColonStar $h $a $n $G $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T ![ break term 46 h ] )"
- non associative with precedence 45
- for @{ 'Exclaim $h $G $L $T }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T ![ break term 46 a, break term 46 h ] )"
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T ![ break term 46 h, break term 46 a ] )"
non associative with precedence 45
- for @{ 'Exclaim $a $h $G $L $T }.
+ for @{ 'Exclaim $h $a $G $L $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T !*[ break term 46 h ] )"
- non associative with precedence 45
- for @{ 'ExclaimStar $h $G $L $T }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( G ⊢ break term 46 L1 ⫃![ break term 46 a, break term 46 h ] break term 46 L2 )"
+notation "hvbox( G ⊢ break term 46 L1 ⫃![ break term 46 h, break term 46 a ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'LRSubEqV $a $h $G $L1 $L2 }.
+ for @{ 'LRSubEqV $h $a $G $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( G ⊢ break term 46 L1 ⊆ⓧ [ break term 46 h, break term 46 o, break term 46 f ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LSubEqX $h $o $f $G $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ⬌* [ break term 46 h, break term 46 n1, break term 46 n2 ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PConvStar $h $n1 $n2 $G $L $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡* [ break term 46 h ] 𝐍 ⦃ break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedEval $h $G $L $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡* [ break term 46 h, break term 46 n ] 𝐍 ⦃ break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedEval $h $n $G $L $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡*𝐍𝐖*[ break term 46 h, break term 46 n ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PRedEvalWStar $h $n $G $L $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡𝐍𝐖*[ break term 46 h ] break term 46 T )"
+ non associative with precedence 45
+ for @{ 'PRedITNormal $h $G $L $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡ [ break term 46 h ] 𝐍 ⦃ break term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedNormal $h $G $L $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ➡* break term 46 L2 )"
- non associative with precedence 45
- for @{ 'PRedSnStar $G $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≽ [ break term 46 h ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedSubTy $h $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≽ [ break term 46 h, break term 46 o ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'PRedSubTy $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≻ [ break term 46 h ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedSubTyProper $h $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≻ [ break term 46 h, break term 46 o ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'PRedSubTyProper $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≥ [ break term 46 h ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedSubTyStar $h $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≥ [ break term 46 h, break term 46 o ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'PRedSubTyStar $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ > [ break term 46 h ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedSubTyStarProper $h $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ > [ break term 46 h, break term 46 o ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'PRedSubTyStarProper $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ≥ [ term 46 h ] 𝐒 ⦃ break term 46 G, break term 46 L, break term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedSubTyStrong $h $G $L $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ≥ [ term 46 h, break term 46 o ] 𝐒 ⦃ break term 46 G, break term 46 L, break term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'PRedSubTyStrong $h $o $G $L $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬈ [ break term 46 h ] 𝐍 ⦃ break term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedTyNormal $h $G $L $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬈ [ break term 46 h, break term 46 o ] 𝐍 ⦃ break term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'PRedTyNormal $h $o $G $L $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ ⬈ * [ break term 46 h, break term 46 T ] 𝐒 ⦃ break term 46 L ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedTySNStrong $h $T $G $L }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( G ⊢ ⬈ * [ break term 46 h, break term 46 o, break term 46 T ] 𝐒 ⦃ break term 46 L ⦄ )"
- non associative with precedence 45
- for @{ 'PRedTySNStrong $h $o $T $G $L }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬈ * [ break term 46 h] 𝐒 ⦃ break term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedTyStrong $h $G $L $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬈ * [ break term 46 h, break term 46 o ] 𝐒 ⦃ break term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'PRedTyStrong $h $o $G $L $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ break term 46 L1 ⊒[ break term 46 h] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'ToPRedTySNStrong $h $G $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/preditnormal_4.ma".
+include "static_2/syntax/tweq.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND WHD RT-TRANSITION ***************************)
+
+definition cnuw (h) (G) (L): predicate term ≝
+ λT1. ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → T1 ≅ T2.
+
+interpretation
+ "normality for t-unbound weak head context-sensitive parallel rt-transition (term)"
+ 'PRedITNormal h G L T = (cnuw h G L T).
+
+(* Basic properties *********************************************************)
+
+lemma cnuw_sort (h) (G) (L): ∀s. ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] ⋆s.
+#h #G #L #s1 #n #X #H
+lapply (cpms_inv_sort1 … H) -H #H destruct //
+qed.
+
+lemma cnuw_ctop (h) (G): ∀i. ⦃G,⋆⦄ ⊢ ➡𝐍𝐖*[h] #i.
+#h #G #i #n #X #H
+elim (cpms_inv_lref1_ctop … H) -H #H #_ destruct //
+qed.
+
+lemma cnuw_zero_unit (h) (G) (L): ∀I. ⦃G,L.ⓤ{I}⦄ ⊢ ➡𝐍𝐖*[h] #0.
+#h #G #L #I #n #X #H
+elim (cpms_inv_zero1_unit … H) -H #H #_ destruct //
+qed.
+
+lemma cnuw_gref (h) (G) (L): ∀l. ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] §l.
+#h #G #L #l1 #n #X #H
+elim (cpms_inv_gref1 … H) -H #H #_ destruct //
+qed.
+
+(* Basic_inversion lemmas ***************************************************)
+
+lemma cnuw_inv_zero_pair (h) (I) (G) (L): ∀V. ⦃G,L.ⓑ{I}V⦄ ⊢ ➡𝐍𝐖*[h] #0 → ⊥.
+#h * #G #L #V #H
+elim (lifts_total V (𝐔❴1❵)) #W #HVW
+[ lapply (H 0 W ?) [ /3 width=3 by cpm_cpms, cpm_delta/ ]
+| lapply (H 1 W ?) [ /3 width=3 by cpm_cpms, cpm_ell/ ]
+] -H #HW
+lapply (tweq_inv_lref_sn … HW) -HW #H destruct
+/2 width=5 by lifts_inv_lref2_uni_lt/
+qed-.
+
+lemma cnuw_inv_cast (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] ⓝV.T → ⊥.
+#h #G #L #V #T #H
+lapply (H 0 T ?) [ /3 width=1 by cpm_cpms, cpm_eps/ ] -H #H
+/2 width=3 by tweq_inv_cast_xy_y/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cnuw_fwd_appl (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] ⓐV.T → ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] T.
+#h #G #L #V #T1 #HT1 #n #T2 #HT12
+lapply (HT1 n (ⓐV.T2) ?) -HT1
+/2 width=3 by cpms_appl_dx, tweq_inv_appl_bi/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cnuw_simple.ma".
+include "basic_2/rt_computation/cnuw_drops.ma".
+include "basic_2/rt_computation/cprs_tweq.ma".
+include "basic_2/rt_computation/lprs_cpms.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND WHD RT-TRANSITION ***************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cnuw_inv_abbr_pos (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] +ⓓV.T → ⊥.
+#h #G #L #V #T1 #H
+elim (cprs_abbr_pos_twneq h G L V T1) #T2 #HT12 #HnT12
+/3 width=2 by/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cnuw_abbr_neg (h) (G) (L): ∀V,T. ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] -ⓓV.T.
+#h #G #L #V1 #T1 #n #X #H
+elim (cpms_inv_abbr_sn_dx … H) -H *
+[ #V2 #T2 #_ #_ #H destruct /1 width=1 by tweq_abbr_neg/
+| #X1 #_ #_ #H destruct
+]
+qed.
+
+lemma cnuw_abst (h) (p) (G) (L): ∀W,T. ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] ⓛ{p}W.T.
+#h #p #G #L #W1 #T1 #n #X #H
+elim (cpms_inv_abst_sn … H) -H #W2 #T2 #_ #_ #H destruct
+/1 width=1 by tweq_abst/
+qed.
+
+lemma cnuw_cpms_trans (h) (n) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] T2.
+#h #n1 #G #L #T1 #HT1 #T2 #HT12 #n2 #T3 #HT23
+/4 width=5 by cpms_trans, tweq_canc_sn/
+qed-.
+
+lemma cnuw_dec_ex (h) (G) (L):
+ ∀T1. ∨∨ ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] T1
+ | ∃∃n,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & (T1 ≅ T2 → ⊥).
+#h #G #L #T1 elim T1 -T1 *
+[ #s /3 width=5 by cnuw_sort, or_introl/
+| #i elim (drops_F_uni L i)
+ [ /3 width=7 by cnuw_atom_drops, or_introl/
+ | * * [ #I | * #V ] #K #HLK
+ [ /3 width=8 by cnuw_unit_drops, or_introl/
+ | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
+ @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpms_delta_drops/ ] #H
+ lapply (tweq_inv_lref_sn … H) -H #H destruct
+ /2 width=5 by lifts_inv_lref2_uni_lt/
+ | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
+ @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpms_ell_drops/ ] #H
+ lapply (tweq_inv_lref_sn … H) -H #H destruct
+ /2 width=5 by lifts_inv_lref2_uni_lt/
+ ]
+ ]
+| #l /3 width=5 by cnuw_gref, or_introl/
+| #p * [ cases p ] #V1 #T1 #_ #_
+ [ elim (cprs_abbr_pos_twneq h G L V1 T1) #T2 #HT12 #HnT12
+ /4 width=4 by ex2_2_intro, or_intror/
+ | /3 width=5 by cnuw_abbr_neg, or_introl/
+ | /3 width=5 by cnuw_abst, or_introl/
+ ]
+| * #V1 #T1 #_ #IH
+ [ elim (simple_dec_ex T1) [ #HT1 | * #p * #W1 #U1 #H destruct ]
+ [ elim IH -IH
+ [ /3 width=6 by cnuw_appl_simple, or_introl/
+ | * #n #T2 #HT12 #HnT12 -HT1
+ @or_intror @(ex2_2_intro … n (ⓐV1.T2)) [ /2 width=1 by cpms_appl_dx/ ] #H
+ lapply (tweq_inv_appl_bi … H) -H /2 width=1 by/
+ ]
+ | elim (lifts_total V1 𝐔❴1❵) #X1 #HVX1
+ @or_intror @(ex2_2_intro … (ⓓ{p}W1.ⓐX1.U1)) [1,2: /2 width=3 by cpms_theta/ ] #H
+ elim (tweq_inv_appl_sn … H) -H #X1 #X2 #_ #H destruct
+ | @or_intror @(ex2_2_intro … (ⓓ{p}ⓝW1.V1.U1)) [1,2: /2 width=2 by cpms_beta/ ] #H
+ elim (tweq_inv_appl_sn … H) -H #X1 #X2 #_ #H destruct
+ ]
+ | @or_intror @(ex2_2_intro … T1) [1,2: /2 width=2 by cpms_eps/ ] #H
+ /2 width=4 by tweq_inv_cast_xy_y/
+ ]
+]
+qed-.
+
+lemma cnuw_dec (h) (G) (L): ∀T. Decidable (⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] T).
+#h #G #L #T1
+elim (cnuw_dec_ex h G L T1)
+[ /2 width=1 by or_introl/
+| * #n #T2 #HT12 #nT12 /4 width=2 by or_intror/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/lifts_tweq.ma".
+include "basic_2/rt_computation/cpms_drops.ma".
+include "basic_2/rt_computation/cnuw.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND WHD RT-TRANSITION ***************************)
+
+(* Properties with generic relocation ***************************************)
+
+lemma cnuw_lifts (h) (G): d_liftable1 … (cnuw h G).
+#h #G #K #T #HT #b #f #L #HLK #U #HTU #n #U0 #H
+elim (cpms_inv_lifts_sn … H … HLK … HTU) -b -L #T0 #HTU0 #HT0
+lapply (HT … HT0) -G -K /2 width=6 by tweq_lifts_bi/
+qed-.
+
+(* Inversion lemmas with generic relocation *********************************)
+
+lemma cnuw_inv_lifts (h) (G): d_deliftable1 … (cnuw h G).
+#h #G #L #U #HU #b #f #K #HLK #T #HTU #n #T0 #H
+elim (cpms_lifts_sn … H … HLK … HTU) -b -K #U0 #HTU0 #HU0
+lapply (HU … HU0) -G -L /2 width=6 by tweq_inv_lifts_bi/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cnuw_lref (h) (I) (G) (L):
+ ∀i. ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] #i → ⦃G,L.ⓘ{I}⦄ ⊢ ➡𝐍𝐖*[h] #↑i.
+#h #I #G #L #i #Hi #n #X2 #H
+elim (cpms_inv_lref_sn … H) -H *
+[ #H #_ destruct //
+| #T2 #HT2 #HTX2
+ lapply (Hi … HT2) -Hi -HT2 #H
+ lapply (tweq_inv_lref_sn … H) -H #H destruct
+ lapply (lifts_inv_lref1_uni … HTX2) -HTX2 #H destruct //
+]
+qed.
+
+lemma cnuw_atom_drops (h) (b) (G) (L):
+ ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] #i.
+#h #b #G #L #i #Hi #n #X #H
+elim (cpms_inv_lref1_drops … H) -H * [ // || #m ] #K #V1 #V2 #HLK
+lapply (drops_gen b … HLK) -HLK #HLK
+lapply (drops_mono … Hi … HLK) -L #H destruct
+qed.
+
+lemma cnuw_unit_drops (h) (I) (G) (L):
+ ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] #i.
+#h #I #G #L #K #i #HLK #n #X #H
+elim (cpms_inv_lref1_drops … H) -H * [ // || #m ] #Y #V1 #V2 #HLY
+lapply (drops_mono … HLK … HLY) -L #H destruct
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tweq_simple.ma".
+include "basic_2/rt_computation/cpms_cpms.ma".
+include "basic_2/rt_computation/cnuw.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND WHD RT-TRANSITION ***************************)
+
+(* Advanced forward lemma with with simple terms ****************************)
+(*
+lemma cnuw_fwd_appl_simple (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] ⓐV.T → 𝐒⦃T⦄.
+#h #G #L #V #T #HT
+elim (simple_dec_ex T) [ // ] * #p #I #W #U #H destruct
+*)
+(* Advanced properties with simple terms ************************************)
+
+lemma cnuw_appl_simple (h) (G) (L):
+ ∀V,T. 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] T → ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] ⓐV.T.
+#h #G #L #V1 #T1 #H1T1 #H2T1 #n #X #H
+elim (cpms_inv_appl_sn … H) -H *
+[ #V2 #T2 #_ #HT12 #H destruct -H1T1
+ /3 width=2 by tweq_appl/
+| #n1 #n2 #p #V2 #T2 #HT12 #_ #_ -n -n2
+ lapply (H2T1 … HT12) -H2T1 -n1 #H
+ lapply (tweq_simple_trans … H H1T1) -H -H1T1 #H
+ elim (simple_inv_bind … H)
+| #n1 #n2 #p #V2 #W2 #W #T2 #_ #_ #HT12 #_ #_ -n -n2 -V2 -W2
+ lapply (H2T1 … HT12) -H2T1 -n1 #H
+ lapply (tweq_simple_trans … H H1T1) -H -H1T1 #H
+ elim (simple_inv_bind … H)
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/predeval_6.ma".
+include "basic_2/rt_transition/cnr.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* EVALUATION FOR T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ON TERMS *)
+
+(* Basic_2A1: uses: cpre *)
+definition cpme (h) (n) (G) (L): relation2 term term ≝
+ λT1,T2. ∧∧ ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T2⦄.
+
+interpretation "evaluation for t-bound context-sensitive parallel rt-transition (term)"
+ 'PRedEval h n G L T1 T2 = (cpme h n G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cpme_intro (h) (n) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T2⦄ → ⦃G,L⦄⊢T1➡*[h,n]𝐍⦃T2⦄.
+/2 width=1 by conj/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpme_fwd_cpms (h) (n) (G) (L):
+ ∀T1,T2. ⦃G,L⦄⊢T1➡*[h,n]𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2.
+#h #n #G #L #T1 #T2 * //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/csx_aaa.ma".
+include "basic_2/rt_computation/cpms_aaa.ma".
+include "basic_2/rt_computation/cpre_csx.ma".
+include "basic_2/rt_computation/cpre_cpms.ma".
+
+(* EVALUATION FOR T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ON TERMS *)
+
+(* Properties with atomic atomic arity assignment on terms ******************)
+
+lemma cpme_total_aaa (h) (n) (A) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A → ∃T2. ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍⦃T2⦄.
+#h #n #A #G #L #T1 #HT1
+elim (cpms_total_aaa h … n … HT1) #T0 #HT10
+elim (cpre_total_csx h G L T0)
+[ #T2 /3 width=4 by cpms_cpre_trans, ex_intro/
+| /4 width=4 by cpms_fwd_cpxs, aaa_csx, csx_cpxs_trans/
+]
+qed-.
lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …):
Q 0 T2 →
- (∀n1,n2,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T → ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) →
- ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T1.
+ (∀n1,n2,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → Q n2 T → Q (n1+n2) T1) →
+ ∀n,T1. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → Q n T1.
#h #G #L #T2 #Q @ltc_ind_sn_refl //
qed-.
lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …):
Q 0 T1 →
- (∀n1,n2,T,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → Q n1 T → ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → Q (n1+n2) T2) →
- ∀n,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T2.
+ (∀n1,n2,T,T2. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T → Q n1 T → ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → Q (n1+n2) T2) →
+ ∀n,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → Q n T2.
#h #G #L #T1 #Q @ltc_ind_dx_refl //
qed-.
(* Basic_1: includes: pr1_pr0 *)
(* Basic_1: uses: pr3_pr2 *)
(* Basic_2A1: includes: cpr_cprs *)
-lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2.
+lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2.
/2 width=1 by ltc_rc/ qed.
-lemma cpms_step_sn (h) (G) (L): ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T →
- ∀n2,T2. ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2.
+lemma cpms_step_sn (h) (G) (L): ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2.
/2 width=3 by ltc_sn/ qed-.
-lemma cpms_step_dx (h) (G) (L): ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T →
- ∀n2,T2. ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2.
+lemma cpms_step_dx (h) (G) (L): ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2.
/2 width=3 by ltc_dx/ qed-.
(* Basic_2A1: uses: cprs_bind_dx *)
lemma cpms_bind_dx (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2.
#n #h #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind/ qed.
lemma cpms_appl_dx (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] ⓐV2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2.
#n #h #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/
qed.
lemma cpms_zeta (n) (h) (G) (L):
∀T1,T. ⬆*[1] T ≘ T1 →
- ∀V,T2. ⦃G, L⦄ ⊢ T ➡*[n, h] T2 → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
+ ∀V,T2. ⦃G,L⦄ ⊢ T ➡*[n,h] T2 → ⦃G,L⦄ ⊢ +ⓓV.T1 ➡*[n,h] T2.
#n #h #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2
/3 width=3 by cpms_step_dx, cpm_cpms, cpm_zeta/
qed.
(* Basic_2A1: uses: cprs_zeta *)
lemma cpms_zeta_dx (n) (h) (G) (L):
∀T2,T. ⬆*[1] T2 ≘ T →
- ∀V,T1. ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[n, h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
+ ∀V,T1. ⦃G,L.ⓓV⦄ ⊢ T1 ➡*[n,h] T → ⦃G,L⦄ ⊢ +ⓓV.T1 ➡*[n,h] T2.
#n #h #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind, cpm_zeta/
qed.
(* Basic_2A1: uses: cprs_eps *)
lemma cpms_eps (n) (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[n, h] T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀V. ⦃G,L⦄ ⊢ ⓝV.T1 ➡*[n,h] T2.
#n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_eps/
qed.
lemma cpms_ee (n) (h) (G) (L):
- ∀U1,U2. ⦃G, L⦄ ⊢ U1 ➡*[n, h] U2 →
- ∀T. ⦃G, L⦄ ⊢ ⓝU1.T ➡*[↑n, h] U2.
+ ∀U1,U2. ⦃G,L⦄ ⊢ U1 ➡*[n,h] U2 →
+ ∀T. ⦃G,L⦄ ⊢ ⓝU1.T ➡*[↑n,h] U2.
#n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
[ /3 width=1 by cpm_cpms, cpm_ee/
| #n1 #n2 #U1 #U #HU1 #HU2 #_ #T >plus_S1
(* Basic_2A1: uses: cprs_beta_dx *)
lemma cpms_beta_dx (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
- ∀T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
+ ∀T1,T2. ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n,h] ⓓ{p}ⓝW2.V2.T2.
#n #h #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
/4 width=7 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_beta/
qed.
(* Basic_2A1: uses: cprs_theta_dx *)
lemma cpms_theta_dx (n) (h) (G) (L):
- ∀V1,V. ⦃G, L⦄ ⊢ V1 ➡[h] V →
+ ∀V1,V. ⦃G,L⦄ ⊢ V1 ➡[h] V →
∀V2. ⬆*[1] V ≘ V2 →
- ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
- ∀T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2.
+ ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
+ ∀T1,T2. ⦃G,L.ⓓW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n,h] ⓓ{p}W2.ⓐV2.T2.
#n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
/4 width=9 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_theta/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[n, h] X2 → X2 = ⋆(((next h)^n) s).
+lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G,L⦄ ⊢ ⋆s ➡*[n,h] X2 → X2 = ⋆(((next h)^n) s).
#n #h #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 //
#n1 #n2 #X #X2 #_ #IH #HX2 destruct
elim (cpm_inv_sort1 … HX2) -HX2 #H #_ destruct //
qed-.
+lemma cpms_inv_lref1_ctop (n) (h) (G):
+ ∀X2,i. ⦃G,⋆⦄ ⊢ #i ➡*[n,h] X2 → ∧∧ X2 = #i & n = 0.
+#n #h #G #X2 #i #H @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+ elim (cpm_inv_lref1_ctop … HX2) -HX2 #H1 #H2 destruct
+ /2 width=1 by conj/
+]
+qed-.
+
+lemma cpms_inv_zero1_unit (n) (h) (I) (K) (G):
+ ∀X2. ⦃G,K.ⓤ{I}⦄ ⊢ #0 ➡*[n,h] X2 → ∧∧ X2 = #0 & n = 0.
+#n #h #I #G #K #X2 #H @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+ elim (cpm_inv_zero1_unit … HX2) -HX2 #H1 #H2 destruct
+ /2 width=1 by conj/
+]
+qed-.
+
+lemma cpms_inv_gref1 (n) (h) (G) (L):
+ ∀X2,l. ⦃G,L⦄ ⊢ §l ➡*[n,h] X2 → ∧∧ X2 = §l & n = 0.
+#n #h #G #L #X2 #l #H @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+ elim (cpm_inv_gref1 … HX2) -HX2 #H1 #H2 destruct
+ /2 width=1 by conj/
+]
+qed-.
+
lemma cpms_inv_cast1 (h) (n) (G) (L):
- ∀W1,T1,X2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 →
- ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
- | ⦃G, L⦄ ⊢ T1 ➡*[n,h] X2
- | ∃∃m. ⦃G, L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m.
+ ∀W1,T1,X2. ⦃G,L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 →
+ ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
+ | ⦃G,L⦄ ⊢ T1 ➡*[n,h] X2
+ | ∃∃m. ⦃G,L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m.
#h #n #G #L #W1 #T1 #X2 #H @(cpms_ind_dx … H) -n -X2
[ /3 width=5 by or3_intro0, ex3_2_intro/
| #n1 #n2 #X #X2 #_ * [ * || * ]
include "basic_2/rt_transition/cpm_aaa.ma".
include "basic_2/rt_computation/cpxs_aaa.ma".
include "basic_2/rt_computation/cpms_cpxs.ma".
+include "basic_2/rt_computation/lprs_cpms.ma".
(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
(* Basic_2A1: uses: scpds_aaa_conf *)
(* Basic_2A1: includes: cprs_aaa_conf *)
-lemma cpms_aaa_conf (n) (h): ∀G,L. Conf3 … (aaa G L) (cpms h G L n).
+lemma cpms_aaa_conf (h) (G) (L) (n): Conf3 … (aaa G L) (cpms h G L n).
/3 width=5 by cpms_fwd_cpxs, cpxs_aaa_conf/ qed-.
-lemma aaa_cpms_total (h) (G) (L) (n) (A):
- ∀T. ⦃G, L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
+lemma cpms_total_aaa (h) (G) (L) (n) (A):
+ ∀T. ⦃G,L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡*[n,h] U.
#h #G #L #n elim n -n
[ /2 width=3 by ex_intro/
| #n #IH #A #T1 #HT1 <plus_SO
/3 width=4 by cpms_step_dx, ex_intro/
]
qed-.
+
+lemma cpms_abst_dx_le_aaa (h) (G) (L) (T) (W) (p):
+ ∀A. ⦃G,L⦄ ⊢ T ⁝ A →
+ ∀n1,U1. ⦃G,L⦄ ⊢ T ➡*[n1,h] ⓛ{p}W.U1 → ∀n2. n1 ≤ n2 →
+ ∃∃U2. ⦃G,L⦄ ⊢ T ➡*[n2,h] ⓛ{p}W.U2 & ⦃G,L.ⓛW⦄ ⊢ U1 ➡*[n2-n1,h] U2.
+#h #G #L #T #W #p #A #HA #n1 #U1 #HTU1 #n2 #Hn12
+lapply (cpms_aaa_conf … HA … HTU1) -HA #HA
+elim (cpms_total_aaa h … (n2-n1) … HA) -HA #X #H
+elim (cpms_inv_abst_sn … H) -H #W0 #U2 #_ #HU12 #H destruct -W0
+>(plus_minus_m_m_commutative … Hn12) in ⊢ (??%?); -Hn12
+/4 width=5 by cpms_trans, cpms_bind_dx, ex2_intro/
+qed-.
(* Basic_2A1: includes: cprs_bind *)
theorem cpms_bind (n) (h) (G) (L):
- ∀I,V1,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+ ∀I,V1,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2.
#n #h #G #L #I #V1 #T1 #T2 #HT12 #V2 #H @(cprs_ind_dx … H) -V2
[ /2 width=1 by cpms_bind_dx/
| #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
qed.
theorem cpms_appl (n) (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] ⓐV2.T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2.
#n #h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
[ /2 width=1 by cpms_appl_dx/
| #V #V2 #_ #HV2 #IH >(plus_n_O … n) -HT12
(* Basic_2A1: includes: cprs_beta_rc *)
theorem cpms_beta_rc (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀W1,T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀W1,T1,T2. ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n,h] ⓓ{p}ⓝW2.V2.T2.
#n #h #G #L #V1 #V2 #HV12 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
[ /2 width=1 by cpms_beta_dx/
| #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
(* Basic_2A1: includes: cprs_beta *)
theorem cpms_beta (n) (h) (G) (L):
- ∀W1,T1,T2. ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 →
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n, h] ⓓ{p}ⓝW2.V2.T2.
+ ∀W1,T1,T2. ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 →
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡*[n,h] ⓓ{p}ⓝW2.V2.T2.
#n #h #G #L #W1 #T1 #T2 #HT12 #W2 #HW12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
[ /2 width=1 by cpms_beta_rc/
| #V #V2 #_ #HV2 #IH #p >(plus_n_O … n) -HT12
(* Basic_2A1: includes: cprs_theta_rc *)
theorem cpms_theta_rc (n) (h) (G) (L):
- ∀V1,V. ⦃G, L⦄ ⊢ V1 ➡[h] V → ∀V2. ⬆*[1] V ≘ V2 →
- ∀W1,T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2.
+ ∀V1,V. ⦃G,L⦄ ⊢ V1 ➡[h] V → ∀V2. ⬆*[1] V ≘ V2 →
+ ∀W1,T1,T2. ⦃G,L.ⓓW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n,h] ⓓ{p}W2.ⓐV2.T2.
#n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #T1 #T2 #HT12 #W2 #H @(cprs_ind_dx … H) -W2
[ /2 width=3 by cpms_theta_dx/
| #W #W2 #_ #HW2 #IH #p >(plus_n_O … n) -HT12
(* Basic_2A1: includes: cprs_theta *)
theorem cpms_theta (n) (h) (G) (L):
- ∀V,V2. ⬆*[1] V ≘ V2 → ∀W1,W2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 →
- ∀T1,T2. ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀V1. ⦃G, L⦄ ⊢ V1 ➡*[h] V →
- ∀p. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n, h] ⓓ{p}W2.ⓐV2.T2.
+ ∀V,V2. ⬆*[1] V ≘ V2 → ∀W1,W2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 →
+ ∀T1,T2. ⦃G,L.ⓓW1⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀V1. ⦃G,L⦄ ⊢ V1 ➡*[h] V →
+ ∀p. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡*[n,h] ⓓ{p}W2.ⓐV2.T2.
#n #h #G #L #V #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #HT12 #V1 #H @(cprs_ind_sn … H) -V1
[ /2 width=3 by cpms_theta_rc/
| #V1 #V0 #HV10 #_ #IH #p >(plus_O_n … n) -HT12
(* Basic_2A1: uses: lstas_scpds_trans scpds_strap2 *)
theorem cpms_trans (h) (G) (L):
- ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T →
- ∀n2,T2. ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2.
+ ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T ➡*[n2,h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2.
/2 width=3 by ltc_trans/ qed-.
(* Basic_2A1: uses: scpds_cprs_trans *)
theorem cpms_cprs_trans (n) (h) (G) (L):
- ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2.
+ ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2.
#n #h #G #L #T1 #T #HT1 #T2 #HT2 >(plus_n_O … n)
/2 width=3 by cpms_trans/ qed-.
(* Advanced inversion lemmas ************************************************)
lemma cpms_inv_appl_sn (n) (h) (G) (L):
- ∀V1,T1,X2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[n, h] X2 →
+ ∀V1,T1,X2. ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[n,h] X2 →
∨∨ ∃∃V2,T2.
- ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 &
+ ⦃G,L⦄ ⊢ V1 ➡*[h] V2 & ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 &
X2 = ⓐV2.T2
| ∃∃n1,n2,p,W,T.
- ⦃G, L⦄ ⊢ T1 ➡*[n1, h] ⓛ{p}W.T & ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[n2, h] X2 &
+ ⦃G,L⦄ ⊢ T1 ➡*[n1,h] ⓛ{p}W.T & ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[n2,h] X2 &
n1 + n2 = n
| ∃∃n1,n2,p,V0,V2,V,T.
- ⦃G, L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 &
- ⦃G, L⦄ ⊢ T1 ➡*[n1, h] ⓓ{p}V.T & ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[n2, h] X2 &
+ ⦃G,L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 &
+ ⦃G,L⦄ ⊢ T1 ➡*[n1,h] ⓓ{p}V.T & ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[n2,h] X2 &
n1 + n2 = n.
#n #h #G #L #V1 #T1 #U2 #H
@(cpms_ind_dx … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
]
qed-.
-lemma cpms_inv_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T & ⦃G, L⦄ ⊢ T ➡*[n2, h] T2.
+lemma cpms_inv_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n1+n2,h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T & ⦃G,L⦄ ⊢ T ➡*[n2,h] T2.
#h #G #L #n1 elim n1 -n1 /2 width=3 by ex2_intro/
#n1 #IH #n2 #T1 #T2 <plus_S1 #H
elim (cpms_inv_succ_sn … H) -H #T0 #HT10 #HT02
(* Advanced main properties *************************************************)
theorem cpms_cast (n) (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀U1,U2. ⦃G, L⦄ ⊢ U1 ➡*[n, h] U2 →
- ⦃G, L⦄ ⊢ ⓝU1.T1 ➡*[n, h] ⓝU2.T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀U1,U2. ⦃G,L⦄ ⊢ U1 ➡*[n,h] U2 →
+ ⦃G,L⦄ ⊢ ⓝU1.T1 ➡*[n,h] ⓝU2.T2.
#n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1 -n
[ /3 width=3 by cpms_cast_sn/
| #n1 #n2 #T1 #T #HT1 #_ #IH #U1 #U2 #H
(* Forward lemmas with unbound context-sensitive rt-computation for terms ***)
(* Basic_2A1: includes: scpds_fwd_cpxs cprs_cpxs *)
-lemma cpms_fwd_cpxs (n) (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2.
+lemma cpms_fwd_cpxs (n) (h): ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ T1 ⬈*[h] T2.
#n #h #G #L #T1 #T2 #H @(cpms_ind_dx … H) -T2
/3 width=4 by cpxs_strap1, cpm_fwd_cpx/
qed-.
(* Advanced properties ******************************************************)
-lemma cpms_delta (n) (h) (G): ∀K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡*[n, h] V2 →
- ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓓV1⦄ ⊢ #0 ➡*[n, h] W2.
+lemma cpms_delta (n) (h) (G): ∀K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡*[n,h] V2 →
+ ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓓV1⦄ ⊢ #0 ➡*[n,h] W2.
#n #h #G #K #V1 #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=3 by cpm_cpms, cpm_delta/
| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
]
qed.
-lemma cpms_ell (n) (h) (G): ∀K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡*[n, h] V2 →
- ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓛV1⦄ ⊢ #0 ➡*[↑n, h] W2.
+lemma cpms_ell (n) (h) (G): ∀K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡*[n,h] V2 →
+ ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓛV1⦄ ⊢ #0 ➡*[↑n,h] W2.
#n #h #G #K #V1 #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=3 by cpm_cpms, cpm_ell/
| #n1 #n2 #V #V2 #_ #IH #HV2 #W2 #HVW2
]
qed.
-lemma cpms_lref (n) (h) (I) (G): ∀K,T,i. ⦃G, K⦄ ⊢ #i ➡*[n, h] T →
- ∀U. ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ➡*[n, h] U.
+lemma cpms_lref (n) (h) (I) (G): ∀K,T,i. ⦃G,K⦄ ⊢ #i ➡*[n,h] T →
+ ∀U. ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ➡*[n,h] U.
#n #h #I #G #K #T #i #H @(cpms_ind_dx … H) -T
[ /3 width=3 by cpm_cpms, cpm_lref/
| #n1 #n2 #T #T2 #_ #IH #HT2 #U2 #HTU2
qed.
lemma cpms_cast_sn (n) (h) (G) (L):
- ∀U1,U2. ⦃G, L⦄ ⊢ U1 ➡*[n, h] U2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓝU1.T1 ➡*[n, h] ⓝU2.T2.
+ ∀U1,U2. ⦃G,L⦄ ⊢ U1 ➡*[n,h] U2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓝU1.T1 ➡*[n,h] ⓝU2.T2.
#n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
[ /3 width=3 by cpm_cpms, cpm_cast/
| #n1 #n2 #U1 #U #HU1 #_ #IH #T1 #T2 #H
(* Basic_2A1: uses: cprs_delta *)
lemma cpms_delta_drops (n) (h) (G):
∀L,K,V,i. ⬇*[i] L ≘ K.ⓓV →
- ∀V2. ⦃G, K⦄ ⊢ V ➡*[n, h] V2 →
- ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ➡*[n, h] W2.
+ ∀V2. ⦃G,K⦄ ⊢ V ➡*[n,h] V2 →
+ ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ➡*[n,h] W2.
#n #h #G #L #K #V #i #HLK #V2 #H @(cpms_ind_dx … H) -V2
[ /3 width=6 by cpm_cpms, cpm_delta_drops/
| #n1 #n2 #V1 #V2 #_ #IH #HV12 #W2 #HVW2
lemma cpms_ell_drops (n) (h) (G):
∀L,K,W,i. ⬇*[i] L ≘ K.ⓛW →
- ∀W2. ⦃G, K⦄ ⊢ W ➡*[n, h] W2 →
- ∀V2. ⬆*[↑i] W2 ≘ V2 → ⦃G, L⦄ ⊢ #i ➡*[↑n, h] V2.
+ ∀W2. ⦃G,K⦄ ⊢ W ➡*[n,h] W2 →
+ ∀V2. ⬆*[↑i] W2 ≘ V2 → ⦃G,L⦄ ⊢ #i ➡*[↑n,h] V2.
#n #h #G #L #K #W #i #HLK #W2 #H @(cpms_ind_dx … H) -W2
[ /3 width=6 by cpm_cpms, cpm_ell_drops/
| #n1 #n2 #W1 #W2 #_ #IH #HW12 #V2 #HWV2
(* Advanced inversion lemmas ************************************************)
lemma cpms_inv_lref1_drops (n) (h) (G):
- ∀L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[n, h] T2 →
+ ∀L,T2,i. ⦃G,L⦄ ⊢ #i ➡*[n,h] T2 →
∨∨ ∧∧ T2 = #i & n = 0
- | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡*[n, h] V2 &
+ | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡*[n,h] V2 &
⬆*[↑i] V2 ≘ T2
- | ∃∃m,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ➡*[m, h] V2 &
+ | ∃∃m,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ➡*[m,h] V2 &
⬆*[↑i] V2 ≘ T2 & n = ↑m.
#n #h #G #L #T2 #i #H @(cpms_ind_dx … H) -T2
[ /3 width=1 by or3_intro0, conj/
qed-.
fact cpms_inv_succ_sn (n) (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[↑n, h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡*[1, h] T & ⦃G, L⦄ ⊢ T ➡*[n, h] T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[↑n,h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[1,h] T & ⦃G,L⦄ ⊢ T ➡*[n,h] T2.
#n #h #G #L #T1 #T2
@(insert_eq_0 … (↑n)) #m #H
@(cpms_ind_sn … H) -T1 -m
(**************************************************************************)
include "basic_2/rt_computation/fpbg_fqup.ma".
-include "basic_2/rt_computation/fpbg_fpbs.ma".
+include "basic_2/rt_computation/fpbg_cpxs.ma".
include "basic_2/rt_computation/cpms_fpbs.ma".
(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
(* Forward lemmas with proper parallel rst-computation for closures *********)
-lemma fpbg_cpms_trans (h) (o) (n): ∀G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ >[h,o] ⦃G2, L2, T⦄ →
- ∀T2. ⦃G2, L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1, L1, T1⦄ >[h,o] ⦃G2, L2, T2⦄.
+lemma cpms_tdneq_fwd_fpbg (h) (n):
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 →
+ (T1 ≛ T2 → ⊥) → ⦃G,L,T1⦄ >[h] ⦃G,L,T2⦄.
+/3 width=2 by cpms_fwd_cpxs, cpxs_tdneq_fpbg/ qed-.
+
+lemma fpbg_cpms_trans (h) (n):
+ ∀G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T⦄ →
+ ∀T2. ⦃G2,L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbg_fpbs_trans, cpms_fwd_fpbs/ qed-.
-lemma cpms_fpbg_trans (h) (o) (n): ∀G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ➡*[n,h] T →
- ∀G2,L2,T2. ⦃G1, L1, T⦄ >[h,o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h,o] ⦃G2, L2, T2⦄.
+lemma cpms_fpbg_trans (h) (n):
+ ∀G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ➡*[n,h] T →
+ ∀G2,L2,T2. ⦃G1,L1,T⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fpbg_trans, cpms_fwd_fpbs/ qed-.
-lemma fqup_cpms_fwd_fpbg (h) (o): ∀G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T⦄ →
- ∀n,T2. ⦃G2, L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1, L1, T1⦄ >[h,o] ⦃G2, L2, T2⦄.
-/3 width=5 by cpms_fwd_fpbs, fqup_fpbg,fpbg_fpbs_trans/ qed-.
+lemma fqup_cpms_fwd_fpbg (h):
+ ∀G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ⬂+ ⦃G2,L2,T⦄ →
+ ∀n,T2. ⦃G2,L2⦄ ⊢ T ➡*[n,h] T2 → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
+/3 width=5 by cpms_fwd_fpbs, fqup_fpbg, fpbg_fpbs_trans/ qed-.
-lemma cpm_tdneq_cpm_cpms_tdeq_sym_fwd_fpbg (h) (o) (G) (L) (T1):
- ∀n1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛[h,o] T → ⊥) →
- ∀n2,T2. ⦃G,L⦄⊢ T ➡*[n2,h] T2 → T1 ≛[h,o] T2 → ⦃G,L,T1⦄ >[h,o] ⦃G,L,T1⦄.
-#h #o #G #L #T1 #n1 #T #H1T1 #H2T1 #n2 #T2 #H1T2 #H2T12
+lemma cpm_tdneq_cpm_cpms_tdeq_sym_fwd_fpbg (h) (G) (L) (T1):
+ ∀n1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) →
+ ∀n2,T2. ⦃G,L⦄⊢ T ➡*[n2,h] T2 → T1 ≛ T2 → ⦃G,L,T1⦄ >[h] ⦃G,L,T1⦄.
+#h #G #L #T1 #n1 #T #H1T1 #H2T1 #n2 #T2 #H1T2 #H2T12
/4 width=7 by cpms_fwd_fpbs, cpm_fpb, fpbs_tdeq_trans, tdeq_sym, ex2_3_intro/
qed-.
(* Forward lemmas with parallel rst-computation for closures ****************)
(* Basic_2A1: uses: cprs_fpbs *)
-lemma cpms_fwd_fpbs (n) (h) (o): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G, L, T1⦄ ≥[h,o] ⦃G, L, T2⦄.
+lemma cpms_fwd_fpbs (n) (h):
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L,T1⦄ ≥[h] ⦃G,L,T2⦄.
/3 width=2 by cpms_fwd_cpxs, cpxs_fpbs/ qed-.
(* Properties with parallel rt-transition for full local environments *******)
lemma lpr_cpm_trans (n) (h) (G):
- ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[n, h] T2 →
- ∀L1. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+ ∀L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀L1. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2.
#n #h #G #L2 #T1 #T2 #H @(cpm_ind … H) -n -G -L2 -T1 -T2
[ /2 width=3 by/
| /3 width=2 by cpm_cpms/
qed-.
lemma lpr_cpms_trans (n) (h) (G):
- ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 →
- ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[n, h] T2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+ ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 →
+ ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2.
#n #h #G #L1 #L2 #HL12 #T1 #T2 #H @(cpms_ind_sn … H) -n -T1
/3 width=3 by lpr_cpm_trans, cpms_trans/
qed-.
(* Basic_2A1: includes cpr_bind2 *)
lemma cpm_bind2 (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2.
/4 width=5 by lpr_cpm_trans, cpms_bind_dx, lpr_pair/ qed.
(* Basic_2A1: includes cprs_bind2_dx *)
lemma cpms_bind2_dx (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2.
/4 width=5 by lpr_cpms_trans, cpms_bind_dx, lpr_pair/ qed.
(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
-(* Properties with degree-based equivalence for local environments **********)
+(* Properties with sort-irrelevant equivalence for local environments *******)
-lemma cpms_rdeq_conf_sn (h) (n) (o) (G) (L1) (L2):
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[n,h] T2 →
- L1 ≛[h,o,T1] L2 → L1 ≛[h,o,T2] L2.
-/3 width=4 by cpms_fwd_cpxs, cpxs_rdeq_conf_sn/ qed-.
+lemma cpms_rdeq_conf_sn (h) (n) (G) (L1) (L2):
+ ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2 →
+ L1 ≛[T1] L2 → L1 ≛[T2] L2.
+/3 width=5 by cpms_fwd_cpxs, cpxs_rdeq_conf_sn/ qed-.
-lemma cpms_rdeq_conf_dx (h) (n) (o) (G) (L1) (L2):
- ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[n,h] T2 →
- L1 ≛[h,o,T1] L2 → L1 ≛[h,o,T2] L2.
-/3 width=4 by cpms_fwd_cpxs, cpxs_rdeq_conf_dx/ qed-.
+lemma cpms_rdeq_conf_dx (h) (n) (G) (L1) (L2):
+ ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ➡*[n,h] T2 →
+ L1 ≛[T1] L2 → L1 ≛[T2] L2.
+/3 width=5 by cpms_fwd_cpxs, cpxs_rdeq_conf_dx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/predevalwstar_6.ma".
+include "basic_2/rt_computation/cnuw.ma".
+
+(* T-UNBOUND WHD EVALUATION FOR T-BOUND RT-TRANSITION ON TERMS **************)
+
+definition cpmuwe (h) (n) (G) (L): relation2 term term ≝
+ λT1,T2. ∧∧ ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 & ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] T2.
+
+interpretation "t-unbound whd evaluation for t-bound context-sensitive parallel rt-transition (term)"
+ 'PRedEvalWStar h n G L T1 T2 = (cpmuwe h n G L T1 T2).
+
+definition R_cpmuwe (h) (G) (L) (T): predicate nat ≝
+ λn. ∃U. ⦃G,L⦄ ⊢ T ➡*𝐍𝐖*[h,n] U.
+
+(* Basic properties *********************************************************)
+
+lemma cpmuwe_intro (h) (n) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2 → ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] T2 → ⦃G,L⦄ ⊢ T1 ➡*𝐍𝐖*[h,n] T2.
+/2 width=1 by conj/ qed.
+
+(* Advanced properties ******************************************************)
+
+lemma cpmuwe_sort (h) (n) (G) (L) (T):
+ ∀s. ⦃G,L⦄ ⊢ T ➡*[n,h] ⋆s → ⦃G,L⦄ ⊢ T ➡*𝐍𝐖*[h,n] ⋆s.
+/3 width=5 by cnuw_sort, cpmuwe_intro/ qed.
+
+lemma cpmuwe_ctop (h) (n) (G) (T):
+ ∀i. ⦃G,⋆⦄ ⊢ T ➡*[n,h] #i → ⦃G,⋆⦄ ⊢ T ➡*𝐍𝐖*[h,n] #i.
+/3 width=5 by cnuw_ctop, cpmuwe_intro/ qed.
+
+lemma cpmuwe_zero_unit (h) (n) (G) (L) (T):
+ ∀I. ⦃G,L.ⓤ{I}⦄ ⊢ T ➡*[n,h] #0 → ⦃G,L.ⓤ{I}⦄ ⊢ T ➡*𝐍𝐖*[h,n] #0.
+/3 width=6 by cnuw_zero_unit, cpmuwe_intro/ qed.
+
+lemma cpmuwe_gref (h) (n) (G) (L) (T):
+ ∀l. ⦃G,L⦄ ⊢ T ➡*[n,h] §l → ⦃G,L⦄ ⊢ T ➡*𝐍𝐖*[h,n] §l.
+/3 width=5 by cnuw_gref, cpmuwe_intro/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpmuwe_fwd_cpms (h) (n) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*𝐍𝐖*[h,n] T2 → ⦃G,L⦄ ⊢ T1 ➡*[n,h] T2.
+#h #n #G #L #T1 #T2 * #HT12 #_ //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cnuw_cnuw.ma".
+include "basic_2/rt_computation/cpmuwe.ma".
+
+(* T-UNBOUND WHD EVALUATION FOR T-BOUND RT-TRANSITION ON TERMS **************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpmuwe_abbr_neg (h) (n) (G) (L) (T):
+ ∀V,U. ⦃G,L⦄ ⊢ T ➡*[n,h] -ⓓV.U → ⦃G,L⦄ ⊢ T ➡*𝐍𝐖*[h,n] -ⓓV.U.
+/3 width=5 by cnuw_abbr_neg, cpmuwe_intro/ qed.
+
+lemma cpmuwe_abst (h) (n) (p) (G) (L) (T):
+ ∀W,U. ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W.U → ⦃G,L⦄ ⊢ T ➡*𝐍𝐖*[h,n] ⓛ{p}W.U.
+/3 width=5 by cnuw_abst, cpmuwe_intro/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tweq_tdeq.ma".
+include "basic_2/rt_computation/csx_cpxs.ma".
+include "basic_2/rt_computation/cpms_cpxs.ma".
+include "basic_2/rt_computation/cnuw_cnuw.ma".
+include "basic_2/rt_computation/cpmuwe.ma".
+
+(* T-UNBOUND WHD EVALUATION FOR T-BOUND RT-TRANSITION ON TERMS **************)
+
+(* Properties with strong normalization for unbound rt-transition for terms *)
+
+lemma cpmuwe_total_csx (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃∃T2,n. ⦃G,L⦄ ⊢ T1 ➡*𝐍𝐖*[h,n] T2.
+#h #G #L #T1 #H
+@(csx_ind_cpxs … H) -T1 #T1 #_ #IHT1
+elim (cnuw_dec_ex h G L T1)
+[ -IHT1 #HT1 /3 width=4 by cpmuwe_intro, ex1_2_intro/
+| * #n1 #T0 #HT10 #HnT10
+ elim (IHT1 … T0) -IHT1
+ [ #T2 #n2 * #HT02 #HT2 /4 width=5 by cpms_trans, cpmuwe_intro, ex1_2_intro/
+ | /3 width=1 by tdeq_tweq/
+ | /2 width=2 by cpms_fwd_cpxs/
+ ]
+]
+qed-.
+
+lemma R_cpmuwe_total_csx (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃n. R_cpmuwe h G L T1 n.
+#h #G #L #T1 #H
+elim (cpmuwe_total_csx … H) -H #T2 #n #HT12
+/3 width=3 by ex_intro (* 2x *)/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/predeval_5.ma".
+include "basic_2/rt_computation/cpme.ma".
+include "basic_2/rt_computation/cprs.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL R-TRANSITION ON TERMS ***********)
+
+interpretation "evaluation for context-sensitive parallel r-transition (term)"
+ 'PRedEval h G L T1 T2 = (cpme h O G L T1 T2).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpms_cpms.ma".
+include "basic_2/rt_computation/cpre.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL R-TRANSITION ON TERMS **********)
+
+(* Properties with t-bound rt-computarion on terms **************************)
+
+lemma cpms_cpre_trans (h) (n) (G) (L):
+ ∀T1,T0. ⦃G,L⦄ ⊢T1 ➡*[n,h] T0 →
+ ∀T2. ⦃G,L⦄ ⊢ T0 ➡*[h] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T1 ➡*[h,n] 𝐍⦃T2⦄.
+#h #n #G #L #T1 #T0 #HT10 #T2 * #HT02 #HT2
+/3 width=3 by cpms_cprs_trans, cpme_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cprs_cnr.ma".
+include "basic_2/rt_computation/cprs_cprs.ma".
+include "basic_2/rt_computation/cpre.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL R-TRANSITION ON TERMS *********)
+
+(* Properties with context-sensitive parallel r-computation for terms ******)
+
+lemma cpre_cprs_conf (h) (G) (L) (T):
+ ∀T1. ⦃G,L⦄ ⊢ T ➡*[h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] 𝐍⦃T2⦄ → ⦃G,L⦄ ⊢ T1 ➡*[h] 𝐍⦃T2⦄.
+#h #G #L #T0 #T1 #HT01 #T2 * #HT02 #HT2
+elim (cprs_conf … HT01 … HT02) -T0 #T0 #HT10 #HT20
+lapply (cprs_inv_cnr_sn … HT20 HT2) -HT20 #H destruct
+/2 width=1 by cpme_intro/
+qed-.
+
+(* Main properties *********************************************************)
+
+(* Basic_1: was: nf2_pr3_confluence *)
+theorem cpre_mono (h) (G) (L) (T):
+ ∀T1. ⦃G,L⦄ ⊢ T ➡*[h] 𝐍⦃T1⦄ → ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] 𝐍⦃T2⦄ → T1 = T2.
+#h #G #L #T0 #T1 * #HT01 #HT1 #T2 * #HT02 #HT2
+elim (cprs_conf … HT01 … HT02) -T0 #T0 #HT10 #HT20
+>(cprs_inv_cnr_sn … HT10 HT1) -T1
+>(cprs_inv_cnr_sn … HT20 HT2) -T2 //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cpm_cpx.ma".
+include "basic_2/rt_transition/cnr_tdeq.ma".
+include "basic_2/rt_computation/csx.ma".
+include "basic_2/rt_computation/cpre.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL R-TRANSITION ON TERMS **********)
+
+(* Properties with strong normalization for unbound rt-transition for terms *)
+
+(* Basic_1: was just: nf2_sn3 *)
+lemma cpre_total_csx (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∃T2. ⦃G,L⦄ ⊢ T1 ➡*[h] 𝐍⦃T2⦄.
+#h #G #L #T1 #H
+@(csx_ind … H) -T1 #T1 #_ #IHT1
+elim (cnr_dec_tdeq h G L T1) [ /3 width=3 by ex_intro, cpme_intro/ ] *
+#T0 #HT10 #HnT10
+elim (IHT1 … HnT10) -IHT1 -HnT10 [| /2 width=2 by cpm_fwd_cpx/ ]
+#T2 * /4 width=3 by cprs_step_sn, ex_intro, cpme_intro/
+qed-.
(* Basic_2A1: was: cprs_ind_dx *)
lemma cprs_ind_sn (h) (G) (L) (T2) (Q:predicate …):
Q T2 →
- (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ⦃G, L⦄ ⊢ T ➡*[h] T2 → Q T → Q T1) →
- ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → Q T1.
+ (∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T → ⦃G,L⦄ ⊢ T ➡*[h] T2 → Q T → Q T1) →
+ ∀T1. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → Q T1.
#h #G #L #T2 #Q #IH1 #IH2 #T1
@(insert_eq_0 … 0) #n #H
@(cpms_ind_sn … H) -n -T1 //
(* Basic_2A1: was: cprs_ind *)
lemma cprs_ind_dx (h) (G) (L) (T1) (Q:predicate …):
Q T1 →
- (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T → ⦃G, L⦄ ⊢ T ➡[h] T2 → Q T → Q T2) →
- ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → Q T2.
+ (∀T,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T → ⦃G,L⦄ ⊢ T ➡[h] T2 → Q T → Q T2) →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → Q T2.
#h #G #L #T1 #Q #IH1 #IH2 #T2
@(insert_eq_0 … 0) #n #H
@(cpms_ind_dx … H) -n -T2 //
(* Basic_1: was: pr3_step *)
(* Basic_2A1: was: cprs_strap2 *)
lemma cprs_step_sn (h) (G) (L):
- ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2.
+ ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[h] T2.
/2 width=3 by cpms_step_sn/ qed-.
(* Basic_2A1: was: cprs_strap1 *)
lemma cprs_step_dx (h) (G) (L):
- ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h] T2.
+ ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ➡*[h] T2.
/2 width=3 by cpms_step_dx/ qed-.
(* Basic_1: was only: pr3_thin_dx *)
lemma cprs_flat_dx (h) (I) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
+ ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2.
#h #I #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cprs_ind_sn … H) -T1
/3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/
qed.
lemma cprs_flat_sn (h) (I) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h] ⓕ{I} V2. T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ⦃G,L⦄ ⊢ ⓕ{I} V1. T1 ➡*[h] ⓕ{I} V2. T2.
#h #I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_sn … H) -V1
/3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/
qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_1: was: pr3_gen_sort *)
-lemma cprs_inv_sort1 (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h] X2 → X2 = ⋆s.
+lemma cprs_inv_sort1 (h) (G) (L): ∀X2,s. ⦃G,L⦄ ⊢ ⋆s ➡*[h] X2 → X2 = ⋆s.
/2 width=4 by cpms_inv_sort1/ qed-.
(* Basic_1: was: pr3_gen_cast *)
-lemma cprs_inv_cast1 (h) (G) (L): ∀W1,T1,X2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h] X2 →
- ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h] T2 & X2 = ⓝW2.T2
- | ⦃G, L⦄ ⊢ T1 ➡*[h] X2.
+lemma cprs_inv_cast1 (h) (G) (L): ∀W1,T1,X2. ⦃G,L⦄ ⊢ ⓝW1.T1 ➡*[h] X2 →
+ ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ➡*[h] W2 & ⦃G,L⦄ ⊢ T1 ➡*[h] T2 & X2 = ⓝW2.T2
+ | ⦃G,L⦄ ⊢ T1 ➡*[h] X2.
#h #G #L #W1 #T1 #X2 #H
elim (cpms_inv_cast1 … H) -H
[ /2 width=1 by or_introl/
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cnr.ma".
+include "basic_2/rt_computation/cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL R-COMPUTATION FOR TERMS ***********************)
+
+(* Inversion lemmas with normal terms for r-transition **********************)
+
+(* Basic_1: was: nf2_pr3_unfold *)
+(* Basic_2A1: was: cprs_inv_cnr1 *)
+lemma cprs_inv_cnr_sn (h) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T1⦄ → T1 = T2.
+#h #G #L #T1 #T2 #H @(cprs_ind_sn … H) -T1 //
+#T1 #T0 #HT10 #_ #IH #HT1
+lapply (HT1 … HT10) -HT10 #H destruct /2 width=1 by/
+qed-.
(* Basic_1: was: pr3_flat *)
theorem cprs_flat (h) (G) (L):
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 →
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2.
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 →
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h] ⓕ{I}V2.T2.
#h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_dx … H) -V2
[ /2 width=3 by cprs_flat_dx/
| /3 width=3 by cpr_pair_sn, cprs_step_dx/
(* Basic_1: was pr3_gen_appl *)
(* Basic_2A1: was: cprs_inv_appl1 *)
lemma cprs_inv_appl_sn (h) (G) (L):
- ∀V1,T1,X2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[h] X2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 &
- ⦃G, L⦄ ⊢ T1 ➡*[h] T2 &
+ ∀V1,T1,X2. ⦃G,L⦄ ⊢ ⓐV1.T1 ➡*[h] X2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 &
+ ⦃G,L⦄ ⊢ T1 ➡*[h] T2 &
X2 = ⓐV2. T2
- | ∃∃p,W,T. ⦃G, L⦄ ⊢ T1 ➡*[h] ⓛ{p}W.T &
- ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[h] X2
- | ∃∃p,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 &
- ⦃G, L⦄ ⊢ T1 ➡*[h] ⓓ{p}V.T &
- ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[h] X2.
+ | ∃∃p,W,T. ⦃G,L⦄ ⊢ T1 ➡*[h] ⓛ{p}W.T &
+ ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V1.T ➡*[h] X2
+ | ∃∃p,V0,V2,V,T. ⦃G,L⦄ ⊢ V1 ➡*[h] V0 & ⬆*[1] V0 ≘ V2 &
+ ⦃G,L⦄ ⊢ T1 ➡*[h] ⓓ{p}V.T &
+ ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ➡*[h] X2.
#h #G #L #V1 #T1 #X2 #H elim (cpms_inv_appl_sn … H) -H *
[ /3 width=5 by or3_intro0, ex3_2_intro/
| #n1 #n2 #p #V2 #T2 #HT12 #HTX2 #H
(* Basic_1: was: pr3_gen_lref *)
(* Basic_2A1: was: cprs_inv_lref1 *)
-lemma cprs_inv_lref1_drops (h) (G): ∀L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[h] T2 →
+lemma cprs_inv_lref1_drops (h) (G): ∀L,T2,i. ⦃G,L⦄ ⊢ #i ➡*[h] T2 →
∨∨ T2 = #i
- | ∃∃K,V1,T1. ⬇*[i] L ≘ K.ⓓV1 & ⦃G, K⦄ ⊢ V1 ➡*[h] T1 &
+ | ∃∃K,V1,T1. ⬇*[i] L ≘ K.ⓓV1 & ⦃G,K⦄ ⊢ V1 ➡*[h] T1 &
⬆*[↑i] T1 ≘ T2.
#h #G #L #T2 #i #H elim (cpms_inv_lref1_drops … H) -H *
[ /2 width=1 by or_introl/
qed-.
lemma cprs_lpr_conf_dx (h) (G):
- ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+ ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 → ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
#h #G #L0 #T0 #T1 #H
@(cprs_ind_dx … H) -T1 /2 width=3 by ex2_intro/
#T #T1 #_ #HT1 #IHT0 #L1 #HL01
qed-.
lemma cprs_lpr_conf_sn (h) (G):
- ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+ ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 →
+ ∃∃T. ⦃G,L0⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01 #T #HT1 #HT0
/3 width=3 by lpr_cpms_trans, ex2_intro/
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tweq_tweq.ma".
+include "static_2/relocation/lifts_tweq.ma".
+include "basic_2/rt_transition/cpr_drops_basic.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL R-COMPUTATION FOR TERMS ***********************)
+
+(* Properties with sort-irrelevant whd equivalence on terms *****************)
+
+lemma cprs_abbr_pos_twneq (h) (G) (L) (V) (T1):
+ ∃∃T2. ⦃G,L⦄ ⊢ +ⓓV.T1 ➡*[h] T2 & (+ⓓV.T1 ≅ T2 → ⊥).
+#h #G #L #V #U1
+elim (cpr_subst h G (L.ⓓV) U1 … 0) [|*: /2 width=4 by drops_refl/ ] #U2 #T2 #HU12 #HTU2
+elim (tweq_dec U1 U2) [ #HpU12 | -HTU2 #HnU12 ]
+[ @(ex2_intro … T2) (**) (* full auto not tried *)
+ [ /3 width=6 by cpms_zeta, cpms_step_sn, cpm_bind/
+ | /4 width=6 by tweq_inv_abbr_pos_x_lifts_y_y, tweq_canc_sn, tweq_abbr_pos/
+ ]
+| /4 width=3 by cpm_cpms, cpm_bind, tweq_inv_abbr_pos_bi, ex2_intro/
+]
+qed-.
(* Basic eliminators ********************************************************)
lemma cpxs_ind: ∀h,G,L,T1. ∀Q:predicate term. Q T1 →
- (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T → ⦃G, L⦄ ⊢ T ⬈[h] T2 → Q T → Q T2) →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → Q T2.
+ (∀T,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T → ⦃G,L⦄ ⊢ T ⬈[h] T2 → Q T → Q T2) →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → Q T2.
#h #L #G #T1 #Q #HT1 #IHT1 #T2 #HT12
@(TC_star_ind … HT1 IHT1 … HT12) //
qed-.
lemma cpxs_ind_dx: ∀h,G,L,T2. ∀Q:predicate term. Q T2 →
- (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬈[h] T → ⦃G, L⦄ ⊢ T ⬈*[h] T2 → Q T → Q T1) →
- ∀T1. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → Q T1.
+ (∀T1,T. ⦃G,L⦄ ⊢ T1 ⬈[h] T → ⦃G,L⦄ ⊢ T ⬈*[h] T2 → Q T → Q T1) →
+ ∀T1. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → Q T1.
#h #G #L #T2 #Q #HT2 #IHT2 #T1 #HT12
@(TC_star_ind_dx … HT2 IHT2 … HT12) //
qed-.
(* Basic properties *********************************************************)
-lemma cpxs_refl: ∀h,G,L,T. ⦃G, L⦄ ⊢ T ⬈*[h] T.
+lemma cpxs_refl: ∀h,G,L,T. ⦃G,L⦄ ⊢ T ⬈*[h] T.
/2 width=1 by inj/ qed.
-lemma cpx_cpxs: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2.
+lemma cpx_cpxs: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → ⦃G,L⦄ ⊢ T1 ⬈*[h] T2.
/2 width=1 by inj/ qed.
-lemma cpxs_strap1: ∀h,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ⬈[h] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2.
+lemma cpxs_strap1: ∀h,G,L,T1,T. ⦃G,L⦄ ⊢ T1 ⬈*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬈[h] T2 → ⦃G,L⦄ ⊢ T1 ⬈*[h] T2.
normalize /2 width=3 by step/ qed-.
-lemma cpxs_strap2: ∀h,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ⬈*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬈*[h] T2.
+lemma cpxs_strap2: ∀h,G,L,T1,T. ⦃G,L⦄ ⊢ T1 ⬈[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬈*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬈*[h] T2.
normalize /2 width=3 by TC_strap/ qed-.
(* Basic_2A1: was just: cpxs_sort *)
-lemma cpxs_sort: ∀h,G,L,s,n. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] ⋆((next h)^n s).
+lemma cpxs_sort: ∀h,G,L,s,n. ⦃G,L⦄ ⊢ ⋆s ⬈*[h] ⋆((next h)^n s).
#h #G #L #s #n elim n -n /2 width=1 by cpx_cpxs/
#n >iter_S /2 width=3 by cpxs_strap1/
qed.
-lemma cpxs_bind_dx: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+lemma cpxs_bind_dx: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I,T1,T2. ⦃G,L. ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
#h #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_pair_sn, cpx_bind/
qed.
-lemma cpxs_flat_dx: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
+lemma cpxs_flat_dx: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
#h #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2
/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
qed.
-lemma cpxs_flat_sn: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
+lemma cpxs_flat_sn: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
#h #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2
/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
qed.
-lemma cpxs_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈*[h] ②{I}V2.T.
+lemma cpxs_pair_sn: ∀h,I,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀T. ⦃G,L⦄ ⊢ ②{I}V1.T ⬈*[h] ②{I}V2.T.
#h #I #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
/3 width=3 by cpxs_strap1, cpx_pair_sn/
qed.
lemma cpxs_zeta (h) (G) (L) (V):
∀T1,T. ⬆*[1] T ≘ T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ⬈*[h] T2 → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2.
+ ∀T2. ⦃G,L⦄ ⊢ T ⬈*[h] T2 → ⦃G,L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2.
#h #G #L #V #T1 #T #HT1 #T2 #H @(cpxs_ind … H) -T2
/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_zeta/
qed.
(* Basic_2A1: was: cpxs_zeta *)
lemma cpxs_zeta_dx (h) (G) (L) (V):
∀T2,T. ⬆*[1] T2 ≘ T →
- ∀T1. ⦃G, L.ⓓV⦄ ⊢ T1 ⬈*[h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2.
+ ∀T1. ⦃G,L.ⓓV⦄ ⊢ T1 ⬈*[h] T → ⦃G,L⦄ ⊢ +ⓓV.T1 ⬈*[h] T2.
#h #G #L #V #T2 #T #HT2 #T1 #H @(cpxs_ind_dx … H) -T1
/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_bind, cpx_zeta/
qed.
-lemma cpxs_eps: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ⬈*[h] T2.
+lemma cpxs_eps: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀V. ⦃G,L⦄ ⊢ ⓝV.T1 ⬈*[h] T2.
#h #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_eps/
qed.
(* Basic_2A1: was: cpxs_ct *)
-lemma cpxs_ee: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ⬈*[h] V2.
+lemma cpxs_ee: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀T. ⦃G,L⦄ ⊢ ⓝV1.T ⬈*[h] V2.
#h #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ee/
qed.
lemma cpxs_beta_dx: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2
/4 width=7 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/
qed.
lemma cpxs_theta_dx: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 →
- ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 →
+ ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2
/4 width=9 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/
qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: wa just: cpxs_inv_sort1 *)
-lemma cpxs_inv_sort1: ∀h,G,L,X2,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] X2 →
+lemma cpxs_inv_sort1: ∀h,G,L,X2,s. ⦃G,L⦄ ⊢ ⋆s ⬈*[h] X2 →
∃n. X2 = ⋆((next h)^n s).
#h #G #L #X2 #s #H @(cpxs_ind … H) -X2 /2 width=2 by ex_intro/
#X #X2 #_ #HX2 * #n #H destruct
@(ex_intro … (↑n)) >iter_S //
qed-.
-lemma cpxs_inv_cast1: ∀h,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ⬈*[h] U2 →
- ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 & ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 & U2 = ⓝW2.T2
- | ⦃G, L⦄ ⊢ T1 ⬈*[h] U2
- | ⦃G, L⦄ ⊢ W1 ⬈*[h] U2.
+lemma cpxs_inv_cast1: ∀h,G,L,W1,T1,U2. ⦃G,L⦄ ⊢ ⓝW1.T1 ⬈*[h] U2 →
+ ∨∨ ∃∃W2,T2. ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 & ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 & U2 = ⓝW2.T2
+ | ⦃G,L⦄ ⊢ T1 ⬈*[h] U2
+ | ⦃G,L⦄ ⊢ W1 ⬈*[h] U2.
#h #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
#U2 #U #_ #HU2 * /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
#W #T #HW1 #HT1 #H destruct
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+(* Properties with normal forms *********************************************)
+
+lemma cpxs_cnx (h) (G) (L) (T1):
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → T1 ≛ T2) → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄.
+/3 width=1 by cpx_cpxs/ qed.
+
(* Inversion lemmas with normal terms ***************************************)
-lemma cpxs_inv_cnx1: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T1⦄ →
- T1 ≛[h, o] T2.
-#h #o #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
-/5 width=8 by cnx_tdeq_trans, tdeq_trans/
+lemma cpxs_inv_cnx1 (h) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ → T1 ≛ T2.
+#h #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
+/5 width=9 by cnx_tdeq_trans, tdeq_trans/
qed-.
theorem cpxs_trans: ∀h,G,L. Transitive … (cpxs h G L).
normalize /2 width=3 by trans_TC/ qed-.
-theorem cpxs_bind: ∀h,p,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 →
- ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+theorem cpxs_bind: ∀h,p,I,G,L,V1,V2,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈*[h] T2 →
+ ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
#h #p #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
/3 width=5 by cpxs_trans, cpxs_bind_dx/
qed.
-theorem cpxs_flat: ∀h,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
+theorem cpxs_flat: ∀h,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈*[h] ⓕ{I}V2.T2.
#h #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
/3 width=5 by cpxs_trans, cpxs_flat_dx/
qed.
theorem cpxs_beta_rc: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2
/4 width=5 by cpxs_trans, cpxs_beta_dx, cpxs_bind_dx, cpx_pair_sn/
qed.
theorem cpxs_beta: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 → ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 → ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈*[h] ⓓ{p}ⓝW2.V2.T2.
#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2
/4 width=5 by cpxs_trans, cpxs_beta_rc, cpxs_bind_dx, cpx_flat/
qed.
theorem cpxs_theta_rc: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 →
- ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 →
+ ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cpxs_ind … H) -W2
/3 width=5 by cpxs_trans, cpxs_theta_dx, cpxs_bind_dx/
qed.
theorem cpxs_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- ⬆*[1] V ≘ V2 → ⦃G, L⦄ ⊢ W1 ⬈*[h] W2 →
- ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ V1 ⬈*[h] V →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
+ ⬆*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ⬈*[h] W2 →
+ ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ V1 ⬈*[h] V →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈*[h] ⓓ{p}W2.ⓐV2.T2.
#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
/3 width=5 by cpxs_trans, cpxs_theta_rc, cpxs_flat_dx/
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpxs_inv_appl1: ∀h,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈*[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 &
+lemma cpxs_inv_appl1: ∀h,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓐV1.T1 ⬈*[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 &
U2 = ⓐV2.T2
- | ∃∃p,W,T. ⦃G, L⦄ ⊢ T1 ⬈*[h] ⓛ{p}W.T & ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V1.T ⬈*[h] U2
- | ∃∃p,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ⬈*[h] V0 & ⬆*[1] V0 ≘ V2 &
- ⦃G, L⦄ ⊢ T1 ⬈*[h] ⓓ{p}V.T & ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] U2.
+ | ∃∃p,W,T. ⦃G,L⦄ ⊢ T1 ⬈*[h] ⓛ{p}W.T & ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V1.T ⬈*[h] U2
+ | ∃∃p,V0,V2,V,T. ⦃G,L⦄ ⊢ V1 ⬈*[h] V0 & ⬆*[1] V0 ≘ V2 &
+ ⦃G,L⦄ ⊢ T1 ⬈*[h] ⓓ{p}V.T & ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] U2.
#h #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5 by or3_intro0, ex3_2_intro/ ]
#U #U2 #_ #HU2 * *
[ #V0 #T0 #HV10 #HT10 #H destruct
(* Advanced properties ******************************************************)
-lemma cpxs_delta: ∀h,I,G,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 →
- ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈*[h] W2.
+lemma cpxs_delta: ∀h,I,G,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀W2. ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈*[h] W2.
#h #I #G #K #V1 #V2 #H @(cpxs_ind … H) -V2
[ /3 width=3 by cpx_cpxs, cpx_delta/
| #V #V2 #_ #HV2 #IH #W2 #HVW2
]
qed.
-lemma cpxs_lref: ∀h,I,G,K,T,i. ⦃G, K⦄ ⊢ #i ⬈*[h] T →
- ∀U. ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈*[h] U.
+lemma cpxs_lref: ∀h,I,G,K,T,i. ⦃G,K⦄ ⊢ #i ⬈*[h] T →
+ ∀U. ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈*[h] U.
#h #I #G #K #T #i #H @(cpxs_ind … H) -T
[ /3 width=3 by cpx_cpxs, cpx_lref/
| #T0 #T #_ #HT2 #IH #U #HTU
(* Basic_2A1: was: cpxs_delta *)
lemma cpxs_delta_drops: ∀h,I,G,L,K,V1,V2,i.
- ⬇*[i] L ≘ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 →
- ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ⬈*[h] W2.
+ ⬇*[i] L ≘ K.ⓑ{I}V1 → ⦃G,K⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ⬈*[h] W2.
#h #I #G #L #K #V1 #V2 #i #HLK #H @(cpxs_ind … H) -V2
[ /3 width=7 by cpx_cpxs, cpx_delta_drops/
| #V #V2 #_ #HV2 #IH #W2 #HVW2
(* Advanced inversion lemmas ************************************************)
-lemma cpxs_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈*[h] T2 →
+lemma cpxs_inv_zero1: ∀h,G,L,T2. ⦃G,L⦄ ⊢ #0 ⬈*[h] T2 →
T2 = #0 ∨
- ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 & ⬆*[1] V2 ≘ T2 &
+ ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈*[h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓑ{I}V1.
#h #G #L #T2 #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
#T #T2 #_ #HT2 *
]
qed-.
-lemma cpxs_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈*[h] T2 →
+lemma cpxs_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈*[h] T2 →
T2 = #(↑i) ∨
- ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈*[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
+ ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬈*[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
#h #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
#T #T2 #_ #HT2 *
[ #H destruct
qed-.
(* Basic_2A1: was: cpxs_inv_lref1 *)
-lemma cpxs_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ⬈*[h] T2 →
+lemma cpxs_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ⬈*[h] T2 →
T2 = #i ∨
- ∃∃I,K,V1,T1. ⬇*[i] L ≘ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ⬈*[h] T1 &
+ ∃∃I,K,V1,T1. ⬇*[i] L ≘ K.ⓑ{I}V1 & ⦃G,K⦄ ⊢ V1 ⬈*[h] T1 &
⬆*[↑i] T1 ≘ T2.
#h #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
#T #T2 #_ #HT2 *
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
-(* Properties with degree-based equivalence for closures ********************)
+(* Properties with sort-irrelevant equivalence for closures *****************)
-lemma fdeq_cpxs_trans: ∀h,o,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T⦄ →
- ∀T2. ⦃G2, L2⦄ ⊢ T ⬈*[h] T2 →
- ∃∃T0. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T0 & ⦃G1, L1, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
+lemma fdeq_cpxs_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T⦄ →
+ ∀T2. ⦃G2,L2⦄ ⊢ T ⬈*[h] T2 →
+ ∃∃T0. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T0 & ⦃G1,L1,T0⦄ ≛ ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
elim (fdeq_inv_gen_dx … H) -H #H #HL12 #HT1 destruct
elim (rdeq_cpxs_trans … HT2 … HL12) #T0 #HT0 #HT02
lapply (cpxs_rdeq_conf_dx … HT2 … HL12) -HL12 #HL12
(* Properties on supclosure *************************************************)
-lemma fqu_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+lemma fqu_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 →
+ ∀T1. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⬂[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
#T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
qed-.
-lemma fquq_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+lemma fquq_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 →
+ ∀T1. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⬂⸮[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fquq_cpx_trans … HT1 … HT2) -T
#T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
qed-.
-lemma fqup_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄.
+lemma fqup_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 →
+ ∀T1. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⬂+[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
qed-.
-lemma fqus_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄.
+lemma fqus_cpxs_trans: ∀h,b,G1,G2,L1,L2,T2,U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 →
+ ∀T1. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & ⦃G1,L1,U1⦄ ⬂*[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqus_cpx_trans … HT1 … HT2) -T
#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
(* Note: a proof based on fqu_cpx_trans_tdneq might exist *)
(* Basic_2A1: uses: fqu_cpxs_trans_neq *)
-lemma fqu_cpxs_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma fqu_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⬂[b] ⦃G2,L2,U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❴1❵)
#U2 #HVU2 @(ex3_intro … U2)
[1,3: /3 width=7 by cpxs_delta, fqu_drop/
[1,3: /2 width=4 by fqu_pair_sn, cpxs_pair_sn/
| #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
]
-| #p #I #G #L #V #T1 #T2 #HT12 #H0 @(ex3_intro … (ⓑ{p,I}V.T2))
+| #p #I #G #L #V #T1 #Hb #T2 #HT12 #H0 @(ex3_intro … (ⓑ{p,I}V.T2))
[1,3: /2 width=4 by fqu_bind_dx, cpxs_bind/
| #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
]
qed-.
(* Basic_2A1: uses: fquq_cpxs_trans_neq *)
-lemma fquq_cpxs_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
+lemma fquq_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⬂⸮[b] ⦃G2,L2,U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
[ #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fquq, ex3_intro/
| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
qed-.
(* Basic_2A1: uses: fqup_cpxs_trans_neq *)
-lemma fqup_cpxs_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+lemma fqup_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⬂+[b] ⦃G2,L2,U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fqup, ex3_intro/
| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
qed-.
(* Basic_2A1: uses: fqus_cpxs_trans_neq *)
-lemma fqus_cpxs_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
+lemma fqus_cpxs_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈*[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⬂*[b] ⦃G2,L2,U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
[ #H12 elim (fqup_cpxs_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqup_fqus, ex3_intro/
| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
(* Advanced properties ******************************************************)
-lemma cpx_bind2: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+lemma cpx_bind2: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬈[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
/4 width=5 by lpx_cpx_trans, cpxs_bind_dx, lpx_pair/ qed.
-lemma cpxs_bind2_dx: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+lemma cpxs_bind2_dx: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
/4 width=5 by lpx_cpxs_trans, cpxs_bind_dx, lpx_pair/ qed.
(* Properties with plus-iterated structural successor for closures **********)
(* Basic_2A1: uses: lpx_fqup_trans *)
-lemma lpx_fqup_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐+[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+lemma lpx_fqup_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 →
+ ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1,K1,T⦄ ⬂+[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1
/3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
(* Properties with star-iterated structural successor for closures **********)
(* Basic_2A1: uses: lpx_fqus_trans *)
-lemma lpx_fqus_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, K1, T⦄ ⊐*[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+lemma lpx_fqus_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 →
+ ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈*[h] T & ⦃G1,K1,T⦄ ⬂*[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fqus_inv_fqup … H) -H
[ #H12 elim (lpx_fqup_trans … H12 … HKL1) -L1 /3 width=5 by fqup_fqus, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
-(* Properties with degree-based equivalence for local environments **********)
+(* Properties with sort-irrelevant equivalence for local environments *******)
(* Basic_2A1: was just: lleq_cpxs_trans *)
-lemma rdeq_cpxs_trans: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
- ∀L2. L2 ≛[h, o, T0] L0 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛[h, o] T1.
-#h #o #G #L0 #T0 #T1 #H @(cpxs_ind_dx … H) -T0 /2 width=3 by ex2_intro/
+lemma rdeq_cpxs_trans: ∀h,G,L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈*[h] T1 →
+ ∀L2. L2 ≛[T0] L0 →
+ ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈*[h] T & T ≛ T1.
+#h #G #L0 #T0 #T1 #H @(cpxs_ind_dx … H) -T0 /2 width=3 by ex2_intro/
#T0 #T #HT0 #_ #IH #L2 #HL2
elim (rdeq_cpx_trans … HL2 … HT0) #U1 #H1 #H2
-elim (IH L2) -IH /2 width=4 by cpx_rdeq_conf_dx/ -L0 #U2 #H3 #H4
+elim (IH L2) -IH /2 width=5 by cpx_rdeq_conf_dx/ -L0 #U2 #H3 #H4
elim (tdeq_cpxs_trans … H2 … H3) -T #U0 #H2 #H3
/3 width=5 by cpxs_strap2, tdeq_trans, ex2_intro/
qed-.
(* Basic_2A1: was just: cpxs_lleq_conf *)
-lemma cpxs_rdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
- ∀L2. L0 ≛[h, o, T0] L2 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛[h, o] T1.
+lemma cpxs_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈*[h] T1 →
+ ∀L2. L0 ≛[T0] L2 →
+ ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈*[h] T & T ≛ T1.
/3 width=3 by rdeq_cpxs_trans, rdeq_sym/ qed-.
(* Basic_2A1: was just: cpxs_lleq_conf_dx *)
-lemma cpxs_rdeq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈*[h] T2 →
- ∀L1. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
-#h #o #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_rdeq_conf_dx/
+lemma cpxs_rdeq_conf_dx: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀L1. L1 ≛[T1] L2 → L1 ≛[T2] L2.
+#h #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_rdeq_conf_dx/
qed-.
(* Basic_2A1: was just: lleq_conf_sn *)
-lemma cpxs_rdeq_conf_sn: ∀h,o,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬈*[h] T2 →
- ∀L2. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
+lemma cpxs_rdeq_conf_sn: ∀h,G,L1,T1,T2. ⦃G,L1⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀L2. L1 ≛[T1] L2 → L1 ≛[T2] L2.
/4 width=6 by cpxs_rdeq_conf_dx, rdeq_sym/ qed-.
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
-(* Properties with degree-based equivalence for terms ***********************)
+(* Properties with sort-irrelevant equivalence for terms ********************)
-lemma tdeq_cpxs_trans: ∀h,o,U1,T1. U1 ≛[h, o] T1 → ∀G,L,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- ∃∃U2. ⦃G, L⦄ ⊢ U1 ⬈*[h] U2 & U2 ≛[h, o] T2.
-#h #o #U1 #T1 #HUT1 #G #L #T2 #HT12 @(cpxs_ind … HT12) -T2 /2 width=3 by ex2_intro/
+lemma tdeq_cpxs_trans: ∀h,U1,T1. U1 ≛ T1 → ∀G,L,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ ∃∃U2. ⦃G,L⦄ ⊢ U1 ⬈*[h] U2 & U2 ≛ T2.
+#h #U1 #T1 #HUT1 #G #L #T2 #HT12 @(cpxs_ind … HT12) -T2 /2 width=3 by ex2_intro/
#T #T2 #_ #HT2 * #U #HU1 #HUT elim (tdeq_cpx_trans … HUT … HT2) -T -T1
/3 width=3 by ex2_intro, cpxs_strap1/
qed-.
(* Note: this requires tdeq to be symmetric *)
(* Nasic_2A1: uses: cpxs_neq_inv_step_sn *)
-lemma cpxs_tdneq_fwd_step_sn: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) →
- ∃∃T,T0. ⦃G, L⦄ ⊢ T1 ⬈[h] T & T1 ≛[h, o] T → ⊥ & ⦃G, L⦄ ⊢ T ⬈*[h] T0 & T0 ≛[h, o] T2.
-#h #o #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
+lemma cpxs_tdneq_fwd_step_sn: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) →
+ ∃∃T,T0. ⦃G,L⦄ ⊢ T1 ⬈[h] T & T1 ≛ T → ⊥ & ⦃G,L⦄ ⊢ T ⬈*[h] T0 & T0 ≛ T2.
+#h #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
[ #H elim H -H //
| #T1 #T0 #HT10 #HT02 #IH #Hn12
- elim (tdeq_dec h o T1 T0) [ -HT10 -HT02 #H10 | -IH #Hn10 ]
+ elim (tdeq_dec T1 T0) [ -HT10 -HT02 #H10 | -IH #Hn10 ]
[ elim IH -IH /3 width=3 by tdeq_trans/ -Hn12
#T3 #T4 #HT03 #Hn03 #HT34 #H42
elim (tdeq_cpx_trans … H10 … HT03) -HT03 #T5 #HT15 #H53
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/syntax/theq_tdeq.ma".
-include "basic_2/rt_computation/cpxs_lsubr.ma".
-include "basic_2/rt_computation/cpxs_cnx.ma".
-include "basic_2/rt_computation/lpxs_cpxs.ma".
-
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
-
-(* Forward lemmas with head equivalence for terms ***************************)
-
-lemma cpxs_fwd_sort: ∀h,o,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ⬈*[h] U →
- ⋆s ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ⬈*[h] U.
-#h #o #G #L #U #s #H elim (cpxs_inv_sort1 … H) -H *
-[ #H destruct /2 width=1 by or_introl/
-| #n #H destruct
- @or_intror >iter_S <(iter_n_Sm … (next h)) // (**)
-]
-qed-.
-
-(* Note: probably this is an inversion lemma *)
-(* Basic_2A1: was: cpxs_fwd_delta *)
-lemma cpxs_fwd_delta_drops: ∀h,o,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
- ∀V2. ⬆*[↑i] V1 ≘ V2 →
- ∀U. ⦃G, L⦄ ⊢ #i ⬈*[h] U →
- #i ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ V2 ⬈*[h] U.
-#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #U #H
-elim (cpxs_inv_lref1_drops … H) -H /2 width=1 by or_introl/
-* #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
-lapply (drops_mono … HLK0 … HLK) -HLK0 #H destruct
-/4 width=9 by cpxs_lifts_bi, drops_isuni_fwd_drop2, or_intror/
-qed-.
-
-(* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: ∀h,o,p,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] U →
- ⓐV.ⓛ{p}W.T ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⓓ{p}ⓝW.V.T ⬈*[h] U.
-#h #o #p #G #L #V #W #T #U #H elim (cpxs_inv_appl1 … H) -H *
-[ #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
-| #b #W0 #T0 #HT0 #HU
- elim (cpxs_inv_abst1 … HT0) -HT0 #W1 #T1 #HW1 #HT1 #H destruct
- lapply (lsubr_cpxs_trans … HT1 (L.ⓓⓝW.V) ?) -HT1
- /5 width=3 by cpxs_trans, cpxs_bind, cpxs_pair_sn, lsubr_beta, or_intror/
-| #b #V1 #V2 #V0 #T1 #_ #_ #HT1 #_
- elim (cpxs_inv_abst1 … HT1) -HT1 #W2 #T2 #_ #_ #H destruct
-]
-qed-.
-
-lemma cpxs_fwd_theta: ∀h,o,p,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}V.T ⬈*[h] U →
- ∀V2. ⬆*[1] V1 ≘ V2 → ⓐV1.ⓓ{p}V.T ⩳[h, o] U ∨
- ⦃G, L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] U.
-#h #o #p #G #L #V1 #V #T #U #H #V2 #HV12
-elim (cpxs_inv_appl1 … H) -H *
-[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
-| #q #W #T0 #HT0 #HU
- elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
- [ #V3 #T3 #_ #_ #H destruct
- | #X #HT2 #H #H0 destruct
- elim (lifts_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct
- @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
- @(cpxs_trans … (+ⓓV.ⓐV2.ⓛ{q}W2.T2)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
- @(cpxs_strap2 … (ⓐV1.ⓛ{q}W.T0)) [2: /2 width=1 by cpxs_beta_dx/ ]
- /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/
- ]
-| #q #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU
- @or_intror @(cpxs_trans … HU) -U (**) (* explicit constructor *)
- elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
- [ #V5 #T5 #HV5 #HT5 #H destruct
- /6 width=9 by cpxs_lifts_bi, drops_refl, drops_drop, cpxs_flat, cpxs_bind/
- | #X #HT1 #H #H0 destruct
- elim (lifts_inv_bind1 … H) -H #V5 #T5 #HV05 #HT05 #H destruct
- lapply (cpxs_lifts_bi … HV13 (Ⓣ) … (L.ⓓV0) … HV12 … HV34) -V3 /3 width=1 by drops_refl, drops_drop/ #HV24
- @(cpxs_trans … (+ⓓV.ⓐV2.ⓓ{q}V5.T5)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
- @(cpxs_strap2 … (ⓐV1.ⓓ{q}V0.T0)) [ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/ ] -V -V5 -T5
- @(cpxs_strap2 … (ⓓ{q}V0.ⓐV2.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpx_theta/
- ]
-]
-qed-.
-
-lemma cpxs_fwd_cast: ∀h,o,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ⬈*[h] U →
- ∨∨ ⓝW. T ⩳[h, o] U | ⦃G, L⦄ ⊢ T ⬈*[h] U | ⦃G, L⦄ ⊢ W ⬈*[h] U.
-#h #o #G #L #W #T #U #H
-elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
-#W0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or3_intro0/
-qed-.
-
-lemma cpxs_fwd_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
- ∀U. ⦃G, L⦄ ⊢ T ⬈*[h] U → T ⩳[h, o] U.
-/3 width=4 by cpxs_inv_cnx1, tdeq_theq/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/syntax/theq_simple_vector.ma".
-include "static_2/relocation/lifts_vector.ma".
-include "basic_2/rt_computation/cpxs_theq.ma".
-
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
-
-(* Vector form of forward lemmas with head equivalence for terms ************)
-
-lemma cpxs_fwd_sort_vector: ∀h,o,G,L,s,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆s ⬈*[h] U →
- ⒶVs.⋆s ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h s) ⬈*[h] U.
-#h #o #G #L #s #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
-#V #Vs #IHVs #U #H
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
-| #p #W1 #T1 #HT1 #HU
- elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (theq_inv_applv_bind_simple … HT1) //
- | @or_intror (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{p}W1.T1)) /3 width=1 by cpxs_flat_dx, cpx_beta/
- ]
-| #p #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
- elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (theq_inv_applv_bind_simple … HT1) //
- | @or_intror (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV1.ⓓ{p}V3.T1)) /3 width=3 by cpxs_flat, cpx_theta/
- ]
-]
-qed-.
-
-(* Basic_2A1: was: cpxs_fwd_delta_vector *)
-lemma cpxs_fwd_delta_drops_vector: ∀h,o,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
- ∀V2. ⬆*[↑i] V1 ≘ V2 →
- ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ⬈*[h] U →
- ⒶVs.#i ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ⬈*[h] U.
-#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta_drops/
-#V #Vs #IHVs #U #H -K -V1
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
-| #q #W0 #T0 #HT0 #HU
- elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (theq_inv_applv_bind_simple … HT0) //
- | @or_intror -i (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /3 width=1 by cpxs_flat_dx, cpx_beta/
- ]
-| #q #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
- elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (theq_inv_applv_bind_simple … HT0) //
- | @or_intror -i (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓓ{q}V3.T0)) /3 width=3 by cpxs_flat, cpx_theta/
- ]
-]
-qed-.
-
-(* Basic_1: was just: pr3_iso_appls_beta *)
-lemma cpxs_fwd_beta_vector: ∀h,o,p,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] U →
- ⒶVs.ⓐV.ⓛ{p}W. T ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] U.
-#h #o #p #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
-#V0 #Vs #IHVs #V #W #T #U #H
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
-| #q #W1 #T1 #HT1 #HU
- elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (theq_inv_applv_bind_simple … HT1) //
- | @or_intror (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓛ{q}W1.T1)) /3 width=1 by cpxs_flat_dx, cpx_beta/
- ]
-| #q #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
- elim (IHVs … HT1) -IHVs -HT1 #HT1
- [ elim (theq_inv_applv_bind_simple … HT1) //
- | @or_intror (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV1.ⓓ{q}V3.T1)) /3 width=3 by cpxs_flat, cpx_theta/
- ]
-]
-qed-.
-
-(* Basic_1: was just: pr3_iso_appls_abbr *)
-lemma cpxs_fwd_theta_vector: ∀h,o,G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
- ∀p,V,T,U. ⦃G, L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] U →
- ⒶV1b.ⓓ{p}V.T ⩳[h, o] U ∨ ⦃G, L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] U.
-#h #o #G #L #V1b #V2b * -V1b -V2b /3 width=1 by or_intror/
-#V1b #V2b #V1a #V2a #HV12a #HV12b #p
-generalize in match HV12a; -HV12a
-generalize in match V2a; -V2a
-generalize in match V1a; -V1a
-elim HV12b -V1b -V2b /2 width=1 by cpxs_fwd_theta/
-#V1b #V2b #V1b #V2b #HV12b #_ #IHV12b #V1a #V2a #HV12a #V #T #U #H
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHV12b -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or_introl/
-| #q #W0 #T0 #HT0 #HU
- elim (IHV12b … HV12b … HT0) -IHV12b -HT0 #HT0
- [ -HV12a -HV12b -HU
- elim (theq_inv_pair1 … HT0) #V1 #T1 #H destruct
- | @or_intror -V1b (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
- [ -HV12a #V1 #T1 #_ #_ #H destruct
- | -V1b #X #HT1 #H #H0 destruct
- elim (lifts_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
- @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{q}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
- @(cpxs_strap2 … (ⓐV1a.ⓛ{q}W0.T0))
- /4 width=7 by cpxs_beta_dx, cpx_zeta, lifts_bind, lifts_flat/
- ]
- ]
-| #q #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
- elim (IHV12b … HV12b … HT0) -HV12b -IHV12b -HT0 #HT0
- [ -HV12a -HV10a -HV0a -HU
- elim (theq_inv_pair1 … HT0) #V1 #T1 #H destruct
- | @or_intror -V1b -V1b (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
- [ #V1 #T1 #HV1 #HT1 #H destruct
- lapply (cpxs_lifts_bi … HV10a (Ⓣ) … (L.ⓓV) … HV12a … HV0a) -V1a -V0a /3 width=1 by drops_refl, drops_drop/ #HV2a
- @(cpxs_trans … (ⓓ{p}V.ⓐV2a.T1)) /3 width=1 by cpxs_bind, cpxs_pair_sn, cpxs_flat_dx, cpxs_bind_dx/
- | #X #HT1 #H #H0 destruct
- elim (lifts_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
- lapply (cpxs_lifts_bi … HV10a (Ⓣ) … (L.ⓓV0) … HV12a … HV0a) -V0a /3 width=1 by drops_refl, drops_drop/ #HV2a
- @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{q}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
- @(cpxs_strap2 … (ⓐV1a.ⓓ{q}V0.T0)) [ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/ ] -V -V1 -T1
- @(cpxs_strap2 … (ⓓ{q}V0.ⓐV2a.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpx_theta/
- ]
- ]
-]
-qed-.
-
-(* Basic_1: was just: pr3_iso_appls_cast *)
-lemma cpxs_fwd_cast_vector: ∀h,o,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] U →
- ∨∨ ⒶVs. ⓝW. T ⩳[h, o] U
- | ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] U
- | ⦃G, L⦄ ⊢ ⒶVs.W ⬈*[h] U.
-#h #o #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
-#V #Vs #IHVs #W #T #U #H
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair, or3_intro0/
-| #q #W0 #T0 #HT0 #HU elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (theq_inv_applv_bind_simple … HT0) //
- | @or3_intro1 -W (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
- | @or3_intro2 -T (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
- ]
-| #q #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
- elim (IHVs … HT0) -IHVs -HT0 #HT0
- [ elim (theq_inv_applv_bind_simple … HT0) //
- | @or3_intro1 -W (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
- | @or3_intro2 -T (**) (* explicit constructor *)
- @(cpxs_trans … HU) -U
- @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
- ]
-]
-qed-.
-
-(* Basic_1: was just: nf2_iso_appls_lref *)
-lemma cpxs_fwd_cnx_vector: ∀h,o,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
- ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ⬈*[h] U → ⒶVs.T ⩳[h, o] U.
-#h #o #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
-#V #Vs #IHVs #U #H
-elim (cpxs_inv_appl1 … H) -H *
-[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by theq_pair/
-| #p #W0 #T0 #HT0 #HU
- lapply (IHVs … HT0) -IHVs -HT0 #HT0
- elim (theq_inv_applv_bind_simple … HT0) //
-| #p #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
- lapply (IHVs … HT0) -IHVs -HT0 #HT0
- elim (theq_inv_applv_bind_simple … HT0) //
-]
-qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/toeq_tdeq.ma".
+include "basic_2/rt_computation/cpxs_lsubr.ma".
+include "basic_2/rt_computation/cpxs_cnx.ma".
+include "basic_2/rt_computation/lpxs_cpxs.ma".
+
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+
+(* Forward lemmas with sort-irrelevant outer equivalence for terms **********)
+
+lemma cpxs_fwd_sort (h) (G) (L):
+ ∀X2,s1. ⦃G,L⦄ ⊢ ⋆s1 ⬈*[h] X2 → ⋆s1 ⩳ X2.
+#h #G #L #X2 #s1 #H
+elim (cpxs_inv_sort1 … H) -H #s2 #H destruct //
+qed-.
+
+(* Note: probably this is an inversion lemma *)
+(* Basic_2A1: was: cpxs_fwd_delta *)
+lemma cpxs_fwd_delta_drops (h) (I) (G) (L) (K):
+ ∀V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
+ ∀V2. ⬆*[↑i] V1 ≘ V2 →
+ ∀X2. ⦃G,L⦄ ⊢ #i ⬈*[h] X2 →
+ ∨∨ #i ⩳ X2 | ⦃G,L⦄ ⊢ V2 ⬈*[h] X2.
+#h #I #G #L #K #V1 #i #HLK #V2 #HV12 #X2 #H
+elim (cpxs_inv_lref1_drops … H) -H /2 width=1 by or_introl/
+* #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
+lapply (drops_mono … HLK0 … HLK) -HLK0 #H destruct
+/4 width=9 by cpxs_lifts_bi, drops_isuni_fwd_drop2, or_intror/
+qed-.
+
+(* Basic_1: was just: pr3_iso_beta *)
+lemma cpxs_fwd_beta (h) (p) (G) (L):
+ ∀V,W,T,X2. ⦃G,L⦄ ⊢ ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
+ ∨∨ ⓐV.ⓛ{p}W.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}ⓝW.V.T ⬈*[h] X2.
+#h #p #G #L #V #W #T #X2 #H elim (cpxs_inv_appl1 … H) -H *
+[ #V0 #T0 #_ #_ #H destruct /2 width=1 by toeq_pair, or_introl/
+| #b #W0 #T0 #HT0 #HU
+ elim (cpxs_inv_abst1 … HT0) -HT0 #W1 #T1 #HW1 #HT1 #H destruct
+ lapply (lsubr_cpxs_trans … HT1 (L.ⓓⓝW.V) ?) -HT1
+ /5 width=3 by cpxs_trans, cpxs_bind, cpxs_pair_sn, lsubr_beta, or_intror/
+| #b #V1 #V2 #V0 #T1 #_ #_ #HT1 #_
+ elim (cpxs_inv_abst1 … HT1) -HT1 #W2 #T2 #_ #_ #H destruct
+]
+qed-.
+
+lemma cpxs_fwd_theta (h) (p) (G) (L):
+ ∀V1,V,T,X2. ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}V.T ⬈*[h] X2 →
+ ∀V2. ⬆*[1] V1 ≘ V2 →
+ ∨∨ ⓐV1.ⓓ{p}V.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}V.ⓐV2.T ⬈*[h] X2.
+#h #p #G #L #V1 #V #T #X2 #H #V2 #HV12
+elim (cpxs_inv_appl1 … H) -H *
+[ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by toeq_pair, or_introl/
+| #q #W #T0 #HT0 #HU
+ elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
+ [ #V3 #T3 #_ #_ #H destruct
+ | #X #HT2 #H #H0 destruct
+ elim (lifts_inv_bind1 … H) -H #W2 #T2 #HW2 #HT02 #H destruct
+ @or_intror @(cpxs_trans … HU) -X2 (**) (* explicit constructor *)
+ @(cpxs_trans … (+ⓓV.ⓐV2.ⓛ{q}W2.T2)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
+ @(cpxs_strap2 … (ⓐV1.ⓛ{q}W.T0)) [2: /2 width=1 by cpxs_beta_dx/ ]
+ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/
+ ]
+| #q #V3 #V4 #V0 #T0 #HV13 #HV34 #HT0 #HU
+ @or_intror @(cpxs_trans … HU) -X2 (**) (* explicit constructor *)
+ elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
+ [ #V5 #T5 #HV5 #HT5 #H destruct
+ /6 width=9 by cpxs_lifts_bi, drops_refl, drops_drop, cpxs_flat, cpxs_bind/
+ | #X #HT1 #H #H0 destruct
+ elim (lifts_inv_bind1 … H) -H #V5 #T5 #HV05 #HT05 #H destruct
+ lapply (cpxs_lifts_bi … HV13 (Ⓣ) … (L.ⓓV0) … HV12 … HV34) -V3 /3 width=1 by drops_refl, drops_drop/ #HV24
+ @(cpxs_trans … (+ⓓV.ⓐV2.ⓓ{q}V5.T5)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T
+ @(cpxs_strap2 … (ⓐV1.ⓓ{q}V0.T0)) [ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/ ] -V -V5 -T5
+ @(cpxs_strap2 … (ⓓ{q}V0.ⓐV2.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpx_theta/
+ ]
+]
+qed-.
+
+lemma cpxs_fwd_cast (h) (G) (L):
+ ∀W,T,X2. ⦃G,L⦄ ⊢ ⓝW.T ⬈*[h] X2 →
+ ∨∨ ⓝW. T ⩳ X2 | ⦃G,L⦄ ⊢ T ⬈*[h] X2 | ⦃G,L⦄ ⊢ W ⬈*[h] X2.
+#h #G #L #W #T #X2 #H
+elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
+#W0 #T0 #_ #_ #H destruct /2 width=1 by toeq_pair, or3_intro0/
+qed-.
+
+lemma cpxs_fwd_cnx (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ →
+ ∀X2. ⦃G,L⦄ ⊢ T1 ⬈*[h] X2 → T1 ⩳ X2.
+/3 width=5 by cpxs_inv_cnx1, tdeq_toeq/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/toeq_simple_vector.ma".
+include "static_2/relocation/lifts_vector.ma".
+include "basic_2/rt_computation/cpxs_toeq.ma".
+
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+
+(* Vector form of forward lemmas with outer equivalence for terms ***********)
+
+lemma cpxs_fwd_sort_vector (h) (G) (L):
+ ∀s,Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.⋆s ⬈*[h] X2 → ⒶVs.⋆s ⩳ X2.
+#h #G #L #s #Vs elim Vs -Vs /2 width=4 by cpxs_fwd_sort/
+#V #Vs #IHVs #X2 #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by toeq_pair/
+| #p #W1 #T1 #HT1 #HU
+ lapply (IHVs … HT1) -IHVs -HT1 #HT1
+ elim (toeq_inv_applv_bind_simple … HT1) //
+| #p #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
+ lapply (IHVs … HT1) -IHVs -HT1 #HT1
+ elim (toeq_inv_applv_bind_simple … HT1) //
+]
+qed-.
+
+(* Basic_2A1: was: cpxs_fwd_delta_vector *)
+lemma cpxs_fwd_delta_drops_vector (h) (I) (G) (L) (K):
+ ∀V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
+ ∀V2. ⬆*[↑i] V1 ≘ V2 →
+ ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.#i ⬈*[h] X2 →
+ ∨∨ ⒶVs.#i ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.V2 ⬈*[h] X2.
+#h #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs
+elim Vs -Vs /2 width=5 by cpxs_fwd_delta_drops/
+#V #Vs #IHVs #X2 #H -K -V1
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by toeq_pair, or_introl/
+| #q #W0 #T0 #HT0 #HU
+ elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (toeq_inv_applv_bind_simple … HT0) //
+ | @or_intror -i (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -X2
+ @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /3 width=1 by cpxs_flat_dx, cpx_beta/
+ ]
+| #q #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
+ elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (toeq_inv_applv_bind_simple … HT0) //
+ | @or_intror -i (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -X2
+ @(cpxs_strap1 … (ⓐV0.ⓓ{q}V3.T0)) /3 width=3 by cpxs_flat, cpx_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_appls_beta *)
+lemma cpxs_fwd_beta_vector (h) (p) (G) (L):
+ ∀Vs,V,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
+ ∨∨ ⒶVs.ⓐV.ⓛ{p}W. T ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] X2.
+#h #p #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
+#V0 #Vs #IHVs #V #W #T #X2 #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by toeq_pair, or_introl/
+| #q #W1 #T1 #HT1 #HU
+ elim (IHVs … HT1) -IHVs -HT1 #HT1
+ [ elim (toeq_inv_applv_bind_simple … HT1) //
+ | @or_intror (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -X2
+ @(cpxs_strap1 … (ⓐV0.ⓛ{q}W1.T1)) /3 width=1 by cpxs_flat_dx, cpx_beta/
+ ]
+| #q #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
+ elim (IHVs … HT1) -IHVs -HT1 #HT1
+ [ elim (toeq_inv_applv_bind_simple … HT1) //
+ | @or_intror (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -X2
+ @(cpxs_strap1 … (ⓐV1.ⓓ{q}V3.T1)) /3 width=3 by cpxs_flat, cpx_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_appls_abbr *)
+lemma cpxs_fwd_theta_vector (h) (G) (L):
+ ∀V1b,V2b. ⬆*[1] V1b ≘ V2b →
+ ∀p,V,T,X2. ⦃G,L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] X2 →
+ ∨∨ ⒶV1b.ⓓ{p}V.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] X2.
+#h #G #L #V1b #V2b * -V1b -V2b /3 width=1 by or_intror/
+#V1b #V2b #V1a #V2a #HV12a #HV12b #p
+generalize in match HV12a; -HV12a
+generalize in match V2a; -V2a
+generalize in match V1a; -V1a
+elim HV12b -V1b -V2b /2 width=1 by cpxs_fwd_theta/
+#V1b #V2b #V1b #V2b #HV12b #_ #IHV12b #V1a #V2a #HV12a #V #T #X2 #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHV12b -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by toeq_pair, or_introl/
+| #q #W0 #T0 #HT0 #HU
+ elim (IHV12b … HV12b … HT0) -IHV12b -HT0 #HT0
+ [ -HV12a -HV12b -HU
+ elim (toeq_inv_pair1 … HT0) #V1 #T1 #H destruct
+ | @or_intror -V1b (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -X2
+ elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
+ [ -HV12a #V1 #T1 #_ #_ #H destruct
+ | -V1b #X #HT1 #H #H0 destruct
+ elim (lifts_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+ @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{q}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
+ @(cpxs_strap2 … (ⓐV1a.ⓛ{q}W0.T0))
+ /4 width=7 by cpxs_beta_dx, cpx_zeta, lifts_bind, lifts_flat/
+ ]
+ ]
+| #q #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
+ elim (IHV12b … HV12b … HT0) -HV12b -IHV12b -HT0 #HT0
+ [ -HV12a -HV10a -HV0a -HU
+ elim (toeq_inv_pair1 … HT0) #V1 #T1 #H destruct
+ | @or_intror -V1b -V1b (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -X2
+ elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
+ [ #V1 #T1 #HV1 #HT1 #H destruct
+ lapply (cpxs_lifts_bi … HV10a (Ⓣ) … (L.ⓓV) … HV12a … HV0a) -V1a -V0a /3 width=1 by drops_refl, drops_drop/ #HV2a
+ @(cpxs_trans … (ⓓ{p}V.ⓐV2a.T1)) /3 width=1 by cpxs_bind, cpxs_pair_sn, cpxs_flat_dx, cpxs_bind_dx/
+ | #X #HT1 #H #H0 destruct
+ elim (lifts_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
+ lapply (cpxs_lifts_bi … HV10a (Ⓣ) … (L.ⓓV0) … HV12a … HV0a) -V0a /3 width=1 by drops_refl, drops_drop/ #HV2a
+ @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{q}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
+ @(cpxs_strap2 … (ⓐV1a.ⓓ{q}V0.T0)) [ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/ ] -V -V1 -T1
+ @(cpxs_strap2 … (ⓓ{q}V0.ⓐV2a.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpx_theta/
+ ]
+ ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_appls_cast *)
+lemma cpxs_fwd_cast_vector (h) (G) (L):
+ ∀Vs,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] X2 →
+ ∨∨ ⒶVs. ⓝW. T ⩳ X2
+ | ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2
+ | ⦃G,L⦄ ⊢ ⒶVs.W ⬈*[h] X2.
+#h #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
+#V #Vs #IHVs #W #T #X2 #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by toeq_pair, or3_intro0/
+| #q #W0 #T0 #HT0 #HU elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (toeq_inv_applv_bind_simple … HT0) //
+ | @or3_intro1 -W (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -X2
+ @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
+ | @or3_intro2 -T (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -X2
+ @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
+ ]
+| #q #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
+ elim (IHVs … HT0) -IHVs -HT0 #HT0
+ [ elim (toeq_inv_applv_bind_simple … HT0) //
+ | @or3_intro1 -W (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -X2
+ @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
+ | @or3_intro2 -T (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -X2
+ @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
+ ]
+]
+qed-.
+
+(* Basic_1: was just: nf2_iso_appls_lref *)
+lemma cpxs_fwd_cnx_vector (h) (G) (L):
+ ∀T. 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
+ ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2 → ⒶVs.T ⩳ X2.
+#h #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
+#V #Vs #IHVs #X2 #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by toeq_pair/
+| #p #W0 #T0 #HT0 #HU
+ lapply (IHVs … HT0) -IHVs -HT0 #HT0
+ elim (toeq_inv_applv_bind_simple … HT0) //
+| #p #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
+ lapply (IHVs … HT0) -IHVs -HT0 #HT0
+ elim (toeq_inv_applv_bind_simple … HT0) //
+]
+qed-.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predtystrong_5.ma".
+include "basic_2/notation/relations/predtystrong_4.ma".
include "static_2/syntax/tdeq.ma".
include "basic_2/rt_transition/cpx.ma".
(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
-definition csx: ∀h. sd h → relation3 genv lenv term ≝
- λh,o,G,L. SN … (cpx h G L) (tdeq h o …).
+definition csx: ∀h. relation3 genv lenv term ≝
+ λh,G,L. SN … (cpx h G L) tdeq.
interpretation
"strong normalization for unbound context-sensitive parallel rt-transition (term)"
- 'PRedTyStrong h o G L T = (csx h o G L T).
+ 'PRedTyStrong h G L T = (csx h G L T).
(* Basic eliminators ********************************************************)
-lemma csx_ind: ∀h,o,G,L. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛[h, o] T2 → ⊥) → Q T2) →
+lemma csx_ind: ∀h,G,L. ∀Q:predicate term.
+ (∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) →
Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → Q T.
-#h #o #G #L #Q #H0 #T1 #H elim H -T1
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q T.
+#h #G #L #Q #H0 #T1 #H elim H -T1
/5 width=1 by SN_intro/
qed-.
(* Basic properties *********************************************************)
(* Basic_1: was just: sn3_pr2_intro *)
-lemma csx_intro: ∀h,o,G,L,T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄) →
- ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄.
+lemma csx_intro: ∀h,G,L,T1.
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄) →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄.
/4 width=1 by SN_intro/ qed.
(* Basic forward lemmas *****************************************************)
-fact csx_fwd_pair_sn_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
- ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
-#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+fact csx_fwd_pair_sn_aux: ∀h,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
+ ∀I,V,T. U = ②{I}V.T → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+#h #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
@csx_intro #V2 #HLV2 #HV2
@(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
#H elim (tdeq_inv_pair … H) -H /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_head *)
-lemma csx_fwd_pair_sn: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃②{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
+lemma csx_fwd_pair_sn: ∀h,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃②{I}V.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
/2 width=5 by csx_fwd_pair_sn_aux/ qed-.
-fact csx_fwd_bind_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
- ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
-#h #o #G #L #U #H elim H -H #U0 #_ #IH #p #I #V #T #H destruct
+fact csx_fwd_bind_dx_aux: ∀h,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
+ ∀p,I,V,T. U = ⓑ{p,I}V.T → ⦃G,L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #G #L #U #H elim H -H #U0 #_ #IH #p #I #V #T #H destruct
@csx_intro #T2 #HLT2 #HT2
-@(IH (ⓑ{p,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
+@(IH (ⓑ{p, I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
#H elim (tdeq_inv_pair … H) -H /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_bind *)
-lemma csx_fwd_bind_dx: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_bind_dx: ∀h,p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ → ⦃G,L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/2 width=4 by csx_fwd_bind_dx_aux/ qed-.
-fact csx_fwd_flat_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
- ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
-#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+fact csx_fwd_flat_dx_aux: ∀h,G,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
+ ∀I,V,T. U = ⓕ{I}V.T → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
@csx_intro #T2 #HLT2 #HT2
@(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
#H elim (tdeq_inv_pair … H) -H /2 width=1 by/
qed-.
(* Basic_1: was just: sn3_gen_flat *)
-lemma csx_fwd_flat_dx: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_flat_dx: ∀h,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/2 width=5 by csx_fwd_flat_dx_aux/ qed-.
-lemma csx_fwd_bind: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_bind: ∀h,p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G,L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
-lemma csx_fwd_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓕ{I}V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_flat: ∀h,I,G,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓕ{I}V.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
(* Basic_1: removed theorems 14:
(* Main properties with atomic arity assignment *****************************)
-theorem aaa_csx: ∀h,o,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
-#h #o #G #L #T #A #H
-@(gcr_aaa … (csx_gcp h o) (csx_gcr h o) … H)
+theorem aaa_csx: ∀h,G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #G #L #T #A #H
+@(gcr_aaa … (csx_gcp h) (csx_gcr h) … H)
qed.
(* Advanced eliminators *****************************************************)
-fact aaa_ind_csx_aux: ∀h,o,G,L,A. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛[h, o] T2 → ⊥) → Q T2) → Q T1
+fact aaa_ind_csx_aux: ∀h,G,L,A. ∀Q:predicate term.
+ (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ T ⁝ A → Q T.
-#h #o #G #L #A #Q #IH #T #H @(csx_ind … H) -T /4 width=5 by cpx_aaa_conf/
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ T ⁝ A → Q T.
+#h #G #L #A #Q #IH #T #H @(csx_ind … H) -T /4 width=5 by cpx_aaa_conf/
qed-.
-lemma aaa_ind_csx: ∀h,o,G,L,A. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛[h, o] T2 → ⊥) → Q T2) → Q T1
+lemma aaa_ind_csx: ∀h,G,L,A. ∀Q:predicate term.
+ (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ T ⁝ A → Q T.
+ ∀T. ⦃G,L⦄ ⊢ T ⁝ A → Q T.
/5 width=9 by aaa_ind_csx_aux, aaa_csx/ qed-.
-fact aaa_ind_csx_cpxs_aux: ∀h,o,G,L,A. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → Q T2) → Q T1
+fact aaa_ind_csx_cpxs_aux: ∀h,G,L,A. ∀Q:predicate term.
+ (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ T ⁝ A → Q T.
-#h #o #G #L #A #Q #IH #T #H @(csx_ind_cpxs … H) -T /4 width=5 by cpxs_aaa_conf/
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ T ⁝ A → Q T.
+#h #G #L #A #Q #IH #T #H @(csx_ind_cpxs … H) -T /4 width=5 by cpxs_aaa_conf/
qed-.
(* Basic_2A1: was: aaa_ind_csx_alt *)
-lemma aaa_ind_csx_cpxs: ∀h,o,G,L,A. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → Q T2) → Q T1
+lemma aaa_ind_csx_cpxs: ∀h,G,L,A. ∀Q:predicate term.
+ (∀T1. ⦃G,L⦄ ⊢ T1 ⁝ A →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ T ⁝ A → Q T.
+ ∀T. ⦃G,L⦄ ⊢ T ⁝ A → Q T.
/5 width=9 by aaa_ind_csx_cpxs_aux, aaa_csx/ qed-.
(* Properties with normal terms for unbound parallel rt-transition **********)
(* Basic_1: was just: sn3_nf2 *)
-lemma cnx_csx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
+lemma cnx_csx: ∀h,G,L,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/2 width=1 by NF_to_SN/ qed.
(* Advanced properties ******************************************************)
-lemma csx_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃⋆s⦄.
-#h #o #G #L #s elim (deg_total h o s)
-#d generalize in match s; -s elim d -d
-[ /3 width=3 by cnx_csx, cnx_sort/
-| #d #IH #s #Hsd lapply (deg_next_SO … Hsd) -Hsd
- #Hsd @csx_intro #X #H #HX
- elim (cpx_inv_sort1 … H) -H #H destruct /2 width=1 by/
- elim HX -HX //
-]
-qed.
+lemma csx_sort: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃⋆s⦄.
+/3 width=4 by cnx_csx, cnx_sort/ qed.
(* STRONGLY NORMALIZING TERM VECTORS FOR UNBOUND PARALLEL RT-TRANSITION *****)
-include "basic_2/rt_computation/cpxs_theq_vector.ma".
-include "basic_2/rt_computation/csx_simple_theq.ma".
+include "basic_2/rt_computation/cpxs_toeq_vector.ma".
+include "basic_2/rt_computation/csx_simple_toeq.ma".
include "basic_2/rt_computation/csx_cnx.ma".
include "basic_2/rt_computation/csx_cpxs.ma".
include "basic_2/rt_computation/csx_vector.ma".
(* Properties with normal terms for unbound parallel rt-transition **********)
(* Basic_1: was just: sn3_appls_lref *)
-lemma csx_applv_cnx: ∀h,o,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
- ∀Vs. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃Vs⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.T⦄.
-#h #o #G #L #T #H1T #H2T #Vs elim Vs -Vs
-[ #_ normalize in ⊢ (?????%); /2 width=1/
+lemma csx_applv_cnx (h) (G) (L):
+ ∀T. 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
+ ∀Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄.
+#h #G #L #T #H1T #H2T #Vs elim Vs -Vs
+[ #_ normalize in ⊢ (????%); /2 width=1 by cnx_csx/
| #V #Vs #IHV #H
elim (csxv_inv_cons … H) -H #HV #HVs
- @csx_appl_simple_theq /2 width=1 by applv_simple/ -IHV -HV -HVs
+ @csx_appl_simple_toeq /2 width=1 by applv_simple/ -IHV -HV -HVs
#X #H #H0
- lapply (cpxs_fwd_cnx_vector … o … H) -H // -H1T -H2T #H
+ lapply (cpxs_fwd_cnx_vector … H) -H // -H1T -H2T #H
elim (H0) -H0 //
]
qed.
(* Advanced properties ******************************************************)
-lemma csx_applv_sort: ∀h,o,G,L,s,Vs. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃Vs⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.⋆s⦄.
-#h #o #G #L #s elim (deg_total h o s)
-#d generalize in match s; -s elim d -d
-[ /3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/
-| #d #IHd #s #Hd #Vs elim Vs -Vs /2 width=1 by/
- #V #Vs #IHVs #HVVs
- elim (csxv_inv_cons … HVVs) #HV #HVs
- @csx_appl_simple_theq /2 width=1 by applv_simple, simple_atom/
- #X #H #H0
- elim (cpxs_fwd_sort_vector … o … H) -H #H
- [ elim H0 -H0 //
- | -H0 @(csx_cpxs_trans … (Ⓐ(V⨮Vs).⋆(next h s)))
- /3 width=1 by cpxs_flat_dx, deg_next_SO/
- ]
-]
-qed.
+(* Note: strong normalization does not depend on this any more *)
+lemma csx_applv_sort (h) (G) (L):
+ ∀s,Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.⋆s⦄.
+/3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/ qed.
(* Properties with unbound context-sensitive rt-computation for terms *******)
(* Basic_1: was just: sn3_intro *)
-lemma csx_intro_cpxs: ∀h,o,G,L,T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄) →
- ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄.
+lemma csx_intro_cpxs: ∀h,G,L,T1.
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄) →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄.
/4 width=1 by cpx_cpxs, csx_intro/ qed-.
(* Basic_1: was just: sn3_pr3_trans *)
-lemma csx_cpxs_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2
+lemma csx_cpxs_trans: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2
/2 width=3 by csx_cpx_trans/
qed-.
(* Eliminators with unbound context-sensitive rt-computation for terms ******)
-lemma csx_ind_cpxs_tdeq: ∀h,o,G,L. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → Q T2) → Q T1
+lemma csx_ind_cpxs_tdeq: ∀h,G,L. ∀Q:predicate term.
+ (∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀T0. ⦃G, L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛[h, o] T2 → Q T2.
-#h #o #G #L #Q #IH #T1 #H @(csx_ind … H) -T1
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T0. ⦃G,L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛ T2 → Q T2.
+#h #G #L #Q #IH #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IH1 #T0 #HT10 #T2 #HT02
@IH -IH /3 width=3 by csx_cpxs_trans, csx_tdeq_trans/ -HT1 #V2 #HTV2 #HnTV2
lapply (tdeq_tdneq_trans … HT02 … HnTV2) -HnTV2 #H
elim (tdeq_cpxs_trans … HT02 … HTV2) -T2 #V0 #HTV0 #HV02
lapply (tdneq_tdeq_canc_dx … H … HV02) -H #HnTV0
-elim (tdeq_dec h o T1 T0) #H
+elim (tdeq_dec T1 T0) #H
[ lapply (tdeq_tdneq_trans … H … HnTV0) -H -HnTV0 #Hn10
lapply (cpxs_trans … HT10 … HTV0) -T0 #H10
elim (cpxs_tdneq_fwd_step_sn … H10 … Hn10) -H10 -Hn10
qed-.
(* Basic_2A1: was: csx_ind_alt *)
-lemma csx_ind_cpxs: ∀h,o,G,L. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → Q T2) → Q T1
+lemma csx_ind_cpxs: ∀h,G,L. ∀Q:predicate term.
+ (∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → Q T.
-#h #o #G #L #Q #IH #T #HT
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q T.
+#h #G #L #Q #IH #T #HT
@(csx_ind_cpxs_tdeq … IH … HT) -IH -HT // (**) (* full auto fails *)
qed-.
(* Advanced properties ******************************************************)
-lemma csx_tdeq_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀T2. T1 ≛[h, o] T2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #G #L #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2
+lemma csx_tdeq_trans (h) (G):
+ ∀L,T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T2. T1 ≛ T2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #G #L #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2
@csx_intro #T1 #HT21 #HnT21 elim (tdeq_cpx_trans … HT2 … HT21) -HT21
/4 width=5 by tdeq_repl/
qed-.
-lemma csx_cpx_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
-elim (tdeq_dec h o T1 T2) /3 width=4 by csx_tdeq_trans/
+lemma csx_cpx_trans (h) (G):
+ ∀L,T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+elim (tdeq_dec T1 T2) /3 width=4 by csx_tdeq_trans/
qed-.
(* Basic_1: was just: sn3_cast *)
-lemma csx_cast: ∀h,o,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃W⦄ →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓝW.T⦄.
-#h #o #G #L #W #HW @(csx_ind … HW) -W
+lemma csx_cast (h) (G):
+ ∀L,W. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ →
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓝW.T⦄.
+#h #G #L #W #HW @(csx_ind … HW) -W
#W #HW #IHW #T #HT @(csx_ind … HT) -T
#T #HT #IHT @csx_intro
#X #H1 #H2 elim (cpx_inv_cast1 … H1) -H1
(* Basic_1: was just: sn3_abbr *)
(* Basic_2A1: was: csx_lref_bind *)
-lemma csx_lref_pair: ∀h,o,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V →
- ⦃G, K⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃#i⦄.
-#h #o #I #G #L #K #V #i #HLK #HV
+lemma csx_lref_pair_drops (h) (G):
+ ∀I,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V →
+ ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄.
+#h #G #I #L #K #V #i #HLK #HV
@csx_intro #X #H #Hi elim (cpx_inv_lref1_drops … H) -H
[ #H destruct elim Hi //
| -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
(* Basic_1: was: sn3_gen_def *)
(* Basic_2A1: was: csx_inv_lref_bind *)
-lemma csx_inv_lref_pair: ∀h,o,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V →
- ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃#i⦄ → ⦃G, K⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
-#h #o #I #G #L #K #V #i #HLK #Hi
+lemma csx_inv_lref_pair_drops (h) (G):
+ ∀I,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ → ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+#h #G #I #L #K #V #i #HLK #Hi
elim (lifts_total V (𝐔❴↑i❵))
/4 width=9 by csx_inv_lifts, csx_cpx_trans, cpx_delta_drops, drops_isuni_fwd_drop2/
qed-.
-lemma csx_inv_lref: ∀h,o,G,L,i. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃#i⦄ →
- ∨∨ ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆
- | ∃∃I,K. ⬇*[i] L ≘ K.ⓤ{I}
- | ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
-#h #o #G #L #i #H elim (drops_F_uni L i) /2 width=1 by or3_intro0/
-* * /4 width=9 by csx_inv_lref_pair, ex2_3_intro, ex1_2_intro, or3_intro2, or3_intro1/
+lemma csx_inv_lref_drops (h) (G):
+ ∀L,i. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ →
+ ∨∨ ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆
+ | ∃∃I,K. ⬇*[i] L ≘ K.ⓤ{I}
+ | ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+#h #G #L #i #H elim (drops_F_uni L i) /2 width=1 by or3_intro0/
+* * /4 width=9 by csx_inv_lref_pair_drops, ex2_3_intro, ex1_2_intro, or3_intro2, or3_intro1/
qed-.
(* *)
(**************************************************************************)
-include "basic_2/rt_computation/cpxs_theq_vector.ma".
-include "basic_2/rt_computation/csx_simple_theq.ma".
+include "basic_2/rt_computation/cpxs_toeq_vector.ma".
+include "basic_2/rt_computation/csx_simple_toeq.ma".
include "basic_2/rt_computation/csx_lsubr.ma".
include "basic_2/rt_computation/csx_lpx.ma".
include "basic_2/rt_computation/csx_vector.ma".
(* Advanced properties ************************************* ****************)
(* Basic_1: was just: sn3_appls_beta *)
-lemma csx_applv_beta: ∀h,o,p,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.ⓓ{p}ⓝW.V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.ⓐV.ⓛ{p}W.T⦄.
-#h #o #p #G #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
+lemma csx_applv_beta (h) (G):
+ ∀p,L,Vs,V,W,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓓ{p}ⓝW.V.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓐV.ⓛ{p}W.T⦄.
+#h #G #p #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
#V0 #Vs #IHV #V #W #T #H1T
lapply (csx_fwd_pair_sn … H1T) #HV0
lapply (csx_fwd_flat_dx … H1T) #H2T
-@csx_appl_simple_theq /2 width=1 by applv_simple, simple_flat/ -IHV -HV0 -H2T
+@csx_appl_simple_toeq /2 width=1 by applv_simple, simple_flat/ -IHV -HV0 -H2T
#X #H #H0
-elim (cpxs_fwd_beta_vector … o … H) -H #H
+elim (cpxs_fwd_beta_vector … H) -H #H
[ -H1T elim H0 -H0 //
| -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
]
qed.
-lemma csx_applv_delta: ∀h,o,I,G,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
- ∀V2. ⬆*[↑i] V1 ≘ V2 →
- ∀Vs. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.V2⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.#i⦄.
-#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
-[ /4 width=11 by csx_inv_lifts, csx_lref_pair, drops_isuni_fwd_drop2/
+lemma csx_applv_delta_drops (h) (G):
+ ∀I,L,K,V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
+ ∀V2. ⬆*[↑i] V1 ≘ V2 →
+ ∀Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.V2⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.#i⦄.
+#h #G #I #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
+[ /4 width=11 by csx_inv_lifts, csx_lref_pair_drops, drops_isuni_fwd_drop2/
| #V #Vs #IHV #H1T
lapply (csx_fwd_pair_sn … H1T) #HV
lapply (csx_fwd_flat_dx … H1T) #H2T
- @csx_appl_simple_theq /2 width=1 by applv_simple, simple_atom/ -IHV -HV -H2T
+ @csx_appl_simple_toeq /2 width=1 by applv_simple, simple_atom/ -IHV -HV -H2T
#X #H #H0
- elim (cpxs_fwd_delta_drops_vector … o … HLK … HV12 … H) -HLK -HV12 -H #H
+ elim (cpxs_fwd_delta_drops_vector … HLK … HV12 … H) -HLK -HV12 -H #H
[ -H1T elim H0 -H0 //
| -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
]
qed.
(* Basic_1: was just: sn3_appls_abbr *)
-lemma csx_applv_theta: ∀h,o,p,G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
- ∀V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓓ{p}V.ⒶV2b.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶV1b.ⓓ{p}V.T⦄.
-#h #o #p #G #L #V1b #V2b * -V1b -V2b /2 width=1 by/
+lemma csx_applv_theta (h) (G):
+ ∀p,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
+ ∀V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⒶV2b.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶV1b.ⓓ{p}V.T⦄.
+#h #G #p #L #V1b #V2b * -V1b -V2b /2 width=1 by/
#V1b #V2b #V1 #V2 #HV12 #H
generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
elim H -V1b -V2b /2 width=3 by csx_appl_theta/
lapply (csx_appl_theta … H … HW12) -H -HW12 #H
lapply (csx_fwd_pair_sn … H) #HW1
lapply (csx_fwd_flat_dx … H) #H1
-@csx_appl_simple_theq /2 width=3 by simple_flat/ -IHV12b -HW1 -H1 #X #H1 #H2
-elim (cpxs_fwd_theta_vector … o … (V2⨮V2b) … H1) -H1 /2 width=1 by liftsv_cons/ -HV12b -HV12
+@csx_appl_simple_toeq /2 width=3 by simple_flat/ -IHV12b -HW1 -H1 #X #H1 #H2
+elim (cpxs_fwd_theta_vector … (V2⨮V2b) … H1) -H1 /2 width=1 by liftsv_cons/ -HV12b -HV12
[ -H #H elim H2 -H2 //
| -H2 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
]
qed.
(* Basic_1: was just: sn3_appls_cast *)
-lemma csx_applv_cast: ∀h,o,G,L,Vs,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.U⦄ →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.ⓝU.T⦄.
-#h #o #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
+lemma csx_applv_cast (h) (G):
+ ∀L,Vs,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.U⦄ →
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.ⓝU.T⦄.
+#h #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
#V #Vs #IHV #U #H1U #T #H1T
lapply (csx_fwd_pair_sn … H1U) #HV
lapply (csx_fwd_flat_dx … H1U) #H2U
lapply (csx_fwd_flat_dx … H1T) #H2T
-@csx_appl_simple_theq /2 width=1 by applv_simple, simple_flat/ -IHV -HV -H2U -H2T
+@csx_appl_simple_toeq /2 width=1 by applv_simple, simple_flat/ -IHV -HV -H2U -H2T
#X #H #H0
-elim (cpxs_fwd_cast_vector … o … H) -H #H
+elim (cpxs_fwd_cast_vector … H) -H #H
[ -H1U -H1T elim H0 -H0 //
| -H1U -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
| -H1T -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
(* Basic_1: was just: sn3_lift *)
(* Basic_2A1: was just: csx_lift *)
-lemma csx_lifts: ∀h,o,G. d_liftable1 … (csx h o G).
-#h #o #G #K #T #H @(csx_ind … H) -T
+lemma csx_lifts: ∀h,G. d_liftable1 … (csx h G).
+#h #G #K #T #H @(csx_ind … H) -T
#T1 #_ #IH #b #f #L #HLK #U1 #HTU1
@csx_intro #U2 #HU12 #HnU12
elim (cpx_inv_lifts_sn … HU12 … HLK … HTU1) -HU12
(* Basic_1: was just: sn3_gen_lift *)
(* Basic_2A1: was just: csx_inv_lift *)
-lemma csx_inv_lifts: ∀h,o,G. d_deliftable1 … (csx h o G).
-#h #o #G #L #U #H @(csx_ind … H) -U
+lemma csx_inv_lifts: ∀h,G. d_deliftable1 … (csx h G).
+#h #G #L #U #H @(csx_ind … H) -U
#U1 #_ #IH #b #f #K #HLK #T1 #HTU1
@csx_intro #T2 #HT12 #HnT12
elim (cpx_lifts_sn … HT12 … HLK … HTU1) -HT12
(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
-(* Properties with degree-based equivalence for closures ********************)
+(* Properties with sort-irrelevant equivalence for closures *****************)
-lemma csx_fdeq_conf: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #G1 #L1 #T1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
+lemma csx_fdeq_conf: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #G1 #L1 #T1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
/3 width=3 by csx_rdeq_conf, csx_tdeq_trans/
qed-.
(* Properties with parallel rst-transition for closures *********************)
(* Basic_2A1: was: csx_fpb_conf *)
-lemma csx_fpbq_conf: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #G1 #L1 #T1 #HT1 #G2 #L2 #T2 *
+lemma csx_fpbq_conf: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #G1 #L1 #T1 #HT1 #G2 #L2 #T2 *
/2 width=6 by csx_cpx_trans, csx_fquq_conf, csx_lpx_conf, csx_fdeq_conf/
qed-.
(* Properties with extended supclosure **************************************)
-lemma csx_fqu_conf: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[ /3 width=5 by csx_inv_lref_pair, drops_refl/
+lemma csx_fqu_conf (h) (b):
+ ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ /3 width=5 by csx_inv_lref_pair_drops, drops_refl/
| /2 width=3 by csx_fwd_pair_sn/
| /2 width=2 by csx_fwd_bind_dx/
| /2 width=4 by csx_fwd_bind_dx_unit/
]
qed-.
-lemma csx_fquq_conf: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=6 by csx_fqu_conf/
+lemma csx_fquq_conf (h) (b):
+ ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=6 by csx_fqu_conf/
* #HG #HL #HT destruct //
qed-.
-lemma csx_fqup_conf: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+lemma csx_fqup_conf (h) (b):
+ ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
/3 width=6 by csx_fqu_conf/
qed-.
-lemma csx_fqus_conf: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -H
+lemma csx_fqus_conf (h) (b):
+ ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -H
/3 width=6 by csx_fquq_conf/
qed-.
(* Main properties with generic computation properties **********************)
-theorem csx_gcp: ∀h,o. gcp (cpx h) (tdeq h o) (csx h o).
-#h #o @mk_gcp
+theorem csx_gcp: ∀h. gcp (cpx h) tdeq (csx h).
+#h @mk_gcp
[ normalize /3 width=13 by cnx_lifts/
-| #G #L elim (deg_total h o 0) /3 width=8 by cnx_sort_iter, ex_intro/
+| /2 width=4 by cnx_sort/
| /2 width=8 by csx_lifts/
| /2 width=3 by csx_fwd_flat_dx/
]
(* Main properties with generic candidates of reducibility ******************)
-theorem csx_gcr: ∀h,o. gcr (cpx h) (tdeq h o) (csx h o) (csx h o).
-#h #o @mk_gcr //
-[ /3 width=1 by csx_applv_cnx/
-|2,3,6: /2 width=1 by csx_applv_beta, csx_applv_sort, csx_applv_cast/
-| /2 width=7 by csx_applv_delta/
-| #G #L #V1b #V2b #HV12b #a #V #T #H #HV
- @(csx_applv_theta … HV12b) -HV12b
- @csx_abbr //
+theorem csx_gcr (h): gcr (cpx h) tdeq (csx h) (csx h).
+#h @mk_gcr
+[ //
+| #G #L #Vs #Hvs #T #HT #H
+ @(csx_applv_cnx … H) -H // (**) (* auto fails *)
+| /2 width=1 by csx_applv_beta/
+| /2 width=7 by csx_applv_delta_drops/
+| /3 width=3 by csx_applv_theta, csx_abbr/
+| /2 width=1 by csx_applv_cast/
]
qed.
(* Properties with unbound parallel rt-transition on all entries ************)
-lemma csx_lpx_conf: ∀h,o,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
-#h #o #G #L1 #T #H @(csx_ind_cpxs … H) -T
+lemma csx_lpx_conf (h) (G):
+ ∀L1,T. ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #G #L1 #T #H @(csx_ind_cpxs … H) -T
/4 width=3 by csx_intro, lpx_cpx_trans/
qed-.
(* Advanced properties ******************************************************)
-lemma csx_abst: ∀h,o,p,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃W⦄ →
- ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓛ{p}W.T⦄.
-#h #o #p #G #L #W #HW
+lemma csx_abst (h) (G):
+ ∀p,L,W. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ →
+ ∀T. ⦃G,L.ⓛW⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓛ{p}W.T⦄.
+#h #G #p #L #W #HW
@(csx_ind … HW) -W #W #_ #IHW #T #HT
@(csx_ind … HT) -T #T #HT #IHT
@csx_intro #X #H1 #H2
elim (cpx_inv_abst1 … H1) -H1 #W0 #T0 #HLW0 #HLT0 #H destruct
elim (tdneq_inv_pair … H2) -H2
[ #H elim H -H //
-| -IHT #H lapply (csx_cpx_trans … o … HLT0) // -HT #HT0
+| -IHT #H lapply (csx_cpx_trans … HLT0) // -HT #HT0
/4 width=5 by csx_lpx_conf, lpx_pair/
| -IHW -HT /4 width=3 by csx_cpx_trans, cpx_pair_sn/
]
qed.
-lemma csx_abbr: ∀h,o,p,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ →
- ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓓ{p}V.T⦄.
-#h #o #p #G #L #V #HV
+lemma csx_abbr (h) (G):
+ ∀p,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ →
+ ∀T. ⦃G,L.ⓓV⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.T⦄.
+#h #G #p #L #V #HV
@(csx_ind … HV) -V #V #_ #IHV #T #HT
@(csx_ind_cpxs … HT) -T #T #HT #IHT
@csx_intro #X #H1 #H2
]
qed.
-fact csx_appl_theta_aux: ∀h,o,p,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ → ∀V1,V2. ⬆*[1] V1 ≘ V2 →
- ∀V,T. U = ⓓ{p}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
-#h #o #p #G #L #X #H
+lemma csx_bind (h) (G):
+ ∀p,I,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ →
+ ∀T. ⦃G,L.ⓑ{I}V⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄.
+#h #G #p * #L #V #HV #T #HT
+/2 width=1 by csx_abbr, csx_abst/
+qed.
+
+fact csx_appl_theta_aux (h) (G):
+ ∀p,L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → ∀V1,V2. ⬆*[1] V1 ≘ V2 →
+ ∀V,T. U = ⓓ{p}V.ⓐV2.T → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
+#h #G #p #L #X #H
@(csx_ind_cpxs … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
lapply (csx_fwd_pair_sn … HVT) #HV
lapply (csx_fwd_bind_dx … HVT) -HVT #HVT
elim (cpx_inv_abbr1 … HL) -HL *
[ #V3 #T3 #HV3 #HLT3 #H0 destruct
elim (cpx_lifts_sn … HLV10 (Ⓣ) … (L.ⓓV) … HV12) -HLV10 /3 width=1 by drops_refl, drops_drop/ #V4 #HV04 #HV24
- elim (tdeq_dec h o (ⓓ{p}V.ⓐV2.T) (ⓓ{p}V3.ⓐV4.T3)) #H0
+ elim (tdeq_dec (ⓓ{p}V.ⓐV2.T) (ⓓ{p}V3.ⓐV4.T3)) #H0
[ -IHVT -HV3 -HV24 -HLT3
elim (tdeq_inv_pair … H0) -H0 #_ #HV3 #H0
elim (tdeq_inv_pair … H0) -H0 #_ #HV24 #HT3
elim (tdneq_inv_pair … H) -H #H elim H -H -G -L
/3 width=6 by tdeq_inv_lifts_bi, tdeq_pair/
- | -V1 @(IHVT … H0 … HV04) -o -V0 /4 width=1 by cpx_cpxs, cpx_flat, cpx_bind/
+ | -V1 @(IHVT … H0 … HV04) -V0 /4 width=1 by cpx_cpxs, cpx_flat, cpx_bind/
]
| #T0 #HT0 #HLT0 #H0 destruct -H -IHVT
lapply (csx_inv_lifts … HVT (Ⓣ) … L ???) -HVT
]
qed-.
-lemma csx_appl_theta: ∀h,o,p,G,L,V,V2,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓓ{p}V.ⓐV2.T⦄ →
- ∀V1. ⬆*[1] V1 ≘ V2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
+lemma csx_appl_theta (h) (G):
+ ∀p,L,V,V2,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⓐV2.T⦄ →
+ ∀V1. ⬆*[1] V1 ≘ V2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
/2 width=5 by csx_appl_theta_aux/ qed.
(* Properties with unbound parallel rt-computation on all entries ***********)
-lemma csx_lpxs_conf: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- ⦃G, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
-#h #o #G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2
+lemma csx_lpxs_conf: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2
/3 by lpxs_step_dx, csx_lpx_conf/
qed-.
(* Advanced properties ******************************************************)
-fact csx_appl_beta_aux: ∀h,o,p,G,L,U1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U1⦄ →
- ∀V,W,T1. U1 = ⓓ{p}ⓝW.V.T1 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV.ⓛ{p}W.T1⦄.
-#h #o #p #G #L #X #H @(csx_ind … H) -X
+fact csx_appl_beta_aux (h) (G):
+ ∀p,L,U1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U1⦄ →
+ ∀V,W,T1. U1 = ⓓ{p}ⓝW.V.T1 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.ⓛ{p}W.T1⦄.
+#h #G #p #L #X #H @(csx_ind … H) -X
#X #HT1 #IHT1 #V #W #T1 #H1 destruct
@csx_intro #X #H1 #H2
elim (cpx_inv_appl1 … H1) -H1 *
qed-.
(* Basic_1: was just: sn3_beta *)
-lemma csx_appl_beta: ∀h,o,p,G,L,V,W,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓓ{p}ⓝW.V.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV.ⓛ{p}W.T⦄.
+lemma csx_appl_beta (h) (G):
+ ∀p,L,V,W,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}ⓝW.V.T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.ⓛ{p}W.T⦄.
/2 width=3 by csx_appl_beta_aux/ qed.
(* Advanced forward lemmas **************************************************)
-fact csx_fwd_bind_dx_unit_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ →
- ∀p,I,J,V,T. U = ⓑ{p,I}V.T → ⦃G, L.ⓤ{J}⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
-#h #o #G #L #U #H elim H -H #U0 #_ #IH #p #I #J #V #T #H destruct
+fact csx_fwd_bind_dx_unit_aux (h) (G):
+ ∀L,U. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ →
+ ∀p,I,J,V,T. U = ⓑ{p,I}V.T → ⦃G,L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #G #L #U #H elim H -H #U0 #_ #IH #p #I #J #V #T #H destruct
@csx_intro #T2 #HLT2 #HT2
-@(IH (ⓑ{p,I}V.T2)) -IH /2 width=4 by cpx_bind_unit/ -HLT2
+@(IH (ⓑ{p, I}V.T2)) -IH /2 width=4 by cpx_bind_unit/ -HLT2
#H elim (tdeq_inv_pair … H) -H /2 width=1 by/
qed-.
-lemma csx_fwd_bind_dx_unit: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ →
- ∀J. ⦃G, L.ⓤ{J}⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_bind_dx_unit (h) (G):
+ ∀p,I,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ →
+ ∀J. ⦃G,L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/2 width=6 by csx_fwd_bind_dx_unit_aux/ qed-.
-lemma csx_fwd_bind_unit: ∀h,o,p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓑ{p,I}V.T⦄ →
- ∀J. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ ∧ ⦃G, L.ⓤ{J}⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
+lemma csx_fwd_bind_unit (h) (G):
+ ∀p,I,L,V,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓑ{p,I}V.T⦄ →
+ ∀J. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ ∧ ⦃G,L.ⓤ{J}⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/3 width=4 by csx_fwd_pair_sn, csx_fwd_bind_dx_unit, conj/ qed-.
+
+(* Properties with restricted refinement for local environments *************)
+
+lemma csx_lsubr_conf (h) (G):
+ ∀L1,T. ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ∀L2. L1 ⫃ L2 → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #G #L1 #T #H
+@(csx_ind … H) -T #T1 #_ #IH #L2 #HL12
+@csx_intro #T2 #HT12 #HnT12
+/3 width=3 by lsubr_cpx_trans/
+qed-.
(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
-(* Properties with degree-based equivalence for local environments **********)
+(* Properties with sort-irrelevant equivalence for local environments *******)
(* Basic_2A1: uses: csx_lleq_conf *)
-lemma csx_rdeq_conf: ∀h,o,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ →
- ∀L2. L1 ≛[h, o, T] L2 → ⦃G, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
-#h #o #G #L1 #T #H
+lemma csx_rdeq_conf: ∀h,G,L1,T. ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ →
+ ∀L2. L1 ≛[T] L2 → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #G #L1 #T #H
@(csx_ind … H) -T #T1 #_ #IH #L2 #HL12
@csx_intro #T2 #HT12 #HnT12
elim (rdeq_cpx_trans … HL12 … HT12) -HT12
-/5 width=4 by cpx_rdeq_conf_sn, csx_tdeq_trans, tdeq_trans/
+/5 width=5 by cpx_rdeq_conf_sn, csx_tdeq_trans, tdeq_trans/
qed-.
(* Basic_2A1: uses: csx_lleq_conf *)
-lemma csx_rdeq_trans: ∀h,o,L1,L2,T. L1 ≛[h, o, T] L2 →
- ∀G. ⦃G, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
+lemma csx_rdeq_trans: ∀h,L1,L2,T. L1 ≛[T] L2 →
+ ∀G. ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
/3 width=3 by csx_rdeq_conf, rdeq_sym/ qed-.
(* Properties with simple terms *********************************************)
-lemma csx_appl_simple: ∀h,o,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ → ∀T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV.T2⦄) →
- 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV.T1⦄.
-#h #o #G #L #V #H @(csx_ind … H) -V
+lemma csx_appl_simple: ∀h,G,L,V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ∀T1.
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T2⦄) →
+ 𝐒⦃T1⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T1⦄.
+#h #G #L #V #H @(csx_ind … H) -V
#V #_ #IHV #T1 #IHT1 #HT1
@csx_intro #X #H1 #H2
elim (cpx_inv_appl1_simple … H1) // -H1
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/syntax/theq_simple.ma".
-include "static_2/syntax/theq_theq.ma".
-include "basic_2/rt_transition/cpx_simple.ma".
-include "basic_2/rt_computation/cpxs.ma".
-include "basic_2/rt_computation/csx_csx.ma".
-
-(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
-
-(* Properties with head equivalence for terms *******************************)
-
-(* Basic_1: was just: sn3_appl_appl *)
-(* Basic_2A1: was: csx_appl_simple_tsts *)
-lemma csx_appl_simple_theq: ∀h,o,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ → ∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ⩳[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV.T2⦄) →
- 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV.T1⦄.
-#h #o #G #L #V #H @(csx_ind … H) -V
-#V #_ #IHV #T1 #H @(csx_ind … H) -T1
-#T1 #H1T1 #IHT1 #H2T1 #H3T1
-@csx_intro #X #HL #H
-elim (cpx_inv_appl1_simple … HL) -HL //
-#V0 #T0 #HLV0 #HLT10 #H0 destruct
-elim (tdneq_inv_pair … H) -H
-[ #H elim H -H //
-| -IHT1 #HV0
- @(csx_cpx_trans … (ⓐV0.T1)) /2 width=1 by cpx_flat/ -HLT10
- @IHV -IHV /4 width=3 by csx_cpx_trans, cpx_pair_sn/
-| -IHV -H1T1 #H1T10
- @(csx_cpx_trans … (ⓐV.T0)) /2 width=1 by cpx_flat/ -HLV0
- elim (theq_dec h o T1 T0) #H2T10
- [ @IHT1 -IHT1 /4 width=5 by cpxs_strap2, cpxs_strap1, theq_canc_sn, simple_theq_repl_dx/
- | -IHT1 -H3T1 -H1T10 /3 width=1 by cpx_cpxs/
- ]
-]
-qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/toeq_simple.ma".
+include "static_2/syntax/toeq_toeq.ma".
+include "basic_2/rt_transition/cpx_simple.ma".
+include "basic_2/rt_computation/cpxs.ma".
+include "basic_2/rt_computation/csx_csx.ma".
+
+(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
+
+(* Properties with outer equivalence for terms ******************************)
+
+(* Basic_1: was just: sn3_appl_appl *)
+(* Basic_2A1: was: csx_appl_simple_tsts *)
+lemma csx_appl_simple_toeq (h) (G) (L):
+ ∀V. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ⩳ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T2⦄) →
+ 𝐒⦃T1⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV.T1⦄.
+#h #G #L #V #H @(csx_ind … H) -V
+#V #_ #IHV #T1 #H @(csx_ind … H) -T1
+#T1 #H1T1 #IHT1 #H2T1 #H3T1
+@csx_intro #X #HL #H
+elim (cpx_inv_appl1_simple … HL) -HL //
+#V0 #T0 #HLV0 #HLT10 #H0 destruct
+elim (tdneq_inv_pair … H) -H
+[ #H elim H -H //
+| -IHT1 #HV0
+ @(csx_cpx_trans … (ⓐV0.T1)) /2 width=1 by cpx_flat/ -HLT10
+ @IHV -IHV /4 width=3 by csx_cpx_trans, cpx_pair_sn/
+| -IHV -H1T1 #H1T10
+ @(csx_cpx_trans … (ⓐV.T0)) /2 width=1 by cpx_flat/ -HLV0
+ elim (toeq_dec T1 T0) #H2T10
+ [ @IHT1 -IHT1 /4 width=5 by cpxs_strap2, cpxs_strap1, toeq_canc_sn, simple_toeq_repl_dx/
+ | -IHT1 -H3T1 -H1T10 /3 width=1 by cpx_cpxs/
+ ]
+]
+qed.
(* STRONGLY NORMALIZING TERMS VECTORS FOR UNBOUND PARALLEL RT-TRANSITION ****)
-definition csxv: ∀h. sd h → relation3 genv lenv (list term) ≝
- λh,o,G,L. all … (csx h o G L).
+definition csxv: ∀h. relation3 genv lenv (list term) ≝
+ λh,G,L. all … (csx h G L).
interpretation
"strong normalization for unbound context-sensitive parallel rt-transition (term vector)"
- 'PRedTyStrong h o G L Ts = (csxv h o G L Ts).
+ 'PRedTyStrong h G L Ts = (csxv h G L Ts).
(* Basic inversion lemmas ***************************************************)
-lemma csxv_inv_cons: ∀h,o,G,L,T,Ts. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⨮Ts⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃Ts⦄.
+lemma csxv_inv_cons: ∀h,G,L,T,Ts. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⨮Ts⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ ∧ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Ts⦄.
normalize // qed-.
(* Basic forward lemmas *****************************************************)
-lemma csx_fwd_applv: ∀h,o,G,L,T,Vs. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.T⦄ →
- ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃Vs⦄ ∧ ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
-#h #o #G #L #T #Vs elim Vs -Vs /2 width=1 by conj/
+lemma csx_fwd_applv: ∀h,G,L,T,Vs. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃ⒶVs.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃Vs⦄ ∧ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #G #L #T #Vs elim Vs -Vs /2 width=1 by conj/
#V #Vs #IHVs #HVs
lapply (csx_fwd_pair_sn … HVs) #HV
lapply (csx_fwd_flat_dx … HVs) -HVs #HVs
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predsubtystarproper_8.ma".
+include "basic_2/notation/relations/predsubtystarproper_7.ma".
include "basic_2/rt_transition/fpb.ma".
include "basic_2/rt_computation/fpbs.ma".
(* PROPER PARALLEL RST-COMPUTATION FOR CLOSURES *****************************)
-definition fpbg: ∀h. sd h → tri_relation genv lenv term ≝
- λh,o,G1,L1,T1,G2,L2,T2.
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+definition fpbg: ∀h. tri_relation genv lenv term ≝
+ λh,G1,L1,T1,G2,L2,T2.
+ ∃∃G,L,T. ⦃G1,L1,T1⦄ ≻[h] ⦃G,L,T⦄ & ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄.
interpretation "proper parallel rst-computation (closure)"
- 'PRedSubTyStarProper h o G1 L1 T1 G2 L2 T2 = (fpbg h o G1 L1 T1 G2 L2 T2).
+ 'PRedSubTyStarProper h G1 L1 T1 G2 L2 T2 = (fpbg h G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma fpb_fpbg: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
+lemma fpb_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/2 width=5 by ex2_3_intro/ qed.
-lemma fpbg_fpbq_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ >[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
+lemma fpbg_fpbq_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≽[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
+#h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
/3 width=9 by fpbs_strap1, ex2_3_intro/
qed-.
+lemma fpbg_fqu_trans (h): ∀G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⬂ ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
+#h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
+/4 width=5 by fpbg_fpbq_trans, fpbq_fquq, fqu_fquq/
+qed-.
+
(* Note: this is used in the closure proof *)
-lemma fpbg_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ >[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
-#h #o #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
+lemma fpbg_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
+#h #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
qed-.
(* Basic_2A1: uses: fpbg_fleq_trans *)
-lemma fpbg_fdeq_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >[h, o] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
+lemma fpbg_fdeq_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ >[h] ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbg_fpbq_trans, fpbq_fdeq/ qed-.
(* Properties with t-bound rt-transition for terms **************************)
-lemma cpm_tdneq_cpm_fpbg (h) (o) (G) (L):
- ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛[h,o] T → ⊥) →
- ∀n2,T2. ⦃G, L⦄ ⊢ T ➡[n2,h] T2 →
- ⦃G, L, T1⦄ >[h,o] ⦃G, L, T2⦄.
+lemma cpm_tdneq_cpm_fpbg (h) (G) (L):
+ ∀n1,T1,T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) →
+ ∀n2,T2. ⦃G,L⦄ ⊢ T ➡[n2,h] T2 → ⦃G,L,T1⦄ >[h] ⦃G,L,T2⦄.
/4 width=5 by fpbq_fpbs, cpm_fpbq, cpm_fpb, ex2_3_intro/ qed.
(* Properties with unbound context-sensitive parallel rt-computation ********)
(* Basic_2A1: was: cpxs_fpbg *)
-lemma cpxs_tdneq_fpbg (h) (o): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- (T1 ≛[h, o] T2 → ⊥) → ⦃G, L, T1⦄ >[h, o] ⦃G, L, T2⦄.
-#h #o #G #L #T1 #T2 #H #H0
+lemma cpxs_tdneq_fpbg (h): ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ (T1 ≛ T2 → ⊥) → ⦃G,L,T1⦄ >[h] ⦃G,L,T2⦄.
+#h #G #L #T1 #T2 #H #H0
elim (cpxs_tdneq_fwd_step_sn … H … H0) -H -H0
/4 width=5 by cpxs_tdeq_fpbs, fpb_cpx, ex2_3_intro/
qed.
-lemma cpxs_fpbg_trans (h) (o): ∀G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G2,L2,T2. ⦃G1, L1, T⦄ >[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
+lemma cpxs_fpbg_trans (h): ∀G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G2,L2,T2. ⦃G1,L1,T⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fpbg_trans, cpxs_fpbs/ qed-.
(* Main properties **********************************************************)
-theorem fpbg_trans: ∀h,o. tri_transitive … (fpbg h o).
+theorem fpbg_trans: ∀h. tri_transitive … (fpbg h).
/3 width=5 by fpbg_fpbs_trans, fpbg_fwd_fpbs/ qed-.
(* Advanced forward lemmas **************************************************)
-lemma fpbg_fwd_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ >[h,o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 *
+lemma fpbg_fwd_fpbs: ∀h,G1,G2,L1,L2,T1,T2.
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 *
/3 width=5 by fpbs_strap2, fpb_fpbq/
qed-.
-(* Advanced properties with degree-based equivalence on closures ************)
+(* Advanced properties with sort-irrelevant equivalence on closures *********)
(* Basic_2A1: uses: fleq_fpbg_trans *)
-lemma fdeq_fpbg_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >[h, o] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
-#h #o #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1
+lemma fdeq_fpbg_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
+#h #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1
elim (fdeq_fpb_trans … H1 … H0) -G -L -T
/4 width=9 by fpbs_strap2, fpbq_fdeq, ex2_3_intro/
qed-.
(* Properties with parallel proper rst-reduction on closures ****************)
-lemma fpb_fpbg_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ≻[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ >[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
+lemma fpb_fpbg_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1,L1,T1⦄ ≻[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbg_fwd_fpbs, ex2_3_intro/ qed-.
(* Properties with parallel rst-reduction on closures ***********************)
-lemma fpbq_fpbg_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ >[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
+lemma fpbq_fpbg_trans: ∀h,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1,L1,T1⦄ ≽[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
+#h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
elim (fpbq_inv_fpb … H1) -H1
/2 width=5 by fdeq_fpbg_trans, fpb_fpbg_trans/
qed-.
(* Properties with parallel rst-compuutation on closures ********************)
-lemma fpbs_fpbg_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ >[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/
+lemma fpbs_fpbg_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
+#h #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/
qed-.
(* Advanced properties with plus-iterated structural successor for closures *)
-lemma fqup_fpbg_trans (h) (o):
- â\88\80G1,G,L1,L,T1,T. â¦\83G1,L1,T1â¦\84 â\8a\90+ ⦃G,L,T⦄ →
- ∀G2,L2,T2. ⦃G,L,T⦄ >[h,o] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h,o] ⦃G2,L2,T2⦄.
+lemma fqup_fpbg_trans (h):
+ â\88\80G1,G,L1,L,T1,T. â¦\83G1,L1,T1â¦\84 â¬\82+ ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ >[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fpbg_trans, fqup_fpbs/ qed-.
(* Advanced inversion lemmas of parallel rst-computation on closures ********)
(* Basic_2A1: was: fpbs_fpbg *)
-lemma fpbs_inv_fpbg: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∨∨ ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄
- | ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
+lemma fpbs_inv_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∨∨ ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄
+ | ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
[ /2 width=1 by or_introl/
| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1
elim (fpbq_inv_fpb … H2) -H2 #H2
(* Advanced properties of parallel rst-computation on closures **************)
-lemma fpbs_fpb_trans: ∀h,o,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, o] ⦃F2, K2, T2⦄ →
- ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, o] ⦃G2, L2, U2⦄ →
- ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, o] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, o] ⦃G2, L2, U2⦄.
-#h #o #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_inv_fpbg … H) -H
+lemma fpbs_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ⦃F1,K1,T1⦄ ≥[h] ⦃F2,K2,T2⦄ →
+ ∀G2,L2,U2. ⦃F2,K2,T2⦄ ≻[h] ⦃G2,L2,U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1,K1,T1⦄ ≻[h] ⦃G1,L1,U1⦄ & ⦃G1,L1,U1⦄ ≥[h] ⦃G2,L2,U2⦄.
+#h #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_inv_fpbg … H) -H
[ #H12 #G2 #L2 #U2 #H2 elim (fdeq_fpb_trans … H12 … H2) -F2 -K2 -T2
/3 width=5 by fdeq_fpbs, ex2_3_intro/
| * #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
(* PROPER PARALLEL RST-COMPUTATION FOR CLOSURES *****************************)
-(* Advanced properties with degree-based equivalence for terms **************)
+(* Advanced properties with sort-irrelevant equivalence for terms ***********)
-lemma fpbg_tdeq_div: ∀h,o,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T⦄ →
- ∀T2. T2 ≛[h, o] T → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
+lemma fpbg_tdeq_div: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T⦄ →
+ ∀T2. T2 ≛ T → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
/4 width=5 by fpbg_fdeq_trans, tdeq_fdeq, tdeq_sym/ qed-.
(* Properties with plus-iterated structural successor for closures **********)
(* Note: this is used in the closure proof *)
-lemma fqup_fpbg: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
+lemma fqup_fpbg: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
/3 width=5 by fqus_fpbs, fpb_fqu, ex2_3_intro/
qed.
(* Properties with unbound rt-computation on full local environments ********)
(* Basic_2A1: uses: lpxs_fpbg *)
-lemma lpxs_rdneq_fpbg: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- (L1 ≛[h, o, T] L2 → ⊥) → ⦃G, L1, T⦄ >[h, o] ⦃G, L2, T⦄.
-#h #o #G #L1 #L2 #T #H #H0
+lemma lpxs_rdneq_fpbg: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ (L1 ≛[T] L2 → ⊥) → ⦃G,L1,T⦄ >[h] ⦃G,L2,T⦄.
+#h #G #L1 #L2 #T #H #H0
elim (lpxs_rdneq_inv_step_sn … H … H0) -H -H0
/4 width=7 by fpb_lpx, lpxs_fdeq_fpbs, fdeq_intro_sn, ex2_3_intro/
qed.
(**************************************************************************)
include "ground_2/lib/star.ma".
-include "basic_2/notation/relations/predsubtystar_8.ma".
+include "basic_2/notation/relations/predsubtystar_7.ma".
include "basic_2/rt_transition/fpbq.ma".
(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
-definition fpbs: ∀h. sd h → tri_relation genv lenv term ≝
- λh,o. tri_TC … (fpbq h o).
+definition fpbs: ∀h. tri_relation genv lenv term ≝
+ λh. tri_TC … (fpbq h).
interpretation "parallel rst-computation (closure)"
- 'PRedSubTyStar h o G1 L1 T1 G2 L2 T2 = (fpbs h o G1 L1 T1 G2 L2 T2).
+ 'PRedSubTyStar h G1 L1 T1 G2 L2 T2 = (fpbs h G1 L1 T1 G2 L2 T2).
(* Basic eliminators ********************************************************)
-lemma fpbs_ind: ∀h,o,G1,L1,T1. ∀Q:relation3 genv lenv term. Q G1 L1 T1 →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, o] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
+lemma fpbs_ind: ∀h,G1,L1,T1. ∀Q:relation3 genv lenv term. Q G1 L1 T1 →
+ (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≽[h] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2.
/3 width=8 by tri_TC_star_ind/ qed-.
-lemma fpbs_ind_dx: ∀h,o,G2,L2,T2. ∀Q:relation3 genv lenv term. Q G2 L2 T2 →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → Q G1 L1 T1.
+lemma fpbs_ind_dx: ∀h,G2,L2,T2. ∀Q:relation3 genv lenv term. Q G2 L2 T2 →
+ (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≽[h] ⦃G,L,T⦄ → ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G1 L1 T1.
/3 width=8 by tri_TC_star_ind_dx/ qed-.
(* Basic properties *********************************************************)
-lemma fpbs_refl: ∀h,o. tri_reflexive … (fpbs h o).
+lemma fpbs_refl: ∀h. tri_reflexive … (fpbs h).
/2 width=1 by tri_inj/ qed.
-lemma fpbq_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fpbq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/2 width=1 by tri_inj/ qed.
-lemma fpbs_strap1: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≽[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fpbs_strap1: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ⦃G,L,T⦄ ≽[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/2 width=5 by tri_step/ qed-.
-lemma fpbs_strap2: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fpbs_strap2: ∀h,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G,L,T⦄ →
+ ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/2 width=5 by tri_TC_strap/ qed-.
(* Basic_2A1: uses: lleq_fpbs fleq_fpbs *)
-lemma fdeq_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fdeq_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=1 by fpbq_fpbs, fpbq_fdeq/ qed.
(* Basic_2A1: uses: fpbs_lleq_trans *)
-lemma fpbs_fdeq_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fpbs_fdeq_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=9 by fpbs_strap1, fpbq_fdeq/ qed-.
(* Basic_2A1: uses: lleq_fpbs_trans *)
-lemma fdeq_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fdeq_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_strap2, fpbq_fdeq/ qed-.
-lemma tdeq_rdeq_lpx_fpbs: ∀h,o,T1,T2. T1 ≛[h, o] T2 → ∀L1,L0. L1 ≛[h, o, T2] L0 →
- ∀G,L2. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃G, L1, T1⦄ ≥[h, o] ⦃G, L2, T2⦄.
+lemma tdeq_rdeq_lpx_fpbs: ∀h,T1,T2. T1 ≛ T2 → ∀L1,L0. L1 ≛[T2] L0 →
+ ∀G,L2. ⦃G,L0⦄ ⊢ ⬈[h] L2 → ⦃G,L1,T1⦄ ≥[h] ⦃G,L2,T2⦄.
/4 width=5 by fdeq_fpbs, fpbs_strap1, fpbq_lpx, fdeq_intro_dx/ qed.
(* Basic_2A1: removed theorems 3:
(* Properties with atomic arity assignment for terms ************************)
-lemma fpbs_aaa_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 /2 width=2 by ex_intro/
+lemma fpbs_aaa_conf: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 /2 width=2 by ex_intro/
#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A
/2 width=8 by fpbq_aaa_conf/
qed-.
(* Properties with unbound context-sensitive parallel rt-transition *********)
(* Basic_2A1: uses: fpbs_cpx_trans_neq *)
-lemma fpbs_cpx_tdneq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ≥[h, o] ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2
+lemma fpbs_cpx_tdneq_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ≥[h] ⦃G2,L2,U2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2
elim (fpbs_inv_star … H) -H #G0 #L0 #L3 #T0 #T3 #HT10 #H10 #HL03 #H32
elim (fdeq_cpx_trans … H32 … HTU2) -HTU2 #T4 #HT34 #H42
lapply (fdeq_tdneq_repl_dx … H32 … H42 … HnTU2) -T2 #HnT34
lapply (lpxs_cpx_trans … HT34 … HL03) -HT34 #HT34
elim (fqus_cpxs_trans_tdneq … H10 … HT34 HnT34) -T3 #T2 #HT02 #HnT02 #H24
-elim (tdeq_dec h o T1 T0) [ #H10 | -HnT02 #HnT10 ]
+elim (tdeq_dec T1 T0) [ #H10 | -HnT02 #HnT10 ]
[ lapply (cpxs_trans … HT10 … HT02) -HT10 -HT02 #HT12
- elim (cpxs_tdneq_fwd_step_sn … o … HT12) [2: /3 width=3 by tdeq_canc_sn/ ] -T0 -HT12
+ elim (cpxs_tdneq_fwd_step_sn … HT12) [2: /3 width=3 by tdeq_canc_sn/ ] -T0 -HT12
| elim (cpxs_tdneq_fwd_step_sn … HT10 … HnT10) -HT10 -HnT10
]
/4 width=16 by fpbs_intro_star, cpxs_tdeq_fpbs_trans, ex3_intro/
(* Properties with unbound context-sensitive parallel rt-computation ********)
-lemma cpxs_fpbs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
-#h #o #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
+lemma cpxs_fpbs: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L,T1⦄ ≥[h] ⦃G,L,T2⦄.
+#h #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
/3 width=5 by fpbq_cpx, fpbs_strap1/
qed.
-lemma fpbs_cpxs_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ∀T2. ⦃G, L⦄ ⊢ T ⬈*[h] T2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
-#h #o #G1 #G #L1 #L #T1 #T #H1 #T2 #H @(cpxs_ind … H) -T2
+lemma fpbs_cpxs_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬈*[h] T2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T2⦄.
+#h #G1 #G #L1 #L #T1 #T #H1 #T2 #H @(cpxs_ind … H) -T2
/3 width=5 by fpbs_strap1, fpbq_cpx/
qed-.
-lemma cpxs_fpbs_trans: ∀h,o,G1,G2,L1,L2,T,T2. ⦃G1, L1, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∀T1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T #T2 #H1 #T1 #H @(cpxs_ind_dx … H) -T1
+lemma cpxs_fpbs_trans: ∀h,G1,G2,L1,L2,T,T2. ⦃G1,L1,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀T1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T #T2 #H1 #T1 #H @(cpxs_ind_dx … H) -T1
/3 width=5 by fpbs_strap2, fpbq_cpx/
qed-.
-lemma cpxs_tdeq_fpbs_trans: ∀h,o,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
- ∀T0. T ≛[h, o] T0 →
- ∀G2,L2,T2. ⦃G1, L1, T0⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma cpxs_tdeq_fpbs_trans: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀T0. T ≛ T0 →
+ ∀G2,L2,T2. ⦃G1,L1,T0⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=3 by cpxs_fpbs_trans, tdeq_fpbs_trans/ qed-.
-lemma cpxs_tdeq_fpbs: ∀h,o,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈*[h] T →
- ∀T2. T ≛[h, o] T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
+lemma cpxs_tdeq_fpbs: ∀h,G,L,T1,T. ⦃G,L⦄ ⊢ T1 ⬈*[h] T →
+ ∀T2. T ≛ T2 → ⦃G,L,T1⦄ ≥[h] ⦃G,L,T2⦄.
/4 width=3 by cpxs_fpbs_trans, fdeq_fpbs, tdeq_fdeq/ qed.
(* Properties with star-iterated structural successor for closures **********)
-lemma cpxs_fqus_fpbs: ∀h,o,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G2,L2,T2. ⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma cpxs_fqus_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G2,L2,T2. ⦃G1,L1,T⦄ ⬂* ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.
(* Properties with plus-iterated structural successor for closures **********)
-lemma cpxs_fqup_fpbs: ∀h,o,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G2,L2,T2. ⦃G1, L1, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma cpxs_fqup_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G2,L2,T2. ⦃G1,L1,T⦄ ⬂+ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed.
(* Properties with sn for unbound parallel rt-transition for terms **********)
(* Basic_2A1: was: csx_fpbs_conf *)
-lemma fpbs_csx_conf: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #G1 #L1 #T1 #HT1 #G2 #L2 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
+lemma fpbs_csx_conf: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G2,L2⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #G1 #L1 #T1 #HT1 #G2 #L2 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
/2 width=5 by csx_fpbq_conf/
qed-.
(* Properties with proper parallel rst-reduction on closures ****************)
-lemma fpb_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fpb_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=1 by fpbq_fpbs, fpb_fpbq/ qed.
(* Main properties **********************************************************)
-theorem fpbs_trans: ∀h,o. tri_transitive … (fpbs h o).
+theorem fpbs_trans: ∀h. tri_transitive … (fpbs h).
/2 width=5 by tri_TC_transitive/ qed-.
(* Advanced properties ******************************************************)
-lemma tdeq_fpbs_trans: ∀h,o,T1,T. T1 ≛[h, o] T →
- ∀G1,G2,L1,L2,T2. ⦃G1, L1, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma tdeq_fpbs_trans: ∀h,T1,T. T1 ≛ T →
+ ∀G1,G2,L1,L2,T2. ⦃G1,L1,T⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fdeq_fpbs_trans, tdeq_fdeq/ qed-.
-lemma fpbs_tdeq_trans: ∀h,o,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T⦄ →
- ∀T2. T ≛[h, o] T2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fpbs_tdeq_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T⦄ →
+ ∀T2. T ≛ T2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fdeq_trans, tdeq_fdeq/ qed-.
(* Properties with plus-iterated structural successor for closures **********)
-lemma fqup_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+lemma fqup_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+ ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
/4 width=5 by fqu_fquq, fpbq_fquq, tri_step/
qed.
-lemma fpbs_fqup_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fpbs_fqup_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ ⬂+ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-.
-lemma fqup_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fqup_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⬂+ ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by fqus_fpbs_trans, fqup_fqus/ qed-.
(* Properties with star-iterated structural successor for closures **********)
-lemma fqus_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
+lemma fqus_fpbs: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂* ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
/3 width=5 by fpbq_fquq, tri_step/
qed.
-lemma fpbs_fqus_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
+lemma fpbs_fqus_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ ⬂* ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+#h #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H @(fqus_ind … H) -G2 -L2 -T2
/3 width=5 by fpbs_strap1, fpbq_fquq/
qed-.
-lemma fqus_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G #G2 #L #L2 #T #T2 #H1 #G1 #L1 #T1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
+lemma fqus_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ⦃G,L,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⬂* ⦃G,L,T⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+#h #G #G2 #L #L2 #T #T2 #H1 #G1 #L1 #T1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
/3 width=5 by fpbs_strap2, fpbq_fquq/
qed-.
(* Properties with unbound rt-computation on full local environments *******)
-lemma lpxs_fpbs: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
-#h #o #G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2
+lemma lpxs_fpbs: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → ⦃G,L1,T⦄ ≥[h] ⦃G,L2,T⦄.
+#h #G #L1 #L2 #T #H @(lpxs_ind_dx … H) -L2
/3 width=5 by fpbq_lpx, fpbs_strap1/
qed.
-lemma fpbs_lpxs_trans: ∀h,o,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L, T2⦄ →
- ∀L2. ⦃G2, L⦄ ⊢ ⬈*[h] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L #T1 #T2 #H1 #L2 #H @(lpxs_ind_dx … H) -L2
+lemma fpbs_lpxs_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L,T2⦄ →
+ ∀L2. ⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L #T1 #T2 #H1 #L2 #H @(lpxs_ind_dx … H) -L2
/3 width=5 by fpbs_strap1, fpbq_lpx/
qed-.
-lemma lpxs_fpbs_trans: ∀h,o,G1,G2,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∀L1. ⦃G1, L1⦄ ⊢ ⬈*[h] L → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L #L2 #T1 #T2 #H1 #L1 #H @(lpxs_ind_sn … H) -L1
+lemma lpxs_fpbs_trans: ∀h,G1,G2,L,L2,T1,T2. ⦃G1,L,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∀L1. ⦃G1,L1⦄ ⊢ ⬈*[h] L → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L #L2 #T1 #T2 #H1 #L1 #H @(lpxs_ind_sn … H) -L1
/3 width=5 by fpbs_strap2, fpbq_lpx/
qed-.
(* Basic_2A1: uses: lpxs_lleq_fpbs *)
-lemma lpxs_fdeq_fpbs: ∀h,o,G1,L1,L,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] L →
- ∀G2,L2,T2. ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma lpxs_fdeq_fpbs: ∀h,G1,L1,L,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] L →
+ ∀G2,L2,T2. ⦃G1,L,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=3 by lpxs_fpbs_trans, fdeq_fpbs/ qed.
-lemma fpbs_lpx_trans: ∀h,o,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L, T2⦄ →
- ∀L2. ⦃G2, L⦄ ⊢ ⬈[h] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fpbs_lpx_trans: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L,T2⦄ →
+ ∀L2. ⦃G2,L⦄ ⊢ ⬈[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=3 by fpbs_lpxs_trans, lpx_lpxs/ qed-.
(* Properties with star-iterated structural successor for closures **********)
-lemma fqus_lpxs_fpbs: ∀h,o,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ →
- ∀L2. ⦃G2, L⦄ ⊢ ⬈*[h] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma fqus_lpxs_fpbs: ∀h,G1,G2,L1,L,T1,T2. ⦃G1,L1,T1⦄ ⬂* ⦃G2,L,T2⦄ →
+ ∀L2. ⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
(* Properties with unbound context-sensitive parallel rt-computation ********)
-lemma cpxs_fqus_lpxs_fpbs: ∀h,o,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G2,L,T2. ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ →
- ∀L2.⦃G2, L⦄ ⊢ ⬈*[h] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+lemma cpxs_fqus_lpxs_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G2,L,T2. ⦃G1,L1,T⦄ ⬂* ⦃G2,L,T2⦄ →
+ ∀L2.⦃G2,L⦄ ⊢ ⬈*[h] L2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
/3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
-lemma fpbs_cpxs_tdeq_fqup_lpx_trans: ∀h,o,G1,G3,L1,L3,T1,T3. ⦃G1, L1, T1⦄ ≥ [h, o] ⦃G3, L3, T3⦄ →
- ∀T4. ⦃G3, L3⦄ ⊢ T3 ⬈*[h] T4 → ∀T5. T4 ≛[h, o] T5 →
- ∀G2,L4,T2. ⦃G3, L3, T5⦄ ⊐+ ⦃G2, L4, T2⦄ →
- ∀L2. ⦃G2, L4⦄ ⊢ ⬈[h] L2 → ⦃G1, L1, T1⦄ ≥ [h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G3 #L1 #L3 #T1 #T3 #H13 #T4 #HT34 #T5 #HT45 #G2 #L4 #T2 #H34 #L2 #HL42
+lemma fpbs_cpxs_tdeq_fqup_lpx_trans: ∀h,G1,G3,L1,L3,T1,T3. ⦃G1,L1,T1⦄ ≥ [h] ⦃G3,L3,T3⦄ →
+ ∀T4. ⦃G3,L3⦄ ⊢ T3 ⬈*[h] T4 → ∀T5. T4 ≛ T5 →
+ ∀G2,L4,T2. ⦃G3,L3,T5⦄ ⬂+ ⦃G2,L4,T2⦄ →
+ ∀L2. ⦃G2,L4⦄ ⊢ ⬈[h] L2 → ⦃G1,L1,T1⦄ ≥ [h] ⦃G2,L2,T2⦄.
+#h #G1 #G3 #L1 #L3 #T1 #T3 #H13 #T4 #HT34 #T5 #HT45 #G2 #L4 #T2 #H34 #L2 #HL42
@(fpbs_lpx_trans … HL42) -L2 (**) (* full auto too slow *)
@(fpbs_fqup_trans … H34) -G2 -L4 -T2
/3 width=3 by fpbs_cpxs_trans, fpbs_tdeq_trans/
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: fpbs_intro_alt *)
-lemma fpbs_intro_star: ∀h,o,G1,L1,T1,T. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G,L,T0. ⦃G1, L1, T⦄ ⊐* ⦃G, L, T0⦄ →
- ∀L0. ⦃G, L⦄ ⊢ ⬈*[h] L0 →
- ∀G2,L2,T2. ⦃G, L0, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ .
+lemma fpbs_intro_star: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
+ ∀G,L,T0. ⦃G1,L1,T⦄ ⬂* ⦃G,L,T0⦄ →
+ ∀L0. ⦃G,L⦄ ⊢ ⬈*[h] L0 →
+ ∀G2,L2,T2. ⦃G,L0,T0⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ .
/3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_fdeq/ qed.
(* Advanced inversion lemmas *************************************************)
(* Basic_2A1: uses: fpbs_inv_alt *)
-lemma fpbs_inv_star: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∃∃G,L,L0,T,T0. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T & ⦃G1, L1, T⦄ ⊐* ⦃G, L, T0⦄
- & ⦃G, L⦄ ⊢ ⬈*[h] L0 & ⦃G, L0, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
+lemma fpbs_inv_star: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ →
+ ∃∃G,L,L0,T,T0. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T & ⦃G1,L1,T⦄ ⬂* ⦃G,L,T0⦄
+ & ⦃G,L⦄ ⊢ ⬈*[h] L0 & ⦃G,L0,T0⦄ ≛ ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
[ /2 width=9 by ex4_5_intro/
| #G1 #G0 #L1 #L0 #T1 #T0 * -G0 -L0 -T0
[ #G0 #L0 #T0 #H10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predsubtystrong_5.ma".
+include "basic_2/notation/relations/predsubtystrong_4.ma".
include "basic_2/rt_transition/fpb.ma".
(* STRONGLY NORMALIZING CLOSURES FOR PARALLEL RST-TRANSITION ****************)
-inductive fsb (h) (o): relation3 genv lenv term ≝
+inductive fsb (h): relation3 genv lenv term ≝
| fsb_intro: ∀G1,L1,T1. (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → fsb h o G2 L2 T2
- ) → fsb h o G1 L1 T1
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → fsb h G2 L2 T2
+ ) → fsb h G1 L1 T1
.
interpretation
"strong normalization for parallel rst-transition (closure)"
- 'PRedSubTyStrong h o G L T = (fsb h o G L T).
+ 'PRedSubTyStrong h G L T = (fsb h G L T).
(* Basic eliminators ********************************************************)
(* Note: eliminator with shorter ground hypothesis *)
(* Note: to be named fsb_ind when fsb becomes a definition like csx, lfsx ***)
-lemma fsb_ind_alt: ∀h,o. ∀Q: relation3 …. (
- ∀G1,L1,T1. ≥[h,o] 𝐒⦃G1, L1, T1⦄ → (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2
+lemma fsb_ind_alt: ∀h. ∀Q: relation3 …. (
+ ∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → (
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2
) → Q G1 L1 T1
) →
- ∀G,L,T. ≥[h, o] 𝐒⦃G, L, T⦄ → Q G L T.
-#h #o #Q #IH #G #L #T #H elim H -G -L -T
+ ∀G,L,T. ≥[h] 𝐒⦃G,L,T⦄ → Q G L T.
+#h #Q #IH #G #L #T #H elim H -G -L -T
/4 width=1 by fsb_intro/
qed-.
(* Main properties with atomic arity assignment for terms *******************)
-(* Note: this is the "big tree" theorem *)
-theorem aaa_fsb: ∀h,o,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ≥[h, o] 𝐒⦃G, L, T⦄.
+theorem aaa_fsb: ∀h,G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ≥[h] 𝐒⦃G,L,T⦄.
/3 width=2 by aaa_csx, csx_fsb/ qed.
(* Advanced eliminators with atomic arity assignment for terms **************)
-fact aaa_ind_fpb_aux: ∀h,o. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+fact aaa_ind_fpb_aux: ∀h. ∀Q:relation3 ….
+ (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
-#h #o #R #IH #G #L #T #H @(csx_ind_fpb … H) -G -L -T
+ ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ∀A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T.
+#h #R #IH #G #L #T #H @(csx_ind_fpb … H) -G -L -T
#G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
-#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h o … G2 … L2 … T2 … HTA1) -A1
+#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf … G2 … L2 … T2 … HTA1) -A1
/2 width=2 by fpb_fpbs/
qed-.
-lemma aaa_ind_fpb: ∀h,o. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+lemma aaa_ind_fpb: ∀h. ∀Q:relation3 ….
+ (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
+ ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T.
/4 width=4 by aaa_ind_fpb_aux, aaa_csx/ qed-.
-fact aaa_ind_fpbg_aux: ∀h,o. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+fact aaa_ind_fpbg_aux: ∀h. ∀Q:relation3 ….
+ (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
-#h #o #Q #IH #G #L #T #H @(csx_ind_fpbg … H) -G -L -T
+ ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ∀A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T.
+#h #Q #IH #G #L #T #H @(csx_ind_fpbg … H) -G -L -T
#G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
-#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h o … G2 … L2 … T2 … HTA1) -A1
+#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf … G2 … L2 … T2 … HTA1) -A1
/2 width=2 by fpbg_fwd_fpbs/
qed-.
-lemma aaa_ind_fpbg: ∀h,o. ∀Q:relation3 ….
- (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+lemma aaa_ind_fpbg: ∀h. ∀Q:relation3 ….
+ (∀G1,L1,T1,A. ⦃G1,L1⦄ ⊢ T1 ⁝ A →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
+ ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → Q G L T.
/4 width=4 by aaa_ind_fpbg_aux, aaa_csx/ qed-.
(* *)
(**************************************************************************)
-include "basic_2/rt_computation/rdsx_csx.ma".
+include "basic_2/rt_computation/rsx_csx.ma".
include "basic_2/rt_computation/fpbs_cpx.ma".
include "basic_2/rt_computation/fpbs_csx.ma".
include "basic_2/rt_computation/fsb_fpbg.ma".
(* Inversion lemmas with context-sensitive stringly rt-normalizing terms ****)
-lemma fsb_inv_csx: ∀h,o,G,L,T. ≥[h, o] 𝐒⦃G, L, T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
-#h #o #G #L #T #H @(fsb_ind_alt … H) -G -L -T /5 width=1 by csx_intro, fpb_cpx/
+lemma fsb_inv_csx: ∀h,G,L,T. ≥[h] 𝐒⦃G,L,T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+#h #G #L #T #H @(fsb_ind_alt … H) -G -L -T /5 width=1 by csx_intro, fpb_cpx/
qed-.
(* Propreties with context-sensitive stringly rt-normalizing terms **********)
-lemma csx_fsb_fpbs: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ≥[h, o] 𝐒⦃G2, L2, T2⦄.
-#h #o #G1 #L1 #T1 #H @(csx_ind … H) -T1
+lemma csx_fsb_fpbs: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄.
+#h #G1 #L1 #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind (Ⓣ) … G2 L2 T2) -G2 -L2 -T2
#G0 #L0 #T0 #IHu #H10
lapply (fpbs_csx_conf … H10) // -HT1 #HT0
generalize in match IHu; -IHu generalize in match H10; -H10
-@(rdsx_ind … (csx_rdsx … HT0)) -L0
+@(rsx_ind … (csx_rsx … HT0)) -L0
#L0 #_ #IHd #H10 #IHu @fsb_intro
#G2 #L2 #T2 * -G2 -L2 -T2 [ -IHd -IHc | -IHu -IHd | ]
[ /4 width=5 by fpbs_fqup_trans, fqu_fqup/
[ /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/
| #G3 #L3 #T3 #H03 #_
elim (lpx_fqup_trans … H03 … HL02) -L2 #L4 #T4 #HT04 #H43 #HL43
- elim (tdeq_dec h o T0 T4) [ -IHc -HT04 #HT04 | -IHu #HnT04 ]
+ elim (tdeq_dec T0 T4) [ -IHc -HT04 #HT04 | -IHu #HnT04 ]
[ elim (tdeq_fqup_trans … H43 … HT04) -T4 #L2 #T4 #H04 #HT43 #HL24
/4 width=7 by fsb_fpbs_trans, tdeq_rdeq_lpx_fpbs, fpbs_fqup_trans/
| elim (cpxs_tdneq_fwd_step_sn … HT04 HnT04) -HT04 -HnT04 #T2 #T5 #HT02 #HnT02 #HT25 #HT54
]
qed.
-lemma csx_fsb: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ≥[h, o] 𝐒⦃G, L, T⦄.
+lemma csx_fsb: ∀h,G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ≥[h] 𝐒⦃G,L,T⦄.
/2 width=5 by csx_fsb_fpbs/ qed.
(* Advanced eliminators *****************************************************)
-lemma csx_ind_fpb: ∀h,o. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+lemma csx_ind_fpb: ∀h. ∀Q:relation3 genv lenv term.
+ (∀G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → Q G L T.
+ ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
-lemma csx_ind_fpbg: ∀h,o. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+lemma csx_ind_fpbg: ∀h. ∀Q:relation3 genv lenv term.
+ (∀G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → Q G L T.
+ ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.
(* STRONGLY NORMALIZING CLOSURES FOR PARALLEL RST-TRANSITION ****************)
-(* Properties with degree-based equivalence for closures ********************)
+(* Properties with sort-irrelevant equivalence for closures *****************)
-lemma fsb_fdeq_trans: ∀h,o,G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ≥[h, o] 𝐒⦃G2, L2, T2⦄.
-#h #o #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
+lemma fsb_fdeq_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄.
+#h #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
#G1 #L1 #T1 #_ #IH #G2 #L2 #T2 #H12
@fsb_intro #G #L #T #H2
elim (fdeq_fpb_trans … H12 … H2) -G2 -L2 -T2
(* Properties with parallel rst-computation for closures ********************)
-lemma fsb_fpbs_trans: ∀h,o,G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ≥[h, o] 𝐒⦃G2, L2, T2⦄.
-#h #o #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
+lemma fsb_fpbs_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄.
+#h #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
#G1 #L1 #T1 #H1 #IH #G2 #L2 #T2 #H12
elim (fpbs_inv_fpbg … H12) -H12
[ -IH /2 width=5 by fsb_fdeq_trans/
(* Properties with proper parallel rst-computation for closures *************)
-lemma fsb_intro_fpbg: ∀h,o,G1,L1,T1. (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → ≥[h, o] 𝐒⦃G2, L2, T2⦄
- ) → ≥[h, o] 𝐒⦃G1, L1, T1⦄.
+lemma fsb_intro_fpbg: ∀h,G1,L1,T1. (
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄
+ ) → ≥[h] 𝐒⦃G1,L1,T1⦄.
/4 width=1 by fsb_intro, fpb_fpbg/ qed.
(* Eliminators with proper parallel rst-computation for closures ************)
-lemma fsb_ind_fpbg_fpbs: ∀h,o. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+lemma fsb_ind_fpbg_fpbs: ∀h. ∀Q:relation3 genv lenv term.
+ (∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
-#h #o #Q #IH1 #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
+ ∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2.
+#h #Q #IH1 #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
#G1 #L1 #T1 #H1 #IH #G2 #L2 #T2 #H12
@IH1 -IH1
[ -IH /2 width=5 by fsb_fpbs_trans/
]
qed-.
-lemma fsb_ind_fpbg: ∀h,o. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+lemma fsb_ind_fpbg: ∀h. ∀Q:relation3 genv lenv term.
+ (∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ → Q G1 L1 T1.
-#h #o #Q #IH #G1 #L1 #T1 #H @(fsb_ind_fpbg_fpbs … H) -H
+ ∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → Q G1 L1 T1.
+#h #Q #IH #G1 #L1 #T1 #H @(fsb_ind_fpbg_fpbs … H) -H
/3 width=1 by/
qed-.
(* Inversion lemmas with proper parallel rst-computation for closures *******)
-lemma fsb_fpbg_refl_false (h) (o) (G) (L) (T):
- ≥[h,o] 𝐒⦃G, L, T⦄ → ⦃G, L, T⦄ >[h,o] ⦃G, L, T⦄ → ⊥.
-#h #o #G #L #T #H
+lemma fsb_fpbg_refl_false (h) (G) (L) (T):
+ ≥[h] 𝐒⦃G,L,T⦄ → ⦃G,L,T⦄ >[h] ⦃G,L,T⦄ → ⊥.
+#h #G #L #T #H
@(fsb_ind_fpbg … H) -G -L -T #G1 #L1 #T1 #_ #IH #H
/2 width=5 by/
qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/topredtysnstrong_4.ma".
+include "basic_2/rt_computation/rsx.ma".
+
+(* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
+
+(* Note: this should be an instance of a more general sex *)
+(* Basic_2A1: uses: lcosx *)
+inductive jsx (h) (G): relation lenv ≝
+| jsx_atom: jsx h G (⋆) (⋆)
+| jsx_bind: ∀I,K1,K2. jsx h G K1 K2 →
+ jsx h G (K1.ⓘ{I}) (K2.ⓘ{I})
+| jsx_pair: ∀I,K1,K2,V. jsx h G K1 K2 →
+ G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ → jsx h G (K1.ⓑ{I}V) (K2.ⓧ)
+.
+
+interpretation
+ "strong normalization for unbound parallel rt-transition (compatibility)"
+ 'ToPRedTySNStrong h G L1 L2 = (jsx h G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact jsx_inv_atom_sn_aux (h) (G):
+ ∀L1,L2. G ⊢ L1 ⊒[h] L2 → L1 = ⋆ → L2 = ⋆.
+#h #G #L1 #L2 * -L1 -L2
+[ //
+| #I #K1 #K2 #_ #H destruct
+| #I #K1 #K2 #V #_ #_ #H destruct
+]
+qed-.
+
+lemma jsx_inv_atom_sn (h) (G): ∀L2. G ⊢ ⋆ ⊒[h] L2 → L2 = ⋆.
+/2 width=5 by jsx_inv_atom_sn_aux/ qed-.
+
+fact jsx_inv_bind_sn_aux (h) (G):
+ ∀L1,L2. G ⊢ L1 ⊒[h] L2 →
+ ∀I,K1. L1 = K1.ⓘ{I} →
+ ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ{I}
+ | ∃∃J,K2,V. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & I = BPair J V & L2 = K2.ⓧ.
+#h #G #L1 #L2 * -L1 -L2
+[ #J #L1 #H destruct
+| #I #K1 #K2 #HK12 #J #L1 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #I #K1 #K2 #V #HK12 #HV #J #L1 #H destruct /3 width=7 by ex4_3_intro, or_intror/
+]
+qed-.
+
+lemma jsx_inv_bind_sn (h) (G):
+ ∀I,K1,L2. G ⊢ K1.ⓘ{I} ⊒[h] L2 →
+ ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ{I}
+ | ∃∃J,K2,V. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & I = BPair J V & L2 = K2.ⓧ.
+/2 width=3 by jsx_inv_bind_sn_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_2A1: uses: lcosx_inv_pair *)
+lemma jsx_inv_pair_sn (h) (G):
+ ∀I,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊒[h] L2 →
+ ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓑ{I}V
+ | ∃∃K2. G ⊢ K1 ⊒[h] K2 & G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ & L2 = K2.ⓧ.
+#h #G #I #K1 #L2 #V #H elim (jsx_inv_bind_sn … H) -H *
+[ /3 width=3 by ex2_intro, or_introl/
+| #J #K2 #X #HK12 #HX #H1 #H2 destruct /3 width=4 by ex3_intro, or_intror/
+]
+qed-.
+
+lemma jsx_inv_void_sn (h) (G):
+ ∀K1,L2. G ⊢ K1.ⓧ ⊒[h] L2 →
+ ∃∃K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓧ.
+#h #G #K1 #L2 #H elim (jsx_inv_bind_sn … H) -H *
+/2 width=3 by ex2_intro/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma jsx_fwd_bind_sn (h) (G):
+ ∀I1,K1,L2. G ⊢ K1.ⓘ{I1} ⊒[h] L2 →
+ ∃∃I2,K2. G ⊢ K1 ⊒[h] K2 & L2 = K2.ⓘ{I2}.
+#h #G #I1 #K1 #L2 #H elim (jsx_inv_bind_sn … H) -H *
+/2 width=4 by ex2_2_intro/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: lcosx_O *)
+lemma jsx_refl (h) (G): reflexive … (jsx h G).
+#h #G #L elim L -L /2 width=1 by jsx_atom, jsx_bind/
+qed.
+
+(* Basic_2A1: removed theorems 2:
+ lcosx_drop_trans_lt lcosx_inv_succ
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/rsx_csx.ma".
+include "basic_2/rt_computation/jsx_drops.ma".
+include "basic_2/rt_computation/jsx_lsubr.ma".
+
+(* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
+
+(* Properties with strongly rt-normalizing terms ****************************)
+
+lemma jsx_csx_conf (h) (G):
+ ∀L1,L2. G ⊢ L1 ⊒[h] L2 →
+ ∀T. ⦃G,L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G,L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+/3 width=5 by jsx_fwd_lsubr, csx_lsubr_conf/ qed-.
+
+(* Properties with strongly rt-normalizing referred local environments ******)
+
+(* Note: Try by induction on the 2nd premise by generalizing V with f *)
+lemma rsx_jsx_trans (h) (G):
+ ∀L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀L2. G ⊢ L1 ⊒[h] L2 → G ⊢ ⬈*[h,V] 𝐒⦃L2⦄.
+#h #G #L1 #V @(fqup_wf_ind_eq (Ⓕ) … G L1 V) -G -L1 -V
+#G0 #L0 #V0 #IH #G #L1 * *
+[ //
+| #i #HG #HL #HV #H #L2 #HL12 destruct
+ elim (rsx_inv_lref_drops … H) -H [|*: * ]
+ [ #HL1 -IH
+ lapply (jsx_fwd_drops_atom_sn … HL12 … HL1) -L1
+ /2 width=1 by rsx_lref_atom_drops/
+ | #I #K1 #HLK1 -IH
+ elim (jsx_fwd_drops_unit_sn … HL12 … HLK1) -L1 [| // ] #K2 #HK12 #HLK2
+ /2 width=3 by rsx_lref_unit_drops/
+ | #I #K1 #V1 #HLK1 #HV1 #HK1
+ elim (jsx_fwd_drops_pair_sn … HL12 … HLK1) -HL12 [3: // |*: * ]
+ [ #K2 #HK12 #HLK2
+ /4 width=6 by rsx_lref_pair_drops, jsx_csx_conf, fqup_lref/
+ | #K2 #_ #HLK2 #_
+ /2 width=3 by rsx_lref_unit_drops/
+ ]
+ ]
+| //
+| #p #I #V #T #HG #HL #HV #H #L2 #HL12 destruct
+ elim (rsx_inv_bind_void … H) -H #HV #HT
+ /4 width=4 by jsx_bind, rsx_bind_void/
+| #I #V #T #HG #HL #HV #H #L2 #HL12 destruct
+ elim (rsx_inv_flat … H) -H #HV #HT
+ /3 width=4 by rsx_flat/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/drops.ma".
+include "basic_2/rt_computation/jsx.ma".
+
+(* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
+
+(* Forward lemmas with uniform slicing for local environments ***************)
+
+lemma jsx_fwd_drops_atom_sn (h) (b) (G):
+ ∀L1,L2. G ⊢ L1 ⊒[h] L2 →
+ ∀f. 𝐔⦃f⦄ → ⬇*[b,f]L1 ≘ ⋆ → ⬇*[b,f]L2 ≘ ⋆.
+#h #b #G #L1 #L2 #H elim H -L1 -L2
+[ #f #_ #H //
+| #I #K1 #K2 #_ #IH #f #Hf #H
+| #I #K1 #K2 #V #_ #HV #IH #f #Hf #H
+]
+elim (drops_inv_bind1_isuni … H) -H [3,6: // |*: * -Hf ]
+[1,3: #_ #H destruct
+|2,4: #g #Hg #HK1 #H destruct /3 width=1 by drops_drop/
+]
+qed-.
+
+lemma jsx_fwd_drops_unit_sn (h) (b) (G):
+ ∀L1,L2. G ⊢ L1 ⊒[h] L2 →
+ ∀f. 𝐔⦃f⦄ → ∀I,K1. ⬇*[b,f]L1 ≘ K1.ⓤ{I} →
+ ∃∃K2. G ⊢ K1 ⊒[h] K2 & ⬇*[b,f]L2 ≘ K2.ⓤ{I}.
+#h #b #G #L1 #L2 #H elim H -L1 -L2
+[ #f #_ #J #Y1 #H
+ lapply (drops_inv_atom1 … H) -H * #H #_ destruct
+| #I #K1 #K2 #HK12 #IH #f #Hf #J #Y1 #H
+| #I #K1 #K2 #V #HK12 #HV #IH #f #Hf #J #Y1 #H
+]
+elim (drops_inv_bind1_isuni … H) -H [3,6: // |*: * -Hf ]
+[1,3: #Hf #H destruct -IH /3 width=3 by drops_refl, ex2_intro/
+|2,4:
+ #g #Hg #HK1 #H destruct
+ elim (IH … Hg … HK1) -K1 -Hg #Y2 #HY12 #HKY2
+ /3 width=3 by drops_drop, ex2_intro/
+]
+qed-.
+
+lemma jsx_fwd_drops_pair_sn (h) (b) (G):
+ ∀L1,L2. G ⊢ L1 ⊒[h] L2 →
+ ∀f. 𝐔⦃f⦄ → ∀I,K1,V. ⬇*[b,f]L1 ≘ K1.ⓑ{I}V →
+ ∨∨ ∃∃K2. G ⊢ K1 ⊒[h] K2 & ⬇*[b,f]L2 ≘ K2.ⓑ{I}V
+ | ∃∃K2. G ⊢ K1 ⊒[h] K2 & ⬇*[b,f]L2 ≘ K2.ⓧ & G ⊢ ⬈*[h,V] 𝐒⦃K2⦄.
+#h #b #G #L1 #L2 #H elim H -L1 -L2
+[ #f #_ #J #Y1 #X1 #H
+ lapply (drops_inv_atom1 … H) -H * #H #_ destruct
+| #I #K1 #K2 #HK12 #IH #f #Hf #J #Y1 #X1 #H
+| #I #K1 #K2 #V #HK12 #HV #IH #f #Hf #J #Y1 #X1 #H
+]
+elim (drops_inv_bind1_isuni … H) -H [3,6: // |*: * -Hf ]
+[1,3:
+ #Hf #H destruct -IH
+ /4 width=4 by drops_refl, ex3_intro, ex2_intro, or_introl, or_intror/
+|2,4:
+ #g #Hg #HK1 #H destruct
+ elim (IH … Hg … HK1) -K1 -Hg * #Y2 #HY12 #HKY2 [2,4: #HX1 ]
+ /4 width=4 by drops_drop, ex3_intro, ex2_intro, or_introl, or_intror/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/jsx_csx.ma".
+
+(* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
+
+(* Main properties **********************************************************)
+
+theorem jsx_trans (h) (G): Transitive … (jsx h G).
+#h #G #L1 #L #H elim H -L1 -L
+[ #L2 #H
+ >(jsx_inv_atom_sn … H) -L2 //
+| #I #K1 #K #_ #IH #L2 #H
+ elim (jsx_inv_bind_sn … H) -H *
+ [ #K2 #HK2 #H destruct /3 width=1 by jsx_bind/
+ | #J #K2 #V #HK2 #HV #H1 #H2 destruct /3 width=1 by jsx_pair/
+ ]
+| #I #K1 #K #V #_ #HV #IH #L2 #H
+ elim (jsx_inv_void_sn … H) -H #K2 #HK2 #H destruct
+ /3 width=3 by rsx_jsx_trans, jsx_pair/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/static/lsubr.ma".
+include "basic_2/rt_computation/jsx.ma".
+
+(* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
+
+(* Forward lemmas with restricted refinement for local environments *********)
+
+lemma jsx_fwd_lsubr (h) (G): ∀L1,L2. G ⊢ L1 ⊒[h] L2 → L1 ⫃ L2.
+#h #G #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by lsubr_bind, lsubr_unit/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/rsx_drops.ma".
+include "basic_2/rt_computation/rsx_lpxs.ma".
+include "basic_2/rt_computation/jsx.ma".
+
+(* COMPATIBILITY OF STRONG NORMALIZATION FOR UNBOUND RT-TRANSITION **********)
+
+(* Properties with strongly normalizing referred local environments *********)
+
+(* Basic_2A1: uses: lsx_cpx_trans_lcosx *)
+lemma rsx_cpx_trans_jsx (h) (G):
+ ∀L0,T1,T2. ⦃G,L0⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L. G ⊢ L0 ⊒[h] L → G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
+#h #G #L0 #T1 #T2 #H @(cpx_ind … H) -G -L0 -T1 -T2
+[ //
+| //
+| #I0 #G #K0 #V1 #V2 #W2 #_ #IH #HVW2 #L #HK0 #HL
+ elim (jsx_inv_pair_sn … HK0) -HK0 *
+ [ #K #HK0 #H destruct
+ /4 width=8 by rsx_lifts, rsx_fwd_pair, drops_refl, drops_drop/
+ | #K #HK0 #HV1 #H destruct
+ /4 width=8 by rsx_lifts, drops_refl, drops_drop/
+ ]
+| #I0 #G #K0 #T #U #i #_ #IH #HTU #L #HK0 #HL
+ elim (jsx_fwd_bind_sn … HK0) -HK0 #I #K #HK0 #H destruct
+ /6 width=8 by rsx_inv_lifts, rsx_lifts, drops_refl, drops_drop/
+| #p #I0 #G #L0 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #HL0 #HL
+ elim (rsx_inv_bind_void … HL) -HL
+ /4 width=2 by jsx_pair, rsx_bind_void/
+| #I0 #G #L0 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #HL0 #HL
+ elim (rsx_inv_flat … HL) -HL /3 width=2 by rsx_flat/
+| #G #L0 #V #U1 #T1 #T2 #HTU1 #_ #IHT12 #L #HL0 #HL
+ elim (rsx_inv_bind_void … HL) -HL #HV #HU1
+ /5 width=8 by rsx_inv_lifts, drops_refl, drops_drop/
+| #G #L0 #V #T1 #T2 #_ #IHT12 #L #HL0 #HL
+ elim (rsx_inv_flat … HL) -HL /2 width=2 by/
+| #G #L0 #V1 #V2 #T #_ #IHV12 #L #HL0 #HL
+ elim (rsx_inv_flat … HL) -HL /2 width=2 by/
+| #p #G #L0 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #HL0 #HL
+ elim (rsx_inv_flat … HL) -HL #HV1 #HL
+ elim (rsx_inv_bind_void … HL) -HL #HW1 #HT1
+ /4 width=2 by jsx_pair, rsx_bind_void, rsx_flat/
+| #p #G #L0 #V1 #V2 #U2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #HVU2 #L #HL0 #HL
+ elim (rsx_inv_flat … HL) -HL #HV1 #HL
+ elim (rsx_inv_bind_void … HL) -HL #HW1 #HT1
+ /6 width=8 by jsx_pair, rsx_lifts, rsx_bind_void, rsx_flat, drops_refl, drops_drop/
+]
+qed-.
+
+(* Advanced properties of strongly normalizing referred local environments **)
+
+(* Basic_2A1: uses: lsx_cpx_trans_O *)
+lemma rsx_cpx_trans (h) (G):
+ ∀L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
+ G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
+/3 width=6 by rsx_cpx_trans_jsx, jsx_refl/ qed-.
+
+lemma rsx_cpxs_trans (h) (G):
+ ∀L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 →
+ G ⊢ ⬈*[h,T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T2] 𝐒⦃L⦄.
+#h #G #L #T1 #T2 #H
+@(cpxs_ind_dx ???????? H) -T1 //
+/3 width=3 by rsx_cpx_trans/
+qed-.
(* Basic properties *********************************************************)
(* Basic_2A1: uses: lprs_pair_refl *)
-lemma lprs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀I. ⦃G, L1.ⓘ{I}⦄ ⊢ ➡*[h] L2.ⓘ{I}.
+lemma lprs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀I. ⦃G,L1.ⓘ{I}⦄ ⊢ ➡*[h] L2.ⓘ{I}.
/2 width=1 by lex_bind_refl_dx/ qed.
-lemma lprs_pair (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡*[h] V2 →
- ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2.
+lemma lprs_pair (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀V1,V2. ⦃G,L1⦄ ⊢ V1 ➡*[h] V2 →
+ ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2.
/2 width=1 by lex_pair/ qed.
-lemma lprs_refl (h) (G): ∀L. ⦃G, L⦄ ⊢ ➡*[h] L.
+lemma lprs_refl (h) (G): ∀L. ⦃G,L⦄ ⊢ ➡*[h] L.
/2 width=1 by lex_refl/ qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: uses: lprs_inv_atom1 *)
-lemma lprs_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ➡*[h] L2 → L2 = ⋆.
+lemma lprs_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ➡*[h] L2 → L2 = ⋆.
/2 width=2 by lex_inv_atom_sn/ qed-.
(* Basic_2A1: was: lprs_inv_pair1 *)
lemma lprs_inv_pair_sn (h) (G):
- ∀I,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡*[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 & L2 = K2.ⓑ{I}V2.
+ ∀I,K1,L2,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2 →
+ ∃∃K2,V2. ⦃G,K1⦄ ⊢ ➡*[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 & L2 = K2.ⓑ{I}V2.
/2 width=1 by lex_inv_pair_sn/ qed-.
(* Basic_2A1: uses: lprs_inv_atom2 *)
-lemma lprs_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] ⋆ → L1 = ⋆.
+lemma lprs_inv_atom_dx (h) (G): ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] ⋆ → L1 = ⋆.
/2 width=2 by lex_inv_atom_dx/ qed-.
(* Basic_2A1: was: lprs_inv_pair2 *)
lemma lprs_inv_pair_dx (h) (G):
- ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡*[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡*[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 & L1 = K1.ⓑ{I}V1.
+ ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ➡*[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G,K1⦄ ⊢ ➡*[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 & L1 = K1.ⓑ{I}V1.
/2 width=1 by lex_inv_pair_dx/ qed-.
(* Basic eliminators ********************************************************)
lemma lprs_ind (h) (G): ∀Q:relation lenv.
Q (⋆) (⋆) → (
∀I,K1,K2.
- ⦃G, K1⦄ ⊢ ➡*[h] K2 →
+ ⦃G,K1⦄ ⊢ ➡*[h] K2 →
Q K1 K2 → Q (K1.ⓘ{I}) (K2.ⓘ{I})
) → (
∀I,K1,K2,V1,V2.
- ⦃G, K1⦄ ⊢ ➡*[h] K2 → ⦃G, K1⦄ ⊢ V1 ➡*[h] V2 →
+ ⦃G,K1⦄ ⊢ ➡*[h] K2 → ⦃G,K1⦄ ⊢ V1 ➡*[h] V2 →
Q K1 K2 → Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
) →
- ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L1 L2.
+ ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → Q L1 L2.
/3 width=4 by lex_ind/ qed-.
(* Properties with t-bound context-sensitive rt-computarion for terms *******)
lemma lprs_cpms_trans (n) (h) (G):
- ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+ ∀L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2.
#n #h #G #L2 #T1 #T2 #HT12 #L1 #H
@(lprs_ind_sn … H) -L1 /2 width=3 by lpr_cpms_trans/
qed-.
lemma lprs_cpm_trans (n) (h) (G):
- ∀L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[n, h] T2 →
- ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[n, h] T2.
+ ∀L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ T1 ➡*[n,h] T2.
/3 width=3 by lprs_cpms_trans, cpm_cpms/ qed-.
(* Basic_2A1: includes cprs_bind2 *)
lemma cpms_bind_dx (n) (h) (G) (L):
- ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n, h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n, h] ⓑ{p,I}V2.T2.
+ ∀V1,V2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡*[n,h] ⓑ{p,I}V2.T2.
/4 width=5 by lprs_cpms_trans, lprs_pair, cpms_bind/ qed.
(* Inversion lemmas with t-bound context-sensitive rt-computarion for terms *)
(* Basic_2A1: includes: cprs_inv_abst1 *)
(* Basic_2A1: uses: scpds_inv_abst1 *)
lemma cpms_inv_abst_sn (n) (h) (G) (L):
- ∀p,V1,T1,X2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡*[n, h] X2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[n, h] T2 &
+ ∀p,V1,T1,X2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ➡*[n,h] X2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ➡*[n,h] T2 &
X2 = ⓛ{p}V2.T2.
#n #h #G #L #p #V1 #T1 #X2 #H
@(cpms_ind_dx … H) -X2 /2 width=5 by ex3_2_intro/
qed-.
(* Basic_2A1: includes: cprs_inv_abst *)
-lemma cpms_inv_abst_bi (n) (h) (G) (L):
- ∀p,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ➡*[n, h] ⓛ{p}W2.T2 →
- ∧∧ ⦃G, L⦄ ⊢ W1 ➡*[h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[n, h] T2.
-#n #h #G #L #p #W1 #W2 #T1 #T2 #H
+lemma cpms_inv_abst_bi (n) (h) (p1) (p2) (G) (L):
+ ∀W1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}W1.T1 ➡*[n,h] ⓛ{p2}W2.T2 →
+ ∧∧ p1 = p2 & ⦃G,L⦄ ⊢ W1 ➡*[h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ➡*[n,h] T2.
+#n #h #p1 #p2 #G #L #W1 #W2 #T1 #T2 #H
elim (cpms_inv_abst_sn … H) -H #W #T #HW1 #HT1 #H destruct
-/2 width=1 by conj/
+/2 width=1 by and3_intro/
qed-.
(* Basic_1: was pr3_gen_abbr *)
(* Basic_2A1: includes: cprs_inv_abbr1 *)
lemma cpms_inv_abbr_sn_dx (n) (h) (G) (L):
- ∀p,V1,T1,X2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡*[n, h] X2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n, h] T2 & X2 = ⓓ{p}V2.T2
- | ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n ,h] T2 & ⬆*[1] X2 ≘ T2 & p = Ⓣ.
+ ∀p,V1,T1,X2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ➡*[n,h] X2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡*[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓓ{p}V2.T2
+ | ∃∃T2. ⦃G,L.ⓓV1⦄ ⊢ T1 ➡*[n ,h] T2 & ⬆*[1] X2 ≘ T2 & p = Ⓣ.
#n #h #G #L #p #V1 #T1 #X2 #H
@(cpms_ind_dx … H) -X2 -n /3 width=5 by ex3_2_intro, or_introl/
#n1 #n2 #X #X2 #_ * *
(* Basic_2A1: uses: scpds_inv_abbr_abst *)
lemma cpms_inv_abbr_abst (n) (h) (G) (L):
- ∀p1,p2,V1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓓ{p1}V1.T1 ➡*[n, h] ⓛ{p2}W2.T2 →
- ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[n, h] T & ⬆*[1] ⓛ{p2}W2.T2 ≘ T & p1 = Ⓣ.
+ ∀p1,p2,V1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓓ{p1}V1.T1 ➡*[n,h] ⓛ{p2}W2.T2 →
+ ∃∃T. ⦃G,L.ⓓV1⦄ ⊢ T1 ➡*[n,h] T & ⬆*[1] ⓛ{p2}W2.T2 ≘ T & p1 = Ⓣ.
#n #h #G #L #p1 #p2 #V1 #W2 #T1 #T2 #H
elim (cpms_inv_abbr_sn_dx … H) -H *
[ #V #T #_ #_ #H destruct
(* Advanced properties ******************************************************)
(* Basic_2A1: was: lprs_pair2 *)
-lemma lprs_pair_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡*[h] V2 →
- ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2.
+lemma lprs_pair_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀V1,V2. ⦃G,L2⦄ ⊢ V1 ➡*[h] V2 →
+ ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2.
/3 width=3 by lprs_pair, lprs_cpms_trans/ qed.
(* Properties on context-sensitive parallel r-computation for terms *********)
-lemma lprs_cprs_conf_dx (h) (G): ∀L0.∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+lemma lprs_cprs_conf_dx (h) (G): ∀L0.∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
#h #G #L0 #T0 #T1 #HT01 #L1 #H
@(lprs_ind_dx … H) -L1 /2 width=3 by ex2_intro/
#L #L1 #_ #HL1 * #T #HT1 #HT0 -L0
/3 width=5 by cprs_trans, ex2_intro/
qed-.
-lemma lprs_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+lemma lprs_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
/3 width=3 by lprs_cprs_conf_dx, cpm_cpms/ qed-.
(* Note: this can be proved on its own using lprs_ind_sn *)
-lemma lprs_cprs_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+lemma lprs_cprs_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡*[h] T1 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G,L0⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01
/3 width=3 by lprs_cpms_trans, ex2_intro/
qed-.
-lemma lprs_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+lemma lprs_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G,L0⦄ ⊢ T1 ➡*[h] T & ⦃G,L1⦄ ⊢ T0 ➡*[h] T.
/3 width=3 by lprs_cprs_conf_sn, cpm_cpms/ qed-.
(* Properties with contextual transitive closure ****************************)
lemma lprs_CTC (h) (G):
- ∀L1,L2. L1⪤[CTC … (λL. cpm h G L 0)] L2 → ⦃G, L1⦄⊢ ➡*[h] L2.
+ ∀L1,L2. L1⪤[CTC … (λL. cpm h G L 0)] L2 → ⦃G,L1⦄⊢ ➡*[h] L2.
/3 width=3 by cprs_CTC, lex_co/ qed.
(* Inversion lemmas with contextual transitive closure **********************)
lemma lprs_inv_CTC (h) (G):
- ∀L1,L2. ⦃G, L1⦄⊢ ➡*[h] L2 → L1⪤[CTC … (λL. cpm h G L 0)] L2.
+ ∀L1,L2. ⦃G,L1⦄⊢ ➡*[h] L2 → L1⪤[CTC … (λL. cpm h G L 0)] L2.
/3 width=3 by cprs_inv_CTC, lex_co/ qed-.
(* Forward lemmas with length for local environments ************************)
-lemma lprs_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → |L1| = |L2|.
+lemma lprs_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → |L1| = |L2|.
/2 width=2 by lex_fwd_length/ qed-.
(* Basic_2A1: was: lprs_ind_dx *)
lemma lprs_ind_sn (h) (G) (L2): ∀Q:predicate lenv. Q L2 →
- (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L → ⦃G, L⦄ ⊢ ➡*[h] L2 → Q L → Q L1) →
- ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L1.
+ (∀L1,L. ⦃G,L1⦄ ⊢ ➡[h] L → ⦃G,L⦄ ⊢ ➡*[h] L2 → Q L → Q L1) →
+ ∀L1. ⦃G,L1⦄ ⊢ ➡*[h] L2 → Q L1.
/4 width=8 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, cpr_refl, lex_CTC_ind_sn/ qed-.
(* Basic_2A1: was: lprs_ind *)
lemma lprs_ind_dx (h) (G) (L1): ∀Q:predicate lenv. Q L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h] L → ⦃G, L⦄ ⊢ ➡[h] L2 → Q L → Q L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L2.
+ (∀L,L2. ⦃G,L1⦄ ⊢ ➡*[h] L → ⦃G,L⦄ ⊢ ➡[h] L2 → Q L → Q L2) →
+ ∀L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → Q L2.
/4 width=8 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, cpr_refl, lex_CTC_ind_dx/ qed-.
(* Properties with unbound rt-transition for full local environments ********)
-lemma lpr_lprs (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
+lemma lpr_lprs (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ ➡*[h] L2.
/4 width=3 by lprs_CTC, lpr_cprs_trans, lex_CTC_inj/ qed.
(* Basic_2A1: was: lprs_strap2 *)
-lemma lprs_step_sn (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L →
- ∀L2.⦃G, L⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
+lemma lprs_step_sn (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ➡[h] L →
+ ∀L2.⦃G,L⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ ➡*[h] L2.
/4 width=3 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, lex_CTC_step_sn/ qed-.
(* Basic_2A1: was: lpxs_strap1 *)
-lemma lprs_step_dx (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡*[h] L →
- ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
+lemma lprs_step_dx (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ➡*[h] L →
+ ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ ➡*[h] L2.
/4 width=3 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, lex_CTC_step_dx/ qed-.
lemma lprs_strip (h) (G): confluent2 … (lprs h G) (lpr h G).
(* Basic_2A1: was: lprs_lpxs *)
(* Note: original proof uses lpr_fwd_lpx and monotonic_TC *)
-lemma lprs_fwd_lpxs (h) (G) : ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2.
+lemma lprs_fwd_lpxs (h) (G) : ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2.
/3 width=3 by cpms_fwd_cpxs, lex_co/ qed-.
(* Properties with transitive closure ***************************************)
lemma lprs_TC (h) (G):
- ∀L1,L2. TC … (lex (λL.cpm h G L 0)) L1 L2 → ⦃G, L1⦄⊢ ➡*[h] L2.
+ ∀L1,L2. TC … (lex (λL.cpm h G L 0)) L1 L2 → ⦃G,L1⦄⊢ ➡*[h] L2.
/4 width=3 by lprs_CTC, lex_CTC, lpr_cprs_trans/ qed.
(* Inversion lemmas with transitive closure *********************************)
lemma lprs_inv_TC (h) (G):
- ∀L1,L2. ⦃G, L1⦄⊢ ➡*[h] L2 → TC … (lex (λL.cpm h G L 0)) L1 L2.
+ ∀L1,L2. ⦃G,L1⦄⊢ ➡*[h] L2 → TC … (lex (λL.cpm h G L 0)) L1 L2.
/3 width=3 by lprs_inv_CTC, lex_inv_CTC/ qed-.
(* Basic properties *********************************************************)
(* Basic_2A1: uses: lpxs_pair_refl *)
-lemma lpxs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- ∀I. ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈*[h] L2.ⓘ{I}.
+lemma lpxs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ ∀I. ⦃G,L1.ⓘ{I}⦄ ⊢ ⬈*[h] L2.ⓘ{I}.
/2 width=1 by lex_bind_refl_dx/ qed.
-lemma lpxs_pair (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2.
+lemma lpxs_pair (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ ∀V1,V2. ⦃G,L1⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2.
/2 width=1 by lex_pair/ qed.
lemma lpxs_refl (h) (G): reflexive … (lpxs h G).
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: was: lpxs_inv_atom1 *)
-lemma lpxs_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ⬈*[h] L2 → L2 = ⋆.
+lemma lpxs_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ⬈*[h] L2 → L2 = ⋆.
/2 width=2 by lex_inv_atom_sn/ qed-.
-lemma lpxs_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈*[h] L2 →
- ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈*[h] I2 & L2 = K2.ⓘ{I2}.
+lemma lpxs_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬈*[h] L2 →
+ ∃∃I2,K2. ⦃G,K1⦄ ⊢ ⬈*[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬈*[h] I2 & L2 = K2.ⓘ{I2}.
/2 width=1 by lex_inv_bind_sn/ qed-.
(* Basic_2A1: was: lpxs_inv_pair1 *)
-lemma lpxs_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ{I}V2.
+lemma lpxs_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2 →
+ ∃∃K2,V2. ⦃G,K1⦄ ⊢ ⬈*[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ{I}V2.
/2 width=1 by lex_inv_pair_sn/ qed-.
(* Basic_2A1: was: lpxs_inv_atom2 *)
-lemma lpxs_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ⬈*[h] ⋆ → L1 = ⋆.
+lemma lpxs_inv_atom_dx (h) (G): ∀L1. ⦃G,L1⦄ ⊢ ⬈*[h] ⋆ → L1 = ⋆.
/2 width=2 by lex_inv_atom_dx/ qed-.
(* Basic_2A1: was: lpxs_inv_pair2 *)
-lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈*[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L1 = K1.ⓑ{I}V1.
+lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ⬈*[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G,K1⦄ ⊢ ⬈*[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈*[h] V2 & L1 = K1.ⓑ{I}V1.
/2 width=1 by lex_inv_pair_dx/ qed-.
(* Basic eliminators ********************************************************)
lemma lpxs_ind (h) (G): ∀Q:relation lenv.
Q (⋆) (⋆) → (
∀I,K1,K2.
- ⦃G, K1⦄ ⊢ ⬈*[h] K2 →
+ ⦃G,K1⦄ ⊢ ⬈*[h] K2 →
Q K1 K2 → Q (K1.ⓘ{I}) (K2.ⓘ{I})
) → (
∀I,K1,K2,V1,V2.
- ⦃G, K1⦄ ⊢ ⬈*[h] K2 → ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 →
+ ⦃G,K1⦄ ⊢ ⬈*[h] K2 → ⦃G,K1⦄ ⊢ V1 ⬈*[h] V2 →
Q K1 K2 → Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
) →
- ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → Q L1 L2.
+ ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → Q L1 L2.
/3 width=4 by lex_ind/ qed-.
(* Properties with context-sensitive extended rt-computation for terms ******)
(* Basic_2A1: was: cpxs_bind2 *)
-lemma cpxs_bind_dx (h) (G): ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+lemma cpxs_bind_dx (h) (G): ∀L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
/4 width=5 by lpxs_cpxs_trans, lpxs_pair, cpxs_bind/ qed.
(* Inversion lemmas with context-sensitive ext rt-computation for terms *****)
-lemma cpxs_inv_abst1 (h) (G): ∀p,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈*[h] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈*[h] T2 &
+lemma cpxs_inv_abst1 (h) (G): ∀p,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈*[h] U2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈*[h] T2 &
U2 = ⓛ{p}V2.T2.
#h #G #p #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
(* Basic_2A1: was: cpxs_inv_abbr1 *)
lemma cpxs_inv_abbr1_dx (h) (p) (G) (L):
- ∀V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 &
+ ∀V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 &
U2 = ⓓ{p}V2.T2
- | ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & ⬆*[1] U2 ≘ T2 & p = Ⓣ.
+ | ∃∃T2. ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & ⬆*[1] U2 ≘ T2 & p = Ⓣ.
#h #p #G #L #V1 #T1 #U2 #H
@(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
#U0 #U2 #_ #HU02 * *
(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
-(* Properties with degree-based equivalence on closures *********************)
+(* Properties with sort-irrelevant equivalence on closures ******************)
-lemma fdeq_lpxs_trans (h) (o): ∀G1,G2,L1,L0,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L0, T2⦄ →
- ∀L2. ⦃G2, L0⦄ ⊢⬈*[h] L2 →
- ∃∃L. ⦃G1, L1⦄ ⊢⬈*[h] L & ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L0 #T1 #T2 #H1 #L2 #HL02
+lemma fdeq_lpxs_trans (h): ∀G1,G2,L1,L0,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L0,T2⦄ →
+ ∀L2. ⦃G2,L0⦄ ⊢⬈*[h] L2 →
+ ∃∃L. ⦃G1,L1⦄ ⊢⬈*[h] L & ⦃G1,L,T1⦄ ≛ ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L0 #T1 #T2 #H1 #L2 #HL02
elim (fdeq_inv_gen_dx … H1) -H1 #HG #HL10 #HT12 destruct
elim (rdeq_lpxs_trans … HL02 … HL10) -L0 #L0 #HL10 #HL02
/3 width=3 by fdeq_intro_dx, ex2_intro/
(* Forward lemmas with length for local environments ************************)
-lemma lpxs_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → |L1| = |L2|.
+lemma lpxs_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → |L1| = |L2|.
/2 width=2 by lex_fwd_length/ qed-.
(* Properties with unbound rt-transition for full local environments ********)
-lemma lpx_lpxs (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2.
+lemma lpx_lpxs (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2.
/3 width=3 by lpx_cpxs_trans, lex_CTC_inj/ qed.
(* Basic_2A1: was: lpxs_strap2 *)
-lemma lpxs_step_sn (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ⬈[h] L →
- ∀L2. ⦃G, L⦄ ⊢ ⬈*[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2.
+lemma lpxs_step_sn (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ⬈[h] L →
+ ∀L2. ⦃G,L⦄ ⊢ ⬈*[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2.
/3 width=3 by lpx_cpxs_trans, lex_CTC_step_sn/ qed-.
(* Basic_2A1: was: lpxs_strap1 *)
-lemma lpxs_step_dx (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ⬈*[h] L →
- ∀L2. ⦃G, L⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈*[h] L2.
+lemma lpxs_step_dx (h) (G): ∀L1,L. ⦃G,L1⦄ ⊢ ⬈*[h] L →
+ ∀L2. ⦃G,L⦄ ⊢ ⬈[h] L2 → ⦃G,L1⦄ ⊢ ⬈*[h] L2.
/3 width=3 by lpx_cpxs_trans, lex_CTC_step_dx/ qed-.
(* Eliminators with unbound rt-transition for full local environments *******)
(* Basic_2A1: was: lpxs_ind_dx *)
lemma lpxs_ind_sn (h) (G) (L2): ∀Q:predicate lenv. Q L2 →
- (∀L1,L. ⦃G, L1⦄ ⊢ ⬈[h] L → ⦃G, L⦄ ⊢ ⬈*[h] L2 → Q L → Q L1) →
- ∀L1. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → Q L1.
+ (∀L1,L. ⦃G,L1⦄ ⊢ ⬈[h] L → ⦃G,L⦄ ⊢ ⬈*[h] L2 → Q L → Q L1) →
+ ∀L1. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → Q L1.
/3 width=7 by lpx_cpxs_trans, cpx_refl, lex_CTC_ind_sn/ qed-.
(* Basic_2A1: was: lpxs_ind *)
lemma lpxs_ind_dx (h) (G) (L1): ∀Q:predicate lenv. Q L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L → ⦃G, L⦄ ⊢ ⬈[h] L2 → Q L → Q L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → Q L2.
+ (∀L,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L → ⦃G,L⦄ ⊢ ⬈[h] L2 → Q L → Q L2) →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → Q L2.
/3 width=7 by lpx_cpxs_trans, cpx_refl, lex_CTC_ind_dx/ qed-.
(* Properties with context-sensitive extended rt-transition for terms *******)
(* Advanced properties ******************************************************)
(* Basic_2A1: was: lpxs_pair2 *)
-lemma lpxs_pair_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2.
+lemma lpxs_pair_dx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ ∀V1,V2. ⦃G,L2⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀I. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈*[h] L2.ⓑ{I}V2.
/3 width=3 by lpxs_pair, lpxs_cpxs_trans/ qed.
(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
-(* Properties with degree-based equivalence on referred entries *************)
+(* Properties with sort-irrelevant equivalence on referred entries **********)
(* Basic_2A1: uses: lleq_lpxs_trans *)
-lemma rdeq_lpxs_trans (h) (o) (G) (T:term): ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈*[h] K2 →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ⬈*[h] K1 & K1 ≛[h, o, T] K2.
-#h #o #G #T #L2 #K2 #H @(lpxs_ind_sn … H) -L2 /2 width=3 by ex2_intro/
+lemma rdeq_lpxs_trans (h) (G) (T:term):
+ ∀L2,K2. ⦃G,L2⦄ ⊢ ⬈*[h] K2 →
+ ∀L1. L1 ≛[T] L2 →
+ ∃∃K1. ⦃G,L1⦄ ⊢ ⬈*[h] K1 & K1 ≛[T] K2.
+#h #G #T #L2 #K2 #H @(lpxs_ind_sn … H) -L2 /2 width=3 by ex2_intro/
#L #L2 #HL2 #_ #IH #L1 #HT
elim (rdeq_lpx_trans … HL2 … HT) -L #L #HL1 #HT
elim (IH … HT) -L2 #K #HLK #HT
qed-.
(* Basic_2A1: uses: lpxs_nlleq_inv_step_sn *)
-lemma lpxs_rdneq_inv_step_sn (h) (o) (G) (T:term):
- ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) →
- ∃∃L,L0. ⦃G, L1⦄ ⊢ ⬈[h] L & L1 ≛[h, o, T] L → ⊥ &
- ⦃G, L⦄ ⊢ ⬈*[h] L0 & L0 ≛[h, o, T] L2.
-#h #o #G #T #L1 #L2 #H @(lpxs_ind_sn … H) -L1
+lemma lpxs_rdneq_inv_step_sn (h) (G) (T:term):
+ ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) →
+ ∃∃L,L0. ⦃G,L1⦄ ⊢ ⬈[h] L & L1 ≛[T] L → ⊥ &
+ ⦃G,L⦄ ⊢ ⬈*[h] L0 & L0 ≛[T] L2.
+#h #G #T #L1 #L2 #H @(lpxs_ind_sn … H) -L1
[ #H elim H -H //
-| #L1 #L #H1 #H2 #IH2 #H12 elim (rdeq_dec h o L1 L T) #H
+| #L1 #L #H1 #H2 #IH2 #H12 elim (rdeq_dec L1 L T) #H
[ -H1 -H2 elim IH2 -IH2 /3 width=3 by rdeq_trans/ -H12
#L0 #L3 #H1 #H2 #H3 #H4 lapply (rdeq_rdneq_trans … H … H2) -H2
#H2 elim (rdeq_lpx_trans … H1 … H) -L
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/lsubeqx_6.ma".
-include "basic_2/rt_computation/rdsx.ma".
-
-(* CLEAR OF STRONGLY NORMALIZING ENTRIES FOR UNBOUND RT-TRANSITION **********)
-
-(* Note: this should be an instance of a more general sex *)
-(* Basic_2A1: uses: lcosx *)
-inductive lsubsx (h) (o) (G): rtmap → relation lenv ≝
-| lsubsx_atom: ∀f. lsubsx h o G f (⋆) (⋆)
-| lsubsx_push: ∀f,I,K1,K2. lsubsx h o G f K1 K2 →
- lsubsx h o G (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I})
-| lsubsx_unit: ∀f,I,K1,K2. lsubsx h o G f K1 K2 →
- lsubsx h o G (↑f) (K1.ⓤ{I}) (K2.ⓧ)
-| lsubsx_pair: ∀f,I,K1,K2,V. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ →
- lsubsx h o G f K1 K2 → lsubsx h o G (↑f) (K1.ⓑ{I}V) (K2.ⓧ)
-.
-
-interpretation
- "local environment refinement (clear)"
- 'LSubEqX h o f G L1 L2 = (lsubsx h o G f L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubsx_inv_atom_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
- L1 = ⋆ → L2 = ⋆.
-#h #o #g #G #L1 #L2 * -g -L1 -L2 //
-[ #f #I #K1 #K2 #_ #H destruct
-| #f #I #K1 #K2 #_ #H destruct
-| #f #I #K1 #K2 #V #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubsx_inv_atom_sn: ∀h,o,g,G,L2. G ⊢ ⋆ ⊆ⓧ[h, o, g] L2 → L2 = ⋆.
-/2 width=7 by lsubsx_inv_atom_sn_aux/ qed-.
-
-fact lsubsx_inv_push_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
- ∀f,I,K1. g = ⫯f → L1 = K1.ⓘ{I} →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓘ{I}.
-#h #o #g #G #L1 #L2 * -g -L1 -L2
-[ #f #g #J #L1 #_ #H destruct
-| #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
- <(injective_push … H1) -g /2 width=3 by ex2_intro/
-| #f #I #K1 #K2 #_ #g #J #L1 #H
- elim (discr_next_push … H)
-| #f #I #K1 #K2 #V #_ #_ #g #J #L1 #H
- elim (discr_next_push … H)
-]
-qed-.
-
-lemma lsubsx_inv_push_sn: ∀h,o,f,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⊆ⓧ[h, o, ⫯f] L2 →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓘ{I}.
-/2 width=5 by lsubsx_inv_push_sn_aux/ qed-.
-
-fact lsubsx_inv_unit_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
- ∀f,I,K1. g = ↑f → L1 = K1.ⓤ{I} →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
-#h #o #g #G #L1 #L2 * -g -L1 -L2
-[ #f #g #J #L1 #_ #H destruct
-| #f #I #K1 #K2 #_ #g #J #L1 #H
- elim (discr_push_next … H)
-| #f #I #K1 #K2 #HK12 #g #J #L1 #H1 #H2 destruct
- <(injective_next … H1) -g /2 width=3 by ex2_intro/
-| #f #I #K1 #K2 #V #_ #_ #g #J #L1 #_ #H destruct
-]
-qed-.
-
-lemma lsubsx_inv_unit_sn: ∀h,o,f,I,G,K1,L2. G ⊢ K1.ⓤ{I} ⊆ⓧ[h, o, ↑f] L2 →
- ∃∃K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
-/2 width=6 by lsubsx_inv_unit_sn_aux/ qed-.
-
-fact lsubsx_inv_pair_sn_aux: ∀h,o,g,G,L1,L2. G ⊢ L1 ⊆ⓧ[h, o, g] L2 →
- ∀f,I,K1,V. g = ↑f → L1 = K1.ⓑ{I}V →
- ∃∃K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
-#h #o #g #G #L1 #L2 * -g -L1 -L2
-[ #f #g #J #L1 #W #_ #H destruct
-| #f #I #K1 #K2 #_ #g #J #L1 #W #H
- elim (discr_push_next … H)
-| #f #I #K1 #K2 #_ #g #J #L1 #W #_ #H destruct
-| #f #I #K1 #K2 #V #HV #HK12 #g #J #L1 #W #H1 #H2 destruct
- <(injective_next … H1) -g /2 width=4 by ex3_intro/
-]
-qed-.
-
-(* Basic_2A1: uses: lcosx_inv_pair *)
-lemma lsubsx_inv_pair_sn: ∀h,o,f,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, o, ↑f] L2 →
- ∃∃K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h, o, f] K2 & L2 = K2.ⓧ.
-/2 width=6 by lsubsx_inv_pair_sn_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lsubsx_inv_pair_sn_gen: ∀h,o,g,I,G,K1,L2,V. G ⊢ K1.ⓑ{I}V ⊆ⓧ[h, o, g] L2 →
- ∨∨ ∃∃f,K2. G ⊢ K1 ⊆ⓧ[h, o, f] K2 & g = ⫯f & L2 = K2.ⓑ{I}V
- | ∃∃f,K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ &
- G ⊢ K1 ⊆ⓧ[h, o, f] K2 & g = ↑f & L2 = K2.ⓧ.
-#h #o #g #I #G #K1 #L2 #V #H
-elim (pn_split g) * #f #Hf destruct
-[ elim (lsubsx_inv_push_sn … H) -H /3 width=5 by ex3_2_intro, or_introl/
-| elim (lsubsx_inv_pair_sn … H) -H /3 width=6 by ex4_2_intro, or_intror/
-]
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lsubsx_fwd_bind_sn: ∀h,o,g,I1,G,K1,L2. G ⊢ K1.ⓘ{I1} ⊆ⓧ[h, o, g] L2 →
- ∃∃I2,K2. G ⊢ K1 ⊆ⓧ[h, o, ⫱g] K2 & L2 = K2.ⓘ{I2}.
-#h #o #g #I1 #G #K1 #L2
-elim (pn_split g) * #f #Hf destruct
-[ #H elim (lsubsx_inv_push_sn … H) -H
-| cases I1 -I1 #I1
- [ #H elim (lsubsx_inv_unit_sn … H) -H
- | #V #H elim (lsubsx_inv_pair_sn … H) -H
- ]
-]
-/2 width=4 by ex2_2_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lsubsx_eq_repl_back: ∀h,o,G,L1,L2. eq_repl_back … (λf. G ⊢ L1 ⊆ⓧ[h, o, f] L2).
-#h #o #G #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-[ #f #I #L1 #L2 #_ #IH #x #H
- elim (eq_inv_px … H) -H /3 width=3 by lsubsx_push/
-| #f #I #L1 #L2 #_ #IH #x #H
- elim (eq_inv_nx … H) -H /3 width=3 by lsubsx_unit/
-| #f #I #L1 #L2 #V #HV #_ #IH #x #H
- elim (eq_inv_nx … H) -H /3 width=3 by lsubsx_pair/
-]
-qed-.
-
-lemma lsubsx_eq_repl_fwd: ∀h,o,G,L1,L2. eq_repl_fwd … (λf. G ⊢ L1 ⊆ⓧ[h, o, f] L2).
-#h #o #G #L1 #L2 @eq_repl_sym /2 width=3 by lsubsx_eq_repl_back/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: uses: lcosx_O *)
-lemma lsubsx_refl: ∀h,o,f,G. 𝐈⦃f⦄ → reflexive … (lsubsx h o G f).
-#h #o #f #G #Hf #L elim L -L
-/3 width=3 by lsubsx_eq_repl_back, lsubsx_push, eq_push_inv_isid/
-qed.
-
-(* Basic_2A1: removed theorems 2:
- lcosx_drop_trans_lt lcosx_inv_succ
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/rt_computation/lsubsx.ma".
-
-(* CLEAR OF STRONGLY NORMALIZING ENTRIES FOR UNBOUND RT-TRANSITION **********)
-
-(* Main properties **********************************************************)
-
-theorem lsubsx_fix: ∀h,o,f,G,L1,L. G ⊢ L1 ⊆ⓧ[h, o, f] L →
- ∀L2. G ⊢ L ⊆ⓧ[h, o, f] L2 → L = L2.
-#h #o #f #G #L1 #L #H elim H -f -L1 -L
-[ #f #L2 #H
- >(lsubsx_inv_atom_sn … H) -L2 //
-| #f #I #K1 #K2 #_ #IH #L2 #H
- elim (lsubsx_inv_push_sn … H) -H /3 width=1 by eq_f2/
-| #f #I #K1 #K2 #_ #IH #L2 #H
- elim (lsubsx_inv_unit_sn … H) -H /3 width=1 by eq_f2/
-| #f #I #K1 #K2 #V #_ #_ #IH #L2 #H
- elim (lsubsx_inv_unit_sn … H) -H /3 width=1 by eq_f2/
-]
-qed-.
-
-theorem lsubsx_trans: ∀h,o,f,G. Transitive … (lsubsx h o G f).
-#h #o #f #G #L1 #L #H1 #L2 #H2
-<(lsubsx_fix … H1 … H2) -L2 //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/rt_computation/rdsx_drops.ma".
-include "basic_2/rt_computation/rdsx_lpxs.ma".
-include "basic_2/rt_computation/lsubsx.ma".
-
-(* CLEAR OF STRONGLY NORMALIZING ENTRIES FOR UNBOUND RT-TRANSITION **********)
-
-(* Properties with strongly normalizing referred local environments *********)
-
-(* Basic_2A1: uses: lsx_cpx_trans_lcosx *)
-lemma rdsx_cpx_trans_lsubsx (h) (o): ∀G,L0,T1,T2. ⦃G, L0⦄ ⊢ T1 ⬈[h] T2 →
- ∀f,L. G ⊢ L0 ⊆ⓧ[h, o, f] L →
- G ⊢ ⬈*[h, o, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, o, T2] 𝐒⦃L⦄.
-#h #o #G #L0 #T1 #T2 #H @(cpx_ind … H) -G -L0 -T1 -T2 //
-[ #I0 #G #K0 #V1 #V2 #W2 #_ #IH #HVW2 #g #L #HK0 #HL
- elim (lsubsx_inv_pair_sn_gen … HK0) -HK0 *
- [ #f #K #HK0 #H1 #H2 destruct
- /4 width=8 by rdsx_lifts, rdsx_fwd_pair, drops_refl, drops_drop/
- | #f #K #HV1 #HK0 #H1 #H2 destruct
- /4 width=8 by rdsx_lifts, drops_refl, drops_drop/
- ]
-| #I0 #G #K0 #T #U #i #_ #IH #HTU #g #L #HK0 #HL
- elim (lsubsx_fwd_bind_sn … HK0) -HK0 #I #K #HK0 #H destruct
- /6 width=8 by rdsx_inv_lifts, rdsx_lifts, drops_refl, drops_drop/
-| #p #I0 #G #L0 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #f #L #HL0 #HL
- elim (rdsx_inv_bind … HL) -HL
- /4 width=2 by lsubsx_pair, rdsx_bind_void/
-| #I0 #G #L0 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #f #L #HL0 #HL
- elim (rdsx_inv_flat … HL) -HL /3 width=2 by rdsx_flat/
-| #G #L0 #V #U1 #T1 #T2 #HTU1 #_ #IHT12 #f #L #HL0 #HL
- elim (rdsx_inv_bind … HL) -HL #HV #HU1
- /5 width=8 by rdsx_inv_lifts, drops_refl, drops_drop/
-| #G #L0 #V #T1 #T2 #_ #IHT12 #f #L #HL0 #HL
- elim (rdsx_inv_flat … HL) -HL /2 width=2 by/
-| #G #L0 #V1 #V2 #T #_ #IHV12 #f #L #HL0 #HL
- elim (rdsx_inv_flat … HL) -HL /2 width=2 by/
-| #p #G #L0 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #f #L #HL0 #HL
- elim (rdsx_inv_flat … HL) -HL #HV1 #HL
- elim (rdsx_inv_bind … HL) -HL #HW1 #HT1
- /4 width=2 by lsubsx_pair, rdsx_bind_void, rdsx_flat/
-| #p #G #L0 #V1 #V2 #U2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #HVU2 #f #L #HL0 #HL
- elim (rdsx_inv_flat … HL) -HL #HV1 #HL
- elim (rdsx_inv_bind … HL) -HL #HW1 #HT1
- /6 width=8 by lsubsx_pair, rdsx_lifts, rdsx_bind_void, rdsx_flat, drops_refl, drops_drop/
-]
-qed-.
-
-(* Advanced properties of strongly normalizing referred local environments **)
-
-(* Basic_2A1: uses: lsx_cpx_trans_O *)
-lemma rdsx_cpx_trans (h) (o): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
- G ⊢ ⬈*[h, o, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, o, T2] 𝐒⦃L⦄.
-/3 width=6 by rdsx_cpx_trans_lsubsx, lsubsx_refl/ qed-.
-
-lemma rdsx_cpxs_trans (h) (o): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
- G ⊢ ⬈*[h, o, T1] 𝐒⦃L⦄ → G ⊢ ⬈*[h, o, T2] 𝐒⦃L⦄.
-#h #o #G #L #T1 #T2 #H
-@(cpxs_ind_dx ???????? H) -T1 //
-/3 width=3 by rdsx_cpx_trans/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/predtysnstrong_5.ma".
-include "static_2/static/rdeq.ma".
-include "basic_2/rt_transition/lpx.ma".
-
-(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
-
-definition rdsx (h) (o) (G) (T): predicate lenv ≝
- SN … (lpx h G) (rdeq h o T).
-
-interpretation
- "strong normalization for unbound context-sensitive parallel rt-transition on referred entries (local environment)"
- 'PRedTySNStrong h o T G L = (rdsx h o G T L).
-
-(* Basic eliminators ********************************************************)
-
-(* Basic_2A1: uses: lsx_ind *)
-lemma rdsx_ind (h) (o) (G) (T):
- ∀Q:predicate lenv.
- (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- (∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → Q L2) →
- Q L1
- ) →
- ∀L. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → Q L.
-#h #o #G #T #Q #H0 #L1 #H elim H -L1
-/5 width=1 by SN_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_2A1: uses: lsx_intro *)
-lemma rdsx_intro (h) (o) (G) (T):
- ∀L1.
- (∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
- G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄.
-/5 width=1 by SN_intro/ qed.
-
-(* Basic forward lemmas *****************************************************)
-
-(* Basic_2A1: uses: lsx_fwd_pair_sn lsx_fwd_bind_sn lsx_fwd_flat_sn *)
-lemma rdsx_fwd_pair_sn (h) (o) (G):
- ∀I,L,V,T. G ⊢ ⬈*[h, o, ②{I}V.T] 𝐒⦃L⦄ →
- G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄.
-#h #o #G #I #L #V #T #H
-@(rdsx_ind … H) -L #L1 #_ #IHL1
-@rdsx_intro #L2 #HL12 #HnL12
-/4 width=3 by rdeq_fwd_pair_sn/
-qed-.
-
-(* Basic_2A1: uses: lsx_fwd_flat_dx *)
-lemma rdsx_fwd_flat_dx (h) (o) (G):
- ∀I,L,V,T. G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄ →
- G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
-#h #o #G #I #L #V #T #H
-@(rdsx_ind … H) -L #L1 #_ #IHL1
-@rdsx_intro #L2 #HL12 #HnL12
-/4 width=3 by rdeq_fwd_flat_dx/
-qed-.
-
-fact rdsx_fwd_pair_aux (h) (o) (G): ∀L. G ⊢ ⬈*[h, o, #0] 𝐒⦃L⦄ →
- ∀I,K,V. L = K.ⓑ{I}V → G ⊢ ⬈*[h, o, V] 𝐒⦃K⦄.
-#h #o #G #L #H
-@(rdsx_ind … H) -L #L1 #_ #IH #I #K1 #V #H destruct
-/5 width=5 by lpx_pair, rdsx_intro, rdeq_fwd_zero_pair/
-qed-.
-
-lemma rdsx_fwd_pair (h) (o) (G):
- ∀I,K,V. G ⊢ ⬈*[h, o, #0] 𝐒⦃K.ⓑ{I}V⦄ → G ⊢ ⬈*[h, o, V] 𝐒⦃K⦄.
-/2 width=4 by rdsx_fwd_pair_aux/ qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_2A1: uses: lsx_inv_flat *)
-lemma rdsx_inv_flat (h) (o) (G): ∀I,L,V,T. G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄ →
- ∧∧ G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ & G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
-/3 width=3 by rdsx_fwd_pair_sn, rdsx_fwd_flat_dx, conj/ qed-.
-
-(* Basic_2A1: removed theorems 9:
- lsx_ge_up lsx_ge
- lsxa_ind lsxa_intro lsxa_lleq_trans lsxa_lpxs_trans lsxa_intro_lpx lsx_lsxa lsxa_inv_lsx
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/rt_computation/csx_lsubr.ma".
-include "basic_2/rt_computation/csx_cpxs.ma".
-include "basic_2/rt_computation/lsubsx_rdsx.ma".
-
-(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: uses: lsx_lref_be_lpxs *)
-lemma rdsx_pair_lpxs (h) (o) (G):
- ∀K1,V. ⦃G, K1⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ →
- ∀K2. G ⊢ ⬈*[h, o, V] 𝐒⦃K2⦄ → ⦃G, K1⦄ ⊢ ⬈*[h] K2 →
- ∀I. G ⊢ ⬈*[h, o, #0] 𝐒⦃K2.ⓑ{I}V⦄.
-#h #o #G #K1 #V #H
-@(csx_ind_cpxs … H) -V #V0 #_ #IHV0 #K2 #H
-@(rdsx_ind … H) -K2 #K0 #HK0 #IHK0 #HK10 #I
-@rdsx_intro #Y #HY #HnY
-elim (lpx_inv_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
-elim (tdeq_dec h o V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnY ]
-[ /5 width=5 by rdsx_rdeq_trans, lpxs_step_dx, rdeq_pair/
-| @(IHV0 … HnV02) -IHV0 -HnV02
- [ /2 width=3 by lpxs_cpx_trans/
- | /3 width=3 by rdsx_lpx_trans, rdsx_cpx_trans/
- | /2 width=3 by lpxs_step_dx/
- ]
-]
-qed.
-
-(* Basic_2A1: uses: lsx_lref_be *)
-lemma rdsx_lref_pair_drops (h) (o) (G):
- ∀K,V. ⦃G, K⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ → G ⊢ ⬈*[h, o, V] 𝐒⦃K⦄ →
- ∀I,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h, o, #i] 𝐒⦃L⦄.
-#h #o #G #K #V #HV #HK #I #i elim i -i
-[ #L #H >(drops_fwd_isid … H) -H /2 width=3 by rdsx_pair_lpxs/
-| #i #IH #L #H
- elim (drops_inv_bind2_isuni_next … H) -H // #J #Y #HY #H destruct
- @(rdsx_lifts … (𝐔❴1❵)) /3 width=6 by drops_refl, drops_drop/ (**) (* full auto fails *)
-]
-qed.
-
-(* Main properties **********************************************************)
-
-(* Basic_2A1: uses: csx_lsx *)
-theorem csx_rdsx (h) (o): ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
-#h #o #G #L #T @(fqup_wf_ind_eq (Ⓕ) … G L T) -G -L -T
-#Z #Y #X #IH #G #L * * //
-[ #i #HG #HL #HT #H destruct
- elim (csx_inv_lref … H) -H [ |*: * ]
- [ /2 width=1 by rdsx_lref_atom/
- | /2 width=3 by rdsx_lref_unit/
- | /4 width=6 by rdsx_lref_pair_drops, fqup_lref/
- ]
-| #p #I #V #T #HG #HL #HT #H destruct
- elim (csx_fwd_bind_unit … H Void) -H /3 width=1 by rdsx_bind_void/
-| #I #V #T #HG #HL #HT #H destruct
- elim (csx_fwd_flat … H) -H /3 width=1 by rdsx_flat/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/static/rdeq_drops.ma".
-include "basic_2/rt_transition/lpx_drops.ma".
-include "basic_2/rt_computation/rdsx_length.ma".
-include "basic_2/rt_computation/rdsx_fqup.ma".
-
-(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
-
-(* Properties with generic relocation ***************************************)
-
-(* Note: this uses length *)
-(* Basic_2A1: uses: lsx_lift_le lsx_lift_ge *)
-lemma rdsx_lifts (h) (o) (G): d_liftable1_isuni … (λL,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄).
-#h #o #G #K #T #H @(rdsx_ind … H) -K
-#K1 #_ #IH #b #f #L1 #HLK1 #Hf #U #HTU @rdsx_intro
-#L2 #HL12 #HnL12 elim (lpx_drops_conf … HLK1 … HL12)
-/5 width=9 by rdeq_lifts_bi, lpx_fwd_length/
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-(* Basic_2A1: uses: lsx_inv_lift_le lsx_inv_lift_be lsx_inv_lift_ge *)
-lemma rdsx_inv_lifts (h) (o) (G): d_deliftable1_isuni … (λL,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄).
-#h #o #G #L #U #H @(rdsx_ind … H) -L
-#L1 #_ #IH #b #f #K1 #HLK1 #Hf #T #HTU @rdsx_intro
-#K2 #HK12 #HnK12 elim (drops_lpx_trans … HLK1 … HK12) -HK12
-/4 width=10 by rdeq_inv_lifts_bi/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: uses: lsx_lref_free *)
-lemma rdsx_lref_atom (h) (o) (G): ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ → G ⊢ ⬈*[h, o, #i] 𝐒⦃L⦄.
-#h #o #G #L1 #i #HL1
-@(rdsx_lifts … (#0) … HL1) -HL1 //
-qed.
-
-(* Basic_2A1: uses: lsx_lref_skip *)
-lemma rdsx_lref_unit (h) (o) (G): ∀I,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → G ⊢ ⬈*[h, o, #i] 𝐒⦃L⦄.
-#h #o #G #I #L1 #K1 #i #HL1
-@(rdsx_lifts … (#0) … HL1) -HL1 //
-qed.
-
-(* Advanced forward lemmas **************************************************)
-
-(* Basic_2A1: uses: lsx_fwd_lref_be *)
-lemma rdsx_fwd_lref_pair (h) (o) (G):
- ∀L,i. G ⊢ ⬈*[h, o, #i] 𝐒⦃L⦄ →
- ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h, o, V] 𝐒⦃K⦄.
-#h #o #G #L #i #HL #I #K #V #HLK
-lapply (rdsx_inv_lifts … HL … HLK … (#0) ?) -L
-/2 width=2 by rdsx_fwd_pair/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/static/rdeq_fqup.ma".
-include "basic_2/rt_computation/rdsx.ma".
-
-(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: uses: lsx_atom *)
-lemma lfsx_atom (h) (o) (G) (T): G ⊢ ⬈*[h, o, T] 𝐒⦃⋆⦄.
-#h #o #G #T
-@rdsx_intro #Y #H #HnT
-lapply (lpx_inv_atom_sn … H) -H #H destruct
-elim HnT -HnT //
-qed.
-
-(* Advanced forward lemmas **************************************************)
-
-(* Basic_2A1: uses: lsx_fwd_bind_dx *)
-(* Note: the exclusion binder (ⓧ) makes this more elegant and much simpler *)
-(* Note: the old proof without the exclusion binder requires lreq *)
-lemma rdsx_fwd_bind_dx (h) (o) (G):
- ∀p,I,L,V,T. G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L⦄ →
- G ⊢ ⬈*[h, o, T] 𝐒⦃L.ⓧ⦄.
-#h #o #G #p #I #L #V #T #H
-@(rdsx_ind … H) -L #L1 #_ #IH
-@rdsx_intro #Y #H #HT
-elim (lpx_inv_unit_sn … H) -H #L2 #HL12 #H destruct
-/4 width=4 by rdeq_fwd_bind_dx_void/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_2A1: uses: lsx_inv_bind *)
-lemma rdsx_inv_bind (h) (o) (G): ∀p,I,L,V,T. G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L⦄ →
- ∧∧ G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ & G ⊢ ⬈*[h, o, T] 𝐒⦃L.ⓧ⦄.
-/3 width=4 by rdsx_fwd_pair_sn, rdsx_fwd_bind_dx, conj/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/static/rdeq_length.ma".
-include "basic_2/rt_transition/lpx_length.ma".
-include "basic_2/rt_computation/rdsx.ma".
-
-(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: uses: lsx_sort *)
-lemma rdsx_sort (h) (o) (G): ∀L,s. G ⊢ ⬈*[h, o, ⋆s] 𝐒⦃L⦄.
-#h #o #G #L1 #s @rdsx_intro #L2 #H #Hs
-elim Hs -Hs /3 width=3 by lpx_fwd_length, rdeq_sort_length/
-qed.
-
-(* Basic_2A1: uses: lsx_gref *)
-lemma rdsx_gref (h) (o) (G): ∀L,l. G ⊢ ⬈*[h, o, §l] 𝐒⦃L⦄.
-#h #o #G #L1 #s @rdsx_intro #L2 #H #Hs
-elim Hs -Hs /3 width=3 by lpx_fwd_length, rdeq_gref_length/
-qed.
-
-lemma rdsx_unit (h) (o) (G): ∀I,L. G ⊢ ⬈*[h, o, #0] 𝐒⦃L.ⓤ{I}⦄.
-#h #o #G #I #L1 @rdsx_intro
-#Y #HY #HnY elim HnY -HnY
-elim (lpx_inv_unit_sn … HY) -HY #L2 #HL12 #H destruct
-/3 width=3 by lpx_fwd_length, rdeq_unit_length/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/rt_computation/lpxs_rdeq.ma".
-include "basic_2/rt_computation/lpxs_lpxs.ma".
-include "basic_2/rt_computation/rdsx_rdsx.ma".
-
-(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
-
-(* Properties with unbound rt-computation for full local environments *******)
-
-(* Basic_2A1: uses: lsx_intro_alt *)
-lemma rdsx_intro_lpxs (h) (o) (G):
- ∀L1,T. (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
- G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄.
-/4 width=1 by lpx_lpxs, rdsx_intro/ qed-.
-
-(* Basic_2A1: uses: lsx_lpxs_trans *)
-lemma rdsx_lpxs_trans (h) (o) (G): ∀L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
-#h #o #G #L1 #T #HL1 #L2 #H @(lpxs_ind_dx … H) -L2
-/2 width=3 by rdsx_lpx_trans/
-qed-.
-
-(* Eliminators with unbound rt-computation for full local environments ******)
-
-lemma rdsx_ind_lpxs_rdeq (h) (o) (G):
- ∀T. ∀Q:predicate lenv.
- (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → Q L2) →
- Q L1
- ) →
- ∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- ∀L0. ⦃G, L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[h, o, T] L2 → Q L2.
-#h #o #G #T #Q #IH #L1 #H @(rdsx_ind … H) -L1
-#L1 #HL1 #IH1 #L0 #HL10 #L2 #HL02
-@IH -IH /3 width=3 by rdsx_lpxs_trans, rdsx_rdeq_trans/ -HL1 #K2 #HLK2 #HnLK2
-lapply (rdeq_rdneq_trans … HL02 … HnLK2) -HnLK2 #H
-elim (rdeq_lpxs_trans … HLK2 … HL02) -L2 #K0 #HLK0 #HK02
-lapply (rdneq_rdeq_canc_dx … H … HK02) -H #HnLK0
-elim (rdeq_dec h o L1 L0 T) #H
-[ lapply (rdeq_rdneq_trans … H … HnLK0) -H -HnLK0 #Hn10
- lapply (lpxs_trans … HL10 … HLK0) -L0 #H10
- elim (lpxs_rdneq_inv_step_sn … H10 … Hn10) -H10 -Hn10
- /3 width=8 by rdeq_trans/
-| elim (lpxs_rdneq_inv_step_sn … HL10 … H) -HL10 -H #L #K #HL1 #HnL1 #HLK #HKL0
- elim (rdeq_lpxs_trans … HLK0 … HKL0) -L0
- /3 width=8 by lpxs_trans, rdeq_trans/
-]
-qed-.
-
-(* Basic_2A1: uses: lsx_ind_alt *)
-lemma rdsx_ind_lpxs (h) (o) (G):
- ∀T. ∀Q:predicate lenv.
- (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → Q L2) →
- Q L1
- ) →
- ∀L. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → Q L.
-#h #o #G #T #Q #IH #L #HL
-@(rdsx_ind_lpxs_rdeq … IH … HL) -IH -HL // (**) (* full auto fails *)
-qed-.
-
-(* Advanced properties ******************************************************)
-
-fact rdsx_bind_lpxs_aux (h) (o) (G):
- ∀p,I,L1,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L1⦄ →
- ∀Y,T. G ⊢ ⬈*[h, o, T] 𝐒⦃Y⦄ →
- ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
-#h #o #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
-#L1 #_ #IHL1 #Y #T #H @(rdsx_ind_lpxs … H) -Y
-#Y #HY #IHY #L2 #H #HL12 destruct
-@rdsx_intro_lpxs #L0 #HL20
-lapply (lpxs_trans … HL12 … HL20) #HL10 #H
-elim (rdneq_inv_bind … H) -H [ -IHY | -HY -IHL1 -HL12 ]
-[ #HnV elim (rdeq_dec h o L1 L2 V)
- [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
- /3 width=4 by rdsx_lpxs_trans, lpxs_bind_refl_dx, rdeq_canc_sn/ (**) (* full auto too slow *)
- | -HnV -HL10 /4 width=4 by rdsx_lpxs_trans, lpxs_bind_refl_dx/
- ]
-| /3 width=4 by lpxs_bind_refl_dx/
-]
-qed-.
-
-(* Basic_2A1: uses: lsx_bind *)
-lemma rdsx_bind (h) (o) (G):
- ∀p,I,L,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ →
- ∀T. G ⊢ ⬈*[h, o, T] 𝐒⦃L.ⓑ{I}V⦄ →
- G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L⦄.
-/2 width=3 by rdsx_bind_lpxs_aux/ qed.
-
-(* Basic_2A1: uses: lsx_flat_lpxs *)
-lemma rdsx_flat_lpxs (h) (o) (G):
- ∀I,L1,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L1⦄ →
- ∀L2,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄ → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L2⦄.
-#h #o #G #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
-#L1 #HL1 #IHL1 #L2 #T #H @(rdsx_ind_lpxs … H) -L2
-#L2 #HL2 #IHL2 #HL12 @rdsx_intro_lpxs
-#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
-#HL10 #H elim (rdneq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
-[ #HnV elim (rdeq_dec h o L1 L2 V)
- [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
- /3 width=5 by rdsx_lpxs_trans, rdeq_canc_sn/ (**) (* full auto too slow: 47s *)
- | -HnV -HL10 /3 width=4 by rdsx_lpxs_trans/
- ]
-| /3 width=3 by/
-]
-qed-.
-
-(* Basic_2A1: uses: lsx_flat *)
-lemma rdsx_flat (h) (o) (G):
- ∀I,L,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ →
- ∀T. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄.
-/2 width=3 by rdsx_flat_lpxs/ qed.
-
-fact rdsx_bind_lpxs_void_aux (h) (o) (G):
- ∀p,I,L1,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L1⦄ →
- ∀Y,T. G ⊢ ⬈*[h, o, T] 𝐒⦃Y⦄ →
- ∀L2. Y = L2.ⓧ → ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
-#h #o #G #p #I #L1 #V #H @(rdsx_ind_lpxs … H) -L1
-#L1 #_ #IHL1 #Y #T #H @(rdsx_ind_lpxs … H) -Y
-#Y #HY #IHY #L2 #H #HL12 destruct
-@rdsx_intro_lpxs #L0 #HL20
-lapply (lpxs_trans … HL12 … HL20) #HL10 #H
-elim (rdneq_inv_bind_void … H) -H [ -IHY | -HY -IHL1 -HL12 ]
-[ #HnV elim (rdeq_dec h o L1 L2 V)
- [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
- /3 width=6 by rdsx_lpxs_trans, lpxs_bind_refl_dx, rdeq_canc_sn/ (**) (* full auto too slow *)
- | -HnV -HL10 /4 width=4 by rdsx_lpxs_trans, lpxs_bind_refl_dx/
- ]
-| /3 width=4 by lpxs_bind_refl_dx/
-]
-qed-.
-
-lemma rdsx_bind_void (h) (o) (G):
- ∀p,I,L,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ →
- ∀T. G ⊢ ⬈*[h, o, T] 𝐒⦃L.ⓧ⦄ →
- G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L⦄.
-/2 width=3 by rdsx_bind_lpxs_void_aux/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/rt_transition/lpx_rdeq.ma".
-include "basic_2/rt_computation/rdsx.ma".
-
-(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: uses: lsx_lleq_trans *)
-lemma rdsx_rdeq_trans (h) (o) (G):
- ∀L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- ∀L2. L1 ≛[h, o, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
-#h #o #G #L1 #T #H @(rdsx_ind … H) -L1
-#L1 #_ #IHL1 #L2 #HL12 @rdsx_intro
-#L #HL2 #HnL2 elim (rdeq_lpx_trans … HL2 … HL12) -HL2
-/4 width=5 by rdeq_repl/
-qed-.
-
-(* Basic_2A1: uses: lsx_lpx_trans *)
-lemma rdsx_lpx_trans (h) (o) (G):
- ∀L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
-#h #o #G #L1 #T #H @(rdsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
-elim (rdeq_dec h o L1 L2 T) /3 width=4 by rdsx_rdeq_trans/
-qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/predtysnstrong_4.ma".
+include "static_2/static/rdeq.ma".
+include "basic_2/rt_transition/lpx.ma".
+
+(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+definition rsx (h) (G) (T): predicate lenv ≝
+ SN … (lpx h G) (rdeq T).
+
+interpretation
+ "strong normalization for unbound context-sensitive parallel rt-transition on referred entries (local environment)"
+ 'PRedTySNStrong h T G L = (rsx h G T L).
+
+(* Basic eliminators ********************************************************)
+
+(* Basic_2A1: uses: lsx_ind *)
+lemma rsx_ind (h) (G) (T) (Q:predicate lenv):
+ (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ Q L1
+ ) →
+ ∀L. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → Q L.
+#h #G #T #Q #H0 #L1 #H elim H -L1
+/5 width=1 by SN_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: uses: lsx_intro *)
+lemma rsx_intro (h) (G) (T):
+ ∀L1.
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄) →
+ G ⊢ ⬈*[h,T] 𝐒⦃L1⦄.
+/5 width=1 by SN_intro/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+(* Basic_2A1: uses: lsx_fwd_pair_sn lsx_fwd_bind_sn lsx_fwd_flat_sn *)
+lemma rsx_fwd_pair_sn (h) (G):
+ ∀I,L,V,T. G ⊢ ⬈*[h,②{I}V.T] 𝐒⦃L⦄ →
+ G ⊢ ⬈*[h,V] 𝐒⦃L⦄.
+#h #G #I #L #V #T #H
+@(rsx_ind … H) -L #L1 #_ #IHL1
+@rsx_intro #L2 #HL12 #HnL12
+/4 width=3 by rdeq_fwd_pair_sn/
+qed-.
+
+(* Basic_2A1: uses: lsx_fwd_flat_dx *)
+lemma rsx_fwd_flat_dx (h) (G):
+ ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄ →
+ G ⊢ ⬈*[h,T] 𝐒⦃L⦄.
+#h #G #I #L #V #T #H
+@(rsx_ind … H) -L #L1 #_ #IHL1
+@rsx_intro #L2 #HL12 #HnL12
+/4 width=3 by rdeq_fwd_flat_dx/
+qed-.
+
+fact rsx_fwd_pair_aux (h) (G):
+ ∀L. G ⊢ ⬈*[h,#0] 𝐒⦃L⦄ →
+ ∀I,K,V. L = K.ⓑ{I}V → G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
+#h #G #L #H
+@(rsx_ind … H) -L #L1 #_ #IH #I #K1 #V #H destruct
+/5 width=5 by lpx_pair, rsx_intro, rdeq_fwd_zero_pair/
+qed-.
+
+lemma rsx_fwd_pair (h) (G):
+ ∀I,K,V. G ⊢ ⬈*[h,#0] 𝐒⦃K.ⓑ{I}V⦄ → G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
+/2 width=4 by rsx_fwd_pair_aux/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_2A1: uses: lsx_inv_flat *)
+lemma rsx_inv_flat (h) (G):
+ ∀I,L,V,T. G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄ →
+ ∧∧ G ⊢ ⬈*[h,V] 𝐒⦃L⦄ & G ⊢ ⬈*[h,T] 𝐒⦃L⦄.
+/3 width=3 by rsx_fwd_pair_sn, rsx_fwd_flat_dx, conj/ qed-.
+
+(* Basic_2A1: removed theorems 9:
+ lsx_ge_up lsx_ge
+ lsxa_ind lsxa_intro lsxa_lleq_trans lsxa_lpxs_trans lsxa_intro_lpx lsx_lsxa lsxa_inv_lsx
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/csx_lsubr.ma".
+include "basic_2/rt_computation/csx_cpxs.ma".
+include "basic_2/rt_computation/jsx_rsx.ma".
+
+(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+(* Forward lemmas with strongly rt-normalizing terms ************************)
+
+fact rsx_fwd_lref_pair_csx_aux (h) (G):
+ ∀L. G ⊢ ⬈*[h,#0] 𝐒⦃L⦄ →
+ ∀I,K,V. L = K.ⓑ{I}V → ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+#h #G #L #H
+@(rsx_ind … H) -L #L #_ #IH #I #K #V1 #H destruct
+@csx_intro #V2 #HV12 #HnV12
+@(IH … I) -IH [1,4: // | -HnV12 | -G #H ]
+[ /2 width=1 by lpx_pair/
+| elim (rdeq_inv_zero_pair_sn … H) -H #Y #X #_ #H1 #H2 destruct -I
+ /2 width=1 by/
+]
+qed-.
+
+lemma rsx_fwd_lref_pair_csx (h) (G):
+ ∀I,K,V. G ⊢ ⬈*[h,#0] 𝐒⦃K.ⓑ{I}V⦄ → ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+/2 width=4 by rsx_fwd_lref_pair_csx_aux/ qed-.
+
+lemma rsx_fwd_lref_pair_csx_drops (h) (G):
+ ∀I,K,V,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h,#i] 𝐒⦃L⦄ → ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+#h #G #I #K #V #i elim i -i
+[ #L #H >(drops_fwd_isid … H) -H
+ /2 width=2 by rsx_fwd_lref_pair_csx/
+| #i #IH #L #H1 #H2
+ elim (drops_inv_bind2_isuni_next … H1) -H1 // #J #Y #HY #H destruct
+ lapply (rsx_inv_lifts … H2 … (𝐔❴1❵) ?????) -H2
+ /3 width=6 by drops_refl, drops_drop/
+]
+qed-.
+
+(* Inversion lemmas with strongly rt-normalizing terms **********************)
+
+lemma rsx_inv_lref_pair (h) (G):
+ ∀I,K,V. G ⊢ ⬈*[h,#0] 𝐒⦃K.ⓑ{I}V⦄ →
+ ∧∧ ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ & G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
+/3 width=2 by rsx_fwd_lref_pair_csx, rsx_fwd_pair, conj/ qed-.
+
+lemma rsx_inv_lref_pair_drops (h) (G):
+ ∀I,K,V,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h,#i] 𝐒⦃L⦄ →
+ ∧∧ ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ & G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
+/3 width=5 by rsx_fwd_lref_pair_csx_drops, rsx_fwd_lref_pair_drops, conj/ qed-.
+
+lemma rsx_inv_lref_drops (h) (G):
+ ∀L,i. G ⊢ ⬈*[h,#i] 𝐒⦃L⦄ →
+ ∨∨ ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆
+ | ∃∃I,K. ⬇*[i] L ≘ K.ⓤ{I}
+ | ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ & G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
+#h #G #L #i #H elim (drops_F_uni L i)
+[ /2 width=1 by or3_intro0/
+| * * /4 width=10 by rsx_fwd_lref_pair_csx_drops, rsx_fwd_lref_pair_drops, ex3_3_intro, ex1_2_intro, or3_intro2, or3_intro1/
+]
+qed-.
+
+(* Properties with strongly rt-normalizing terms ****************************)
+
+(* Note: swapping the eliminations to avoid rsx_cpx_trans: no solution found *)
+(* Basic_2A1: uses: lsx_lref_be_lpxs *)
+lemma rsx_lref_pair_lpxs (h) (G):
+ ∀K1,V. ⦃G,K1⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ →
+ ∀K2. G ⊢ ⬈*[h,V] 𝐒⦃K2⦄ → ⦃G,K1⦄ ⊢ ⬈*[h] K2 →
+ ∀I. G ⊢ ⬈*[h,#0] 𝐒⦃K2.ⓑ{I}V⦄.
+#h #G #K1 #V #H
+@(csx_ind_cpxs … H) -V #V0 #_ #IHV0 #K2 #H
+@(rsx_ind … H) -K2 #K0 #HK0 #IHK0 #HK10 #I
+@rsx_intro #Y #HY #HnY
+elim (lpx_inv_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
+elim (tdeq_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnY ]
+[ /5 width=5 by rsx_rdeq_trans, lpxs_step_dx, rdeq_pair/
+| @(IHV0 … HnV02) -IHV0 -HnV02
+ [ /2 width=3 by lpxs_cpx_trans/
+ | /3 width=3 by rsx_lpx_trans, rsx_cpx_trans/
+ | /2 width=3 by lpxs_step_dx/
+ ]
+]
+qed.
+
+lemma rsx_lref_pair (h) (G):
+ ∀K,V. ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → G ⊢ ⬈*[h,V] 𝐒⦃K⦄ → ∀I. G ⊢ ⬈*[h,#0] 𝐒⦃K.ⓑ{I}V⦄.
+/2 width=3 by rsx_lref_pair_lpxs/ qed.
+
+(* Basic_2A1: uses: lsx_lref_be *)
+lemma rsx_lref_pair_drops (h) (G):
+ ∀K,V. ⦃G,K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → G ⊢ ⬈*[h,V] 𝐒⦃K⦄ →
+ ∀I,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h,#i] 𝐒⦃L⦄.
+#h #G #K #V #HV #HK #I #i elim i -i
+[ #L #H >(drops_fwd_isid … H) -H /2 width=1 by rsx_lref_pair/
+| #i #IH #L #H
+ elim (drops_inv_bind2_isuni_next … H) -H // #J #Y #HY #H destruct
+ @(rsx_lifts … (𝐔❴1❵)) /3 width=6 by drops_refl, drops_drop/ (**) (* full auto fails *)
+]
+qed.
+
+(* Main properties with strongly rt-normalizing terms ***********************)
+
+(* Basic_2A1: uses: csx_lsx *)
+theorem csx_rsx (h) (G): ∀L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → G ⊢ ⬈*[h,T] 𝐒⦃L⦄.
+#h #G #L #T @(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T
+#Z #Y #X #IH #G #L * *
+[ //
+| #i #HG #HL #HT #H destruct
+ elim (csx_inv_lref_drops … H) -H [ |*: * ]
+ [ /2 width=1 by rsx_lref_atom_drops/
+ | /2 width=3 by rsx_lref_unit_drops/
+ | /4 width=6 by rsx_lref_pair_drops, fqup_lref/
+ ]
+| //
+| #p #I #V #T #HG #HL #HT #H destruct
+ elim (csx_fwd_bind … H) -H /3 width=1 by rsx_bind/
+| #I #V #T #HG #HL #HT #H destruct
+ elim (csx_fwd_flat … H) -H /3 width=1 by rsx_flat/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/static/rdeq_drops.ma".
+include "basic_2/rt_transition/lpx_drops.ma".
+include "basic_2/rt_computation/rsx_length.ma".
+include "basic_2/rt_computation/rsx_fqup.ma".
+
+(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+(* Properties with generic relocation ***************************************)
+
+(* Note: this uses length *)
+(* Basic_2A1: uses: lsx_lift_le lsx_lift_ge *)
+lemma rsx_lifts (h) (G): d_liftable1_isuni … (λL,T. G ⊢ ⬈*[h,T] 𝐒⦃L⦄).
+#h #G #K #T #H @(rsx_ind … H) -K
+#K1 #_ #IH #b #f #L1 #HLK1 #Hf #U #HTU @rsx_intro
+#L2 #HL12 #HnL12 elim (lpx_drops_conf … HLK1 … HL12)
+/5 width=9 by rdeq_lifts_bi, lpx_fwd_length/
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+(* Basic_2A1: uses: lsx_inv_lift_le lsx_inv_lift_be lsx_inv_lift_ge *)
+lemma rsx_inv_lifts (h) (G): d_deliftable1_isuni … (λL,T. G ⊢ ⬈*[h,T] 𝐒⦃L⦄).
+#h #G #L #U #H @(rsx_ind … H) -L
+#L1 #_ #IH #b #f #K1 #HLK1 #Hf #T #HTU @rsx_intro
+#K2 #HK12 #HnK12 elim (drops_lpx_trans … HLK1 … HK12) -HK12
+/4 width=10 by rdeq_inv_lifts_bi/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: lsx_lref_free *)
+lemma rsx_lref_atom_drops (h) (G): ∀L,i. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ → G ⊢ ⬈*[h,#i] 𝐒⦃L⦄.
+#h #G #L1 #i #HL1
+@(rsx_lifts … (#0) … HL1) -HL1 //
+qed.
+
+(* Basic_2A1: uses: lsx_lref_skip *)
+lemma rsx_lref_unit_drops (h) (G): ∀I,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → G ⊢ ⬈*[h,#i] 𝐒⦃L⦄.
+#h #G #I #L1 #K1 #i #HL1
+@(rsx_lifts … (#0) … HL1) -HL1 //
+qed.
+
+(* Advanced forward lemmas **************************************************)
+
+(* Basic_2A1: uses: lsx_fwd_lref_be *)
+lemma rsx_fwd_lref_pair_drops (h) (G):
+ ∀L,i. G ⊢ ⬈*[h,#i] 𝐒⦃L⦄ →
+ ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → G ⊢ ⬈*[h,V] 𝐒⦃K⦄.
+#h #G #L #i #HL #I #K #V #HLK
+lapply (rsx_inv_lifts … HL … HLK … (#0) ?) -L
+/2 width=2 by rsx_fwd_pair/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/static/rdeq_fqup.ma".
+include "basic_2/rt_computation/rsx.ma".
+
+(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: lsx_atom *)
+lemma lfsx_atom (h) (G) (T): G ⊢ ⬈*[h,T] 𝐒⦃⋆⦄.
+#h #G #T
+@rsx_intro #Y #H #HnT
+lapply (lpx_inv_atom_sn … H) -H #H destruct
+elim HnT -HnT //
+qed.
+
+(* Advanced forward lemmas **************************************************)
+
+(* Basic_2A1: uses: lsx_fwd_bind_dx *)
+(* Note: the exclusion binder (ⓧ) makes this more elegant and much simpler *)
+(* Note: the old proof without the exclusion binder requires lreq *)
+lemma rsx_fwd_bind_dx_void (h) (G):
+ ∀p,I,L,V,T. G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄ → G ⊢ ⬈*[h,T] 𝐒⦃L.ⓧ⦄.
+#h #G #p #I #L #V #T #H
+@(rsx_ind … H) -L #L1 #_ #IH
+@rsx_intro #Y #H #HT
+elim (lpx_inv_unit_sn … H) -H #L2 #HL12 #H destruct
+/4 width=4 by rdeq_fwd_bind_dx_void/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_2A1: uses: lsx_inv_bind *)
+lemma rsx_inv_bind_void (h) (G):
+ ∀p,I,L,V,T. G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄ →
+ ∧∧ G ⊢ ⬈*[h,V] 𝐒⦃L⦄ & G ⊢ ⬈*[h,T] 𝐒⦃L.ⓧ⦄.
+/3 width=4 by rsx_fwd_pair_sn, rsx_fwd_bind_dx_void, conj/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/static/rdeq_length.ma".
+include "basic_2/rt_transition/lpx_length.ma".
+include "basic_2/rt_computation/rsx.ma".
+
+(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: lsx_sort *)
+lemma rsx_sort (h) (G): ∀L,s. G ⊢ ⬈*[h,⋆s] 𝐒⦃L⦄.
+#h #G #L1 #s @rsx_intro #L2 #H #Hs
+elim Hs -Hs /3 width=3 by lpx_fwd_length, rdeq_sort_length/
+qed.
+
+(* Basic_2A1: uses: lsx_gref *)
+lemma rsx_gref (h) (G): ∀L,l. G ⊢ ⬈*[h,§l] 𝐒⦃L⦄.
+#h #G #L1 #s @rsx_intro #L2 #H #Hs
+elim Hs -Hs /3 width=3 by lpx_fwd_length, rdeq_gref_length/
+qed.
+
+lemma rsx_unit (h) (G): ∀I,L. G ⊢ ⬈*[h,#0] 𝐒⦃L.ⓤ{I}⦄.
+#h #G #I #L1 @rsx_intro
+#Y #HY #HnY elim HnY -HnY
+elim (lpx_inv_unit_sn … HY) -HY #L2 #HL12 #H destruct
+/3 width=3 by lpx_fwd_length, rdeq_unit_length/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/lpxs_rdeq.ma".
+include "basic_2/rt_computation/lpxs_lpxs.ma".
+include "basic_2/rt_computation/rsx_rsx.ma".
+
+(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+(* Properties with unbound rt-computation for full local environments *******)
+
+(* Basic_2A1: uses: lsx_intro_alt *)
+lemma rsx_intro_lpxs (h) (G):
+ ∀L1,T. (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄) →
+ G ⊢ ⬈*[h,T] 𝐒⦃L1⦄.
+/4 width=1 by lpx_lpxs, rsx_intro/ qed-.
+
+(* Basic_2A1: uses: lsx_lpxs_trans *)
+lemma rsx_lpxs_trans (h) (G):
+ ∀L1,T. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄.
+#h #G #L1 #T #HL1 #L2 #H @(lpxs_ind_dx … H) -L2
+/2 width=3 by rsx_lpx_trans/
+qed-.
+
+(* Eliminators with unbound rt-computation for full local environments ******)
+
+lemma rsx_ind_lpxs_rdeq (h) (G) (T) (Q:predicate lenv):
+ (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ Q L1
+ ) →
+ ∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L0. ⦃G,L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[T] L2 → Q L2.
+#h #G #T #Q #IH #L1 #H @(rsx_ind … H) -L1
+#L1 #HL1 #IH1 #L0 #HL10 #L2 #HL02
+@IH -IH /3 width=3 by rsx_lpxs_trans, rsx_rdeq_trans/ -HL1 #K2 #HLK2 #HnLK2
+lapply (rdeq_rdneq_trans … HL02 … HnLK2) -HnLK2 #H
+elim (rdeq_lpxs_trans … HLK2 … HL02) -L2 #K0 #HLK0 #HK02
+lapply (rdneq_rdeq_canc_dx … H … HK02) -H #HnLK0
+elim (rdeq_dec L1 L0 T) #H
+[ lapply (rdeq_rdneq_trans … H … HnLK0) -H -HnLK0 #Hn10
+ lapply (lpxs_trans … HL10 … HLK0) -L0 #H10
+ elim (lpxs_rdneq_inv_step_sn … H10 … Hn10) -H10 -Hn10
+ /3 width=8 by rdeq_trans/
+| elim (lpxs_rdneq_inv_step_sn … HL10 … H) -HL10 -H #L #K #HL1 #HnL1 #HLK #HKL0
+ elim (rdeq_lpxs_trans … HLK0 … HKL0) -L0
+ /3 width=8 by lpxs_trans, rdeq_trans/
+]
+qed-.
+
+(* Basic_2A1: uses: lsx_ind_alt *)
+lemma rsx_ind_lpxs (h) (G) (T) (Q:predicate lenv):
+ (∀L1. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G,L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[T] L2 → ⊥) → Q L2) →
+ Q L1
+ ) →
+ ∀L. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → Q L.
+#h #G #T #Q #IH #L #HL
+@(rsx_ind_lpxs_rdeq … IH … HL) -IH -HL // (**) (* full auto fails *)
+qed-.
+
+(* Advanced properties ******************************************************)
+
+fact rsx_bind_lpxs_aux (h) (G):
+ ∀p,I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀Y,T. G ⊢ ⬈*[h,T] 𝐒⦃Y⦄ →
+ ∀L2. Y = L2.ⓑ{I}V → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+#h #G #p #I #L1 #V #H @(rsx_ind_lpxs … H) -L1
+#L1 #_ #IHL1 #Y #T #H @(rsx_ind_lpxs … H) -Y
+#Y #HY #IHY #L2 #H #HL12 destruct
+@rsx_intro_lpxs #L0 #HL20
+lapply (lpxs_trans … HL12 … HL20) #HL10 #H
+elim (rdneq_inv_bind … H) -H [ -IHY | -HY -IHL1 -HL12 ]
+[ #HnV elim (rdeq_dec L1 L2 V)
+ [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
+ /3 width=4 by rsx_lpxs_trans, lpxs_bind_refl_dx, rdeq_canc_sn/ (**) (* full auto too slow *)
+ | -HnV -HL10 /4 width=4 by rsx_lpxs_trans, lpxs_bind_refl_dx/
+ ]
+| /3 width=4 by lpxs_bind_refl_dx/
+]
+qed-.
+
+(* Basic_2A1: uses: lsx_bind *)
+lemma rsx_bind (h) (G):
+ ∀p,I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L.ⓑ{I}V⦄ →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄.
+/2 width=3 by rsx_bind_lpxs_aux/ qed.
+
+(* Basic_2A1: uses: lsx_flat_lpxs *)
+lemma rsx_flat_lpxs (h) (G):
+ ∀I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀L2,T. G ⊢ ⬈*[h,T] 𝐒⦃L2⦄ → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L2⦄.
+#h #G #I #L1 #V #H @(rsx_ind_lpxs … H) -L1
+#L1 #HL1 #IHL1 #L2 #T #H @(rsx_ind_lpxs … H) -L2
+#L2 #HL2 #IHL2 #HL12 @rsx_intro_lpxs
+#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
+#HL10 #H elim (rdneq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
+[ #HnV elim (rdeq_dec L1 L2 V)
+ [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
+ /3 width=5 by rsx_lpxs_trans, rdeq_canc_sn/ (**) (* full auto too slow: 47s *)
+ | -HnV -HL10 /3 width=4 by rsx_lpxs_trans/
+ ]
+| /3 width=3 by/
+]
+qed-.
+
+(* Basic_2A1: uses: lsx_flat *)
+lemma rsx_flat (h) (G):
+ ∀I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L⦄ → G ⊢ ⬈*[h,ⓕ{I}V.T] 𝐒⦃L⦄.
+/2 width=3 by rsx_flat_lpxs/ qed.
+
+fact rsx_bind_lpxs_void_aux (h) (G):
+ ∀p,I,L1,V. G ⊢ ⬈*[h,V] 𝐒⦃L1⦄ →
+ ∀Y,T. G ⊢ ⬈*[h,T] 𝐒⦃Y⦄ →
+ ∀L2. Y = L2.ⓧ → ⦃G,L1⦄ ⊢ ⬈*[h] L2 →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+#h #G #p #I #L1 #V #H @(rsx_ind_lpxs … H) -L1
+#L1 #_ #IHL1 #Y #T #H @(rsx_ind_lpxs … H) -Y
+#Y #HY #IHY #L2 #H #HL12 destruct
+@rsx_intro_lpxs #L0 #HL20
+lapply (lpxs_trans … HL12 … HL20) #HL10 #H
+elim (rdneq_inv_bind_void … H) -H [ -IHY | -HY -IHL1 -HL12 ]
+[ #HnV elim (rdeq_dec L1 L2 V)
+ [ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
+ /3 width=6 by rsx_lpxs_trans, lpxs_bind_refl_dx, rdeq_canc_sn/ (**) (* full auto too slow *)
+ | -HnV -HL10 /4 width=4 by rsx_lpxs_trans, lpxs_bind_refl_dx/
+ ]
+| /3 width=4 by lpxs_bind_refl_dx/
+]
+qed-.
+
+lemma rsx_bind_void (h) (G):
+ ∀p,I,L,V. G ⊢ ⬈*[h,V] 𝐒⦃L⦄ →
+ ∀T. G ⊢ ⬈*[h,T] 𝐒⦃L.ⓧ⦄ →
+ G ⊢ ⬈*[h,ⓑ{p,I}V.T] 𝐒⦃L⦄.
+/2 width=3 by rsx_bind_lpxs_void_aux/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/lpx_rdeq.ma".
+include "basic_2/rt_computation/rsx.ma".
+
+(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: lsx_lleq_trans *)
+lemma rsx_rdeq_trans (h) (G):
+ ∀L1,T. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L2. L1 ≛[T] L2 → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄.
+#h #G #L1 #T #H @(rsx_ind … H) -L1
+#L1 #_ #IHL1 #L2 #HL12 @rsx_intro
+#L #HL2 #HnL2 elim (rdeq_lpx_trans … HL2 … HL12) -HL2
+/4 width=5 by rdeq_repl/
+qed-.
+
+(* Basic_2A1: uses: lsx_lpx_trans *)
+lemma rsx_lpx_trans (h) (G):
+ ∀L1,T. G ⊢ ⬈*[h,T] 𝐒⦃L1⦄ →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → G ⊢ ⬈*[h,T] 𝐒⦃L2⦄.
+#h #G #L1 #T #H @(rsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
+elim (rdeq_dec L1 L2 T) /3 width=4 by rsx_rdeq_trans/
+qed-.
(* CONTEXT-SENSITIVE PARALLEL R-CONVERSION FOR TERMS ************************)
definition cpc: sh → relation4 genv lenv term term ≝
- λh,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 ∨ ⦃G, L⦄ ⊢ T2 ➡[h] T1.
+ λh,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 ∨ ⦃G,L⦄ ⊢ T2 ➡[h] T1.
interpretation
"context-sensitive parallel r-conversion (term)"
(* Basic forward lemmas *****************************************************)
-lemma cpc_fwd_cpr: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌[h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[h] T & ⦃G, L⦄ ⊢ T2 ➡[h] T.
+lemma cpc_fwd_cpr: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬌[h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[h] T & ⦃G,L⦄ ⊢ T2 ➡[h] T.
#h #G #L #T1 #T2 * /2 width=3 by ex2_intro/
qed-.
(* Main properties **********************************************************)
-theorem cpc_conf: ∀h,G,L,T0,T1,T2. ⦃G, L⦄ ⊢ T0 ⬌[h] T1 → ⦃G, L⦄ ⊢ T0 ⬌[h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ⬌[h] T & ⦃G, L⦄ ⊢ T2 ⬌[h] T.
+theorem cpc_conf: ∀h,G,L,T0,T1,T2. ⦃G,L⦄ ⊢ T0 ⬌[h] T1 → ⦃G,L⦄ ⊢ T0 ⬌[h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ⬌[h] T & ⦃G,L⦄ ⊢ T2 ⬌[h] T.
/3 width=3 by cpc_sym, ex2_intro/ qed-.
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_5_7.ma".
include "basic_2/notation/relations/pconveta_5.ma".
include "basic_2/rt_computation/cpms.ma".
(* avtivate genv *)
inductive cpce (h): relation4 genv lenv term term ≝
| cpce_sort: ∀G,L,s. cpce h G L (⋆s) (⋆s)
-| cpce_ldef: ∀G,K,V. cpce h G (K.ⓓV) (#0) (#0)
-| cpce_ldec: ∀n,G,K,V,s. ⦃G,K⦄ ⊢ V ➡*[n,h] ⋆s →
- cpce h G (K.â\93\9bV) (#0) (#0)
-| cpce_eta : ∀n,p,G,K,V,W1,W2,T. ⦃G,K⦄ ⊢ V ➡*[n,h] ⓛ{p}W1.T →
- cpce h G K W1 W2 → cpce h G (K.ⓛV) (#0) (+ⓛW2.ⓐ#0.#1)
+| cpce_atom: ∀G,i. cpce h G (⋆) (#i) (#i)
+| cpce_zero: ∀G,K,I. (∀n,p,W,V,U. I = BPair Abst W → ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥) →
+ cpce h G (K.â\93\98{I}) (#0) (#0)
+| cpce_eta : ∀n,p,G,K,W,V1,V2,W2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V1.U →
+ cpce h G K V1 V2 → ⬆*[1] V2 ≘ W2 → cpce h G (K.ⓛW) (#0) (+ⓛW2.ⓐ#0.#1)
| cpce_lref: ∀I,G,K,T,U,i. cpce h G K (#i) T →
⬆*[1] T ≘ U → cpce h G (K.ⓘ{I}) (#↑i) U
+| cpce_gref: ∀G,L,l. cpce h G L (§l) (§l)
| cpce_bind: ∀p,I,G,K,V1,V2,T1,T2.
cpce h G K V1 V2 → cpce h G (K.ⓑ{I}V1) T1 T2 →
cpce h G K (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
#h #G #Y #X2 #s0
@(insert_eq_0 … (⋆s0)) #X1 * -G -Y -X1 -X2
[ #G #L #s #_ //
-| #G #K #V #_ //
-| #n #G #K #V #s #_ #_ //
-| #n #p #G #K #V #W1 #W2 #T #_ #_ #H destruct
+| #G #i #_ //
+| #G #K #I #_ #_ //
+| #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #H destruct
| #I #G #K #T #U #i #_ #_ #H destruct
+| #G #L #l #_ //
| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H destruct
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
]
qed-.
-lemma cpce_inv_ldef_sn (h) (G) (K) (X2):
- ∀V. ⦃G,K.ⓓV⦄ ⊢ #0 ⬌η[h] X2 → #0 = X2.
-#h #G #Y #X2 #X
-@(insert_eq_0 … (Y.ⓓX)) #Y1
-@(insert_eq_0 … (#0)) #X1
-* -G -Y1 -X1 -X2
+lemma cpce_inv_atom_sn (h) (G) (X2):
+ ∀i. ⦃G,⋆⦄ ⊢ #i ⬌η[h] X2 → #i = X2.
+#h #G #X2 #j
+@(insert_eq_0 … LAtom) #Y
+@(insert_eq_0 … (#j)) #X1
+* -G -Y -X1 -X2
[ #G #L #s #_ #_ //
-| #G #K #V #_ #_ //
-| #n #G #K #V #s #_ #_ #_ //
-| #n #p #G #K #V #W1 #W2 #T #_ #_ #_ #H destruct
-| #I #G #K #T #U #i #_ #_ #H #_ destruct
+| #G #i #_ #_ //
+| #G #K #I #_ #_ #_ //
+| #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #_ #H destruct
+| #I #G #K #T #U #i #_ #_ #_ #H destruct
+| #G #L #l #_ #_ //
| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
]
qed-.
-lemma cpce_inv_ldec_sn (h) (G) (K) (X2):
- ∀V. ⦃G,K.ⓛV⦄ ⊢ #0 ⬌η[h] X2 →
- ∨∨ ∃∃n,s. ⦃G,K⦄ ⊢ V ➡*[n,h] ⋆s & #0 = X2
- | ∃∃n,p,W1,W2,T. ⦃G,K⦄ ⊢ V ➡*[n,h] ⓛ{p}W1.T & ⦃G,K⦄ ⊢ W1 ⬌η[h] W2 & +ⓛW2.ⓐ#0.#1 = X2.
-#h #G #Y #X2 #X
-@(insert_eq_0 … (Y.ⓛX)) #Y1
+lemma cpce_inv_zero_sn (h) (G) (K) (X2):
+ ∀I. ⦃G,K.ⓘ{I}⦄ ⊢ #0 ⬌η[h] X2 →
+ ∨∨ ∧∧ ∀n,p,W,V,U. I = BPair Abst W → ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥ & #0 = X2
+ | ∃∃n,p,W,V1,V2,W2,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V1.U & ⦃G,K⦄ ⊢ V1 ⬌η[h] V2
+ & ⬆*[1] V2 ≘ W2 & I = BPair Abst W & +ⓛW2.ⓐ#0.#1 = X2.
+#h #G #Y0 #X2 #Z
+@(insert_eq_0 … (Y0.ⓘ{Z})) #Y
@(insert_eq_0 … (#0)) #X1
-* -G -Y1 -X1 -X2
+* -G -Y -X1 -X2
[ #G #L #s #H #_ destruct
-| #G #K #V #_ #H destruct
-| #n #G #K #V #s #HV #_ #H destruct /3 width=3 by ex2_2_intro, or_introl/
-| #n #p #G #K #V #W1 #W2 #T #HV #HW #_ #H destruct /3 width=8 by ex3_5_intro, or_intror/
+| #G #i #_ #H destruct
+| #G #K #I #HI #_ #H destruct /4 width=7 by or_introl, conj/
+| #n #p #G #K #W #V1 #V2 #W2 #U #HWU #HV12 #HVW2 #_ #H destruct /3 width=12 by or_intror, ex5_7_intro/
| #I #G #K #T #U #i #_ #_ #H #_ destruct
+| #G #L #l #H #_ destruct
| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
]
lemma cpce_inv_lref_sn (h) (G) (K) (X2):
∀I,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬌η[h] X2 →
∃∃T2. ⦃G,K⦄ ⊢ #i ⬌η[h] T2 & ⬆*[1] T2 ≘ X2.
-#h #G #Y #X2 #Z #j
-@(insert_eq_0 … (Y.ⓘ{Z})) #Y1
+#h #G #Y0 #X2 #Z #j
+@(insert_eq_0 … (Y0.ⓘ{Z})) #Y
@(insert_eq_0 … (#↑j)) #X1
-* -G -Y1 -X1 -X2
+* -G -Y -X1 -X2
[ #G #L #s #H #_ destruct
-| #G #K #V #H #_ destruct
-| #n #G #K #V #s #_ #H #_ destruct
-| #n #p #G #K #V #W1 #W2 #T #_ #_ #H #_ destruct
+| #G #i #_ #H destruct
+| #G #K #I #_ #H #_ destruct
+| #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #H #_ destruct
| #I #G #K #T #U #i #Hi #HTU #H1 #H2 destruct /2 width=3 by ex2_intro/
+| #G #L #l #H #_ destruct
| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H #_ destruct
]
qed-.
+lemma cpce_inv_gref_sn (h) (G) (L) (X2):
+ ∀l. ⦃G,L⦄ ⊢ §l ⬌η[h] X2 → §l = X2.
+#h #G #Y #X2 #k
+@(insert_eq_0 … (§k)) #X1 * -G -Y -X1 -X2
+[ #G #L #s #_ //
+| #G #i #_ //
+| #G #K #I #_ #_ //
+| #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #H destruct
+| #I #G #K #T #U #i #_ #_ #H destruct
+| #G #L #l #_ //
+| #p #I #G #K #V1 #V2 #T1 #T2 #_ #_ #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
+]
+qed-.
+
lemma cpce_inv_bind_sn (h) (G) (K) (X2):
∀p,I,V1,T1. ⦃G,K⦄ ⊢ ⓑ{p,I}V1.T1 ⬌η[h] X2 →
∃∃V2,T2. ⦃G,K⦄ ⊢ V1 ⬌η[h] V2 & ⦃G,K.ⓑ{I}V1⦄ ⊢ T1 ⬌η[h] T2 & ⓑ{p,I}V2.T2 = X2.
#h #G #Y #X2 #q #Z #U #X
@(insert_eq_0 … (ⓑ{q,Z}U.X)) #X1 * -G -Y -X1 -X2
[ #G #L #s #H destruct
-| #G #K #V #H destruct
-| #n #G #K #V #s #_ #H destruct
-| #n #p #G #K #V #W1 #W2 #T #_ #_ #H destruct
+| #G #i #H destruct
+| #G #K #I #_ #H destruct
+| #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #H destruct
| #I #G #K #T #U #i #_ #_ #H destruct
+| #G #L #l #H destruct
| #p #I #G #K #V1 #V2 #T1 #T2 #HV #HT #H destruct /2 width=5 by ex3_2_intro/
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
]
#h #G #Y #X2 #Z #U #X
@(insert_eq_0 … (ⓕ{Z}U.X)) #X1 * -G -Y -X1 -X2
[ #G #L #s #H destruct
-| #G #K #V #H destruct
-| #n #G #K #V #s #_ #H destruct
-| #n #p #G #K #V #W1 #W2 #T #_ #_ #H destruct
+| #G #i #H destruct
+| #G #K #I #_ #H destruct
+| #n #p #G #K #W #V1 #V2 #W2 #U #_ #_ #_ #H destruct
| #I #G #K #T #U #i #_ #_ #H destruct
+| #G #L #l #H destruct
| #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #H destruct
| #I #G #K #V1 #V2 #T1 #T2 #HV #HT #H destruct /2 width=5 by ex3_2_intro/
]
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/drops.ma".
+include "basic_2/rt_conversion/cpce.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL ETA-CONVERSION FOR TERMS **********************)
+
+(* Properties with uniform slicing for local environments *******************)
+
+lemma cpce_eta_drops (h) (n) (G) (K):
+ ∀p,W,V1,U. ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V1.U →
+ ∀V2. ⦃G,K⦄ ⊢ V1 ⬌η[h] V2 →
+ ∀i,L. ⬇*[i] L ≘ K.ⓛW →
+ ∀W2. ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ⬌η[h] +ⓛW2.ⓐ#0.#↑i.
+#h #n #G #K #p #W #V1 #U #HWU #V2 #HV12 #i elim i -i
+[ #L #HLK #W2 #HVW2
+ >(drops_fwd_isid … HLK) -L [| // ] /2 width=8 by cpce_eta/
+| #i #IH #L #HLK #W2 #HVW2
+ elim (drops_inv_succ … HLK) -HLK #I #Y #HYK #H destruct
+ elim (lifts_split_trans … HVW2 (𝐔❴↑i❵) (𝐔❴1❵)) [| // ] #X2 #HVX2 #HXW2
+ /5 width=7 by cpce_lref, lifts_push_lref, lifts_bind, lifts_flat/
+]
+qed.
+
+lemma cpce_zero_drops (h) (G):
+ ∀i,L. (∀n,p,K,W,V,U. ⬇*[i] L ≘ K.ⓛW → ⦃G,K⦄ ⊢ W ➡*[n,h] ⓛ{p}V.U → ⊥) →
+ ⦃G,L⦄ ⊢ #i ⬌η[h] #i.
+#h #G #i elim i -i
+[ * [ #_ // ] #L #I #Hi
+ /4 width=8 by cpce_zero, drops_refl/
+| #i #IH * [ -IH #_ // ] #L #I #Hi
+ /5 width=8 by cpce_lref, drops_drop/
+]
+qed.
lemma lpce_inv_unit_sn (h) (G):
∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ⬌η[h] L2 →
- ∃∃K2. ⦃G, K1⦄ ⊢ ⬌η[h] K2 & L2 = K2.ⓤ{I}.
+ ∃∃K2. ⦃G,K1⦄ ⊢ ⬌η[h] K2 & L2 = K2.ⓤ{I}.
/2 width=1 by lex_inv_unit_sn/ qed-.
lemma lpce_inv_pair_sn (h) (G):
(* Basic_2A1: was: cpcs_ind_dx *)
lemma cpcs_ind_sn (h) (G) (L) (T2):
∀Q:predicate term. Q T2 →
- (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌[h] T → ⦃G, L⦄ ⊢ T ⬌*[h] T2 → Q T → Q T1) →
- ∀T1. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → Q T1.
+ (∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌[h] T → ⦃G,L⦄ ⊢ T ⬌*[h] T2 → Q T → Q T1) →
+ ∀T1. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → Q T1.
normalize /3 width=6 by TC_star_ind_dx/
qed-.
(* Basic_2A1: was: cpcs_ind *)
lemma cpcs_ind_dx (h) (G) (L) (T1):
∀Q:predicate term. Q T1 →
- (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T → ⦃G, L⦄ ⊢ T ⬌[h] T2 → Q T → Q T2) →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 → Q T2.
+ (∀T,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T → ⦃G,L⦄ ⊢ T ⬌[h] T2 → Q T → Q T2) →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 → Q T2.
normalize /3 width=6 by TC_star_ind/
qed-.
/2 width=1 by cpc_sym/
qed-.
-lemma cpc_cpcs (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpc_cpcs (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/2 width=1 by inj/ qed.
(* Basic_2A1: was: cpcs_strap2 *)
-lemma cpcs_step_sn (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_step_sn (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
normalize /2 width=3 by TC_strap/
qed-.
(* Basic_2A1: was: cpcs_strap1 *)
-lemma cpcs_step_dx (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ⬌[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_step_dx (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬌[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
normalize /2 width=3 by step/
qed-.
(* Basic_1: was: pc3_pr2_r *)
-lemma cpr_cpcs_dx (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cpcs_dx (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=1 by cpc_cpcs, or_introl/ qed.
(* Basic_1: was: pc3_pr2_x *)
-lemma cpr_cpcs_sn (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T2 ➡[h] T1 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cpcs_sn (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T2 ➡[h] T1 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=1 by cpc_cpcs, or_intror/ qed.
(* Basic_1: was: pc3_pr2_u *)
(* Basic_2A1: was: cpcs_cpr_strap2 *)
-lemma cpcs_cpr_step_sn (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T → ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_step_sn (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T → ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_sn, or_introl/ qed-.
(* Basic_2A1: was: cpcs_cpr_strap1 *)
-lemma cpcs_cpr_step_dx (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_step_dx (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_dx, or_introl/ qed-.
-lemma cpcs_cpr_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_dx, or_intror/ qed-.
-lemma cpr_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpr_cpcs_dx, cpcs_step_dx, or_intror/ qed-.
(* Basic_1: was: pc3_pr2_u2 *)
-lemma cpcs_cpr_conf (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T ➡[h] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cpr_conf (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T ➡[h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpcs_step_sn, or_intror/ qed-.
(* Basic_1: removed theorems 9:
(* Main inversion lemmas with atomic arity assignment on terms **************)
(* Note: lemma 1500 *)
-theorem cpcs_aaa_mono (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 →
- ∀A1. ⦃G, L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T2 ⁝ A2 →
+theorem cpcs_aaa_mono (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀A1. ⦃G,L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G,L⦄ ⊢ T2 ⁝ A2 →
A1 = A2.
#h #G #L #T1 #T2 #HT12 #A1 #HA1 #A2 #HA2
elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
(* Advanced properties ******************************************************)
-lemma cpcs_bind1 (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2.
+lemma cpcs_bind1 (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2.
/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
-lemma cpcs_bind2 (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2.
+lemma cpcs_bind2 (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 →
+ ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬌*[h] ⓑ{p,I}V2.T2.
/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
(* Advanced properties with r-transition for full local environments ********)
(* Basic_1: was: pc3_wcpr0 *)
-lemma lpr_cpcs_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌*[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2.
+lemma lpr_cpcs_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 →
+ ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
/3 width=5 by cpcs_canc_dx, lpr_cprs_conf/
qed-.
(* Inversion lemmas with context sensitive r-computation on terms ***********)
-lemma cpcs_inv_cprs (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡*[h] T & ⦃G, L⦄ ⊢ T2 ➡*[h] T.
+lemma cpcs_inv_cprs (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[h] T & ⦃G,L⦄ ⊢ T2 ➡*[h] T.
#h #G #L #T1 #T2 #H @(cpcs_ind_dx … H) -T2
[ /3 width=3 by ex2_intro/
| #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
(* Basic_1: was: pc3_gen_sort *)
(* Basic_2A1: was: cpcs_inv_sort *)
-lemma cpcs_inv_sort_bi (h) (G) (L): ∀s1,s2. ⦃G, L⦄ ⊢ ⋆s1 ⬌*[h] ⋆s2 → s1 = s2.
+lemma cpcs_inv_sort_bi (h) (G) (L): ∀s1,s2. ⦃G,L⦄ ⊢ ⋆s1 ⬌*[h] ⋆s2 → s1 = s2.
#h #G #L #s1 #s2 #H elim (cpcs_inv_cprs … H) -H
#T #H1 >(cprs_inv_sort1 … H1) -T #H2
lapply (cprs_inv_sort1 … H2) -L #H destruct //
(* Basic_2A1: was: cpcs_inv_abst1 *)
lemma cpcs_inv_abst_sn (h) (G) (L):
- ∀p,W1,T1,X. ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ⬌*[h] X →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2.
+ ∀p,W1,T1,X. ⦃G,L⦄ ⊢ ⓛ{p}W1.T1 ⬌*[h] X →
+ ∃∃W2,T2. ⦃G,L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G,L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2.
#h #G #L #p #W1 #T1 #T #H
elim (cpcs_inv_cprs … H) -H #X #H1 #H2
elim (cpms_inv_abst_sn … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
(* Basic_2A1: was: cpcs_inv_abst2 *)
lemma cpcs_inv_abst_dx (h) (G) (L):
- ∀p,W1,T1,X. ⦃G, L⦄ ⊢ X ⬌*[h] ⓛ{p}W1.T1 →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2.
+ ∀p,W1,T1,X. ⦃G,L⦄ ⊢ X ⬌*[h] ⓛ{p}W1.T1 →
+ ∃∃W2,T2. ⦃G,L⦄ ⊢ X ➡*[h] ⓛ{p}W2.T2 & ⦃G,L⦄ ⊢ ⓛ{p}W1.T1 ➡*[h] ⓛ{p}W2.T2.
/3 width=1 by cpcs_inv_abst_sn, cpcs_sym/ qed-.
(* Basic_1: was: pc3_gen_sort_abst *)
lemma cpcs_inv_sort_abst (h) (G) (L):
- ∀p,W,T,s. ⦃G, L⦄ ⊢ ⋆s ⬌*[h] ⓛ{p}W.T → ⊥.
+ ∀p,W,T,s. ⦃G,L⦄ ⊢ ⋆s ⬌*[h] ⓛ{p}W.T → ⊥.
#h #G #L #p #W #T #s #H
elim (cpcs_inv_cprs … H) -H #X #H1
>(cprs_inv_sort1 … H1) -X #H2
(* Properties with context sensitive r-computation on terms *****************)
(* Basic_1: was: pc3_pr3_r *)
-lemma cpcs_cprs_dx (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_dx (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T2 #H @(cprs_ind_dx … H) -T2
/3 width=3 by cpcs_cpr_step_dx, cpcs_step_dx, cpc_cpcs/
qed.
(* Basic_1: was: pc3_pr3_x *)
-lemma cpcs_cprs_sn (h) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T1 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_sn (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T1 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T2 #H @(cprs_ind_sn … H) -T2
/3 width=3 by cpcs_cpr_div, cpcs_step_sn, cpcs_cprs_dx/
qed.
(* Basic_2A1: was: cpcs_cprs_strap1 *)
-lemma cpcs_cprs_step_dx (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_step_dx (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=3 by cpcs_cpr_step_dx/
qed-.
(* Basic_2A1: was: cpcs_cprs_strap2 *)
-lemma cpcs_cprs_step_sn (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_step_sn (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T #H #T2 #HT2 @(cprs_ind_sn … H) -T1 /2 width=3 by cpcs_cpr_step_sn/
qed-.
-lemma cpcs_cprs_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ⬌*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_sn … H) -T2 /2 width=3 by cpcs_cpr_div/
qed-.
(* Basic_1: was: pc3_pr3_conf *)
-lemma cpcs_cprs_conf (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T ➡*[h] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ⬌*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_conf (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T ➡*[h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ⬌*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /2 width=3 by cpcs_cpr_conf/
qed-.
(* Basic_1: was: pc3_pr3_t *)
(* Basic_1: note: pc3_pr3_t should be renamed *)
-lemma cprs_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cprs_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_sn … H) -T2
/2 width=3 by cpcs_cpr_div, cpcs_cprs_dx/
qed.
-lemma cprs_cpr_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cprs_cpr_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡*[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=5 by cpm_cpms, cprs_div/ qed-.
-lemma cpr_cprs_div (h) (G) (L): ∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h] T →
- ∀T2. ⦃G, L⦄ ⊢ T2 ➡*[h] T → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cprs_div (h) (G) (L): ∀T1,T. ⦃G,L⦄ ⊢ T1 ➡[h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
/3 width=3 by cpm_cpms, cprs_div/ qed-.
-lemma cpr_cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G, L⦄ ⊢ T ➡*[h] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G,L⦄ ⊢ T ➡*[h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
/2 width=3 by cpr_cprs_div/
qed-.
-lemma cprs_cpr_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G, L⦄ ⊢ T ➡*[h] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 → ⦃G, L⦄ ⊢ T2 ⬌*[h] T1.
+lemma cprs_cpr_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G,L⦄ ⊢ T ➡*[h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 → ⦃G,L⦄ ⊢ T2 ⬌*[h] T1.
#h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
/2 width=3 by cprs_cpr_div/
qed-.
-lemma cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G, L⦄ ⊢ T ➡*[h] T1 →
- ∀T2. ⦃G, L⦄ ⊢ T ➡*[h] T2 → ⦃G, L⦄ ⊢ T1 ⬌*[h] T2.
+lemma cprs_conf_cpcs (h) (G) (L): ∀T,T1. ⦃G,L⦄ ⊢ T ➡*[h] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2
/2 width=3 by cprs_div/
qed-.
(* Basic_1: was only: pc3_thin_dx *)
-lemma cpcs_flat (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2.
+lemma cpcs_flat (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2.
#h #G #L #V1 #V2 #HV12 #T1 #T2 #HT12
elim (cpcs_inv_cprs … HV12) -HV12
elim (cpcs_inv_cprs … HT12) -HT12
/3 width=5 by cprs_flat, cprs_div/
qed.
-lemma cpcs_flat_dx_cpr_rev (h) (G) (L): ∀V1,V2. ⦃G, L⦄ ⊢ V2 ➡[h] V1 →
- ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌*[h] T2 →
- ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2.
+lemma cpcs_flat_dx_cpr_rev (h) (G) (L): ∀V1,V2. ⦃G,L⦄ ⊢ V2 ➡[h] V1 →
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀I. ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬌*[h] ⓕ{I}V2.T2.
/3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed.
-lemma cpcs_bind_dx (h) (G) (L): ∀I,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌*[h] T2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ⬌*[h] ⓑ{p,I}V.T2.
+lemma cpcs_bind_dx (h) (G) (L): ∀I,V,T1,T2. ⦃G,L.ⓑ{I}V⦄ ⊢ T1 ⬌*[h] T2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T1 ⬌*[h] ⓑ{p,I}V.T2.
#h #G #L #I #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
/3 width=5 by cprs_div, cpms_bind/
qed.
-lemma cpcs_bind_sn (h) (G) (L): ∀I,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌*[h] V2 →
- ∀p. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T ⬌*[h] ⓑ{p,I}V2.T.
+lemma cpcs_bind_sn (h) (G) (L): ∀I,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬌*[h] V2 →
+ ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T ⬌*[h] ⓑ{p,I}V2.T.
#h #G #L #I #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12
/3 width=5 by cprs_div, cpms_bind/
qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpre_csx.ma".
+include "basic_2/rt_computation/cpre_cpre.ma".
+include "basic_2/rt_equivalence/cpcs_cprs.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL R-EQUIVALENCE FOR TERMS ***********************)
+
+(* Properties with strongly normalizing terms for unbound rt-transition *****)
+
+(* Basic_1: was: cpcs_dec *)
+lemma csx_cpcs_dec (h) (G) (L):
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ → ∀T2. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄ →
+ Decidable … (⦃G,L⦄ ⊢ T1 ⬌*[h] T2).
+#h #G #L #T1 #HT1 #T2 #HT2
+elim (cpre_total_csx … HT1) -HT1 #U1 #HTU1
+elim (cpre_total_csx … HT2) -HT2 #U2 #HTU2
+elim (eq_term_dec U1 U2) [ #H destruct | #HnU12 ]
+[ cases HTU1 -HTU1 #HTU1 #_
+ cases HTU2 -HTU2 #HTU2 #_
+ /3 width=3 by cprs_div, or_introl/
+| @or_intror #H
+ elim (cpcs_inv_cprs … H) -H #T0 #HT10 #HT20
+ lapply (cpre_cprs_conf … HT10 … HTU1) -T1 #H1
+ lapply (cpre_cprs_conf … HT20 … HTU2) -T2 #H2
+ /3 width=6 by cpre_mono/
+]
+qed-.
(* Properties with parallel r-computation for full local environments *******)
-lemma lpr_cpcs_trans (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 →
- ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G, L1⦄ ⊢ T1 ⬌*[h] T2.
+lemma lpr_cpcs_trans (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 →
+ ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L1⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
/4 width=5 by cprs_div, lpr_cpms_trans/
qed-.
-lemma lprs_cpcs_trans (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G, L1⦄ ⊢ T1 ⬌*[h] T2.
+lemma lprs_cpcs_trans (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀T1,T2. ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2 → ⦃G,L1⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
/4 width=5 by cprs_div, lprs_cpms_trans/
qed-.
-lemma lprs_cprs_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2.
+lemma lprs_cprs_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡*[h] L2 →
+ ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (lprs_cprs_conf_dx … HT12 … HL12) -L1
/2 width=3 by cprs_div/
qed-.
(* Basic_1: was: pc3_wcpr0_t *)
(* Basic_1: note: pc3_wcpr0_t should be renamed *)
(* Note: alternative proof /3 width=5 by lprs_cprs_conf, lpr_lprs/ *)
-lemma lpr_cprs_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2.
+lemma lpr_cprs_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 →
+ ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡*[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2.
#h #G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (cprs_lpr_conf_dx … HT12 … HL12) -L1
/2 width=3 by cprs_div/
qed-.
(* Basic_1: was only: pc3_pr0_pr2_t *)
(* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
-lemma lpr_cpr_conf (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 →
- ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡[h] T2 → ⦃G, L2⦄ ⊢ T1 ⬌*[h] T2.
+lemma lpr_cpr_conf (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 →
+ ∀T1,T2. ⦃G,L1⦄ ⊢ T1 ➡[h] T2 → ⦃G,L2⦄ ⊢ T1 ⬌*[h] T2.
/3 width=5 by lpr_cprs_conf, cpm_cpms/ qed-.
(* Advanced inversion lemmas ************************************************)
(* Note: there must be a proof suitable for lfpr *)
-lemma cpcs_inv_abst_sn (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 →
- ∧∧ ⦃G, L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2.
+lemma cpcs_inv_abst_sn (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 →
+ ∧∧ ⦃G,L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2.
#h #G #L #p1 #p2 #W1 #W2 #T1 #T2 #H
elim (cpcs_inv_cprs … H) -H #T #H1 #H2
elim (cpms_inv_abst_sn … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
/4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
qed-.
-lemma cpcs_inv_abst_dx (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 →
- ∧∧ ⦃G, L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2.
+lemma cpcs_inv_abst_dx (h) (G) (L): ∀p1,p2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h] ⓛ{p2}W2.T2 →
+ ∧∧ ⦃G,L⦄ ⊢ W1 ⬌*[h] W2 & ⦃G,L.ⓛW2⦄ ⊢ T1 ⬌*[h] T2 & p1 = p2.
#h #G #L #p1 #p2 #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12
#HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/
qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/pconvstar_7.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-EQUIVALENCE FOR TERMS **************)
+
+(* Basic_2A1: uses: scpes *)
+definition cpes (h) (n1) (n2): relation4 genv lenv term term ≝
+ λG,L,T1,T2.
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T & ⦃G,L⦄ ⊢ T2 ➡*[n2,h] T.
+
+interpretation "t-bound context-sensitive parallel rt-equivalence (term)"
+ 'PConvStar h n1 n2 G L T1 T2 = (cpes h n1 n2 G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: uses: scpds_div *)
+lemma cpms_div (h) (n1) (n2):
+ ∀G,L,T1,T. ⦃G,L⦄ ⊢ T1 ➡*[n1,h] T →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[n2,h] T → ⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2.
+/2 width=3 by ex2_intro/ qed.
+
+lemma cpes_refl (h): ∀G,L. reflexive … (cpes h 0 0 G L).
+/2 width=3 by cpms_div/ qed.
+
+(* Basic_2A1: uses: scpes_sym *)
+lemma cpes_sym (h) (n1) (n2):
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2 → ⦃G,L⦄ ⊢ T2 ⬌*[h,n2,n1] T1.
+#h #n1 #n2 #G #L #T1 #T2 * /2 width=3 by cpms_div/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cpms_aaa.ma".
+include "basic_2/rt_equivalence/cpes.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-EQUIVALENCE FOR TERMS **************)
+
+(* Properties with atomic arity assignment on terms *************************)
+
+(* Basic_2A1: uses: scpes_refl *)
+lemma cpes_refl_aaa (h) (n):
+ ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ⦃G,L⦄ ⊢ T ⬌*[h,n,n] T.
+#h #n #G #L #T #A #HA
+elim (cpms_total_aaa h … n … HA) #U #HTU
+/2 width=3 by cpms_div/
+qed.
+
+(* Main inversion lemmas with atomic arity assignment on terms **************)
+
+(* Basic_2A1: uses: scpes_aaa_mono *)
+theorem cpes_aaa_mono (h) (n1) (n2):
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬌*[h,n1,n2] T2 →
+ ∀A1. ⦃G,L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G,L⦄ ⊢ T2 ⁝ A2 → A1 = A2.
+#h #n1 #n2 #G #L #T1 #T2 * #T #HT1 #HT2 #A1 #HA1 #A2 #HA2
+lapply (cpms_aaa_conf … HA1 … HT1) -T1 #HA1
+lapply (cpms_aaa_conf … HA2 … HT2) -T2 #HA2
+lapply (aaa_mono … HA1 … HA2) -L -T //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/lprs_cpms.ma".
+include "basic_2/rt_equivalence/cpes_cpms.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-EQUIVALENCE FOR TERMS **************)
+
+(* Advanced forward lemmas **************************************************)
+
+lemma cpes_fwd_abst_bi (h) (n1) (n2) (p1) (p2) (G) (L):
+ ∀W1,W2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}W1.T1 ⬌*[h,n1,n2] ⓛ{p2}W2.T2 →
+ ∧∧ p1 = p2 & ⦃G,L⦄ ⊢ W1 ⬌*[h,0,O] W2.
+#h #n1 #n2 #p1 #p2 #G #L #W1 #W2 #T1 #T2 * #X #H1 #H2
+elim (cpms_inv_abst_sn … H1) #W0 #X0 #HW10 #_ #H destruct
+elim (cpms_inv_abst_bi … H2) #H #HW20 #_ destruct
+/3 width=3 by cpms_div, conj/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem cpes_cpes_trans (h) (n1) (n2) (G) (L) (T):
+ ∀T1. ⦃G,L⦄ ⊢ T ⬌*[h,n1,0] T1 →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬌*[h,0,n2] T2 → ⦃G,L⦄ ⊢ T ⬌*[h,n1,n2] T2.
+#h #n1 #n2 #G #L #T #T1 #HT1 #T2 * #X #HX1 #HX2
+lapply (cpes_cprs_trans … HT1 … HX1) -T1 #HTX
+lapply (cpes_cpms_div … HTX … HX2) -X //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_computation/cprs_cprs.ma".
+include "basic_2/rt_equivalence/cpes.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-EQUIVALENCE FOR TERMS **************)
+
+(* Properties with t-bound rt-computation on terms **************************)
+
+lemma cpes_cprs_trans (h) (n) (G) (L) (T0):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ⬌*[h,n,0] T0 →
+ ∀T2. ⦃G,L⦄ ⊢ T0 ➡*[h] T2 → ⦃G,L⦄ ⊢ T1 ⬌*[h,n,0] T2.
+#h #n #G #L #T0 #T1 * #T #HT1 #HT0 #T2 #HT02
+elim (cprs_conf … HT0 … HT02) -T0 #T0 #HT0 #HT20
+/3 width=3 by cpms_div, cpms_cprs_trans/
+qed-.
+
+lemma cpes_cpms_div (h) (n) (n1) (n2) (G) (L) (T0):
+ ∀T1. ⦃G,L⦄ ⊢ T1 ⬌*[h,n,n1] T0 →
+ ∀T2. ⦃G,L⦄ ⊢ T2 ➡*[n2,h] T0 → ⦃G,L⦄ ⊢ T1 ⬌*[h,n,n2+n1] T2.
+#h #n #n1 #n2 #G #L #T0 #T1 * #T #HT1 #HT0 #T2 #HT20
+lapply (cpms_trans … HT20 … HT0) -T0 #HT2
+/2 width=3 by cpms_div/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/prednormal_4.ma".
+include "basic_2/rt_transition/cpr.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE R-TRANSITION **************************)
+
+definition cnr (h) (G) (L): predicate term ≝ NF … (cpm h G L 0) (eq …).
+
+interpretation
+ "normality for context-sensitive r-transition (term)"
+ 'PRedNormal h G L T = (cnr h G L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cnr_inv_abst (h) (p) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓛ{p}V.T⦄ → ∧∧ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G,L.ⓛV⦄ ⊢ ➡[h] 𝐍⦃T⦄.
+#h #p #G #L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (ⓛ{p}V2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓛ{p}V1.T2) ?) -HVT1 /2 width=2 by cpm_bind/ -HT2 #H destruct //
+]
+qed-.
+
+(* Basic_2A1: was: cnr_inv_abbr *)
+lemma cnr_inv_abbr_neg (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃-ⓓV.T⦄ → ∧∧ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G,L.ⓓV⦄ ⊢ ➡[h] 𝐍⦃T⦄.
+#h #G #L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpr_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpm_bind/ -HT2 #H destruct //
+]
+qed-.
+
+(* Basic_2A1: was: cnr_inv_eps *)
+lemma cnr_inv_cast (h) (G) (L): ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓝV.T⦄ → ⊥.
+#h #G #L #V #T #H lapply (H T ?) -H
+/2 width=4 by cpm_eps, discr_tpair_xy_y/
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: nf2_sort *)
+lemma cnr_sort (h) (G) (L): ∀s. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃⋆s⦄.
+#h #G #L #s #X #H
+>(cpr_inv_sort1 … H) //
+qed.
+
+lemma cnr_gref (h) (G) (L): ∀l. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃§l⦄.
+#h #G #L #l #X #H
+>(cpr_inv_gref1 … H) //
+qed.
+
+(* Basic_1: was: nf2_abst *)
+lemma cnr_abst (h) (p) (G) (L):
+ ∀W,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃W⦄ → ⦃G,L.ⓛW⦄ ⊢ ➡[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓛ{p}W.T⦄.
+#h #p #G #L #W #T #HW #HT #X #H
+elim (cpm_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+<(HW … HW0) -W0 <(HT … HT0) -T0 //
+qed.
+
+lemma cnr_abbr_neg (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G,L.ⓓV⦄ ⊢ ➡[h] 𝐍⦃T⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃-ⓓV.T⦄.
+#h #G #L #V #T #HV #HT #X #H
+elim (cpm_inv_abbr1 … H) -H *
+[ #V0 #T0 #HV0 #HT0 #H destruct
+ <(HV … HV0) -V0 <(HT … HT0) -T0 //
+| #T0 #_ #_ #H destruct
+]
+qed.
+
+
+(* Basic_1: removed theorems 1: nf2_abst_shift *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cpr_drops.ma".
+include "basic_2/rt_transition/cnr.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE R-TRANSITION **************************)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was only: nf2_csort_lref *)
+lemma cnr_lref_atom (h) (b) (G) (L):
+ ∀i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄.
+#h #b #G #L #i #Hi #X #H
+elim (cpr_inv_lref1_drops … H) -H // * #K #V1 #V2 #HLK
+lapply (drops_gen b … HLK) -HLK #HLK
+lapply (drops_mono … Hi … HLK) -L #H destruct
+qed.
+
+(* Basic_1: was: nf2_lref_abst *)
+lemma cnr_lref_abst (h) (G) (L):
+ ∀K,V,i. ⬇*[i] L ≘ K.ⓛV → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄.
+#h #G #L #K #V #i #HLK #X #H
+elim (cpr_inv_lref1_drops … H) -H // *
+#K0 #V1 #V2 #HLK0 #_ #_
+lapply (drops_mono … HLK … HLK0) -L #H destruct
+qed.
+
+lemma cnr_lref_unit (h) (I) (G) (L):
+ ∀K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄.
+#h #I #G #L #K #i #HLK #X #H
+elim (cpr_inv_lref1_drops … H) -H // *
+#K0 #V1 #V2 #HLK0 #_ #_
+lapply (drops_mono … HLK … HLK0) -L #H destruct
+qed.
+
+(* Properties with generic relocation ***************************************)
+
+(* Basic_1: was: nf2_lift *)
+(* Basic_2A1: uses: cnr_lift *)
+lemma cnr_lifts (h) (G): d_liftable1 … (cnr h G).
+#h #G #K #T #HT #b #f #L #HLK #U #HTU #U0 #H
+elim (cpm_inv_lifts_sn … H … HLK … HTU) -b -L #T0 #HTU0 #HT0
+lapply (HT … HT0) -G -K #H destruct /2 width=4 by lifts_mono/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_2A1: was: cnr_inv_delta *)
+lemma cnr_inv_lref_abbr (h) (G) (L):
+ ∀K,V,i. ⬇*[i] L ≘ K.ⓓV → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃#i⦄ → ⊥.
+#h #G #L #K #V #i #HLK #H
+elim (lifts_total V 𝐔❴↑i❵) #W #HVW
+lapply (H W ?) -H [ /3 width=6 by cpm_delta_drops/ ] -HLK #H destruct
+elim (lifts_inv_lref2_uni_lt … HVW) -HVW //
+qed-.
+
+(* Inversion lemmas with generic relocation *********************************)
+
+(* Note: this was missing in Basic_1 *)
+(* Basic_2A1: uses: cnr_inv_lift *)
+lemma cnr_inv_lifts (h) (G): d_deliftable1 … (cnr h G).
+#h #G #L #U #HU #b #f #K #HLK #T #HTU #T0 #H
+elim (cpm_lifts_sn … H … HLK … HTU) -b -K #U0 #HTU0 #HU0
+lapply (HU … HU0) -G -L #H destruct /2 width=4 by lifts_inj/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cpm_simple.ma".
+include "basic_2/rt_transition/cnr.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE R-TRANSITION **************************)
+
+(* Inversion lemmas with simple terms ***************************************)
+
+lemma cnr_inv_appl (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T⦄ & 𝐒⦃T⦄.
+#h #G #L #V1 #T1 #HVT1 @and3_intro
+[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpr_pair_sn/ -HV2 #H destruct //
+| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpr_flat/ -HT2 #H destruct //
+| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
+ [ elim (lifts_total V1 𝐔❴1❵) #V2 #HV12
+ lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /2 width=3 by cpm_theta/ -HV12 #H destruct
+ | lapply (H (ⓓ{a}ⓝW1.V1.U1) ?) -H /2 width=1 by cpm_beta/ #H destruct
+ ]
+]
+qed-.
+
+(* Properties with simple terms *********************************************)
+
+(* Basic_1: was only: nf2_appl_lref *)
+lemma cnr_appl_simple (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄.
+#h #G #L #V #T #HV #HT #HS #X #H
+elim (cpm_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
+<(HV … HV0) -V0 <(HT … HT0) -T0 //
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/lifts_tdeq.ma".
+include "basic_2/rt_transition/cpr_drops_basic.ma".
+include "basic_2/rt_transition/cnr_simple.ma".
+include "basic_2/rt_transition/cnr_drops.ma".
+
+(* NORMAL TERMS FOR CONTEXT-SENSITIVE R-TRANSITION **************************)
+
+(* Properties with context-free sort-irrelevant equivalence for terms *******)
+
+(* Basic_1: was: nf2_dec *)
+(* Basic_2A1: uses: cnr_dec *)
+lemma cnr_dec_tdeq (h) (G) (L):
+ ∀T1. ∨∨ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T1⦄
+ | ∃∃T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 & (T1 ≛ T2 → ⊥).
+#h #G #L #T1
+@(fqup_wf_ind_eq (Ⓣ) … G L T1) -G -L -T1 #G0 #L0 #T0 #IH #G #L * *
+[ #s #HG #HL #HT destruct -IH
+ /3 width=4 by cnr_sort, or_introl/
+| #i #HG #HL #HT destruct -IH
+ elim (drops_F_uni L i)
+ [ /3 width=6 by cnr_lref_atom, or_introl/
+ | * * [ #I | * #V ] #K #HLK
+ [ /3 width=7 by cnr_lref_unit, or_introl/
+ | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
+ @or_intror @(ex2_intro … W) [ /2 width=6 by cpm_delta_drops/ ] #H
+ lapply (tdeq_inv_lref1 … H) -H #H destruct
+ /2 width=5 by lifts_inv_lref2_uni_lt/
+ | /3 width=7 by cnr_lref_abst, or_introl/
+ ]
+ ]
+| #l #HG #HL #HT destruct -IH
+ /3 width=4 by cnr_gref, or_introl/
+| #p * [ cases p ] #V1 #T1 #HG #HL #HT destruct
+ [ elim (cpr_subst h G (L.ⓓV1) T1 0 L V1) [| /2 width=1 by drops_refl/ ] #T2 #X2 #HT12 #HXT2 -IH
+ elim (tdeq_dec T1 T2) [ -HT12 #HT12 | #HnT12 ]
+ [ elim (tdeq_inv_lifts_dx … HT12 … HXT2) -T2 #X1 #HXT1 #_ -X2
+ @or_intror @(ex2_intro … X1) [ /2 width=3 by cpm_zeta/ ] #H
+ /2 width=7 by tdeq_lifts_inv_pair_sn/
+ | @or_intror @(ex2_intro … (+ⓓV1.T2)) [ /2 width=1 by cpm_bind/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ | elim (IH G L V1) [ elim (IH G (L.ⓓV1) T1) [| * | // ] | * | // ] -IH
+ [ #HT1 #HV1 /3 width=6 by cnr_abbr_neg, or_introl/
+ | #T2 #HT12 #HnT12 #_
+ @or_intror @(ex2_intro … (-ⓓV1.T2)) [ /2 width=1 by cpm_bind/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ | #V2 #HV12 #HnV12
+ @or_intror @(ex2_intro … (-ⓓV2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ | elim (IH G L V1) [ elim (IH G (L.ⓛV1) T1) [| * | // ] | * | // ] -IH
+ [ #HT1 #HV1 /3 width=6 by cnr_abst, or_introl/
+ | #T2 #HT12 #HnT12 #_
+ @or_intror @(ex2_intro … (ⓛ{p}V1.T2)) [ /2 width=1 by cpm_bind/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ | #V2 #HV12 #HnV12
+ @or_intror @(ex2_intro … (ⓛ{p}V2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ ]
+| * #V1 #T1 #HG #HL #HT destruct [| -IH ]
+ [ elim (IH G L V1) [ elim (IH G L T1) [| * | // ] | * | // ] -IH
+ [ #HT1 #HV1
+ elim (simple_dec_ex T1) [| * #p * #W1 #U1 #H destruct ]
+ [ /3 width=6 by cnr_appl_simple, or_introl/
+ | elim (lifts_total V1 𝐔❴1❵) #X1 #HVX1
+ @or_intror @(ex2_intro … (ⓓ{p}W1.ⓐX1.U1)) [ /2 width=3 by cpm_theta/ ] #H
+ elim (tdeq_inv_pair … H) -H #H destruct
+ | @or_intror @(ex2_intro … (ⓓ{p}ⓝW1.V1.U1)) [ /2 width=1 by cpm_beta/ ] #H
+ elim (tdeq_inv_pair … H) -H #H destruct
+ ]
+ | #T2 #HT12 #HnT12 #_
+ @or_intror @(ex2_intro … (ⓐV1.T2)) [ /2 width=1 by cpm_appl/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ | #V2 #HV12 #HnV12
+ @or_intror @(ex2_intro … (ⓐV2.T1)) [ /2 width=1 by cpr_pair_sn/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ | @or_intror @(ex2_intro … T1) [ /2 width=1 by cpm_eps/ ] #H
+ /2 width=4 by tdeq_inv_pair_xy_y/
+ ]
+]
+qed-.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predtynormal_5.ma".
+include "basic_2/notation/relations/predtynormal_4.ma".
include "static_2/syntax/tdeq.ma".
include "basic_2/rt_transition/cpx.ma".
(* NORMAL TERMS FOR UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ********)
-definition cnx: ∀h. sd h → relation3 genv lenv term ≝
- λh,o,G,L. NF … (cpx h G L) (tdeq h o …).
+definition cnx: ∀h. relation3 genv lenv term ≝
+ λh,G,L. NF … (cpx h G L) tdeq.
interpretation
"normality for unbound context-sensitive parallel rt-transition (term)"
- 'PRedTyNormal h o G L T = (cnx h o G L T).
+ 'PRedTyNormal h G L T = (cnx h G L T).
(* Basic inversion lemmas ***************************************************)
-lemma cnx_inv_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃⋆s⦄ → deg h o s 0.
-#h #o #G #L #s #H
-lapply (H (⋆(next h s)) ?) -H /2 width=2 by cpx_ess/ -G -L #H
-elim (tdeq_inv_sort1 … H) -H #s0 #d #H1 #H2 #H destruct
-lapply (deg_next … H1) #H0
-lapply (deg_mono … H0 … H2) -H0 -H2 #H
->(pred_inv_fix_sn … H) -H //
-qed-.
-
-lemma cnx_inv_abst: ∀h,o,p,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃ⓛ{p}V.T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄.
-#h #o #p #G #L #V1 #T1 #HVT1 @conj
+lemma cnx_inv_abst: ∀h,p,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓛ{p}V.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ ∧ ⦃G,L.ⓛV⦄ ⊢ ⬈[h] 𝐍⦃T⦄.
+#h #p #G #L #V1 #T1 #HVT1 @conj
[ #V2 #HV2 lapply (HVT1 (ⓛ{p}V2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2
| #T2 #HT2 lapply (HVT1 (ⓛ{p}V1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2
]
qed-.
(* Basic_2A1: was: cnx_inv_abbr *)
-lemma cnx_inv_abbr_neg: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃-ⓓV.T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄.
-#h #o #G #L #V1 #T1 #HVT1 @conj
+lemma cnx_inv_abbr_neg: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃-ⓓV.T⦄ →
+ ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ ∧ ⦃G,L.ⓓV⦄ ⊢ ⬈[h] 𝐍⦃T⦄.
+#h #G #L #V1 #T1 #HVT1 @conj
[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2
| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2
]
qed-.
(* Basic_2A1: was: cnx_inv_eps *)
-lemma cnx_inv_cast: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃ⓝV.T⦄ → ⊥.
-#h #o #G #L #V #T #H lapply (H T ?) -H
+lemma cnx_inv_cast: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓝV.T⦄ → ⊥.
+#h #G #L #V #T #H lapply (H T ?) -H
/2 width=6 by cpx_eps, tdeq_inv_pair_xy_y/
qed-.
(* Basic properties *********************************************************)
-lemma cnx_sort: ∀h,o,G,L,s. deg h o s 0 → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃⋆s⦄.
-#h #o #G #L #s #Hs #X #H elim (cpx_inv_sort1 … H) -H
-/3 width=3 by tdeq_sort, deg_next/
-qed.
-
-lemma cnx_sort_iter: ∀h,o,G,L,s,d. deg h o s d → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃⋆((next h)^d s)⦄.
-#h #o #G #L #s #d #Hs lapply (deg_iter … d Hs) -Hs
-<minus_n_n /2 width=6 by cnx_sort/
+lemma cnx_sort: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃⋆s⦄.
+#h #G #L #s #X #H elim (cpx_inv_sort1 … H) -H
+/2 width=1 by tdeq_sort/
qed.
-lemma cnx_abst: ∀h,o,p,G,L,W,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃ⓛ{p}W.T⦄.
-#h #o #p #G #L #W #T #HW #HT #X #H
+lemma cnx_abst: ∀h,p,G,L,W,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃W⦄ → ⦃G,L.ⓛW⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
+ ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓛ{p}W.T⦄.
+#h #p #G #L #W #T #HW #HT #X #H
elim (cpx_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
@tdeq_pair [ @HW | @HT ] // (**) (* auto fails because δ-expansion gets in the way *)
qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/lifts_tdeq.ma".
+include "basic_2/rt_transition/cpx_drops_basic.ma".
+include "basic_2/rt_transition/cnx.ma".
+
+(* NORMAL TERMS FOR UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ********)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cnx_inv_abbr_pos (h) (G) (L): ∀V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃+ⓓV.T⦄ → ⊥.
+#h #G #L #V #U1 #H
+elim (cpx_subst h G (L.ⓓV) U1 … 0) [|*: /2 width=4 by drops_refl/ ] #U2 #T2 #HU12 #HTU2
+elim (tdeq_dec U1 U2) #HnU12 [ -HU12 | -HTU2 ]
+[ elim (tdeq_inv_lifts_dx … HnU12 … HTU2) -U2 #T1 #HTU1 #_ -T2
+ lapply (H T1 ?) -H [ /2 width=3 by cpx_zeta/ ] #H
+ /2 width=9 by tdeq_lifts_inv_pair_sn/
+| lapply (H (+ⓓV.U2) ?) -H [ /2 width=1 by cpx_bind/ ] -HU12 #H
+ elim (tdeq_inv_pair … H) -H #_ #_ /2 width=1 by/
+]
+qed-.
(* Advanced properties ******************************************************)
-lemma cnx_tdeq_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T1⦄ →
- ∀T2. T1 ≛[h, o] T2 → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T2⦄.
-#h #o #G #L #T1 #HT1 #T2 #HT12 #T #HT2
+lemma cnx_tdeq_trans: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T1⦄ →
+ ∀T2. T1 ≛ T2 → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T2⦄.
+#h #G #L #T1 #HT1 #T2 #HT12 #T #HT2
elim (tdeq_cpx_trans … HT12 … HT2) -HT2 #T0 #HT10 #HT0
lapply (HT1 … HT10) -HT1 -HT10 /2 width=5 by tdeq_repl/ (**) (* full auto fails *)
qed-.
(* Properties with generic slicing ******************************************)
-lemma cnx_lref_atom: ∀h,o,G,L,i. ⬇*[i] L ≘ ⋆ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃#i⦄.
-#h #o #G #L #i #Hi #X #H elim (cpx_inv_lref1_drops … H) -H // *
+lemma cnx_lref_atom: ∀h,G,L,i. ⬇*[i] L ≘ ⋆ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄.
+#h #G #L #i #Hi #X #H elim (cpx_inv_lref1_drops … H) -H // *
#I #K #V1 #V2 #HLK lapply (drops_mono … Hi … HLK) -L #H destruct
qed.
-lemma cnx_lref_unit: ∀h,o,I,G,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃#i⦄.
-#h #o #I #G #L #K #i #HLK #X #H elim (cpx_inv_lref1_drops … H) -H // *
+lemma cnx_lref_unit: ∀h,I,G,L,K,i. ⬇*[i] L ≘ K.ⓤ{I} → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄.
+#h #I #G #L #K #i #HLK #X #H elim (cpx_inv_lref1_drops … H) -H // *
#Z #Y #V1 #V2 #HLY lapply (drops_mono … HLK … HLY) -L #H destruct
qed.
(* Basic_2A1: includes: cnx_lift *)
-lemma cnx_lifts: ∀h,o,G. d_liftable1 … (cnx h o G).
-#h #o #G #K #T #HT #b #f #L #HLK #U #HTU #U0 #H
+lemma cnx_lifts: ∀h,G. d_liftable1 … (cnx h G).
+#h #G #K #T #HT #b #f #L #HLK #U #HTU #U0 #H
elim (cpx_inv_lifts_sn … H … HLK … HTU) -b -L #T0 #HTU0 #HT0
lapply (HT … HT0) -G -K /2 width=6 by tdeq_lifts_bi/
qed-.
(* Inversion lemmas with generic slicing ************************************)
(* Basic_2A1: was: cnx_inv_delta *)
-lemma cnx_inv_lref_pair: ∀h,o,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃#i⦄ → ⊥.
-#h #o #I #G #L #K #V #i #HLK #H
+lemma cnx_inv_lref_pair: ∀h,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃#i⦄ → ⊥.
+#h #I #G #L #K #V #i #HLK #H
elim (lifts_total V (𝐔❴↑i❵)) #W #HVW
lapply (H W ?) -H /2 width=7 by cpx_delta_drops/ -HLK
#H lapply (tdeq_inv_lref1 … H) -H #H destruct
qed-.
(* Basic_2A1: includes: cnx_inv_lift *)
-lemma cnx_inv_lifts: ∀h,o,G. d_deliftable1 … (cnx h o G).
-#h #o #G #L #U #HU #b #f #K #HLK #T #HTU #T0 #H
+lemma cnx_inv_lifts: ∀h,G. d_deliftable1 … (cnx h G).
+#h #G #L #U #HU #b #f #K #HLK #T #HTU #T0 #H
elim (cpx_lifts_sn … H … HLK … HTU) -b -K #U0 #HTU0 #HU0
lapply (HU … HU0) -G -L /2 width=6 by tdeq_inv_lifts_bi/
qed-.
(* Inversion lemmas with simple terms ***************************************)
-lemma cnx_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃ⓐV.T⦄ →
- ∧∧ ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ & 𝐒⦃T⦄.
-#h #o #G #L #V1 #T1 #HVT1 @and3_intro
+lemma cnx_inv_appl: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓐV.T⦄ →
+ ∧∧ ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ & ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ & 𝐒⦃T⦄.
+#h #G #L #V1 #T1 #HVT1 @and3_intro
[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpx_pair_sn/ -HV2
#H elim (tdeq_inv_pair … H) -H //
| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpx_flat/ -HT2
(* Properties with simple terms *********************************************)
-lemma cnx_appl_simple: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ → 𝐒⦃T⦄ →
- ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃ⓐV.T⦄.
-#h #o #G #L #V #T #HV #HT #HS #X #H elim (cpx_inv_appl1_simple … H) -H //
+lemma cnx_appl_simple: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ →
+ ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓐV.T⦄.
+#h #G #L #V #T #HV #HT #HS #X #H elim (cpx_inv_appl1_simple … H) -H //
#V0 #T0 #HV0 #HT0 #H destruct
@tdeq_pair [ @HV | @HT ] // (**) (* auto fails because δ-expansion gets in the way *)
qed.
include "ground_2/steps/rtc_max.ma".
include "ground_2/steps/rtc_plus.ma".
include "basic_2/notation/relations/predty_7.ma".
-include "static_2/syntax/item_sh.ma".
+include "static_2/syntax/sh.ma".
include "static_2/syntax/lenv.ma".
include "static_2/syntax/genv.ma".
include "static_2/relocation/lifts.ma".
(* avtivate genv *)
inductive cpg (Rt:relation rtc) (h): rtc → relation4 genv lenv term term ≝
| cpg_atom : ∀I,G,L. cpg Rt h (𝟘𝟘) G L (⓪{I}) (⓪{I})
-| cpg_ess : ∀G,L,s. cpg Rt h (𝟘𝟙) G L (⋆s) (⋆(next h s))
+| cpg_ess : ∀G,L,s. cpg Rt h (𝟘𝟙) G L (⋆s) (⋆(⫯[h]s))
| cpg_delta: ∀c,G,L,V1,V2,W2. cpg Rt h c G L V1 V2 →
⬆*[1] V2 ≘ W2 → cpg Rt h c G (L.ⓓV1) (#0) W2
| cpg_ell : ∀c,G,L,V1,V2,W2. cpg Rt h c G L V1 V2 →
(* Basic properties *********************************************************)
(* Note: this is "∀Rt. reflexive … Rt → ∀h,g,L. reflexive … (cpg Rt h (𝟘𝟘) L)" *)
-lemma cpg_refl: ∀Rt. reflexive … Rt → ∀h,G,T,L. ⦃G, L⦄ ⊢ T ⬈[Rt, 𝟘𝟘, h] T.
+lemma cpg_refl: ∀Rt. reflexive … Rt → ∀h,G,T,L. ⦃G,L⦄ ⊢ T ⬈[Rt,𝟘𝟘,h] T.
#Rt #HRt #h #G #T elim T -T // * /2 width=1 by cpg_bind/
* /2 width=1 by cpg_appl, cpg_cast/
qed.
(* Basic inversion lemmas ***************************************************)
-fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[Rt, c, h] T2 → ∀J. T1 = ⓪{J} →
+fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[Rt,c,h] T2 → ∀J. T1 = ⓪{J} →
∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘
- | ∃∃s. J = Sort s & T2 = ⋆(next h s) & c = 𝟘𝟙
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃s. J = Sort s & T2 = ⋆(⫯[h]s) & c = 𝟘𝟙
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0 & c = cV
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
#Rt #c #h #G #L #T1 #T2 * -c -G -L -T1 -T2
[ #I #G #L #J #H destruct /3 width=1 by or5_intro0, conj/
]
qed-.
-lemma cpg_inv_atom1: ∀Rt,c,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[Rt, c, h] T2 →
+lemma cpg_inv_atom1: ∀Rt,c,h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ⬈[Rt,c,h] T2 →
∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘
- | ∃∃s. J = Sort s & T2 = ⋆(next h s) & c = 𝟘𝟙
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃s. J = Sort s & T2 = ⋆(⫯[h]s) & c = 𝟘𝟙
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0 & c = cV
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
/2 width=3 by cpg_inv_atom1_aux/ qed-.
-lemma cpg_inv_sort1: ∀Rt,c,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[Rt, c, h] T2 →
- ∨∨ T2 = ⋆s ∧ c = 𝟘𝟘 | T2 = ⋆(next h s) ∧ c = 𝟘𝟙.
+lemma cpg_inv_sort1: ∀Rt,c,h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ⬈[Rt,c,h] T2 →
+ ∨∨ T2 = ⋆s ∧ c = 𝟘𝟘 | T2 = ⋆(⫯[h]s) ∧ c = 𝟘𝟙.
#Rt #c #h #G #L #T2 #s #H
elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
[ #s0 #H destruct /3 width=1 by or_intror, conj/
]
qed-.
-lemma cpg_inv_zero1: ∀Rt,c,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[Rt, c, h] T2 →
+lemma cpg_inv_zero1: ∀Rt,c,h,G,L,T2. ⦃G,L⦄ ⊢ #0 ⬈[Rt,c,h] T2 →
∨∨ T2 = #0 ∧ c = 𝟘𝟘
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1 & c = cV
- | ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓛV1 & c = cV+𝟘𝟙.
#Rt #c #h #G #L #T2 #H
elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/
]
qed-.
-lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈[Rt, c, h] T2 →
+lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈[Rt,c,h] T2 →
∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘
- | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
+ | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
#Rt #c #h #G #L #T2 #i #H
elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
[ #s #H destruct
]
qed-.
-lemma cpg_inv_gref1: ∀Rt,c,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ⬈[Rt, c, h] T2 → T2 = §l ∧ c = 𝟘𝟘.
+lemma cpg_inv_gref1: ∀Rt,c,h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ⬈[Rt,c,h] T2 → T2 = §l ∧ c = 𝟘𝟘.
#Rt #c #h #G #L #T2 #l #H
elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/
[ #s #H destruct
]
qed-.
-fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 →
+fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 →
∀p,J,V1,U1. U = ⓑ{p,J}V1.U1 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ U1 ⬈[Rt, cT, h] T2 &
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓑ{J}V1⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
U2 = ⓑ{p,J}V2.T2 & c = ((↕*cV)∨cT)
- | ∃∃cT,T. ⬆*[1] T ≘ U1 & ⦃G, L⦄ ⊢ T ⬈[Rt, cT, h] U2 &
+ | ∃∃cT,T. ⬆*[1] T ≘ U1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 &
p = true & J = Abbr & c = cT+𝟙𝟘.
#Rt #c #h #G #L #U #U2 * -c -G -L -U -U2
[ #I #G #L #q #J #W #U1 #H destruct
]
qed-.
-lemma cpg_inv_bind1: ∀Rt,c,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt, c, h] U2 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_bind1: ∀Rt,c,h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt,c,h] U2 →
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U2 = ⓑ{p,I}V2.T2 & c = ((↕*cV)∨cT)
- | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[Rt, cT, h] U2 &
+ | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 &
p = true & I = Abbr & c = cT+𝟙𝟘.
/2 width=3 by cpg_inv_bind1_aux/ qed-.
-lemma cpg_inv_abbr1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[Rt, c, h] U2 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_abbr1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈[Rt,c,h] U2 →
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U2 = ⓓ{p}V2.T2 & c = ((↕*cV)∨cT)
- | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[Rt, cT, h] U2 &
+ | ∃∃cT,T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[Rt,cT,h] U2 &
p = true & c = cT+𝟙𝟘.
#Rt #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H *
/3 width=8 by ex4_4_intro, ex4_2_intro, or_introl, or_intror/
qed-.
-lemma cpg_inv_abst1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈[Rt, c, h] U2 →
- ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_abst1: ∀Rt,c,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈[Rt,c,h] U2 →
+ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U2 = ⓛ{p}V2.T2 & c = ((↕*cV)∨cT).
#Rt #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H *
[ /3 width=8 by ex4_4_intro/
]
qed-.
-fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 →
+fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 →
∀V1,U1. U = ⓐV1.U1 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 &
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
- | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+ | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
- | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V & ⬆*[1] V ≘ V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+ | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V & ⬆*[1] V ≘ V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
#Rt #c #h #G #L #U #U2 * -c -G -L -U -U2
[ #I #G #L #W #U1 #H destruct
]
qed-.
-lemma cpg_inv_appl1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ⬈[Rt, c, h] U2 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_appl1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐV1.U1 ⬈[Rt,c,h] U2 →
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
- | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+ | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
- | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V & ⬆*[1] V ≘ V2 & ⦃G, L⦄ ⊢ W1 ⬈[Rt, cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+ | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V & ⬆*[1] V ≘ V2 & ⦃G,L⦄ ⊢ W1 ⬈[Rt,cW,h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
/2 width=3 by cpg_inv_appl1_aux/ qed-.
-fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G, L⦄ ⊢ U ⬈[Rt, c, h] U2 →
+fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 →
∀V1,U1. U = ⓝV1.U1 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 &
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
- | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] U2 & c = cT+𝟙𝟘
- | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] U2 & c = cV+𝟘𝟙.
+ | ∃∃cT. ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘
+ | ∃∃cV. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙.
#Rt #c #h #G #L #U #U2 * -c -G -L -U -U2
[ #I #G #L #W #U1 #H destruct
| #G #L #s #W #U1 #H destruct
]
qed-.
-lemma cpg_inv_cast1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[Rt, c, h] U2 →
- ∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_cast1: ∀Rt,c,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ⬈[Rt,c,h] U2 →
+ ∨∨ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] T2 &
Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
- | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[Rt, cT, h] U2 & c = cT+𝟙𝟘
- | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] U2 & c = cV+𝟘𝟙.
+ | ∃∃cT. ⦃G,L⦄ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘
+ | ∃∃cV. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙.
/2 width=3 by cpg_inv_cast1_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma cpg_inv_zero1_pair: ∀Rt,c,h,I,G,K,V1,T2. ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[Rt, c, h] T2 →
+lemma cpg_inv_zero1_pair: ∀Rt,c,h,I,G,K,V1,T2. ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈[Rt,c,h] T2 →
∨∨ T2 = #0 ∧ c = 𝟘𝟘
- | ∃∃cV,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
I = Abbr & c = cV
- | ∃∃cV,V2. ⦃G, K⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃cV,V2. ⦃G,K⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⬆*[1] V2 ≘ T2 &
I = Abst & c = cV+𝟘𝟙.
#Rt #c #h #I #G #K #V1 #T2 #H elim (cpg_inv_zero1 … H) -H /2 width=1 by or3_intro0/
* #z #Y #X1 #X2 #HX12 #HXT2 #H1 #H2 destruct /3 width=5 by or3_intro1, or3_intro2, ex4_2_intro/
qed-.
-lemma cpg_inv_lref1_bind: ∀Rt,c,h,I,G,K,T2,i. ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[Rt, c, h] T2 →
+lemma cpg_inv_lref1_bind: ∀Rt,c,h,I,G,K,T2,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[Rt,c,h] T2 →
∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘
- | ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[Rt, c, h] T & ⬆*[1] T ≘ T2.
+ | ∃∃T. ⦃G,K⦄ ⊢ #i ⬈[Rt,c,h] T & ⬆*[1] T ≘ T2.
#Rt #c #h #I #G #L #T2 #i #H elim (cpg_inv_lref1 … H) -H /2 width=1 by or_introl/
* #Z #Y #T #HT #HT2 #H destruct /3 width=3 by ex2_intro, or_intror/
qed-.
(* Basic forward lemmas *****************************************************)
-lemma cpg_fwd_bind1_minus: ∀Rt,c,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[Rt, c, h] T → ∀p.
- ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt, c, h] ⓑ{p,I}V2.T2 &
+lemma cpg_fwd_bind1_minus: ∀Rt,c,h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[Rt,c,h] T → ∀p.
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[Rt,c,h] ⓑ{p,I}V2.T2 &
T = -ⓑ{I}V2.T2.
#Rt #c #h #I #G #L #V1 #T1 #T #H #p elim (cpg_inv_bind1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct /3 width=4 by cpg_bind, ex2_2_intro/
(* Advanced properties ******************************************************)
-lemma cpg_delta_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓓV → ⦃G, K⦄ ⊢ V ⬈[Rt, c, h] V2 →
- ⬆*[↑i] V2 ≘ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c, h] T2.
+lemma cpg_delta_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓓV → ⦃G,K⦄ ⊢ V ⬈[Rt,c,h] V2 →
+ ⬆*[↑i] V2 ≘ T2 → ⦃G,L⦄ ⊢ #i ⬈[Rt,c,h] T2.
#Rt #c #h #G #K #V #V2 #i elim i -i
[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/
| #i #IH #L0 #T0 #H0 #HV2 #HVT2
]
qed.
-lemma cpg_ell_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓛV → ⦃G, K⦄ ⊢ V ⬈[Rt,c, h] V2 →
- ⬆*[↑i] V2 ≘ T2 → ⦃G, L⦄ ⊢ #i ⬈[Rt, c+𝟘𝟙, h] T2.
+lemma cpg_ell_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≘ K.ⓛV → ⦃G,K⦄ ⊢ V ⬈[Rt,c,h] V2 →
+ ⬆*[↑i] V2 ≘ T2 → ⦃G,L⦄ ⊢ #i ⬈[Rt,c+𝟘𝟙,h] T2.
#Rt #c #h #G #K #V #V2 #i elim i -i
[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/
| #i #IH #L0 #T0 #H0 #HV2 #HVT2
(* Advanced inversion lemmas ************************************************)
-lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ⬈[Rt,c, h] T2 →
+lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ⦃G,L⦄ ⊢ #i ⬈[Rt,c,h] T2 →
∨∨ T2 = #i ∧ c = 𝟘𝟘
- | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
+ | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ⬈[Rt,cV,h] V2 &
⬆*[↑i] V2 ≘ T2 & c = cV
- | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
+ | ∃∃cV,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ⬈[Rt,cV,h] V2 &
⬆*[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙.
#Rt #c #h #G #i elim i -i
[ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
]
qed-.
-lemma cpg_inv_atom1_drops: ∀Rt,c,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[Rt, c, h] T2 →
+lemma cpg_inv_atom1_drops: ∀Rt,c,h,I,G,L,T2. ⦃G,L⦄ ⊢ ⓪{I} ⬈[Rt,c,h] T2 →
∨∨ T2 = ⓪{I} ∧ c = 𝟘𝟘
- | ∃∃s. T2 = ⋆(next h s) & I = Sort s & c = 𝟘𝟙
- | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
+ | ∃∃s. T2 = ⋆(⫯[h]s) & I = Sort s & c = 𝟘𝟙
+ | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ⬈[Rt,cV,h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i & c = cV
- | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ⬈[Rt, cV, h] V2 &
+ | ∃∃cV,i,K,V,V2. ⬇*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ⬈[Rt,cV,h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i & c = cV + 𝟘𝟙.
#Rt #c #h * #n #G #L #T2 #H
[ elim (cpg_inv_sort1 … H) -H *
(* Properties with simple terms *********************************************)
(* Note: the main property of simple terms *)
-lemma cpg_inv_appl1_simple: ∀Rt,c,h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈[Rt, c, h] U → 𝐒⦃T1⦄ →
- ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[Rt, cV, h] V2 & ⦃G, L⦄ ⊢ T1 ⬈[Rt, cT, h] T2 &
+lemma cpg_inv_appl1_simple: ∀Rt,c,h,G,L,V1,T1,U. ⦃G,L⦄ ⊢ ⓐV1.T1 ⬈[Rt,c,h] U → 𝐒⦃T1⦄ →
+ ∃∃cV,cT,V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[Rt,cV,h] V2 & ⦃G,L⦄ ⊢ T1 ⬈[Rt,cT,h] T2 &
U = ⓐV2.T2 & c = ((↕*cV)∨cT).
#Rt #c #h #G #L #V1 #T1 #U #H #HT1 elim (cpg_inv_appl1 … H) -H *
[ /2 width=8 by ex4_4_intro/
(* Basic_2A1: includes: cpr *)
definition cpm (h) (G) (L) (n): relation2 term term ≝
- λT1,T2. ∃∃c. 𝐑𝐓⦃n, c⦄ & ⦃G, L⦄ ⊢ T1 ⬈[eq_t, c, h] T2.
+ λT1,T2. ∃∃c. 𝐑𝐓⦃n,c⦄ & ⦃G,L⦄ ⊢ T1 ⬈[eq_t,c,h] T2.
interpretation
"t-bound context-sensitive parallel rt-transition (term)"
(* Basic properties *********************************************************)
-lemma cpm_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ➡[1, h] ⋆(next h s).
+lemma cpm_ess: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⋆s ➡[1,h] ⋆(⫯[h]s).
/2 width=3 by cpg_ess, ex2_intro/ qed.
-lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 →
- ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓓV1⦄ ⊢ #0 ➡[n, h] W2.
+lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 →
+ ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓓV1⦄ ⊢ #0 ➡[n,h] W2.
#n #h #G #K #V1 #V2 #W2 *
/3 width=5 by cpg_delta, ex2_intro/
qed.
-lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 →
- ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓛV1⦄ ⊢ #0 ➡[↑n, h] W2.
+lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 →
+ ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓛV1⦄ ⊢ #0 ➡[↑n,h] W2.
#n #h #G #K #V1 #V2 #W2 *
/3 width=5 by cpg_ell, ex2_intro, isrt_succ/
qed.
-lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T →
- ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ➡[n, h] U.
+lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T →
+ ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ➡[n,h] U.
#n #h #I #G #K #T #U #i *
/3 width=5 by cpg_lref, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_bind *)
lemma cpm_bind: ∀n,h,p,I,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{p,I}V2.T2.
#n #h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
/5 width=5 by cpg_bind, isrt_max_O1, isr_shift, ex2_intro/
qed.
lemma cpm_appl: ∀n,h,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[n, h] ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.T1 ➡[n,h] ⓐV2.T2.
#n #h #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
/5 width=5 by isrt_max_O1, isr_shift, cpg_appl, ex2_intro/
qed.
lemma cpm_cast: ∀n,h,G,L,U1,U2,T1,T2.
- ⦃G, L⦄ ⊢ U1 ➡[n, h] U2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓝU1.T1 ➡[n, h] ⓝU2.T2.
+ ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓝU1.T1 ➡[n,h] ⓝU2.T2.
#n #h #G #L #U1 #U2 #T1 #T2 * #cU #HcU #HU12 *
/4 width=6 by cpg_cast, isrt_max_idem1, isrt_mono, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_zeta *)
lemma cpm_zeta (n) (h) (G) (L):
- ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[n,h] T2 →
- ∀V. ⦃G, L⦄ ⊢ +ⓓV.T1 ➡[n, h] T2.
+ ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
+ ∀V. ⦃G,L⦄ ⊢ +ⓓV.T1 ➡[n,h] T2.
#n #h #G #L #T1 #T #HT1 #T2 *
/3 width=5 by cpg_zeta, isrt_plus_O2, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_eps *)
-lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ➡[n, h] T2.
+lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ ⓝV.T1 ➡[n,h] T2.
#n #h #G #L #V #T1 #T2 *
/3 width=3 by cpg_eps, isrt_plus_O2, ex2_intro/
qed.
-lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ➡[↑n, h] V2.
+lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → ⦃G,L⦄ ⊢ ⓝV1.T ➡[↑n,h] V2.
#n #h #G #L #V1 #V2 #T *
/3 width=3 by cpg_ee, isrt_succ, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_beta *)
lemma cpm_beta: ∀n,h,p,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n, h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n,h] ⓓ{p}ⓝW2.V2.T2.
#n #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV12 * #riW #rhW #HW12 *
/6 width=7 by cpg_beta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/
qed.
(* Basic_2A1: includes: cpr_theta *)
lemma cpm_theta: ∀n,h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V → ⬆*[1] V ≘ V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
- ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n, h] ⓓ{p}W2.ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V → ⬆*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
+ ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n,h] ⓓ{p}W2.ⓐV2.T2.
#n #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV1 #HV2 * #riW #rhW #HW12 *
/6 width=9 by cpg_theta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[n, h] T2 →
+lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ➡[n,h] T2 →
∨∨ T2 = ⓪{J} ∧ n = 0
- | ∃∃s. T2 = ⋆(next h s) & J = Sort s & n = 1
- | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s & n = 1
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0
- | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[m,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓛV1 & J = LRef 0 & n = ↑m
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T & ⬆*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
#n #h #J #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H *
[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or5_intro0, conj/
]
qed-.
-lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[n,h] T2 →
+lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ➡[n,h] T2 →
∧∧ T2 = ⋆(((next h)^n) s) & n ≤ 1.
#n #h #G #L #T2 #s * #c #Hc #H
elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct
#H destruct /2 width=1 by conj/
qed-.
-lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[n, h] T2 →
+lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G,L⦄ ⊢ #0 ➡[n,h] T2 →
∨∨ T2 = #0 ∧ n = 0
- | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1
- | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[m,h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓛV1 & n = ↑m.
#n #h #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or3_intro0, conj/
]
qed-.
-lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ➡[n, h] T2 →
+lemma cpm_inv_zero1_unit (n) (h) (I) (K) (G):
+ ∀X2. ⦃G,K.ⓤ{I}⦄ ⊢ #0 ➡[n,h] X2 → ∧∧ X2 = #0 & n = 0.
+#n #h #I #G #K #X2 #H
+elim (cpm_inv_zero1 … H) -H *
+[ #H1 #H2 destruct /2 width=1 by conj/
+| #Y #X1 #X2 #_ #_ #H destruct
+| #m #Y #X1 #X2 #_ #_ #H destruct
+]
+qed.
+
+lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ➡[n,h] T2 →
∨∨ T2 = #(↑i) ∧ n = 0
- | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
+ | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ➡[n,h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
#n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
[ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or_introl, conj/
| #I #K #V2 #HV2 #HVT2 #H destruct
]
qed-.
-lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[n, h] T2 → T2 = §l ∧ n = 0.
+lemma cpm_inv_lref1_ctop (n) (h) (G):
+ ∀X2,i. ⦃G,⋆⦄ ⊢ #i ➡[n,h] X2 → ∧∧ X2 = #i & n = 0.
+#n #h #G #X2 * [| #i ] #H
+[ elim (cpm_inv_zero1 … H) -H *
+ [ #H1 #H2 destruct /2 width=1 by conj/
+ | #Y #X1 #X2 #_ #_ #H destruct
+ | #m #Y #X1 #X2 #_ #_ #H destruct
+ ]
+| elim (cpm_inv_lref1 … H) -H *
+ [ #H1 #H2 destruct /2 width=1 by conj/
+ | #Z #Y #X0 #_ #_ #H destruct
+ ]
+]
+qed.
+
+lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ➡[n,h] T2 → T2 = §l ∧ n = 0.
#n #h #G #L #T2 #l * #c #Hc #H elim (cpg_inv_gref1 … H) -H
#H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/
qed-.
(* Basic_2A1: includes: cpr_inv_bind1 *)
-lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 &
+lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 &
U2 = ⓑ{p,I}V2.T2
- | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ➡[n, h] U2 &
+ | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ➡[n,h] U2 &
p = true & I = Abbr.
#n #h #p #I #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_bind1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
(* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
(* Basic_2A1: includes: cpr_inv_abbr1 *)
-lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T2 &
+lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ➡[n,h] T2 &
U2 = ⓓ{p}V2.T2
- | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ➡[n, h] U2 & p = true.
+ | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ➡[n,h] U2 & p = true.
#n #h #p #G #L #V1 #T1 #U2 #H
elim (cpm_inv_bind1 … H) -H
[ /3 width=1 by or_introl/
(* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
(* Basic_2A1: includes: cpr_inv_abst1 *)
-lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[n, h] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[n, h] T2 &
+lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ➡[n,h] U2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ➡[n,h] T2 &
U2 = ⓛ{p}V2.T2.
#n #h #p #G #L #V1 #T1 #U2 #H
elim (cpm_inv_bind1 … H) -H
(* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
(* Basic_2A1: includes: cpr_inv_appl1 *)
-lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 &
+lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐ V1.U1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ U1 ➡[n,h] T2 &
U2 = ⓐV2.T2
- | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 &
+ | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ W1 ➡[h] W2 &
+ ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 &
U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2
- | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 &
- ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 &
+ | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 &
+ ⦃G,L⦄ ⊢ W1 ➡[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 &
U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
#n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
]
qed-.
-lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[n, h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 &
+lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ➡[n,h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 & ⦃G,L⦄ ⊢ U1 ➡[n,h] T2 &
U2 = ⓝV2.T2
- | ⦃G, L⦄ ⊢ U1 ➡[n, h] U2
- | ∃∃m. ⦃G, L⦄ ⊢ V1 ➡[m, h] U2 & n = ↑m.
+ | ⦃G,L⦄ ⊢ U1 ➡[n,h] U2
+ | ∃∃m. ⦃G,L⦄ ⊢ V1 ➡[m,h] U2 & n = ↑m.
#n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H *
[ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: cpr_fwd_bind1_minus *)
-lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n, h] T → ∀p.
- ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2 &
+lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n,h] T → ∀p.
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{p,I}V2.T2 &
T = -ⓑ{I}V2.T2.
#n #h #I #G #L #V1 #T1 #T * #c #Hc #H #p elim (cpg_fwd_bind1_minus … H p) -H
/3 width=4 by ex2_2_intro, ex2_intro/
lemma cpm_ind (h): ∀Q:relation5 nat genv lenv term term.
(∀I,G,L. Q 0 G L (⓪{I}) (⓪{I})) →
- (∀G,L,s. Q 1 G L (⋆s) (⋆(next h s))) →
- (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → Q n G K V1 V2 →
+ (∀G,L,s. Q 1 G L (⋆s) (⋆(⫯[h]s))) →
+ (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
⬆*[1] V2 ≘ W2 → Q n G (K.ⓓV1) (#0) W2
- ) → (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → Q n G K V1 V2 →
+ ) → (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
⬆*[1] V2 ≘ W2 → Q (↑n) G (K.ⓛV1) (#0) W2
- ) → (∀n,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T → Q n G K (#i) T →
+ ) → (∀n,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T → Q n G K (#i) T →
⬆*[1] T ≘ U → Q n G (K.ⓘ{I}) (#↑i) (U)
- ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 →
Q 0 G L V1 V2 → Q n G (L.ⓑ{I}V1) T1 T2 → Q n G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
- ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
Q 0 G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓐV1.T1) (ⓐV2.T2)
- ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
Q n G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓝV1.T1) (ⓝV2.T2)
- ) → (∀n,G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ➡[n, h] T2 →
+ ) → (∀n,G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
Q n G L T T2 → Q n G L (+ⓓV.T1) T2
- ) → (∀n,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
Q n G L T1 T2 → Q n G L (ⓝV.T1) T2
- ) → (∀n,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 →
+ ) → (∀n,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 →
Q n G L V1 V2 → Q (↑n) G L (ⓝV1.T) V2
- ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 →
Q 0 G L V1 V2 → Q 0 G L W1 W2 → Q n G (L.ⓛW1) T1 T2 →
Q n G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
- ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 →
+ ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 →
Q 0 G L V1 V → Q 0 G L W1 W2 → Q n G (L.ⓓW1) T1 T2 →
⬆*[1] V ≘ V2 → Q n G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
) →
- ∀n,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → Q n G L T1 T2.
+ ∀n,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → Q n G L T1 T2.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #IH12 #IH13 #n #G #L #T1 #T2
* #c #HC #H generalize in match HC; -HC generalize in match n; -n
elim H -c -G -L -T1 -T2
(* Note: one of these U is the inferred type of T *)
lemma aaa_cpm_SO (h) (G) (L) (A):
- ∀T. ⦃G, L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U.
+ ∀T. ⦃G,L⦄ ⊢ T ⁝ A → ∃U. ⦃G,L⦄ ⊢ T ➡[1,h] U.
#h #G #L #A #T #H elim H -G -L -T -A
[ /3 width=2 by ex_intro/
| * #G #L #V #B #_ * #V0 #HV0
(* Forward lemmas with unbound context-sensitive rt-transition for terms ****)
(* Basic_2A1: includes: cpr_cpx *)
-lemma cpm_fwd_cpx: ∀n,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2.
+lemma cpm_fwd_cpx: ∀n,h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2.
#n #h #G #L #T1 #T2 * #c #Hc #H elim H -L -T1 -T2
/2 width=3 by cpx_theta, cpx_beta, cpx_ee, cpx_eps, cpx_zeta, cpx_flat, cpx_bind, cpx_lref, cpx_delta/
qed-.
(* Basic_1: includes: pr2_delta1 *)
(* Basic_2A1: includes: cpr_delta *)
lemma cpm_delta_drops: ∀n,h,G,L,K,V,V2,W2,i.
- ⬇*[i] L ≘ K.ⓓV → ⦃G, K⦄ ⊢ V ➡[n, h] V2 →
- ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ➡[n, h] W2.
+ ⬇*[i] L ≘ K.ⓓV → ⦃G,K⦄ ⊢ V ➡[n,h] V2 →
+ ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ➡[n,h] W2.
#n #h #G #L #K #V #V2 #W2 #i #HLK *
/3 width=8 by cpg_delta_drops, ex2_intro/
qed.
lemma cpm_ell_drops: ∀n,h,G,L,K,V,V2,W2,i.
- ⬇*[i] L ≘ K.ⓛV → ⦃G, K⦄ ⊢ V ➡[n, h] V2 →
- ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ➡[↑n, h] W2.
+ ⬇*[i] L ≘ K.ⓛV → ⦃G,K⦄ ⊢ V ➡[n,h] V2 →
+ ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ➡[↑n,h] W2.
#n #h #G #L #K #V #V2 #W2 #i #HLK *
/3 width=8 by cpg_ell_drops, isrt_succ, ex2_intro/
qed.
(* Advanced inversion lemmas ************************************************)
-lemma cpm_inv_atom1_drops: ∀n,h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡[n, h] T2 →
+lemma cpm_inv_atom1_drops: ∀n,h,I,G,L,T2. ⦃G,L⦄ ⊢ ⓪{I} ➡[n,h] T2 →
∨∨ T2 = ⓪{I} ∧ n = 0
- | ∃∃s. T2 = ⋆(next h s) & I = Sort s & n = 1
- | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 &
+ | ∃∃s. T2 = ⋆(⫯[h]s) & I = Sort s & n = 1
+ | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡[n,h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i
- | ∃∃m,K,V,V2,i. ⬇*[i] L ≘ K.ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 &
+ | ∃∃m,K,V,V2,i. ⬇*[i] L ≘ K.ⓛV & ⦃G,K⦄ ⊢ V ➡[m,h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i & n = ↑m.
#n #h #I #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1_drops … H) -H *
[ #H1 #H2 destruct lapply (isrt_inv_00 … Hc) -Hc
]
qed-.
-lemma cpm_inv_lref1_drops: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[n, h] T2 →
+lemma cpm_inv_lref1_drops: ∀n,h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ➡[n,h] T2 →
∨∨ T2 = #i ∧ n = 0
- | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[n, h] V2 &
+ | ∃∃K,V,V2. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡[n,h] V2 &
⬆*[↑i] V2 ≘ T2
- | ∃∃m,K,V,V2. ⬇*[i] L ≘ K. ⓛV & ⦃G, K⦄ ⊢ V ➡[m, h] V2 &
+ | ∃∃m,K,V,V2. ⬇*[i] L ≘ K. ⓛV & ⦃G,K⦄ ⊢ V ➡[m,h] V2 &
⬆*[↑i] V2 ≘ T2 & n = ↑m.
#n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1_drops … H) -H *
[ #H1 #H2 destruct lapply (isrt_inv_00 … Hc) -Hc
(* Advanced forward lemmas **************************************************)
-fact cpm_fwd_plus_aux (n) (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+fact cpm_fwd_plus_aux (n) (h): ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
∀n1,n2. n1+n2 = n →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T & ⦃G, L⦄ ⊢ T ➡[n2, h] T2.
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T & ⦃G,L⦄ ⊢ T ➡[n2,h] T2.
#n #h #G #L #T1 #T2 #H @(cpm_ind … H) -G -L -T1 -T2 -n
[ #I #G #L #n1 #n2 #H
elim (plus_inv_O3 … H) -H #H1 #H2 destruct
]
qed-.
-lemma cpm_fwd_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n1+n2, h] T2 →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T & ⦃G, L⦄ ⊢ T ➡[n2, h] T2.
+lemma cpm_fwd_plus (h) (G) (L): ∀n1,n2,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n1+n2,h] T2 →
+ ∃∃T. ⦃G,L⦄ ⊢ T1 ➡[n1,h] T & ⦃G,L⦄ ⊢ T ➡[n2,h] T2.
/2 width=3 by cpm_fwd_plus_aux/ qed-.
#n #h #G #L1 #T1 #T2 * /3 width=5 by lsubr_cpg_trans, ex2_intro/
qed-.
-lemma cpm_bind_unit (n) (h) (G): ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀J,T1,T2. ⦃G, L.ⓤ{J}⦄ ⊢ T1 ➡[n, h] T2 →
- ∀p,I. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2.
+lemma cpm_bind_unit (n) (h) (G): ∀L,V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀J,T1,T2. ⦃G,L.ⓤ{J}⦄ ⊢ T1 ➡[n,h] T2 →
+ ∀p,I. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{p,I}V2.T2.
/4 width=4 by lsubr_cpm_trans, cpm_bind, lsubr_unit/ qed.
(* Properties with simple terms *********************************************)
(* Basic_2A1: includes: cpr_inv_appl1_simple *)
-lemma cpm_inv_appl1_simple: ∀n,h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[n, h] U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 &
+lemma cpm_inv_appl1_simple: ∀n,h,G,L,V1,T1,U. ⦃G,L⦄ ⊢ ⓐV1.T1 ➡[n,h] U → 𝐒⦃T1⦄ →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 &
U = ⓐV2.T2.
#n #h #G #L #V1 #T1 #U * #c #Hc #H #HT1 elim (cpg_inv_appl1_simple … H HT1) -H -HT1
#cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct elim (isrt_inv_max … Hc) -Hc
(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
-(* Inversion lemmas with degree-based equivalence for terms *****************)
+(* Inversion lemmas with sort-irrelevant equivalence for terms **************)
-lemma cpm_tdeq_inv_lref_sn (n) (h) (o) (G) (L) (i):
- ∀X. ⦃G,L⦄ ⊢ #i ➡[n,h] X → #i ≛[h,o] X →
+lemma cpm_tdeq_inv_lref_sn (n) (h) (G) (L) (i):
+ ∀X. ⦃G,L⦄ ⊢ #i ➡[n,h] X → #i ≛ X →
∧∧ X = #i & n = 0.
-#n #h #o #G #L #i #X #H1 #H2
+#n #h #G #L #i #X #H1 #H2
lapply (tdeq_inv_lref1 … H2) -H2 #H destruct
elim (cpm_inv_lref1_drops … H1) -H1 // * [| #m ]
#K #V1 #V2 #_ #_ #H -V1
elim (lifts_inv_lref2_uni_lt … H) -H //
qed-.
-lemma cpm_tdeq_inv_atom_sn (n) (h) (o) (I) (G) (L):
- ∀X. ⦃G,L⦄ ⊢ ⓪{I} ➡[n,h] X → ⓪{I} ≛[h,o] X →
+lemma cpm_tdeq_inv_atom_sn (n) (h) (I) (G) (L):
+ ∀X. ⦃G,L⦄ ⊢ ⓪{I} ➡[n,h] X → ⓪{I} ≛ X →
∨∨ ∧∧ X = ⓪{I} & n = 0
- | ∃∃s. X = ⋆(next h s) & I = Sort s & n = 1 & deg h o s 0.
-#n #h #o * #s #G #L #X #H1 #H2
+ | ∃∃s. X = ⋆(⫯[h]s) & I = Sort s & n = 1.
+#n #h * #s #G #L #X #H1 #H2
[ elim (cpm_inv_sort1 … H1) -H1
- cases n -n [| #n ] #H #Hn destruct
+ cases n -n [| #n ] #H #Hn destruct -H2
[ /3 width=1 by or_introl, conj/
- | elim (tdeq_inv_sort1 … H2) -H2 #x #d #Hs
- <(le_n_O_to_eq n) [| /2 width=3 by le_S_S_to_le/ ] -n #Hx #H destruct
- lapply (deg_next … Hs) #H
- lapply (deg_mono … H Hx) -H -Hx #Hd
- lapply (pred_inv_fix_sn … Hd) -Hd #H destruct
- /3 width=4 by refl, ex4_intro, or_intror/
+ | <(le_n_O_to_eq n) [| /2 width=3 by le_S_S_to_le/ ] -n
+ /3 width=3 by ex3_intro, or_intror/
]
| elim (cpm_tdeq_inv_lref_sn … H1 H2) -H1 -H2 /3 width=1 by or_introl, conj/
| elim (cpm_inv_gref1 … H1) -H1 -H2 /3 width=1 by or_introl, conj/
(* Note: cpr_flat: does not hold in basic_1 *)
(* Basic_1: includes: pr2_thin_dx *)
lemma cpr_flat: ∀h,I,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡[h] ⓕ{I}V2.T2.
+ ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[h] T2 →
+ ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ➡[h] ⓕ{I}V2.T2.
#h * /2 width=1 by cpm_cast, cpm_appl/
qed.
(* Basic_1: was: pr2_head_1 *)
-lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
- ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h] ②{I}V2.T.
+lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 →
+ ∀T. ⦃G,L⦄ ⊢ ②{I}V1.T ➡[h] ②{I}V2.T.
#h * /2 width=1 by cpm_bind, cpr_flat/
qed.
(* Basic inversion properties ***********************************************)
-lemma cpr_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h] T2 →
+lemma cpr_inv_atom1: ∀h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ➡[h] T2 →
∨∨ T2 = ⓪{J}
- | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1 & J = LRef 0
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
#h #J #G #L #T2 #H elim (cpm_inv_atom1 … H) -H *
[2,4:|*: /3 width=8 by or3_intro0, or3_intro1, or3_intro2, ex4_4_intro, ex4_3_intro/ ]
qed-.
(* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
-lemma cpr_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[h] T2 → T2 = ⋆s.
+lemma cpr_inv_sort1: ∀h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ➡[h] T2 → T2 = ⋆s.
#h #G #L #T2 #s #H elim (cpm_inv_sort1 … H) -H //
qed-.
-lemma cpr_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[h] T2 →
+lemma cpr_inv_zero1: ∀h,G,L,T2. ⦃G,L⦄ ⊢ #0 ➡[h] T2 →
∨∨ T2 = #0
- | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓓV1.
#h #G #L #T2 #H elim (cpm_inv_zero1 … H) -H *
/3 width=6 by ex3_3_intro, or_introl, or_intror/
#n #K #V1 #V2 #_ #_ #_ #H destruct
qed-.
-lemma cpr_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ➡[h] T2 →
+lemma cpr_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ➡[h] T2 →
∨∨ T2 = #(↑i)
- | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
+ | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
#h #G #L #T2 #i #H elim (cpm_inv_lref1 … H) -H *
/3 width=6 by ex3_3_intro, or_introl, or_intror/
qed-.
-lemma cpr_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[h] T2 → T2 = §l.
+lemma cpr_inv_gref1: ∀h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ➡[h] T2 → T2 = §l.
#h #G #L #T2 #l #H elim (cpm_inv_gref1 … H) -H //
qed-.
(* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
-lemma cpr_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1.U1 ➡[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
+lemma cpr_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝ V1.U1 ➡[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ U1 ➡[h] T2 &
U2 = ⓝV2.T2
- | ⦃G, L⦄ ⊢ U1 ➡[h] U2.
+ | ⦃G,L⦄ ⊢ U1 ➡[h] U2.
#h #G #L #V1 #U1 #U2 #H elim (cpm_inv_cast1 … H) -H
/2 width=1 by or_introl, or_intror/ * #n #_ #H destruct
qed-.
-lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
+lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ U1 ➡[h] T2 &
U2 = ⓕ{I}V2.T2
- | (⦃G, L⦄ ⊢ U1 ➡[h] U2 ∧ I = Cast)
- | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 & U1 = ⓛ{p}W1.T1 &
+ | (⦃G,L⦄ ⊢ U1 ➡[h] U2 ∧ I = Cast)
+ | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ W1 ➡[h] W2 &
+ ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[h] T2 & U1 = ⓛ{p}W1.T1 &
U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
- | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 &
- ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 &
+ | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≘ V2 &
+ ⦃G,L⦄ ⊢ W1 ➡[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[h] T2 &
U1 = ⓓ{p}W1.T1 &
U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
#h * #G #L #V1 #U1 #U2 #H
lemma cpr_ind (h): ∀Q:relation4 genv lenv term term.
(∀I,G,L. Q G L (⓪{I}) (⓪{I})) →
- (∀G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 → Q G K V1 V2 →
+ (∀G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[h] V2 → Q G K V1 V2 →
⬆*[1] V2 ≘ W2 → Q G (K.ⓓV1) (#0) W2
- ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[h] T → Q G K (#i) T →
+ ) → (∀I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[h] T → Q G K (#i) T →
⬆*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
- ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h] T2 →
+ ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[h] T2 →
Q G L V1 V2 → Q G (L.ⓑ{I}V1) T1 T2 → Q G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
- ) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2 →
+ ) → (∀I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[h] T2 →
Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ➡[h] T2 →
+ ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[h] T2 →
Q G L T T2 → Q G L (+ⓓV.T1) T2
- ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2 →
+ ) → (∀G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2 →
Q G L (ⓝV.T1) T2
- ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 →
+ ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[h] T2 →
Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
Q G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
- ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 →
+ ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[h] T2 →
Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
⬆*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
) →
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[h] T2 → Q G L T1 T2.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T1 #T2
@(insert_eq_0 … 0) #n #H
@(cpm_ind … H) -G -L -T1 -T2 -n [2,4,11:|*: /3 width=4 by/ ]
(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: includes: cpr_inv_atom1 *)
-lemma cpr_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡[h] T2 →
+lemma cpr_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G,L⦄ ⊢ ⓪{I} ➡[h] T2 →
∨∨ T2 = ⓪{I}
- | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[h] V2 &
+ | ∃∃K,V,V2,i. ⬇*[i] L ≘ K.ⓓV & ⦃G,K⦄ ⊢ V ➡[h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i.
#h #I #G #L #T2 #H elim (cpm_inv_atom1_drops … H) -H *
[ /2 width=1 by or_introl/
(* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
(* Basic_2A1: includes: cpr_inv_lref1 *)
-lemma cpr_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h] T2 →
+lemma cpr_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ➡[h] T2 →
∨∨ T2 = #i
- | ∃∃K,V,V2. ⬇*[i] L ≘ K. ⓓV & ⦃G, K⦄ ⊢ V ➡[h] V2 &
+ | ∃∃K,V,V2. ⬇*[i] L ≘ K. ⓓV & ⦃G,K⦄ ⊢ V ➡[h] V2 &
⬆*[↑i] V2 ≘ T2.
#h #G #L #T2 #i #H elim (cpm_inv_lref1_drops … H) -H *
[ /2 width=1 by or_introl/
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/lifts_basic.ma".
+include "basic_2/rt_transition/cpm_drops.ma".
+include "basic_2/rt_transition/cpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL R-TRANSITION FOR TERMS ************************)
+
+(* Properties with basic relocation *****************************************)
+
+lemma cpr_subst (h) (G) (L) (U1) (i):
+ ∀K,V. ⬇*[i] L ≘ K.ⓓV →
+ ∃∃U2,T2. ⦃G,L⦄ ⊢ U1 ➡[h] U2 & ⬆[i,1] T2 ≘ U2.
+#h #G #L #U1 @(fqup_wf_ind_eq (Ⓣ) … G L U1) -G -L -U1
+#G0 #L0 #U0 #IH #G #L * *
+[ #s #HG #HL #HT #i #K #V #_ destruct -IH
+ /2 width=4 by lifts_sort, ex2_2_intro/
+| #j #HG #HL #HT #i #K #V #HLK destruct -IH
+ elim (lt_or_eq_or_gt i j) #Hij
+ [ /3 width=4 by lifts_lref_ge_minus, cpr_refl, ex2_2_intro/
+ | elim (lifts_total V (𝐔❴↑i❵)) #U2 #HU2
+ elim (lifts_split_trans … HU2 (𝐔❴i❵) (𝐁❴i,1❵)) [2: @(after_basic_rc i 0) ]
+ /3 width=7 by cpm_delta_drops, ex2_2_intro/
+ | /3 width=4 by lifts_lref_lt, cpr_refl, ex2_2_intro/
+ ]
+| #l #HG #HL #HT #i #K #V #_ destruct -IH
+ /2 width=4 by lifts_gref, ex2_2_intro/
+| #p #J #W1 #U1 #HG #HL #HT #i #K #V #HLK destruct
+ elim (IH G L W1 … HLK) [| // ] #W2 #V2 #HW12 #HVW2
+ elim (IH G (L.ⓑ{J}W1) U1 … (↑i)) [|*: /3 width=4 by drops_drop/ ] #U2 #T2 #HU12 #HTU2
+ /3 width=9 by cpm_bind, lifts_bind, ex2_2_intro/
+| #J #W1 #U1 #HG #HL #HT #i #K #V #HLK destruct
+ elim (IH G L W1 … HLK) [| // ] #W2 #V2 #HW12 #HVW2
+ elim (IH G L U1 … HLK) [| // ] #U2 #T2 #HU12 #HTU2
+ /3 width=8 by cpr_flat, lifts_flat, ex2_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/lifts_tdeq.ma".
+include "basic_2/rt_transition/cpr_drops_basic.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL R-TRANSITION FOR TERMS ************************)
+
+(* Properties with context-free sort-irrelevant equivalence *****************)
+
+lemma cpr_abbr_pos_tdneq (h) (G) (L) (V) (T1):
+ ∃∃T2. ⦃G,L⦄ ⊢ +ⓓV.T1 ➡[h] T2 & (+ⓓV.T1 ≛ T2 → ⊥).
+#h #G #L #V #U1
+elim (cpr_subst h G (L.ⓓV) U1 … 0) [|*: /2 width=4 by drops_refl/ ] #U2 #T2 #HU12 #HTU2
+elim (tdeq_dec U1 U2) [ -HU12 #HU12 | -HTU2 #HnU12 ]
+[ elim (tdeq_inv_lifts_dx … HU12 … HTU2) -U2 #T1 #HTU1 #_ -T2
+ /3 width=9 by cpm_zeta, tdeq_lifts_inv_pair_sn, ex2_intro/
+| @(ex2_intro … (+ⓓV.U2)) [ /2 width=1 by cpm_bind/ ] -HU12 #H
+ elim (tdeq_inv_pair … H) -H #_ #_ /2 width=1 by/
+]
+qed-.
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
definition cpx (h): relation4 genv lenv term term ≝
- λG,L,T1,T2. ∃c. ⦃G, L⦄ ⊢ T1 ⬈[eq_f, c, h] T2.
+ λG,L,T1,T2. ∃c. ⦃G,L⦄ ⊢ T1 ⬈[eq_f,c,h] T2.
interpretation
"unbound context-sensitive parallel rt-transition (term)"
(* Basic properties *********************************************************)
(* Basic_2A1: was: cpx_st *)
-lemma cpx_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] ⋆(next h s).
+lemma cpx_ess: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⋆s ⬈[h] ⋆(⫯[h]s).
/2 width=2 by cpg_ess, ex_intro/ qed.
-lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 →
- ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2.
+lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 →
+ ⬆*[1] V2 ≘ W2 → ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2.
#h * #G #K #V1 #V2 #W2 *
/3 width=4 by cpg_delta, cpg_ell, ex_intro/
qed.
-lemma cpx_lref: ∀h,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T →
- ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] U.
+lemma cpx_lref: ∀h,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ⬈[h] T →
+ ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] U.
#h #I #G #K #T #U #i *
/3 width=4 by cpg_lref, ex_intro/
qed.
lemma cpx_bind: ∀h,p,I,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
- ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
#h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
/3 width=2 by cpg_bind, ex_intro/
qed.
lemma cpx_flat: ∀h,I,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
#h * #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
/3 width=5 by cpg_appl, cpg_cast, ex_intro/
qed.
lemma cpx_zeta (h) (G) (L):
- ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬈[h] T2 →
- ∀V. ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
+ ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G,L⦄ ⊢ T ⬈[h] T2 →
+ ∀V. ⦃G,L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
#h #G #L #T1 #T #HT1 #T2 *
/3 width=4 by cpg_zeta, ex_intro/
qed.
-lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ⬈[h] T2.
+lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → ⦃G,L⦄ ⊢ ⓝV.T1 ⬈[h] T2.
#h #G #L #V #T1 #T2 *
/3 width=2 by cpg_eps, ex_intro/
qed.
(* Basic_2A1: was: cpx_ct *)
-lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ⬈[h] V2.
+lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ ⓝV1.T ⬈[h] V2.
#h #G #L #V1 #V2 #T *
/3 width=2 by cpg_ee, ex_intro/
qed.
lemma cpx_beta: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈[h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈[h] ⓓ{p}ⓝW2.V2.T2.
#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 *
/3 width=2 by cpg_beta, ex_intro/
qed.
lemma cpx_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 →
- ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 →
+ ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2.
#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 *
/3 width=4 by cpg_theta, ex_intro/
qed.
(* Advanced properties ******************************************************)
-lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈[h] ②{I}V2.T.
+lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀T. ⦃G,L⦄ ⊢ ②{I}V1.T ⬈[h] ②{I}V2.T.
#h * /2 width=2 by cpx_flat, cpx_bind/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[h] T2 →
+lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ⬈[h] T2 →
∨∨ T2 = ⓪{J}
- | ∃∃s. T2 = ⋆(next h s) & J = Sort s
- | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s
+ | ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓑ{I}V1 & J = LRef 0
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 &
L = K.ⓘ{I} & J = LRef (↑i).
#h #J #G #L #T2 * #c #H elim (cpg_inv_atom1 … H) -H *
/4 width=8 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_4_intro, ex2_intro, ex_intro/
qed-.
-lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] T2 →
- ∨∨ T2 = ⋆s | T2 = ⋆(next h s).
+lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ⬈[h] T2 →
+ ∨∨ T2 = ⋆s | T2 = ⋆(⫯[h]s).
#h #G #L #T2 #s * #c #H elim (cpg_inv_sort1 … H) -H *
/2 width=1 by or_introl, or_intror/
qed-.
-lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[h] T2 →
+lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G,L⦄ ⊢ #0 ⬈[h] T2 →
∨∨ T2 = #0
- | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓑ{I}V1.
#h #G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H *
/4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈[h] T2 →
+lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈[h] T2 →
∨∨ T2 = #(↑i)
- | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
+ | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
#h #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
/4 width=6 by ex3_3_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ⬈[h] T2 → T2 = §l.
+lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ⬈[h] T2 → T2 = §l.
#h #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
qed-.
-lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 &
+lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 &
U2 = ⓑ{p,I}V2.T2
- | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[h] U2 &
+ | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[h] U2 &
p = true & I = Abbr.
#h #p #I #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H *
/4 width=5 by ex4_intro, ex3_2_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 &
+lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 &
U2 = ⓓ{p}V2.T2
- | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[h] U2 & p = true.
+ | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[h] U2 & p = true.
#h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H *
/4 width=5 by ex3_2_intro, ex3_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈[h] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈[h] T2 &
+lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈[h] U2 →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈[h] T2 &
U2 = ⓛ{p}V2.T2.
#h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abst1 … H) -H
/3 width=5 by ex3_2_intro, ex_intro/
qed-.
-lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
+lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐ V1.U1 ⬈[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[h] T2 &
U2 = ⓐV2.T2
- | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
+ | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[h] W2 &
+ ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2
- | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≘ V2 &
- ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
+ | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≘ V2 &
+ ⦃G,L⦄ ⊢ W1 ⬈[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
#h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
/4 width=13 by or3_intro0, or3_intro1, or3_intro2, ex6_7_intro, ex5_6_intro, ex3_2_intro, ex_intro/
qed-.
-lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
+lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ⬈[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[h] T2 &
U2 = ⓝV2.T2
- | ⦃G, L⦄ ⊢ U1 ⬈[h] U2
- | ⦃G, L⦄ ⊢ V1 ⬈[h] U2.
+ | ⦃G,L⦄ ⊢ U1 ⬈[h] U2
+ | ⦃G,L⦄ ⊢ V1 ⬈[h] U2.
#h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_cast1 … H) -H *
/4 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex_intro/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] T2 →
+lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] T2 →
∨∨ T2 = #0
- | ∃∃V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2.
+ | ∃∃V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2.
#h #I #G #L #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H *
/4 width=3 by ex2_intro, ex_intro, or_intror, or_introl/
qed-.
-lemma cpx_inv_lref1_bind: ∀h,I,G,K,T2,i. ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] T2 →
+lemma cpx_inv_lref1_bind: ∀h,I,G,K,T2,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] T2 →
∨∨ T2 = #(↑i)
- | ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2.
+ | ∃∃T. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2.
#h #I #G #L #T2 #i * #c #H elim (cpg_inv_lref1_bind … H) -H *
/4 width=3 by ex2_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
+lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[h] T2 &
U2 = ⓕ{I}V2.T2
- | (⦃G, L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
- | (⦃G, L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
- | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
+ | (⦃G,L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
+ | (⦃G,L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
+ | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[h] W2 &
+ ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
U1 = ⓛ{p}W1.T1 &
U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
- | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≘ V2 &
- ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
+ | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≘ V2 &
+ ⦃G,L⦄ ⊢ W1 ⬈[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
U1 = ⓓ{p}W1.T1 &
U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
#h * #G #L #V1 #U1 #U2 #H
(* Basic forward lemmas *****************************************************)
-lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p.
- ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2 &
+lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p.
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2 &
T = -ⓑ{I}V2.T2.
#h #I #G #L #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
/3 width=4 by ex2_2_intro, ex_intro/
lemma cpx_ind: ∀h. ∀Q:relation4 genv lenv term term.
(∀I,G,L. Q G L (⓪{I}) (⓪{I})) →
- (∀G,L,s. Q G L (⋆s) (⋆(next h s))) →
- (∀I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 → Q G K V1 V2 →
+ (∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
+ (∀I,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 → Q G K V1 V2 →
⬆*[1] V2 ≘ W2 → Q G (K.ⓑ{I}V1) (#0) W2
- ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T → Q G K (#i) T →
+ ) → (∀I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ⬈[h] T → Q G K (#i) T →
⬆*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
- ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
+ ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
Q G L V1 V2 → Q G (L.ⓑ{I}V1) T1 T2 → Q G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
- ) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
+ ) → (∀I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ⬈[h] T2 → Q G L T T2 →
+ ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ⬈[h] T2 → Q G L T T2 →
Q G L (+ⓓV.T1) T2
- ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2 →
+ ) → (∀G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2 →
Q G L (ⓝV.T1) T2
- ) → (∀G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
+ ) → (∀G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
Q G L (ⓝV1.T) V2
- ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
+ ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
Q G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
- ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
+ ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
⬆*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
) →
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2.
+ ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2.
#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2
* #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/
qed-.
(* Basic_2A1: was: cpx_delta *)
lemma cpx_delta_drops: ∀h,I,G,L,K,V,V2,W2,i.
- ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⬈[h] V2 →
- ⬆*[↑i] V2 ≘ W2 → ⦃G, L⦄ ⊢ #i ⬈[h] W2.
+ ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ⬈[h] V2 →
+ ⬆*[↑i] V2 ≘ W2 → ⦃G,L⦄ ⊢ #i ⬈[h] W2.
#h * #G #L #K #V #V2 #W2 #i #HLK *
/3 width=7 by cpg_ell_drops, cpg_delta_drops, ex_intro/
qed.
(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: was: cpx_inv_atom1 *)
-lemma cpx_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ⬈[h] T2 →
+lemma cpx_inv_atom1_drops: ∀h,I,G,L,T2. ⦃G,L⦄ ⊢ ⓪{I} ⬈[h] T2 →
∨∨ T2 = ⓪{I}
- | ∃∃s. T2 = ⋆(next h s) & I = Sort s
- | ∃∃J,K,V,V2,i. ⬇*[i] L ≘ K.ⓑ{J}V & ⦃G, K⦄ ⊢ V ⬈[h] V2 &
+ | ∃∃s. T2 = ⋆(⫯[h]s) & I = Sort s
+ | ∃∃J,K,V,V2,i. ⬇*[i] L ≘ K.ⓑ{J}V & ⦃G,K⦄ ⊢ V ⬈[h] V2 &
⬆*[↑i] V2 ≘ T2 & I = LRef i.
#h #I #G #L #T2 * #c #H elim (cpg_inv_atom1_drops … H) -H *
/4 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex2_intro, ex_intro/
qed-.
(* Basic_2A1: was: cpx_inv_lref1 *)
-lemma cpx_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #i ⬈[h] T2 →
+lemma cpx_inv_lref1_drops: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #i ⬈[h] T2 →
T2 = #i ∨
- ∃∃J,K,V,V2. ⬇*[i] L ≘ K. ⓑ{J}V & ⦃G, K⦄ ⊢ V ⬈[h] V2 &
+ ∃∃J,K,V,V2. ⬇*[i] L ≘ K. ⓑ{J}V & ⦃G,K⦄ ⊢ V ⬈[h] V2 &
⬆*[↑i] V2 ≘ T2.
#h #G #L #T1 #i * #c #H elim (cpg_inv_lref1_drops … H) -H *
/4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/lifts_basic.ma".
+include "basic_2/rt_transition/cpx_drops.ma".
+
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+
+(* Properties with basic relocation *****************************************)
+
+lemma cpx_subst (h) (G) (L) (U1) (i):
+ ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V →
+ ∃∃U2,T2. ⦃G,L⦄ ⊢ U1 ⬈[h] U2 & ⬆[i,1] T2 ≘ U2.
+#h #G #L #U1 @(fqup_wf_ind_eq (Ⓣ) … G L U1) -G -L -U1
+#G0 #L0 #U0 #IH #G #L * *
+[ #s #HG #HL #HT #i #I #K #V #_ destruct -IH
+ /2 width=4 by lifts_sort, ex2_2_intro/
+| #j #HG #HL #HT #i #I #K #V #HLK destruct -IH
+ elim (lt_or_eq_or_gt i j) #Hij
+ [ /3 width=4 by lifts_lref_ge_minus, cpx_refl, ex2_2_intro/
+ | elim (lifts_total V (𝐔❴↑i❵)) #U2 #HU2
+ elim (lifts_split_trans … HU2 (𝐔❴i❵) (𝐁❴i,1❵)) [2: @(after_basic_rc i 0) ]
+ /3 width=7 by cpx_delta_drops, ex2_2_intro/
+ | /3 width=4 by lifts_lref_lt, cpx_refl, ex2_2_intro/
+ ]
+| #l #HG #HL #HT #i #I #K #V #_ destruct -IH
+ /2 width=4 by lifts_gref, ex2_2_intro/
+| #p #J #W1 #U1 #HG #HL #HT #i #I #K #V #HLK destruct
+ elim (IH G L W1 … HLK) [| // ] #W2 #V2 #HW12 #HVW2
+ elim (IH G (L.ⓑ{J}W1) U1 … (↑i)) [|*: /3 width=4 by drops_drop/ ] #U2 #T2 #HU12 #HTU2
+ /3 width=9 by cpx_bind, lifts_bind, ex2_2_intro/
+| #J #W1 #U1 #HG #HL #HT #i #I #K #V #HLK destruct
+ elim (IH G L W1 … HLK) [| // ] #W2 #V2 #HW12 #HVW2
+ elim (IH G L U1 … HLK) [| // ] #U2 #T2 #HU12 #HTU2
+ /3 width=8 by cpx_flat, lifts_flat, ex2_2_intro/
+]
+qed-.
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
-(* Properties with degree-based equivalence for closures ********************)
+(* Properties with sort-irrelevant equivalence for closures *****************)
-lemma fdeq_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T⦄ →
- ∀T2. ⦃G2, L2⦄ ⊢ T ⬈[h] T2 →
- ∃∃T0. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T0 & ⦃G1, L1, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
+lemma fdeq_cpx_trans: ∀h,G1,G2,L1,L2,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T⦄ →
+ ∀T2. ⦃G2,L2⦄ ⊢ T ⬈[h] T2 →
+ ∃∃T0. ⦃G1,L1⦄ ⊢ T1 ⬈[h] T0 & ⦃G1,L1,T0⦄ ≛ ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
elim (fdeq_inv_gen_dx … H) -H #H #HL12 #HT1 destruct
elim (rdeq_cpx_trans … HL12 … HT2) #T0 #HT0 #HT02
lapply (cpx_rdeq_conf_dx … HT2 … HL12) -HL12 #HL12
(* Properties on supclosure *************************************************)
-lemma fqu_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+lemma fqu_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⬂[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=3 by cpx_pair_sn, cpx_bind, cpx_flat, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, ex2_intro/
[ #I #G #L2 #V2 #X2 #HVX2
]
qed-.
-lemma fquq_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+lemma fquq_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⬂⸮[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
[ #HT12 #U2 #HTU2 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma fqup_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄.
+lemma fqup_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⬂+[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
[ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
/3 width=3 by fqu_fqup, ex2_intro/
]
qed-.
-lemma fqus_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄.
+lemma fqus_cpx_trans: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1,L1,U1⦄ ⬂*[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H
[ #HT12 #U2 #HTU2 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma fqu_cpx_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma fqu_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⬂[b] ⦃G2,L2,U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #V1 #V2 #HV12 #_ elim (lifts_total V2 𝐔❴1❵)
#U2 #HVU2 @(ex3_intro … U2)
[1,3: /3 width=7 by cpx_delta, fqu_drop/
[1,3: /2 width=4 by fqu_pair_sn, cpx_pair_sn/
| #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
]
-| #p #I #G #L #V #T1 #T2 #HT12 #H0 @(ex3_intro … (ⓑ{p,I}V.T2))
+| #p #I #G #L #V #T1 #Hb #T2 #HT12 #H0 @(ex3_intro … (ⓑ{p,I}V.T2))
[1,3: /2 width=4 by fqu_bind_dx, cpx_bind/
| #H elim (tdeq_inv_pair … H) -H /2 width=1 by/
]
]
qed-.
-lemma fquq_cpx_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
+lemma fquq_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⬂⸮[b] ⦃G2,L2,U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 elim H12 -H12
[ #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fquq, ex3_intro/
| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
]
qed-.
-lemma fqup_cpx_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, U2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+lemma fqup_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⬂+[b] ⦃G2,L2,U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fqup, ex3_intro/
| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
]
qed-.
-lemma fqus_cpx_trans_tdneq: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛[h, o] U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, U2⦄.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
+lemma fqus_cpx_trans_tdneq: ∀h,b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ⬈[h] U2 → (T2 ≛ U2 → ⊥) →
+ ∃∃U1. ⦃G1,L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛ U1 → ⊥ & ⦃G1,L1,U1⦄ ⬂*[b] ⦃G2,L2,U2⦄.
+#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_fqup … H12) -H12
[ #H12 elim (fqup_cpx_trans_tdneq … H12 … HTU2 H) -T2
/3 width=4 by fqup_fqus, ex3_intro/
| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
#h #G #L1 #T1 #T2 * /3 width=4 by lsubr_cpg_trans, ex_intro/
qed-.
-lemma cpx_bind_unit (h) (G): ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀J,T1,T2. ⦃G, L.ⓤ{J}⦄ ⊢ T1 ⬈[h] T2 →
- ∀p,I. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
+lemma cpx_bind_unit (h) (G): ∀L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀J,T1,T2. ⦃G,L.ⓤ{J}⦄ ⊢ T1 ⬈[h] T2 →
+ ∀p,I. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
/4 width=4 by lsubr_cpx_trans, cpx_bind, lsubr_unit/ qed.
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
-(* Properties with degree-based equivalence for local environments **********)
+(* Properties with sort-irrelevant equivalence for local environments *******)
(* Basic_2A1: was just: cpx_lleq_conf_sn *)
-lemma cpx_rdeq_conf_sn: ∀h,o,G. s_r_confluent1 … (cpx h G) (rdeq h o).
+lemma cpx_rdeq_conf_sn: ∀h,G. s_r_confluent1 … (cpx h G) rdeq.
/3 width=6 by cpx_rex_conf/ qed-.
(* Basic_2A1: was just: cpx_lleq_conf_dx *)
-lemma cpx_rdeq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
- ∀L1. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
-/4 width=4 by cpx_rdeq_conf_sn, rdeq_sym/ qed-.
+lemma cpx_rdeq_conf_dx: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L1. L1 ≛[T1] L2 → L1 ≛[T2] L2.
+/4 width=5 by cpx_rdeq_conf_sn, rdeq_sym/ qed-.
qed-.
(*
(* Basic_2A1: was: cpx_lleq_conf *)
-lemma cpx_req_conf: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
- ∀L1. L2 ≘[T1] L1 → ⦃G, L1⦄ ⊢ T1 ⬈[h] T2.
+lemma cpx_req_conf: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L1. L2 ≘[T1] L1 → ⦃G,L1⦄ ⊢ T1 ⬈[h] T2.
/3 width=3 by req_cpx_trans, req_sym/ qed-.
*)
(* Basic_2A1: was: cpx_lleq_conf_sn *)
/2 width=5 by cpx_rex_conf/ qed-.
(*
(* Basic_2A1: was: cpx_lleq_conf_dx *)
-lemma cpx_req_conf_dx: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
+lemma cpx_req_conf_dx: ∀h,G,L2,T1,T2. ⦃G,L2⦄ ⊢ T1 ⬈[h] T2 →
∀L1. L1 ≘[T1] L2 → L1 ≘[T2] L2.
/4 width=6 by cpx_req_conf_sn, req_sym/ qed-.
*)
(* Inversion lemmas with simple terms ***************************************)
-lemma cpx_inv_appl1_simple: ∀h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ⬈[h] U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ T1 ⬈[h] T2 &
+lemma cpx_inv_appl1_simple: ∀h,G,L,V1,T1,U. ⦃G,L⦄ ⊢ ⓐV1.T1 ⬈[h] U → 𝐒⦃T1⦄ →
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ T1 ⬈[h] T2 &
U = ⓐV2.T2.
#h #G #L #V1 #T1 #U * #c #H #HT1 elim (cpg_inv_appl1_simple … H) -H
/3 width=5 by ex3_2_intro, ex_intro/
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predsubtyproper_8.ma".
+include "basic_2/notation/relations/predsubtyproper_7.ma".
include "static_2/s_transition/fqu.ma".
include "static_2/static/rdeq.ma".
include "basic_2/rt_transition/lpr_lpx.ma".
(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
-inductive fpb (h) (o) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpb_fqu: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → fpb h o G1 L1 T1 G2 L2 T2
-| fpb_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛[h, o] T2 → ⊥) → fpb h o G1 L1 T1 G1 L1 T2
-| fpb_lpx: ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[h, o, T1] L2 → ⊥) → fpb h o G1 L1 T1 G1 L2 T1
+inductive fpb (h) (G1) (L1) (T1): relation3 genv lenv term ≝
+| fpb_fqu: ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⬂ ⦃G2,L2,T2⦄ → fpb h G1 L1 T1 G2 L2 T2
+| fpb_cpx: ∀T2. ⦃G1,L1⦄ ⊢ T1 ⬈[h] T2 → (T1 ≛ T2 → ⊥) → fpb h G1 L1 T1 G1 L1 T2
+| fpb_lpx: ∀L2. ⦃G1,L1⦄ ⊢ ⬈[h] L2 → (L1 ≛[T1] L2 → ⊥) → fpb h G1 L1 T1 G1 L2 T1
.
interpretation
"proper parallel rst-transition (closure)"
- 'PRedSubTyProper h o G1 L1 T1 G2 L2 T2 = (fpb h o G1 L1 T1 G2 L2 T2).
+ 'PRedSubTyProper h G1 L1 T1 G2 L2 T2 = (fpb h G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
(* Basic_2A1: includes: cpr_fpb *)
-lemma cpm_fpb (n) (h) (o) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → (T1 ≛[h, o] T2 → ⊥) →
- ⦃G, L, T1⦄ ≻[h, o] ⦃G, L, T2⦄.
+lemma cpm_fpb (n) (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → (T1 ≛ T2 → ⊥) →
+ ⦃G,L,T1⦄ ≻[h] ⦃G,L,T2⦄.
/3 width=2 by fpb_cpx, cpm_fwd_cpx/ qed.
-lemma lpr_fpb (h) (o) (G) (T): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) →
- ⦃G, L1, T⦄ ≻[h, o] ⦃G, L2, T⦄.
+lemma lpr_fpb (h) (G) (T): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → (L1 ≛[T] L2 → ⊥) →
+ ⦃G,L1,T⦄ ≻[h] ⦃G,L2,T⦄.
/3 width=1 by fpb_lpx, lpr_fwd_lpx/ qed.
(* Properties with degree-based equivalence for closures ********************)
(* Basic_2A1: uses: fleq_fpb_trans *)
-lemma fdeq_fpb_trans: ∀h,o,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≛[h, o] ⦃F2, K2, T2⦄ →
- ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, o] ⦃G2, L2, U2⦄ →
- ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, o] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≛[h, o] ⦃G2, L2, U2⦄.
-#h #o #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
+lemma fdeq_fpb_trans: ∀h,F1,F2,K1,K2,T1,T2. ⦃F1,K1,T1⦄ ≛ ⦃F2,K2,T2⦄ →
+ ∀G2,L2,U2. ⦃F2,K2,T2⦄ ≻[h] ⦃G2,L2,U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1,K1,T1⦄ ≻[h] ⦃G1,L1,U1⦄ & ⦃G1,L1,U1⦄ ≛ ⦃G2,L2,U2⦄.
+#h #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
#K2 #T2 #HK12 #HT12 #G2 #L2 #U2 #H12
elim (tdeq_fpb_trans … HT12 … H12) -T2 #K0 #T0 #H #HT0 #HK0
elim (rdeq_fpb_trans … HK12 … H) -K2 #L0 #U0 #H #HUT0 #HLK0
(* Inversion lemmas with degree-based equivalence for closures **************)
(* Basic_2A1: uses: fpb_inv_fleq *)
-lemma fpb_inv_fdeq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+lemma fpb_inv_fdeq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⊥.
+#h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
[ #G2 #L2 #T2 #H12 #H elim (fdeq_inv_gen_sn … H) -H
/3 width=11 by rdeq_fwd_length, fqu_inv_tdeq/
| #T2 #_ #HnT #H elim (fdeq_inv_gen_sn … H) -H /2 width=1 by/
(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
-(* Properties with degree-based equivalence for local environments **********)
+(* Properties with sort-irrelevant equivalence for local environments *******)
-lemma tdeq_fpb_trans: ∀h,o,U2,U1. U2 ≛[h, o] U1 →
- ∀G1,G2,L1,L2,T1. ⦃G1, L1, U1⦄ ≻[h, o] ⦃G2, L2, T1⦄ →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ≻[h, o] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
-#h #o #U2 #U1 #HU21 #G1 #G2 #L1 #L2 #T1 * -G2 -L2 -T1
+lemma tdeq_fpb_trans: ∀h,U2,U1. U2 ≛ U1 →
+ ∀G1,G2,L1,L2,T1. ⦃G1,L1,U1⦄ ≻[h] ⦃G2,L2,T1⦄ →
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ≻[h] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+#h #U2 #U1 #HU21 #G1 #G2 #L1 #L2 #T1 * -G2 -L2 -T1
[ #G2 #L2 #T1 #H
elim (tdeq_fqu_trans … H … HU21) -H
/3 width=5 by fpb_fqu, ex3_2_intro/
qed-.
(* Basic_2A1: was just: lleq_fpb_trans *)
-lemma rdeq_fpb_trans: ∀h,o,F,K1,K2,T. K1 ≛[h, o, T] K2 →
- ∀G,L2,U. ⦃F, K2, T⦄ ≻[h, o] ⦃G, L2, U⦄ →
- ∃∃L1,U0. ⦃F, K1, T⦄ ≻[h, o] ⦃G, L1, U0⦄ & U0 ≛[h, o] U & L1 ≛[h, o, U] L2.
-#h #o #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
+lemma rdeq_fpb_trans: ∀h,F,K1,K2,T. K1 ≛[T] K2 →
+ ∀G,L2,U. ⦃F,K2,T⦄ ≻[h] ⦃G,L2,U⦄ →
+ ∃∃L1,U0. ⦃F,K1,T⦄ ≻[h] ⦃G,L1,U0⦄ & U0 ≛ U & L1 ≛[U] L2.
+#h #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
[ #G #L2 #U #H2 elim (rdeq_fqu_trans … H2 … HT) -K2
/3 width=5 by fpb_fqu, ex3_2_intro/
| #U #HTU #HnTU elim (rdeq_cpx_trans … HT … HTU) -HTU
- /5 width=10 by fpb_cpx, cpx_rdeq_conf_sn, tdeq_trans, tdeq_rdeq_conf, ex3_2_intro/
+ /5 width=11 by fpb_cpx, cpx_rdeq_conf_sn, tdeq_trans, tdeq_rdeq_conf, ex3_2_intro/ (**) (* time: 36s on dev *)
| #L2 #HKL2 #HnKL2 elim (rdeq_lpx_trans … HKL2 … HT) -HKL2
/6 width=5 by fpb_lpx, (* 2x *) rdeq_canc_sn, ex3_2_intro/
]
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predsubty_8.ma".
+include "basic_2/notation/relations/predsubty_7.ma".
include "static_2/static/fdeq.ma".
include "static_2/s_transition/fquq.ma".
include "basic_2/rt_transition/lpr_lpx.ma".
(* PARALLEL RST-TRANSITION FOR CLOSURES *************************************)
(* Basic_2A1: includes: fleq_fpbq fpbq_lleq *)
-inductive fpbq (h) (o) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpbq_fquq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
-| fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → fpbq h o G1 L1 T1 G1 L1 T2
-| fpbq_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h] L2 → fpbq h o G1 L1 T1 G1 L2 T1
-| fpbq_fdeq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
+inductive fpbq (h) (G1) (L1) (T1): relation3 genv lenv term ≝
+| fpbq_fquq: ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⬂⸮ ⦃G2,L2,T2⦄ → fpbq h G1 L1 T1 G2 L2 T2
+| fpbq_cpx : ∀T2. ⦃G1,L1⦄ ⊢ T1 ⬈[h] T2 → fpbq h G1 L1 T1 G1 L1 T2
+| fpbq_lpx : ∀L2. ⦃G1,L1⦄ ⊢ ⬈[h] L2 → fpbq h G1 L1 T1 G1 L2 T1
+| fpbq_fdeq: ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → fpbq h G1 L1 T1 G2 L2 T2
.
interpretation
"parallel rst-transition (closure)"
- 'PRedSubTy h o G1 L1 T1 G2 L2 T2 = (fpbq h o G1 L1 T1 G2 L2 T2).
+ 'PRedSubTy h G1 L1 T1 G2 L2 T2 = (fpbq h G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma fpbq_refl (h) (o): tri_reflexive … (fpbq h o).
+lemma fpbq_refl (h): tri_reflexive … (fpbq h).
/2 width=1 by fpbq_cpx/ qed.
(* Basic_2A1: includes: cpr_fpbq *)
-lemma cpm_fpbq (n) (h) (o) (G) (L): ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L, T1⦄ ≽[h, o] ⦃G, L, T2⦄.
+lemma cpm_fpbq (n) (h) (G) (L): ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L,T1⦄ ≽[h] ⦃G,L,T2⦄.
/3 width=2 by fpbq_cpx, cpm_fwd_cpx/ qed.
-lemma lpr_fpbq (h) (o) (G) (T): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1, T⦄ ≽[h, o] ⦃G, L2, T⦄.
+lemma lpr_fpbq (h) (G) (T): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1,T⦄ ≽[h] ⦃G,L2,T⦄.
/3 width=1 by fpbq_lpx, lpr_fwd_lpx/ qed.
(* Basic_2A1: removed theorems 2:
(* Properties with atomic arity assignment for terms ************************)
-lemma fpbq_aaa_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+lemma fpbq_aaa_conf: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
+#h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
/3 width=8 by lpx_aaa_conf, cpx_aaa_conf, aaa_fdeq_conf, aaa_fquq_conf, ex_intro/
qed-.
(* Properties with proper parallel rst-transition for closures **************)
-lemma fpb_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+lemma fpb_fpbq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
/3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lpx, fqu_fquq/
qed.
(* Basic_2A1: fpb_fpbq_alt *)
-lemma fpb_fpbq_ffdneq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
- ∧∧ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ & (⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥).
+lemma fpb_fpbq_ffdneq: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ →
+ ∧∧ ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ & (⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⊥).
/3 width=10 by fpb_fpbq, fpb_inv_fdeq, conj/ qed-.
(* Inversrion lemmas with proper parallel rst-transition for closures *******)
(* Basic_2A1: uses: fpbq_ind_alt *)
-lemma fpbq_inv_fpb: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
- ∨∨ ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄
- | ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+lemma fpbq_inv_fpb: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ →
+ ∨∨ ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄
+ | ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
[ #G2 #L2 #T2 * [2: * #H1 #H2 #H3 destruct ]
/3 width=1 by fpb_fqu, fdeq_intro_sn, or_intror, or_introl/
-| #T2 #H elim (tdeq_dec h o T1 T2)
+| #T2 #H elim (tdeq_dec T1 T2)
/4 width=1 by fpb_cpx, fdeq_intro_sn, or_intror, or_introl/
-| #L2 elim (rdeq_dec h o L1 L2 T1)
+| #L2 elim (rdeq_dec L1 L2 T1)
/4 width=1 by fpb_lpx, fdeq_intro_sn, or_intror, or_introl/
| /2 width=1 by or_introl/
]
qed-.
(* Basic_2A1: fpbq_inv_fpb_alt *)
-lemma fpbq_ffdneq_inv_fpb: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
- (⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥) → ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #H0
+lemma fpbq_ffdneq_inv_fpb: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≽[h] ⦃G2,L2,T2⦄ →
+ (⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⊥) → ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄.
+#h #G1 #G2 #L1 #L2 #T1 #T2 #H #H0
elim (fpbq_inv_fpb … H) -H // #H elim H0 -H0 //
qed-.
(* Basic properties *********************************************************)
-lemma lpr_bind (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 →
- ∀I1,I2. ⦃G, K1⦄ ⊢ I1 ➡[h] I2 → ⦃G, K1.ⓘ{I1}⦄ ⊢ ➡[h] K2.ⓘ{I2}.
+lemma lpr_bind (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ➡[h] K2 →
+ ∀I1,I2. ⦃G,K1⦄ ⊢ I1 ➡[h] I2 → ⦃G,K1.ⓘ{I1}⦄ ⊢ ➡[h] K2.ⓘ{I2}.
/2 width=1 by lex_bind/ qed.
(* Note: lemma 250 *)
(* Advanced properties ******************************************************)
-lemma lpr_bind_refl_dx (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 →
- ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ➡[h] K2.ⓘ{I}.
+lemma lpr_bind_refl_dx (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ➡[h] K2 →
+ ∀I. ⦃G,K1.ⓘ{I}⦄ ⊢ ➡[h] K2.ⓘ{I}.
/2 width=1 by lex_bind_refl_dx/ qed.
-lemma lpr_pair (h) (G): ∀K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h] V2 →
- ∀I. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h] K2.ⓑ{I}V2.
+lemma lpr_pair (h) (G): ∀K1,K2,V1,V2. ⦃G,K1⦄ ⊢ ➡[h] K2 → ⦃G,K1⦄ ⊢ V1 ➡[h] V2 →
+ ∀I. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ➡[h] K2.ⓑ{I}V2.
/2 width=1 by lex_pair/ qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: was: lpr_inv_atom1 *)
(* Basic_1: includes: wcpr0_gen_sort *)
-lemma lpr_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ➡[h] L2 → L2 = ⋆.
+lemma lpr_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ➡[h] L2 → L2 = ⋆.
/2 width=2 by lex_inv_atom_sn/ qed-.
-lemma lpr_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ➡[h] L2 →
- ∃∃I2,K2. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ I1 ➡[h] I2 &
+lemma lpr_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ➡[h] L2 →
+ ∃∃I2,K2. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ I1 ➡[h] I2 &
L2 = K2.ⓘ{I2}.
/2 width=1 by lex_inv_bind_sn/ qed-.
(* Basic_2A1: was: lpr_inv_atom2 *)
-lemma lpr_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ➡[h] ⋆ → L1 = ⋆.
+lemma lpr_inv_atom_dx (h) (G): ∀L1. ⦃G,L1⦄ ⊢ ➡[h] ⋆ → L1 = ⋆.
/2 width=2 by lex_inv_atom_dx/ qed-.
-lemma lpr_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓘ{I2} →
- ∃∃I1,K1. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ I1 ➡[h] I2 &
+lemma lpr_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G,L1⦄ ⊢ ➡[h] K2.ⓘ{I2} →
+ ∃∃I1,K1. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ I1 ➡[h] I2 &
L1 = K1.ⓘ{I1}.
/2 width=1 by lex_inv_bind_dx/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lpr_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G, K1.ⓤ{I}⦄ ⊢ ➡[h] L2 →
- ∃∃K2. ⦃G, K1⦄ ⊢ ➡[h] K2 & L2 = K2.ⓤ{I}.
+lemma lpr_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ➡[h] L2 →
+ ∃∃K2. ⦃G,K1⦄ ⊢ ➡[h] K2 & L2 = K2.ⓤ{I}.
/2 width=1 by lex_inv_unit_sn/ qed-.
(* Basic_2A1: was: lpr_inv_pair1 *)
(* Basic_1: includes: wcpr0_gen_head *)
-lemma lpr_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2 &
+lemma lpr_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ➡[h] L2 →
+ ∃∃K2,V2. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡[h] V2 &
L2 = K2.ⓑ{I}V2.
/2 width=1 by lex_inv_pair_sn/ qed-.
-lemma lpr_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓤ{I} →
- ∃∃K1. ⦃G, K1⦄ ⊢ ➡[h] K2 & L1 = K1.ⓤ{I}.
+lemma lpr_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G,L1⦄ ⊢ ➡[h] K2.ⓤ{I} →
+ ∃∃K1. ⦃G,K1⦄ ⊢ ➡[h] K2 & L1 = K1.ⓤ{I}.
/2 width=1 by lex_inv_unit_dx/ qed-.
(* Basic_2A1: was: lpr_inv_pair2 *)
-lemma lpr_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h] V2 &
+lemma lpr_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ➡[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G,K1⦄ ⊢ ➡[h] K2 & ⦃G,K1⦄ ⊢ V1 ➡[h] V2 &
L1 = K1.ⓑ{I}V1.
/2 width=1 by lex_inv_pair_dx/ qed-.
-lemma lpr_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h] L2.ⓑ{I2}V2 →
- ∧∧ ⦃G, L1⦄ ⊢ ➡[h] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h] V2 & I1 = I2.
+lemma lpr_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G,L1.ⓑ{I1}V1⦄ ⊢ ➡[h] L2.ⓑ{I2}V2 →
+ ∧∧ ⦃G,L1⦄ ⊢ ➡[h] L2 & ⦃G,L1⦄ ⊢ V1 ➡[h] V2 & I1 = I2.
/2 width=1 by lex_inv_pair/ qed-.
(* Basic_1: removed theorems 3: wcpr0_getl wcpr0_getl_back
(* Properties with extended structural successor for closures ***************)
-lemma fqu_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+lemma fqu_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⬂[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ /3 width=5 by lpr_pair, fqu_lref_O, ex3_2_intro/
| /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/
]
qed-.
-lemma fqu_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐[b] ⦃G2, L2, U2⦄.
+lemma fqu_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⬂[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ /3 width=5 by lpr_pair, fqu_lref_O, ex3_2_intro/
| /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/
]
qed-.
-lemma fqu_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ➡[h] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄.
+lemma fqu_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ∀K2. ⦃G2,L2⦄ ⊢ ➡[h] K2 →
+ ∃∃K1,T. ⦃G1,L1⦄ ⊢ ➡[h] K1 & ⦃G1,L1⦄ ⊢ T1 ➡[h] T & ⦃G1,K1,T⦄ ⬂[b] ⦃G2,K2,T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ /3 width=5 by lpr_bind_refl_dx, fqu_lref_O, ex3_2_intro/
| /3 width=5 by cpr_pair_sn, fqu_pair_sn, ex3_2_intro/
-| #p #I #G2 #L2 #V2 #T2 #X #H
+| #p #I #G2 #L2 #V2 #T2 #Hb #X #H
elim (lpr_inv_pair_sn … H) -H #K2 #W2 #HLK2 #HVW2 #H destruct
/3 width=5 by cpr_pair_sn, fqu_bind_dx, ex3_2_intro/
| #p #I #G2 #L2 #V2 #T2 #Hb #X #H
(* Note: does not hold in Basic_2A1 because it requires cpm *)
(* Note: L1 = K0.ⓛV0 and T1 = #0 require n = 1 *)
-lemma lpr_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h] L1 →
- ∃∃n,K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[n, h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h] L2 & n ≤ 1.
+lemma lpr_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ➡[h] L1 →
+ ∃∃n,K2,T. ⦃G1,K1⦄ ⊢ T1 ➡[n,h] T & ⦃G1,K1,T⦄ ⬂[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ➡[h] L2 & n ≤ 1.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ * #G #K #V #K1 #H
elim (lpr_inv_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
(* Properties with extended optional structural successor for closures ******)
-lemma fquq_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+lemma fquq_cpr_trans_sn (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⬂⸮[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 cases H -H
[ #HT12 elim (fqu_cpr_trans_sn … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma fquq_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
- ∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h] L & ⦃G1, L⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮[b] ⦃G2, L2, U2⦄.
+lemma fquq_cpr_trans_dx (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ∀U2. ⦃G2,L2⦄ ⊢ T2 ➡[h] U2 →
+ ∃∃L,U1. ⦃G1,L1⦄ ⊢ ➡[h] L & ⦃G1,L⦄ ⊢ T1 ➡[h] U1 & ⦃G1,L,U1⦄ ⬂⸮[b] ⦃G2,L2,U2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 cases H -H
[ #HT12 elim (fqu_cpr_trans_dx … HT12 … HTU2) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma fquq_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ➡[h] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄.
+lemma fquq_lpr_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ∀K2. ⦃G2,L2⦄ ⊢ ➡[h] K2 →
+ ∃∃K1,T. ⦃G1,L1⦄ ⊢ ➡[h] K1 & ⦃G1,L1⦄ ⊢ T1 ➡[h] T & ⦃G1,K1,T⦄ ⬂⸮[b] ⦃G2,K2,T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 cases H -H
[ #H12 elim (fqu_lpr_trans … H12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma lpr_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h] L1 →
- ∃∃n,K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[n, h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h] L2 & n ≤ 1.
+lemma lpr_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ➡[h] L1 →
+ ∃∃n,K2,T. ⦃G1,K1⦄ ⊢ T1 ➡[n,h] T & ⦃G1,K1,T⦄ ⬂⸮[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ➡[h] L2 & n ≤ 1.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
[ #H12 elim (lpr_fqu_trans … H12 … HKL1) -L1 /3 width=7 by fqu_fquq, ex4_3_intro/
| * #H1 #H2 #H3 destruct /2 width=7 by ex4_3_intro/
(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
-lemma lpr_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → |L1| = |L2|.
+lemma lpr_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → |L1| = |L2|.
/2 width=2 by lex_fwd_length/ qed-.
(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
definition IH_cpr_conf_lpr (h): relation3 genv lenv term ≝ λG,L,T.
- ∀T1. ⦃G, L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡[h] T2 →
- ∀L1. ⦃G, L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 →
- ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡[h] T0 & ⦃G, L2⦄ ⊢ T2 ➡[h] T0.
+ ∀T1. ⦃G,L⦄ ⊢ T ➡[h] T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[h] T2 →
+ ∀L1. ⦃G,L⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 →
+ ∃∃T0. ⦃G,L1⦄ ⊢ T1 ➡[h] T0 & ⦃G,L2⦄ ⊢ T2 ➡[h] T0.
(* Main properties with context-sensitive parallel reduction for terms ******)
fact cpr_conf_lpr_atom_atom (h):
- ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡[h] T & ⦃G, L2⦄ ⊢ ⓪{I} ➡[h] T.
+ ∀I,G,L1,L2. ∃∃T. ⦃G,L1⦄ ⊢ ⓪{I} ➡[h] T & ⦃G,L2⦄ ⊢ ⓪{I} ➡[h] T.
/2 width=3 by cpr_refl, ex2_intro/ qed-.
fact cpr_conf_lpr_atom_delta (h):
∀G0,L0,i. (
- ∀G,L,T. ⦃G0, L0, #i⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,#i⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
- ∀V2. ⦃G0, K0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ #i ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T.
+ ∀V2. ⦃G0,K0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ #i ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T.
#h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
elim (lpr_drops_conf … HLK0 … HL01) -HL01 // #X1 #H1 #HLK1
elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
fact cpr_conf_lpr_delta_delta (h):
∀G0,L0,i. (
- ∀G,L,T. ⦃G0, L0, #i⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,#i⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
- ∀V1. ⦃G0, K0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⬆*[↑i] V1 ≘ T1 →
+ ∀V1. ⦃G0,K0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⬆*[↑i] V1 ≘ T1 →
∀KX,VX. ⬇*[i] L0 ≘ KX.ⓓVX →
- ∀V2. ⦃G0, KX⦄ ⊢ VX ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ T1 ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T.
+ ∀V2. ⦃G0,KX⦄ ⊢ VX ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T.
#h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
lapply (drops_mono … H … HLK0) -H #H destruct
fact cpr_conf_lpr_bind_bind (h):
∀p,I,G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓑ{p,I}V0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓑ{p,I}V2.T2 ➡[h] T.
#h #p #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HV01 … HV02 … HL01 … HL02) //
fact cpr_conf_lpr_bind_zeta (h):
∀G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,+ⓓV0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 →
- ∀T2. ⬆*[1]T2 ≘ T0 → ∀X2. ⦃G0, L0⦄ ⊢ T2 ➡[h] X2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ +ⓓV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ X2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 →
+ ∀T2. ⬆*[1]T2 ≘ T0 → ∀X2. ⦃G0,L0⦄ ⊢ T2 ➡[h] X2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ +ⓓV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ X2 ➡[h] T.
#h #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
#T2 #HT20 #X2 #HTX2 #L1 #HL01 #L2 #HL02
elim (cpm_inv_lifts_sn … HT01 (Ⓣ) … L0 … HT20) -HT01 [| /3 width=1 by drops_refl, drops_drop/ ] #T #HT1 #HT2
fact cpr_conf_lpr_zeta_zeta (h):
∀G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,+ⓓV0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀T1. ⬆*[1] T1 ≘ T0 → ∀X1. ⦃G0, L0⦄ ⊢ T1 ➡[h] X1 →
- ∀T2. ⬆*[1] T2 ≘ T0 → ∀X2. ⦃G0, L0⦄ ⊢ T2 ➡[h] X2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ X1 ➡[h] T & ⦃G0, L2⦄ ⊢ X2 ➡[h] T.
+ ∀T1. ⬆*[1] T1 ≘ T0 → ∀X1. ⦃G0,L0⦄ ⊢ T1 ➡[h] X1 →
+ ∀T2. ⬆*[1] T2 ≘ T0 → ∀X2. ⦃G0,L0⦄ ⊢ T2 ➡[h] X2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ X1 ➡[h] T & ⦃G0,L2⦄ ⊢ X2 ➡[h] T.
#h #G0 #L0 #V0 #T0 #IH #T1 #HT10 #X1 #HTX1
#T2 #HT20 #X2 #HTX2 #L1 #HL01 #L2 #HL02
lapply (lifts_inj … HT20 … HT10) -HT20 #H destruct
fact cpr_conf_lpr_flat_flat (h):
∀I,G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓕ{I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓕ{I}V0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0, L0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓕ{I}V1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓕ{I}V2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓕ{I}V1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓕ{I}V2.T2 ➡[h] T.
#h #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HV01 … HV02 … HL01 … HL02) //
fact cpr_conf_lpr_flat_eps (h):
∀G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓝV0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1,T1. ⦃G0, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0, L0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓝV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T.
+ ∀V1,T1. ⦃G0,L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓝV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T.
#h #G0 #L0 #V0 #T0 #IH #V1 #T1 #HT01
#T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
fact cpr_conf_lpr_eps_eps (h):
∀G0,L0,V0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓝV0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀T1. ⦃G0, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0, L0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ T1 ➡[h] T & ⦃G0, L2⦄ ⊢ T2 ➡[h] T.
+ ∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ T1 ➡[h] T & ⦃G0,L2⦄ ⊢ T2 ➡[h] T.
#h #G0 #L0 #V0 #T0 #IH #T1 #HT01
#T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
fact cpr_conf_lpr_flat_beta (h):
∀p,G0,L0,V0,W0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓐV0.ⓛ{p}W0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0⦄ ⊢ ⓛ{p}W0.T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0⦄ ⊢ ⓛ{p}W0.T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (cpm_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
*)
fact cpr_conf_lpr_flat_theta (h):
∀p,G0,L0,V0,W0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓐV0.ⓓ{p}W0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0, L0⦄ ⊢ ⓓ{p}W0.T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
- ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G0,L0⦄ ⊢ ⓓ{p}W0.T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
+ ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
fact cpr_conf_lpr_beta_beta (h):
∀p,G0,L0,V0,W0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓐV0.ⓛ{p}W0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀W1. ⦃G0, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0, L0.ⓛW0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀W1. ⦃G0,L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0,L0.ⓛW0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}ⓝW2.V2.T2 ➡[h] T.
#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
(* Basic_1: was: pr0_upsilon_upsilon *)
fact cpr_conf_lpr_theta_theta (h):
∀p,G0,L0,V0,W0,T0. (
- ∀G,L,T. ⦃G0, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, L, T⦄ → IH_cpr_conf_lpr h G L T
+ ∀G,L,T. ⦃G0,L0,ⓐV0.ⓓ{p}W0.T0⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
) →
- ∀V1. ⦃G0, L0⦄ ⊢ V0 ➡[h] V1 → ∀U1. ⬆*[1] V1 ≘ U1 →
- ∀W1. ⦃G0, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0, L0.ⓓW0⦄ ⊢ T0 ➡[h] T1 →
- ∀V2. ⦃G0, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
- ∀W2. ⦃G0, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
- ∀L1. ⦃G0, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G0, L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡[h] T & ⦃G0, L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
+ ∀V1. ⦃G0,L0⦄ ⊢ V0 ➡[h] V1 → ∀U1. ⬆*[1] V1 ≘ U1 →
+ ∀W1. ⦃G0,L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G0,L0.ⓓW0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G0,L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
+ ∀W2. ⦃G0,L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G0,L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡[h] T & ⦃G0,L2⦄ ⊢ ⓓ{p}W2.ⓐU2.T2 ➡[h] T.
#h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
(* Properties with context-sensitive parallel reduction for terms ***********)
-lemma lpr_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L1⦄ ⊢ T1 ➡[h] T.
+lemma lpr_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T0 ➡[h] T & ⦃G,L1⦄ ⊢ T1 ➡[h] T.
#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) -HT01 -HL01
/2 width=3 by ex2_intro/
qed-.
-lemma lpr_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡[h] T & ⦃G, L0⦄ ⊢ T1 ➡[h] T.
+lemma lpr_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G,L0⦄ ⊢ T0 ➡[h] T1 → ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T0 ➡[h] T & ⦃G,L0⦄ ⊢ T1 ➡[h] T.
#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) -HT01 -HL01
/2 width=3 by ex2_intro/
(* Forward lemmas with unbound parallel rt-transition for ref local envs ****)
(* Basic_2A1: was: lpr_lpx *)
-lemma lpr_fwd_lpx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ⬈[h] L2.
+lemma lpr_fwd_lpx (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ➡[h] L2 → ⦃G,L1⦄ ⊢ ⬈[h] L2.
/3 width=3 by cpm_fwd_cpx, lex_co/ qed-.
(* Basic properties *********************************************************)
-lemma lpx_bind (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 →
- ∀I1,I2. ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 → ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] K2.ⓘ{I2}.
+lemma lpx_bind (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 →
+ ∀I1,I2. ⦃G,K1⦄ ⊢ I1 ⬈[h] I2 → ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬈[h] K2.ⓘ{I2}.
/2 width=1 by lex_bind/ qed.
lemma lpx_refl (h) (G): reflexive … (lpx h G).
(* Advanced properties ******************************************************)
-lemma lpx_bind_refl_dx (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 →
- ∀I. ⦃G, K1.ⓘ{I}⦄ ⊢ ⬈[h] K2.ⓘ{I}.
+lemma lpx_bind_refl_dx (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 →
+ ∀I. ⦃G,K1.ⓘ{I}⦄ ⊢ ⬈[h] K2.ⓘ{I}.
/2 width=1 by lex_bind_refl_dx/ qed.
-lemma lpx_pair (h) (G): ∀K1,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 → ∀V1,V2. ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 →
- ∀I.⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2.
+lemma lpx_pair (h) (G): ∀K1,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 → ∀V1,V2. ⦃G,K1⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I.⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2.
/2 width=1 by lex_pair/ qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: was: lpx_inv_atom1 *)
-lemma lpx_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ⬈[h] L2 → L2 = ⋆.
+lemma lpx_inv_atom_sn (h) (G): ∀L2. ⦃G,⋆⦄ ⊢ ⬈[h] L2 → L2 = ⋆.
/2 width=2 by lex_inv_atom_sn/ qed-.
-lemma lpx_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢ ⬈[h] L2 →
- ∃∃I2,K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
+lemma lpx_inv_bind_sn (h) (G): ∀I1,L2,K1. ⦃G,K1.ⓘ{I1}⦄ ⊢ ⬈[h] L2 →
+ ∃∃I2,K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬈[h] I2 &
L2 = K2.ⓘ{I2}.
/2 width=1 by lex_inv_bind_sn/ qed-.
(* Basic_2A1: was: lpx_inv_atom2 *)
-lemma lpx_inv_atom_dx: ∀h,G,L1. ⦃G, L1⦄ ⊢ ⬈[h] ⋆ → L1 = ⋆.
+lemma lpx_inv_atom_dx: ∀h,G,L1. ⦃G,L1⦄ ⊢ ⬈[h] ⋆ → L1 = ⋆.
/2 width=2 by lex_inv_atom_dx/ qed-.
-lemma lpx_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓘ{I2} →
- ∃∃I1,K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈[h] I2 &
+lemma lpx_inv_bind_dx (h) (G): ∀I2,L1,K2. ⦃G,L1⦄ ⊢ ⬈[h] K2.ⓘ{I2} →
+ ∃∃I1,K1. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ I1 ⬈[h] I2 &
L1 = K1.ⓘ{I1}.
/2 width=1 by lex_inv_bind_dx/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lpx_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G, K1.ⓤ{I}⦄ ⊢ ⬈[h] L2 →
- ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & L2 = K2.ⓤ{I}.
+lemma lpx_inv_unit_sn (h) (G): ∀I,L2,K1. ⦃G,K1.ⓤ{I}⦄ ⊢ ⬈[h] L2 →
+ ∃∃K2. ⦃G,K1⦄ ⊢ ⬈[h] K2 & L2 = K2.ⓤ{I}.
/2 width=1 by lex_inv_unit_sn/ qed-.
(* Basic_2A1: was: lpx_inv_pair1 *)
-lemma lpx_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ⬈[h] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
+lemma lpx_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ⦃G,K1.ⓑ{I}V1⦄ ⊢ ⬈[h] L2 →
+ ∃∃K2,V2. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈[h] V2 &
L2 = K2.ⓑ{I}V2.
/2 width=1 by lex_inv_pair_sn/ qed-.
-lemma lpx_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓤ{I} →
- ∃∃K1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & L1 = K1.ⓤ{I}.
+lemma lpx_inv_unit_dx (h) (G): ∀I,L1,K2. ⦃G,L1⦄ ⊢ ⬈[h] K2.ⓤ{I} →
+ ∃∃K1. ⦃G,K1⦄ ⊢ ⬈[h] K2 & L1 = K1.ⓤ{I}.
/2 width=1 by lex_inv_unit_dx/ qed-.
(* Basic_2A1: was: lpx_inv_pair2 *)
-lemma lpx_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G, L1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ⬈[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈[h] V2 &
+lemma lpx_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ⦃G,L1⦄ ⊢ ⬈[h] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G,K1⦄ ⊢ ⬈[h] K2 & ⦃G,K1⦄ ⊢ V1 ⬈[h] V2 &
L1 = K1.ⓑ{I}V1.
/2 width=1 by lex_inv_pair_dx/ qed-.
-lemma lpx_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ⬈[h] L2.ⓑ{I2}V2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 & I1 = I2.
+lemma lpx_inv_pair (h) (G): ∀I1,I2,L1,L2,V1,V2. ⦃G,L1.ⓑ{I1}V1⦄ ⊢ ⬈[h] L2.ⓑ{I2}V2 →
+ ∧∧ ⦃G,L1⦄ ⊢ ⬈[h] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 & I1 = I2.
/2 width=1 by lex_inv_pair/ qed-.
(* Note: lemma 500 *)
(* Basic_2A1: was: cpx_lpx_aaa_conf *)
-lemma cpx_aaa_conf_lpx (h): ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+lemma cpx_aaa_conf_lpx (h): ∀G,L1,T1,A. ⦃G,L1⦄ ⊢ T1 ⁝ A →
+ ∀T2. ⦃G,L1⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L2⦄ ⊢ T2 ⁝ A.
#h #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
[ #G #L1 #s #X #H
elim (cpx_inv_sort1 … H) -H #H destruct //
(* Properties with extended structural successor for closures ***************)
-lemma lpx_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+lemma lpx_fqu_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 →
+ ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⬂[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #K #V #K1 #H
elim (lpx_inv_pair_dx … H) -H #K0 #V0 #HK0 #HV0 #H destruct
]
qed-.
-lemma fqu_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐[b] ⦃G2, K2, T2⦄.
+lemma fqu_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ∀K2. ⦃G2,L2⦄ ⊢ ⬈[h] K2 →
+ ∃∃K1,T. ⦃G1,L1⦄ ⊢ ⬈[h] K1 & ⦃G1,L1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⬂[b] ⦃G2,K2,T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ /3 width=5 by lpx_bind_refl_dx, fqu_lref_O, ex3_2_intro/
| /3 width=5 by cpx_pair_sn, fqu_pair_sn, ex3_2_intro/
-| #p #I #G2 #L2 #V2 #T2 #X #H
+| #p #I #G2 #L2 #V2 #T2 #Hb #X #H
elim (lpx_inv_pair_sn … H) -H #K2 #W2 #HLK2 #HVW2 #H destruct
/3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
| #p #I #G2 #L2 #V2 #T2 #Hb #X #H
(* Properties with extended optional structural successor for closures ******)
-lemma lpx_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ⬈[h] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ⬈[h] L2.
+lemma lpx_fquq_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ∀K1. ⦃G1,K1⦄ ⊢ ⬈[h] L1 →
+ ∃∃K2,T. ⦃G1,K1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⬂⸮[b] ⦃G2,K2,T2⦄ & ⦃G2,K2⦄ ⊢ ⬈[h] L2.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 cases H -H
[ #H12 elim (lpx_fqu_trans … H12 … HKL1) -L1 /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma fquq_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ⬈[h] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ⬈[h] K1 & ⦃G1, L1⦄ ⊢ T1 ⬈[h] T & ⦃G1, K1, T⦄ ⊐⸮[b] ⦃G2, K2, T2⦄.
+lemma fquq_lpx_trans (h) (b): ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ∀K2. ⦃G2,L2⦄ ⊢ ⬈[h] K2 →
+ ∃∃K1,T. ⦃G1,L1⦄ ⊢ ⬈[h] K1 & ⦃G1,L1⦄ ⊢ T1 ⬈[h] T & ⦃G1,K1,T⦄ ⬂⸮[b] ⦃G2,K2,T2⦄.
#h #b #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 cases H -H
[ #H12 elim (fqu_lpx_trans … H12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
(* Forward lemmas with free variables inclusion for restricted closures *****)
(* Basic_2A1: uses: lpx_cpx_frees_trans *)
-lemma lpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
+lemma lpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G,L0⦄ ⊢ ⬈[h] L2 → ⦃L2,T1⦄ ⊆ ⦃L0,T0⦄.
/3 width=4 by rpx_cpx_conf_fsge, lpx_rpx/ qed-.
(* Basic_2A1: uses: lpx_frees_trans *)
-lemma lpx_fsge_comp (h) (G): ∀L0,L2,T0. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃L2, T0⦄ ⊆ ⦃L0, T0⦄.
+lemma lpx_fsge_comp (h) (G): ∀L0,L2,T0. ⦃G,L0⦄ ⊢ ⬈[h] L2 → ⦃L2,T0⦄ ⊆ ⦃L0,T0⦄.
/2 width=4 by lpx_cpx_conf_fsge/ qed-.
(* Forward lemmas with length for local environments ************************)
-lemma lpx_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → |L1| = |L2|.
+lemma lpx_fwd_length (h) (G): ∀L1,L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → |L1| = |L2|.
/2 width=2 by lex_fwd_length/ qed-.
(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
-(* Properties with degree-based equivalence for local environments **********)
+(* Properties with sort-irrelevant equivalence for local environments *******)
(* Basic_2A1: uses: lleq_lpx_trans *)
-lemma rdeq_lpx_trans (h) (o) (G): ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈[h] K2 →
- ∀L1. ∀T:term. L1 ≛[h, o, T] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h] K1 & K1 ≛[h, o, T] K2.
-#h #o #G #L2 #K2 #HLK2 #L1 #T #HL12
+lemma rdeq_lpx_trans (h) (G): ∀L2,K2. ⦃G,L2⦄ ⊢ ⬈[h] K2 →
+ ∀L1. ∀T:term. L1 ≛[T] L2 →
+ ∃∃K1. ⦃G,L1⦄ ⊢ ⬈[h] K1 & K1 ≛[T] K2.
+#h #G #L2 #K2 #HLK2 #L1 #T #HL12
lapply (lpx_rpx … T HLK2) -HLK2 #HLK2
elim (rdeq_rpx_trans … HLK2 … HL12) -L2 #K #H #HK2
elim (rpx_inv_lpx_req … H) -H #K1 #HLK1 #HK1
(* Basic properties ***********************************************************)
-lemma rpx_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] ⋆.
+lemma rpx_atom: ∀h,I,G. ⦃G,⋆⦄ ⊢ ⬈[h,⓪{I}] ⋆.
/2 width=1 by rex_atom/ qed.
lemma rpx_sort: ∀h,I1,I2,G,L1,L2,s.
- ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2}.
+ ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,⋆s] L2.ⓘ{I2}.
/2 width=1 by rex_sort/ qed.
lemma rpx_pair: ∀h,I,G,L1,L2,V1,V2.
- ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 → ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2.
+ ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 → ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,#0] L2.ⓑ{I}V2.
/2 width=1 by rex_pair/ qed.
lemma rpx_lref: ∀h,I1,I2,G,L1,L2,i.
- ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2}.
+ ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,#↑i] L2.ⓘ{I2}.
/2 width=1 by rex_lref/ qed.
lemma rpx_gref: ∀h,I1,I2,G,L1,L2,l.
- ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2}.
+ ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 → ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,§l] L2.ⓘ{I2}.
/2 width=1 by rex_gref/ qed.
lemma rpx_bind_repl_dx: ∀h,I,I1,G,L1,L2,T.
- ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I1} →
- ∀I2. ⦃G, L1⦄ ⊢ I ⬈[h] I2 →
- ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I2}.
+ ⦃G,L1.ⓘ{I}⦄ ⊢ ⬈[h,T] L2.ⓘ{I1} →
+ ∀I2. ⦃G,L1⦄ ⊢ I ⬈[h] I2 →
+ ⦃G,L1.ⓘ{I}⦄ ⊢ ⬈[h,T] L2.ⓘ{I2}.
/2 width=2 by rex_bind_repl_dx/ qed-.
(* Basic inversion lemmas ***************************************************)
-lemma rpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G, ⋆⦄ ⊢ ⬈[h, T] Y2 → Y2 = ⋆.
+lemma rpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G,⋆⦄ ⊢ ⬈[h,T] Y2 → Y2 = ⋆.
/2 width=3 by rex_inv_atom_sn/ qed-.
-lemma rpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G, Y1⦄ ⊢ ⬈[h, T] ⋆ → Y1 = ⋆.
+lemma rpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G,Y1⦄ ⊢ ⬈[h,T] ⋆ → Y1 = ⋆.
/2 width=3 by rex_inv_atom_dx/ qed-.
-lemma rpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] Y2 →
+lemma rpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G,Y1⦄ ⊢ ⬈[h,⋆s] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 &
+ | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
/2 width=1 by rex_inv_sort/ qed-.
-lemma rpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] Y2 →
+lemma rpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G,Y1⦄ ⊢ ⬈[h,#↑i] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 &
+ | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
/2 width=1 by rex_inv_lref/ qed-.
-lemma rpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] Y2 →
+lemma rpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G,Y1⦄ ⊢ ⬈[h,§l] Y2 →
∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 &
+ | ∃∃I1,I2,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
/2 width=1 by rex_inv_gref/ qed-.
-lemma rpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
+lemma rpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 →
+ ∧∧ ⦃G,L1⦄ ⊢ ⬈[h,V] L2 & ⦃G,L1.ⓑ{I}V⦄ ⊢ ⬈[h,T] L2.ⓑ{I}V.
/2 width=2 by rex_inv_bind/ qed-.
-lemma rpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+lemma rpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L2 →
+ ∧∧ ⦃G,L1⦄ ⊢ ⬈[h,V] L2 & ⦃G,L1⦄ ⊢ ⬈[h,T] L2.
/2 width=2 by rex_inv_flat/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma rpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] Y2 →
- ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
+lemma rpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,⋆s] Y2 →
+ ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 & Y2 = L2.ⓘ{I2}.
/2 width=2 by rex_inv_sort_bind_sn/ qed-.
-lemma rpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2} →
- ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
+lemma rpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G,Y1⦄ ⊢ ⬈[h,⋆s] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,⋆s] L2 & Y1 = L1.ⓘ{I1}.
/2 width=2 by rex_inv_sort_bind_dx/ qed-.
-lemma rpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] Y2 →
- ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
+lemma rpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,#0] Y2 →
+ ∃∃L2,V2. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 &
Y2 = L2.ⓑ{I}V2.
/2 width=1 by rex_inv_zero_pair_sn/ qed-.
-lemma rpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2 →
- ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
+lemma rpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G,Y1⦄ ⊢ ⬈[h,#0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 &
Y1 = L1.ⓑ{I}V1.
/2 width=1 by rex_inv_zero_pair_dx/ qed-.
-lemma rpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] Y2 →
- ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y2 = L2.ⓘ{I2}.
+lemma rpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,#↑i] Y2 →
+ ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 & Y2 = L2.ⓘ{I2}.
/2 width=2 by rex_inv_lref_bind_sn/ qed-.
-lemma rpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2} →
- ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y1 = L1.ⓘ{I1}.
+lemma rpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ⦃G,Y1⦄ ⊢ ⬈[h,#↑i] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,#i] L2 & Y1 = L1.ⓘ{I1}.
/2 width=2 by rex_inv_lref_bind_dx/ qed-.
-lemma rpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] Y2 →
- ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y2 = L2.ⓘ{I2}.
+lemma rpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G,L1.ⓘ{I1}⦄ ⊢ ⬈[h,§l] Y2 →
+ ∃∃I2,L2. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 & Y2 = L2.ⓘ{I2}.
/2 width=2 by rex_inv_gref_bind_sn/ qed-.
-lemma rpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2} →
- ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y1 = L1.ⓘ{I1}.
+lemma rpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G,Y1⦄ ⊢ ⬈[h,§l] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G,L1⦄ ⊢ ⬈[h,§l] L2 & Y1 = L1.ⓘ{I1}.
/2 width=2 by rex_inv_gref_bind_dx/ qed-.
(* Basic forward lemmas *****************************************************)
lemma rpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
+ ⦃G,L1⦄ ⊢ ⬈[h,②{I}V.T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,V] L2.
/2 width=3 by rex_fwd_pair_sn/ qed-.
lemma rpx_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
+ ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 → ⦃G,L1.ⓑ{I}V⦄ ⊢ ⬈[h,T] L2.ⓑ{I}V.
/2 width=2 by rex_fwd_bind_dx/ qed-.
lemma rpx_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+ ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T] L2.
/2 width=3 by rex_fwd_flat_dx/ qed-.
lemma rpx_refl: ∀h,G,T. reflexive … (rpx h G T).
/2 width=1 by rex_refl/ qed.
-lemma rpx_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L.ⓑ{I}V2.
+lemma rpx_pair_refl: ∀h,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I,T. ⦃G,L.ⓑ{I}V1⦄ ⊢ ⬈[h,T] L.ⓑ{I}V2.
/2 width=1 by rex_pair_refl/ qed.
(* Advanced inversion lemmas ************************************************)
-lemma rpx_inv_bind_void: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ.
+lemma rpx_inv_bind_void: ∀h,p,I,G,L1,L2,V,T. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 →
+ ∧∧ ⦃G,L1⦄ ⊢ ⬈[h,V] L2 & ⦃G,L1.ⓧ⦄ ⊢ ⬈[h,T] L2.ⓧ.
/2 width=3 by rex_inv_bind_void/ qed-.
(* Advanced forward lemmas **************************************************)
lemma rpx_fwd_bind_dx_void: ∀h,p,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ.
+ ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2 → ⦃G,L1.ⓧ⦄ ⊢ ⬈[h,T] L2.ⓧ.
/2 width=4 by rex_fwd_bind_dx_void/ qed-.
(* Note: "⦃L2, T1⦄ ⊆ ⦃L2, T0⦄" does not hold *)
(* Note: Take L0 = K0.ⓓ(ⓝW.V), L2 = K0.ⓓW, T0 = #0, T1 = ⬆*[1]V *)
-(* Note: This invalidates rpxs_cpx_conf: "∀h,G. s_r_confluent1 … (cpx h G) (rpxs h G)" *)
-lemma rpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
-#h #G0 #L0 #T0 @(fqup_wf_ind_eq (â\92») … G0 L0 T0) -G0 -L0 -T0
+(* Note: This invalidates rpxs_cpx_conf: "∀h, G. s_r_confluent1 … (cpx h G) (rpxs h G)" *)
+lemma rpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G,L0⦄ ⊢⬈[h,T0] L2 → ⦃L2,T1⦄ ⊆ ⦃L0,T0⦄.
+#h #G0 #L0 #T0 @(fqup_wf_ind_eq (â\93\89) … G0 L0 T0) -G0 -L0 -T0
#G #L #T #IH #G0 #L0 * *
[ #s #HG #HL #HT #X #HX #Y #HY destruct -IH
elim (cpx_inv_sort1 … HX) -HX #H destruct
lemma rpx_cpx_conf (h) (G): s_r_confluent1 … (cpx h G) (rpx h G).
/2 width=5 by cpx_rex_conf/ qed-.
-lemma rpx_cpx_conf_fsge_dx (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T1⦄.
+lemma rpx_cpx_conf_fsge_dx (h) (G): ∀L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G,L0⦄ ⊢⬈[h,T0] L2 → ⦃L2,T1⦄ ⊆ ⦃L0,T1⦄.
/3 width=5 by rpx_cpx_conf, rpx_fsge_comp/ qed-.
(* Forward lemmas with length for local environments ************************)
-lemma rpx_fwd_length: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → |L1| = |L2|.
+lemma rpx_fwd_length: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 → |L1| = |L2|.
/2 width=3 by rex_fwd_length/ qed-.
(* Inversion lemmas with length for local environments **********************)
-lemma rpx_inv_zero_length: ∀h,G,Y1,Y2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] Y2 →
+lemma rpx_inv_zero_length: ∀h,G,Y1,Y2. ⦃G,Y1⦄ ⊢ ⬈[h,#0] Y2 →
∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 &
- ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
+ | ∃∃I,L1,L2,V1,V2. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 &
+ ⦃G,L1⦄ ⊢ V1 ⬈[h] V2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
| ∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
/2 width=1 by rex_inv_zero_length/ qed-.
(* Properties with syntactic equivalence for referred local environments ****)
-lemma fleq_rpx (h) (G): ∀L1,L2,T. L1 ≡[T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+lemma fleq_rpx (h) (G): ∀L1,L2,T. L1 ≡[T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T] L2.
/2 width=1 by req_fwd_rex/ qed.
(* Properties with unbound parallel rt-transition for full local envs *******)
-lemma lpx_rpx: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+lemma lpx_rpx: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T] L2.
/2 width=1 by rex_lex/ qed.
(* Inversion lemmas with unbound parallel rt-transition for full local envs *)
-lemma rpx_inv_lpx_req: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 →
- ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h] L & L ≡[T] L2.
+lemma rpx_inv_lpx_req: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 →
+ ∃∃L. ⦃G,L1⦄ ⊢ ⬈[h] L & L ≡[T] L2.
/3 width=3 by rpx_fsge_comp, rex_inv_lex_req/ qed-.
(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-(* Properties with degree-based equivalence for local environments **********)
+(* Properties with sort-irrelevant equivalence for local environments *******)
-lemma rpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ∀o,I,T.
- ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≛[h, o, V] L2.
+lemma rpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G,L1⦄ ⊢ ⬈[h,V] L2 → ∀I,T.
+ ∃∃L. ⦃G,L1⦄ ⊢ ⬈[h,②{I}V.T] L & L ≛[V] L2.
/3 width=5 by rpx_fsge_comp, rex_pair_sn_split/ qed-.
-lemma rpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ∀o,I,V.
- ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≛[h, o, T] L2.
+lemma rpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 → ∀I,V.
+ ∃∃L. ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L & L ≛[T] L2.
/3 width=5 by rpx_fsge_comp, rex_flat_dx_split/ qed-.
-lemma rpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 → ∀o,p.
- ∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[h, o, T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V.
+lemma rpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,T] L2 → ∀p.
+ ∃∃L,V. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[T] L2 & ⦃G,L1⦄ ⊢ V1 ⬈[h] V.
/3 width=5 by rpx_fsge_comp, rex_bind_dx_split/ qed-.
-lemma rpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G, K1.ⓧ⦄ ⊢ ⬈[h, T] L2 → ∀o,p,I,V.
- ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[h, o, T] L2.
+lemma rpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G,K1.ⓧ⦄ ⊢ ⬈[h,T] L2 → ∀p,I,V.
+ ∃∃K2. ⦃G,K1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[T] L2.
/3 width=5 by rpx_fsge_comp, rex_bind_dx_split_void/ qed-.
-lemma rpx_tdeq_conf: ∀h,o,G. s_r_confluent1 … (cdeq h o) (rpx h G).
+lemma rpx_tdeq_conf: ∀h,G. s_r_confluent1 … cdeq (rpx h G).
/2 width=5 by tdeq_rex_conf/ qed-.
-lemma rpx_tdeq_div: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
- ∀G,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T2] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T1] L2.
+lemma rpx_tdeq_div: ∀h,T1,T2. T1 ≛ T2 →
+ ∀G,L1,L2. ⦃G,L1⦄ ⊢ ⬈[h,T2] L2 → ⦃G,L1⦄ ⊢ ⬈[h,T1] L2.
/2 width=5 by tdeq_rex_div/ qed-.
-lemma cpx_tdeq_conf_sex: ∀h,o,G. R_confluent2_rex … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
-#h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
+lemma cpx_tdeq_conf_rex: ∀h,G. R_confluent2_rex … (cpx h G) cdeq (cpx h G) cdeq.
+#h #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
[ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
- elim (tdeq_inv_sort1 … H0) -H0 #s1 #d1 #Hs0 #Hs1 #H destruct
- /4 width=3 by tdeq_sort, deg_next, ex2_intro/
+ elim (tdeq_inv_sort1 … H0) -H0 #s1 #H destruct
+ /3 width=3 by tdeq_sort, ex2_intro/
| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
>(tdeq_inv_lref1 … H0) -H0
elim (rpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
]
qed-.
-lemma cpx_tdeq_conf: ∀h,o,G,L. ∀T0:term. ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
- ∀T2. T0 ≛[h, o] T2 →
- ∃∃T. T1 ≛[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
-#h #o #G #L #T0 #T1 #HT01 #T2 #HT02
-elim (cpx_tdeq_conf_sex … HT01 … HT02 L … L) -HT01 -HT02
+lemma cpx_tdeq_conf: ∀h,G,L. ∀T0:term. ∀T1. ⦃G,L⦄ ⊢ T0 ⬈[h] T1 →
+ ∀T2. T0 ≛ T2 →
+ ∃∃T. T1 ≛ T & ⦃G,L⦄ ⊢ T2 ⬈[h] T.
+#h #G #L #T0 #T1 #HT01 #T2 #HT02
+elim (cpx_tdeq_conf_rex … HT01 … HT02 L … L) -HT01 -HT02
/2 width=3 by rex_refl, ex2_intro/
qed-.
-lemma tdeq_cpx_trans: ∀h,o,G,L,T2. ∀T0:term. T2 ≛[h, o] T0 →
- ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
- ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≛[h, o] T1.
-#h #o #G #L #T2 #T0 #HT20 #T1 #HT01
+lemma tdeq_cpx_trans: ∀h,G,L,T2. ∀T0:term. T2 ≛ T0 →
+ ∀T1. ⦃G,L⦄ ⊢ T0 ⬈[h] T1 →
+ ∃∃T. ⦃G,L⦄ ⊢ T2 ⬈[h] T & T ≛ T1.
+#h #G #L #T2 #T0 #HT20 #T1 #HT01
elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
qed-.
(* Basic_2A1: uses: cpx_lleq_conf *)
-lemma cpx_rdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. L0 ≛[h, o, T0] L2 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≛[h, o] T.
-#h #o #G #L0 #T0 #T1 #HT01 #L2 #HL02
-elim (cpx_tdeq_conf_sex … HT01 T0 … L0 … HL02) -HT01 -HL02
+lemma cpx_rdeq_conf: ∀h,G,L0,T0,T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. L0 ≛[T0] L2 →
+ ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈[h] T & T1 ≛ T.
+#h #G #L0 #T0 #T1 #HT01 #L2 #HL02
+elim (cpx_tdeq_conf_rex … HT01 T0 … L0 … HL02) -HT01 -HL02
/2 width=3 by rex_refl, ex2_intro/
qed-.
(* Basic_2A1: uses: lleq_cpx_trans *)
-lemma rdeq_cpx_trans: ∀h,o,G,L2,L0,T0. L2 ≛[h, o, T0] L0 →
- ∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≛[h, o] T1.
-#h #o #G #L2 #L0 #T0 #HL20 #T1 #HT01
-elim (cpx_rdeq_conf … o … HT01 L2) -HT01
+lemma rdeq_cpx_trans: ∀h,G,L2,L0,T0. L2 ≛[T0] L0 →
+ ∀T1. ⦃G,L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∃∃T. ⦃G,L2⦄ ⊢ T0 ⬈[h] T & T ≛ T1.
+#h #G #L2 #L0 #T0 #HL20 #T1 #HT01
+elim (cpx_rdeq_conf … HT01 L2) -HT01
/3 width=3 by rdeq_sym, tdeq_sym, ex2_intro/
qed-.
-lemma rpx_rdeq_conf: ∀h,o,G,T. confluent2 … (rpx h G T) (rdeq h o T).
-/3 width=6 by rpx_fsge_comp, rdeq_fsge_comp, cpx_tdeq_conf_sex, rex_conf/ qed-.
+lemma rpx_rdeq_conf: ∀h,G,T. confluent2 … (rpx h G T) (rdeq T).
+/3 width=6 by rpx_fsge_comp, rdeq_fsge_comp, cpx_tdeq_conf_rex, rex_conf/ qed-.
-lemma rdeq_rpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≛[h, o, T] K2.
-#h #o #G #T #L2 #K2 #HLK2 #L1 #HL12
-elim (rpx_rdeq_conf … o … HLK2 L1)
+lemma rdeq_rpx_trans: ∀h,G,T,L2,K2. ⦃G,L2⦄ ⊢ ⬈[h,T] K2 →
+ ∀L1. L1 ≛[T] L2 →
+ ∃∃K1. ⦃G,L1⦄ ⊢ ⬈[h,T] K1 & K1 ≛[T] K2.
+#h #G #T #L2 #K2 #HLK2 #L1 #HL12
+elim (rpx_rdeq_conf … HLK2 L1)
/3 width=3 by rdeq_sym, ex2_intro/
qed-.
(* Main properties **********************************************************)
-theorem rpx_bind: ∀h,G,L1,L2,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 →
- ∀I,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V2 →
- ∀p. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L2.
+theorem rpx_bind: ∀h,G,L1,L2,V1. ⦃G,L1⦄ ⊢ ⬈[h,V1] L2 →
+ ∀I,V2,T. ⦃G,L1.ⓑ{I}V1⦄ ⊢ ⬈[h,T] L2.ⓑ{I}V2 →
+ ∀p. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V1.T] L2.
/2 width=2 by rex_bind/ qed.
-theorem rpx_flat: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 →
- ∀I,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2.
+theorem rpx_flat: ∀h,G,L1,L2,V. ⦃G,L1⦄ ⊢ ⬈[h,V] L2 →
+ ∀I,T. ⦃G,L1⦄ ⊢ ⬈[h,T] L2 → ⦃G,L1⦄ ⊢ ⬈[h,ⓕ{I}V.T] L2.
/2 width=1 by rex_flat/ qed.
-theorem rpx_bind_void: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 →
- ∀T. ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ →
- ∀p,I. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2.
+theorem rpx_bind_void: ∀h,G,L1,L2,V. ⦃G,L1⦄ ⊢ ⬈[h,V] L2 →
+ ∀T. ⦃G,L1.ⓧ⦄ ⊢ ⬈[h,T] L2.ⓧ →
+ ∀p,I. ⦃G,L1⦄ ⊢ ⬈[h,ⓑ{p,I}V.T] L2.
/2 width=1 by rex_bind_void/ qed.
<table name="basic_2_sum"/>
<subsection name="B">Stage "B"</subsection>
+ <news class="beta" date="2019 September 3.">
+ Applicability condition is now parametrized
+ with a generic subset of numbers.
+ </news>
+ <news class="beta" date="2019 June 2.">
+ Applicability condition parametrized
+ with an initial interval of numbers
+ allows λδ-2B to generalize both λδ-2A and λδ-1B.
+ </news>
+ <news class="beta" date="2019 April 16.">
+ Extended (λδ-2A) and restricted (λδ-1B) validity is decidable
+ (anniversary milestone).
+ </news>
+ <news class="beta" date="2019 March 25.">
+ Preservation of validity for rt-computation
+ does not need the sort degree parameter
+ (i.e. no induction on the degree).
+ </news>
<news class="beta" date="2018 November 1.">
- Extended (λδ-2) and restricted (λδ-1) type rules justified.
+ Extended (λδ-2A) and restricted (λδ-1A) type rules justified.
</news>
<news class="alpha" date="2018 September 21.">
λδ-2A completed with
class "wine"
[ { "iterated dynamic typing" * } {
[ { "context-sensitive iterated native type assignment" * } {
- [ [ "for terms" ] "ntas" + "( ⦃?,?⦄ ⊢ ? :[?,?,?] ? )" + "( ⦃?,?⦄ ⊢ ? :[?,?] ? )" + "( ⦃?,?⦄ ⊢ ? :*[?,?] ? )" * ]
+ [ [ "for terms" ] "ntas" + "( ⦃?,?⦄ ⊢ ? :*[?,?,?] ? )" "ntas_nta" + "ntas_preserve" * ]
}
]
}
class "magenta"
[ { "dynamic typing" * } {
[ { "context-sensitive native type assignment" * } {
- [ [ "for terms" ] "nta" + "( ⦃?,?⦄ ⊢ ? :[?,?] ? )" + "( ⦃?,?⦄ ⊢ ? :[?] ? )" + "( ⦃?,?⦄ ⊢ ? :*[?] ? )" "nta_drops" + "nta_aaa" + "nta_fsb" + "nta_cpms" + "nta_cpcs" + "nta_preserve" + "nta_preserve_cpcs" + "nta_ind" * ]
+ [ [ "for terms" ] "nta" + "( ⦃?,?⦄ ⊢ ? :[?,?] ? )" "nta_drops" + "nta_aaa" + "nta_fsb" + "nta_cpms" + "nta_cpcs" + "nta_preserve" + "nta_preserve_cpcs" + "nta_ind" + "nta_eval" * ]
}
]
[ { "context-sensitive native validity" * } {
[ [ "restricted refinement for lenvs" ] "lsubv ( ? ⊢ ? ⫃![?,?] ? )" "lsubv_drops" + "lsubv_lsubr" + "lsubv_lsuba" + "lsubv_cpms" + "lsubv_cpcs" + "lsubv_cnv" + "lsubv_lsubv" * ]
- [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" + "( ⦃?,?⦄ ⊢ ? ![?] )" + "( ⦃?,?⦄ ⊢ ? !*[?] )" "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tdeq_trans" + "cnv_cpm_tdeq_conf" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_cpms_tdeq_conf" + "cnv_cpcs" + "cnv_preserve_sub" + "cnv_preserve" * ]
+ [ [ "for terms" ] "cnv" + "( ⦃?,?⦄ ⊢ ? ![?,?] )" "cnv_acle" + "cnv_drops" + "cnv_fqus" + "cnv_aaa" + "cnv_fsb" + "cnv_cpm_trans" + "cnv_cpm_conf" + "cnv_cpm_tdeq" + "cnv_cpm_tdeq_trans" + "cnv_cpm_tdeq_conf" + "cnv_cpms_tdeq" + "cnv_cpms_conf" + "cnv_cpms_tdeq_conf" + "cnv_cpme" + "cnv_cpmuwe" + "cnv_cpmuwe_cpme" + "cnv_eval" + "cnv_cpce" + "cnv_cpes" + "cnv_cpcs" + "cnv_preserve_sub" + "cnv_preserve" + "cnv_preserve_cpes" + "cnv_preserve_cpcs" * ]
}
]
}
class "prune"
[ { "rt-equivalence" * } {
[ { "context-sensitive parallel r-equivalence" * } {
- [ [ "for terms" ] "cpcs ( ⦃?,?⦄ ⊢ ? ⬌*[?] ? )" "cpcs_drops" + "cpcs_lsubr" + "cpcs_aaa" + "cpcs_cprs" + "cpcs_lprs" + "cpcs_cpc" + "cpcs_cpcs" * ]
+ [ [ "for terms" ] "cpcs ( ⦃?,?⦄ ⊢ ? ⬌*[?] ? )" "cpcs_drops" + "cpcs_lsubr" + "cpcs_aaa" + "cpcs_csx" + "cpcs_cprs" + "cpcs_lprs" + "cpcs_cpc" + "cpcs_cpcs" * ]
+ }
+ ]
+ [ { "t-bound context-sensitive parallel rt-equivalence" * } {
+ [ [ "for terms" ] "cpes ( ⦃?,?⦄ ⊢ ? ⬌*[?,?,?] ? )" "cpes_aaa" + "cpes_cpms" + "cpes_cpes" * ]
}
]
}
[ { "context-sensitive parallel eta-conversion" * } {
[ [ "for lenvs on all entries" ] "lpce ( ⦃?,?⦄ ⊢ ⬌η[?] ? )" * ]
[ [ "for binders" ] "cpce_ext" + "( ⦃?,?⦄ ⊢ ? ⬌η[?] ? )" * ]
- [ [ "for terms" ] "cpce" + "( ⦃?,?⦄ ⊢ ? ⬌η[?] ? )" * ]
+ [ [ "for terms" ] "cpce" + "( ⦃?,?⦄ ⊢ ? ⬌η[?] ? )" "cpce_drops" * ]
}
]
[ { "context-sensitive parallel r-conversion" * } {
class "sky"
[ { "rt-computation" * } {
[ { "context-sensitive parallel r-computation" * } {
+ [ [ "evaluation for terms" ] "cpre ( ⦃?,?⦄ ⊢ ? ➡*[?] 𝐍⦃?⦄ )" "cpre_csx" + "cpre_cpms" + "cpre_cpre" * ]
[ [ "for lenvs on all entries" ] "lprs ( ⦃?,?⦄ ⊢ ➡*[?] ? )" "lprs_tc" + "lprs_ctc" + "lprs_length" + "lprs_drops" + "lprs_aaa" + "lprs_lpr" + "lprs_lpxs" + "lprs_cpms" + "lprs_cprs" + "lprs_lprs" * ]
[ [ "for binders" ] "cprs_ext" + "( ⦃?,?⦄ ⊢ ? ➡*[?] ?)" * ]
- [ [ "for terms" ] "cprs" + "( ⦃?,?⦄ ⊢ ? ➡*[?] ?)" "cprs_ctc" + "cprs_drops" + "cprs_cpr" + "cprs_lpr" + "cprs_cprs" * ]
+ [ [ "for terms" ] "cprs" + "( ⦃?,?⦄ ⊢ ? ➡*[?] ?)" "cprs_ctc" + "cprs_tweq" + "cprs_drops" + "cprs_cpr" + "cprs_lpr" + "cprs_cnr" + "cprs_cprs" * ]
}
]
[ { "t-bound context-sensitive parallel rt-computation" * } {
+ [ [ "t-unbound whd evaluation for terms" ] "cpmuwe ( ⦃?,?⦄ ⊢ ? ⬌*𝐍𝐖*[?,?] ? )" "cpmuwe_csx" + "cpmuwe_cpmuwe" * ]
+ [ [ "t-unbound whd normal form for terms" ] "cnuw ( ⦃?,?⦄ ⊢ ⬌𝐍𝐖*[?] ? )" "cnuw_drops" + "cnuw_simple" + "cnuw_cnuw" * ]
+ [ [ "t-bpund evaluation for terms" ] "cpme ( ⦃?,?⦄ ⊢ ? ➡*[?,?] 𝐍⦃?⦄ )" "cpme_aaa" * ]
[ [ "for terms" ] "cpms" + "( ⦃?,?⦄ ⊢ ? ➡*[?,?] ? )" "cpms_drops" + "cpms_lsubr" + "cpms_rdeq" + "cpms_aaa" + "cpms_lpr" + "cpms_cpxs" + "cpms_fpbs" + "cpms_fpbg" + "cpms_cpms" * ]
}
]
[ { "unbound context-sensitive parallel rst-computation" * } {
- [ [ "strongly normalizing for closures" ] "fsb" + "( ≥[?,?] 𝐒⦃?,?,?⦄ )" "fsb_fdeq" + "fsb_aaa" + "fsb_csx" + "fsb_fpbg" * ]
- [ [ "proper for closures" ] "fpbg" + "( ⦃?,?,?⦄ >[?,?] ⦃?,?,?⦄ )" "fpbg_fqup" + "fpbg_cpxs" + "fpbg_lpxs" + "fpbg_fpbs" + "fpbg_fpbg" * ]
- [ [ "for closures" ] "fpbs" + "( ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_fqup" + "fpbs_fqus" + "fpbs_aaa" + "fpbs_cpx" + "fpbs_fpb" + "fpbs_cpxs" + "fpbs_lpxs" + "fpbs_csx" + "fpbs_fpbs" * ]
+ [ [ "strongly normalizing for closures" ] "fsb" + "( ≥[?] 𝐒⦃?,?,?⦄ )" "fsb_fdeq" + "fsb_aaa" + "fsb_csx" + "fsb_fpbg" * ]
+ [ [ "proper for closures" ] "fpbg" + "( ⦃?,?,?⦄ >[?] ⦃?,?,?⦄ )" "fpbg_fqup" + "fpbg_cpxs" + "fpbg_lpxs" + "fpbg_fpbs" + "fpbg_fpbg" * ]
+ [ [ "for closures" ] "fpbs" + "( ⦃?,?,?⦄ ≥[?] ⦃?,?,?⦄ )" "fpbs_fqup" + "fpbs_fqus" + "fpbs_aaa" + "fpbs_cpx" + "fpbs_fpb" + "fpbs_cpxs" + "fpbs_lpxs" + "fpbs_csx" + "fpbs_fpbs" * ]
}
]
[ { "unbound context-sensitive parallel rt-computation" * } {
- [ [ "refinement for lenvs on selected entries" ] "lsubsx" + "( ? ⊢ ? ⊆ⓧ[?,?,?] ? )" "lsubsx_lfsx" + "lsubsx_lsubsx" * ]
- [ [ "strongly normalizing for lenvs on referred entries" ] "rdsx" + "( ? ⊢ ⬈*[?,?,?] 𝐒⦃?⦄ )" "rdsx_length" + "rdsx_drops" + "rdsx_fqup" + "rdsx_cpxs" + "rdsx_csx" + "rdsx_rdsx" * ]
- [ [ "strongly normalizing for term vectors" ] "csx_vector" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_cnx_vector" + "csx_csx_vector" * ]
- [ [ "strongly normalizing for terms" ] "csx" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_simple" + "csx_simple_theq" + "csx_drops" + "csx_fqus" + "csx_lsubr" + "csx_rdeq" + "csx_fdeq" + "csx_aaa" + "csx_gcp" + "csx_gcr" + "csx_lpx" + "csx_cnx" + "csx_fpbq" + "csx_cpxs" + "csx_lpxs" + "csx_csx" * ]
+ [ [ "compatibility for lenvs" ] "jsx" + "( ? ⊢ ? ⊒[?] ? )" "jsx_drops" + "jsx_lsubr" + "jsx_csx" + "jsx_rsx" + "jsx_jsx" * ]
+ [ [ "strongly normalizing for lenvs on referred entries" ] "rsx" + "( ? ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "rsx_length" + "rsx_drops" + "rsx_fqup" + "rsx_cpxs" + "rsx_csx" + "rsx_rsx" * ]
+ [ [ "strongly normalizing for term vectors" ] "csx_vector" + "( ⦃?,?⦄ ⊢ ⬈*[?] 𝐒⦃?⦄ )" "csx_cnx_vector" + "csx_csx_vector" * ]
+ [ [ "strongly normalizing for terms" ] "csx" + "( ⦃?,?⦄ ⊢ ⬈*[?] 𝐒⦃?⦄ )" "csx_simple" + "csx_simple_toeq" + "csx_drops" + "csx_fqus" + "csx_lsubr" + "csx_rdeq" + "csx_fdeq" + "csx_aaa" + "csx_gcp" + "csx_gcr" + "csx_lpx" + "csx_cnx" + "csx_fpbq" + "csx_cpxs" + "csx_lpxs" + "csx_csx" * ]
[ [ "for lenvs on all entries" ] "lpxs" + "( ⦃?,?⦄ ⊢ ⬈*[?] ? )" "lpxs_length" + "lpxs_drops" + "lpxs_rdeq" + "lpxs_fdeq" + "lpxs_aaa" + "lpxs_lpx" + "lpxs_cpxs" + "lpxs_lpxs" * ]
[ [ "for binders" ] "cpxs_ext" + "( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" * ]
- [ [ "for terms" ] "cpxs" + "( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_theq" + "cpxs_theq_vector" + "cpxs_drops" + "cpxs_fqus" + "cpxs_lsubr" + "cpxs_rdeq" + "cpxs_fdeq" + "cpxs_aaa" + "cpxs_lpx" + "cpxs_cnx" + "cpxs_cpxs" * ]
+ [ [ "for terms" ] "cpxs" + "( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_toeq" + "cpxs_toeq_vector" + "cpxs_drops" + "cpxs_fqus" + "cpxs_lsubr" + "cpxs_rdeq" + "cpxs_fdeq" + "cpxs_aaa" + "cpxs_lpx" + "cpxs_cnx" + "cpxs_cpxs" * ]
}
]
}
]
class "cyan"
[ { "rt-transition" * } {
- [ { "unbound parallel rst-transition" * } {
- [ [ "for closures" ] "fpbq" + "( ⦃?,?,?⦄ ≽[?,?] ⦃?,?,?⦄ )" "fpbq_aaa" + "fpbq_fpb" * ]
- [ [ "proper for closures" ] "fpb" + "( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" "fpb_rdeq" + "fpb_fdeq" * ]
- }
- ]
[ { "context-sensitive parallel r-transition" * } {
+ [ [ "normal form for terms" ] "cnr ( ⦃?,?⦄ ⊢ ➡[?] 𝐍⦃?⦄ )" "cnr_simple" + "cnr_tdeq" + "cnr_drops" * ]
[ [ "for lenvs on all entries" ] "lpr" + "( ⦃?,?⦄ ⊢ ➡[?] ? )" "lpr_length" + "lpr_drops" + "lpr_fquq" + "lpr_aaa" + "lpr_lpx" + "lpr_lpr" * ]
[ [ "for binders" ] "cpr_ext" + "( ⦃?,?⦄ ⊢ ? ➡[?] ? )" * ]
- [ [ "for terms" ] "cpr" + "( ⦃?,?⦄ ⊢ ? ➡[?] ? )" "cpr_drops" + "cpr_cpr" * ]
+ [ [ "for terms" ] "cpr" + "( ⦃?,?⦄ ⊢ ? ➡[?] ? )" "cpr_drops" + "cpr_drops_basic" + "cpr_tdeq" + "cpr_cpr" * ]
}
]
[ { "t-bound context-sensitive parallel rt-transition" * } {
[ [ "for terms" ] "cpm" + "( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpm_simple" + "cpm_tdeq" + "cpm_drops" + "cpm_lsubr" + "cpm_fsle" + "cpm_aaa" + "cpm_cpx" * ]
}
]
+ [ { "unbound parallel rst-transition" * } {
+ [ [ "for closures" ] "fpbq" + "( ⦃?,?,?⦄ ≽[?] ⦃?,?,?⦄ )" "fpbq_aaa" + "fpbq_fpb" * ]
+ [ [ "proper for closures" ] "fpb" + "( ⦃?,?,?⦄ ≻[?] ⦃?,?,?⦄ )" "fpb_rdeq" + "fpb_fdeq" * ]
+ }
+ ]
[ { "unbound context-sensitive parallel rt-transition" * } {
- [ [ "normal form for terms" ] "cnx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] 𝐍⦃?⦄ )" "cnx_simple" + "cnx_drops" + "cnx_cnx" * ]
+ [ [ "normal form for terms" ] "cnx" + "( ⦃?,?⦄ ⊢ ⬈[?] 𝐍⦃?⦄ )" "cnx_simple" + "cnx_drops" + "cnx_basic" + "cnx_cnx" * ]
[ [ "for lenvs on referred entries" ] "rpx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "rpx_length" + "rpx_drops" + "rpx_fqup" + "rpx_fsle" + "rpx_rdeq" + "rpx_lpx" + "rpx_rpx" * ]
[ [ "for lenvs on all entries" ] "lpx" + "( ⦃?,?⦄ ⊢ ⬈[?] ? )" "lpx_length" + "lpx_drops" + "lpx_fquq" + "lpx_fsle" + "lpx_rdeq" + "lpx_aaa" * ]
[ [ "for binders" ] "cpx_ext" + "( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" * ]
- [ [ "for terms" ] "cpx" + "( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_fqus" + "cpx_lsubr" + "cpx_req" + "cpx_rdeq" + "cpx_fdeq" * ]
+ [ [ "for terms" ] "cpx" + "( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_drops_basic" + "cpx_fqus" + "cpx_lsubr" + "cpx_req" + "cpx_rdeq" + "cpx_fdeq" * ]
}
]
[ { "bound context-sensitive parallel rt-transition" * } {
]
[ [ "" ] "shnv ( ⦃?,?⦄ ⊢ ? ¡[?,?,?] )" * ]
[ { "decomposed rt-equivalence" * } {
- [ [ "" ] "scpes ( ⦃?,?⦄ ⊢ ? •*⬌*[?,?,?,?] ? )" "scpes_aaa" + "scpes_cpcs" + "scpes_scpes" * ]
+ "scpes_cpcs" + "scpes_scpes"
}
]
[ [ "for lenvs on referred entries" ] "rpxs" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] ? )" "rpxs_length" + "rpxs_drops" + "rpxs_fqup" + "rpxs_rdeq" + "rpxs_fdeq" + "rpxs_aaa" + "rpxs_cpxs" + "rpxs_lpxs" + "rpxs_rpxs" * ]
[ [ "" ] "cpxe ( ⦃?,?⦄ ⊢ ➡*[?,?] 𝐍⦃?⦄ )" * ]
}
]
- [ { "evaluation for context-sensitive reduction" * } {
- [ [ "" ] "cpre ( ⦃?,?⦄ ⊢ ➡* 𝐍⦃?⦄ )" "cpre_cpre" * ]
- }
- ]
[ { "normal forms for context-sensitive rt-reduction" * } {
[ [ "" ] "cnx_crx" + "cnx_cix" * ]
}
}
]
[ { "normal forms for context-sensitive reduction" * } {
- [ [ "" ] "cnr ( ⦃?,?⦄ ⊢ ➡ 𝐍⦃?⦄ )" "cnr_lift" + "cnr_crr" + "cnr_cir" * ]
+ "cnr_crr" + "cnr_cir"
}
]
[ { "irreducible forms for context-sensitive reduction" * } {
-../../matitac.opt `cat partial.txt`
+../../matitac.opt ground_2
+../../matitac.opt static_2
+../../matitac.opt basic_2
+../../matitac.opt apps_2
#x #y #H elim (decidable_lt x y) /2 width=1 by not_lt_to_le/
#Hxy elim (H Hxy)
qed-.
+
+lemma arith_m2 (x) (y): x < y → x+(y-↑x) = ↓y.
+#x #y #Hxy >minus_minus [|*: // ] <minus_Sn_n //
+qed-.
lemma minus_plus_m_m_commutative: ∀n,m:nat. n = m + n - m.
// qed-.
+lemma plus_minus_m_m_commutative (n) (m): m ≤ n → n = m+(n-m).
+/2 width=1 by plus_minus_associative/ qed-.
+
lemma plus_to_minus_2: ∀m1,m2,n1,n2. n1 ≤ m1 → n2 ≤ m2 →
m1+n2 = m2+n1 → m1-n1 = m2-n2.
#m1 #m2 #n1 #n2 #H1 #H2 #H
lemma le_plus_to_minus_comm: ∀n,m,p. n ≤ p+m → n-p ≤ m.
/2 width=1 by le_plus_to_minus/ qed-.
+lemma le_inv_S1: ∀m,n. ↑m ≤ n → ∃∃p. m ≤ p & ↑p = n.
+#m *
+[ #H lapply (le_n_O_to_eq … H) -H
+ #H destruct
+| /3 width=3 by monotonic_pred, ex2_intro/
+]
+qed-.
+
(* Note: this might interfere with nat.ma *)
lemma monotonic_lt_pred: ∀m,n. m < n → 0 < m → pred m < pred n.
#m #n #Hmn #Hm whd >(S_pred … Hm)
lemma lt_le_false: ∀x,y. x < y → y ≤ x → ⊥.
/3 width=4 by lt_refl_false, lt_to_le_to_lt/ qed-.
+lemma le_dec (n) (m): Decidable (n≤m).
+#n elim n -n [ /2 width=1 by or_introl/ ]
+#n #IH * [ /3 width=2 by lt_zero_false, or_intror/ ]
+#m elim (IH m) -IH
+[ /3 width=1 by or_introl, le_S_S/
+| /4 width=1 by or_intror, le_S_S_to_le/
+]
+qed-.
+
lemma succ_inv_refl_sn: ∀x. ↑x = x → ⊥.
#x #H @(lt_le_false x (↑x)) //
qed-.
| #n1 #IH #n2 elim n2 -n2 /3 width=1 by monotonic_lt_pred/
]
qed.
+
+(* Decidability of predicates ***********************************************)
+
+lemma dec_lt (R:predicate nat):
+ (∀n. Decidable … (R n)) →
+ ∀n. Decidable … (∃∃m. m < n & R m).
+#R #HR #n elim n -n [| #n * ]
+[ @or_intror * /2 width=2 by lt_zero_false/
+| * /4 width=3 by lt_S, or_introl, ex2_intro/
+| #H0 elim (HR n) -HR
+ [ /3 width=3 by or_introl, ex2_intro/
+ | #Hn @or_intror * #m #Hmn #Hm
+ elim (le_to_or_lt_eq … Hmn) -Hmn #H destruct [ -Hn | -H0 ]
+ /4 width=3 by lt_S_S_to_lt, ex2_intro/
+ ]
+]
+qed-.
+
+lemma dec_min (R:predicate nat):
+ (∀n. Decidable … (R n)) → ∀n. R n →
+ ∃∃m. m ≤ n & R m & (∀p. p < m → R p → ⊥).
+#R #HR #n
+@(nat_elim1 n) -n #n #IH #Hn
+elim (dec_lt … HR n) -HR [ -Hn | -IH ]
+[ * #p #Hpn #Hp
+ elim (IH … Hpn Hp) -IH -Hp #m #Hmp #Hm #HNm
+ @(ex3_intro … Hm HNm) -HNm
+ /3 width=3 by lt_to_le, le_to_lt_to_lt/
+| /4 width=4 by ex3_intro, ex2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/notation/relations/ringeq_3.ma".
+include "ground_2/lib/list.ma".
+
+(* EXTENSIONAL EQUIVALENCE OF LISTS *****************************************)
+
+rec definition eq_list A (l1,l2:list A) on l1 ≝
+match l1 with
+[ nil ⇒
+ match l2 with
+ [ nil ⇒ ⊤
+ | cons _ _ ⇒ ⊥
+ ]
+| cons a1 l1 ⇒
+ match l2 with
+ [ nil ⇒ ⊥
+ | cons a2 l2 ⇒ a1 = a2 ∧ eq_list A l1 l2
+ ]
+].
+
+interpretation "extensional equivalence (list)"
+ 'RingEq A l1 l2 = (eq_list A l1 l2).
+
+(* Basic properties *********************************************************)
+
+lemma eq_list_refl (A): reflexive … (eq_list A).
+#A #l elim l -l /2 width=1 by conj/
+qed.
+
+(* Main properties **********************************************************)
+
+theorem eq_eq_list (A,l1,l2): l1 = l2 → l1 ≗{A} l2.
+// qed.
+
+(* Main inversion propertiess ***********************************************)
+
+theorem eq_list_inv_eq (A,l1,l2): l1 ≗{A} l2 → l1 = l2.
+#A #l1 elim l1 -l1 [| #a1 #l1 #IH ] *
+[ //
+| #a2 #l2 #H elim H
+| #H elim H
+| #a2 #l2 * #Ha #Hl /3 width=1 by eq_f2/
+]
+qed-.
(* GENERIC RELATIONS ********************************************************)
definition replace_2 (A) (B): relation3 (relation2 A B) (relation A) (relation B) ≝
- λR,Sa,Sb. ∀a1,b1. R a1 b1 → ∀a2. Sa a1 a2 → ∀b2. Sb b1 b2 → R a2 b2.
+ λR,Sa,Sb. ∀a1,b1. R a1 b1 → ∀a2. Sa a1 a2 → ∀b2. Sb b1 b2 → R a2 b2.
(* Inclusion ****************************************************************)
(* Properties of relations **************************************************)
-definition relation5: Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
-≝ λA,B,C,D,E.A→B→C→D→E→Prop.
+definition relation5: Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] ≝
+ λA,B,C,D,E.A→B→C→D→E→Prop.
-definition relation6: Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
-≝ λA,B,C,D,E,F.A→B→C→D→E→F→Prop.
+definition relation6: Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] ≝
+ λA,B,C,D,E,F.A→B→C→D→E→F→Prop.
-(**) (* we dont use "∀a. reflexive … (R a)" since auto seems to dislike repeatd δ-expansion *)
+(**) (* we don't use "∀a. reflexive … (R a)" since auto seems to dislike repeatd δ-expansion *)
definition c_reflexive (A) (B): predicate (relation3 A B B) ≝
- λR. ∀a,b. R a b b.
+ λR. ∀a,b. R a b b.
definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
-definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
- ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
+definition Transitive (A) (R:relation A): Prop ≝
+ ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
-definition left_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R.
- ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → R a1 a2.
+definition left_cancellable (A) (R:relation A): Prop ≝
+ ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 → R a1 a2.
-definition right_cancellable: ∀A. ∀R: relation A. Prop ≝ λA,R.
- ∀a1,a0. R a1 a0 → ∀a2. R a2 a0 → R a1 a2.
+definition right_cancellable (A) (R:relation A): Prop ≝
+ ∀a1,a0. R a1 a0 → ∀a2. R a2 a0 → R a1 a2.
-definition pw_confluent2: ∀A. relation A → relation A → predicate A ≝ λA,R1,R2,a0.
- ∀a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
- ∃∃a. R2 a1 a & R1 a2 a.
+definition pw_confluent2 (A) (R1,R2:relation A): predicate A ≝
+ λa0.
+ ∀a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
+ ∃∃a. R2 a1 a & R1 a2 a.
-definition confluent2: ∀A. relation (relation A) ≝ λA,R1,R2.
- ∀a0. pw_confluent2 A R1 R2 a0.
+definition confluent2 (A): relation (relation A) ≝
+ λR1,R2.
+ ∀a0. pw_confluent2 A R1 R2 a0.
-definition transitive2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
- ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
- ∃∃a. R2 a1 a & R1 a a2.
+definition transitive2 (A) (R1,R2:relation A): Prop ≝
+ ∀a1,a0. R1 a1 a0 → ∀a2. R2 a0 a2 →
+ ∃∃a. R2 a1 a & R1 a a2.
-definition bi_confluent: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
- ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
- ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
+definition bi_confluent (A) (B) (R: bi_relation A B): Prop ≝
+ ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
+ ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
-definition lsub_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
- ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2.
+definition lsub_trans (A) (B): relation2 (A→relation B) (relation A) ≝
+ λR1,R2.
+ ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → R1 L1 T1 T2.
-definition s_r_confluent1: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
- ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2.
+definition s_r_confluent1 (A) (B): relation2 (A→relation B) (B→relation A) ≝
+ λR1,R2.
+ ∀L1,T1,T2. R1 L1 T1 T2 → ∀L2. R2 T1 L1 L2 → R2 T2 L1 L2.
-definition is_mono: ∀B:Type[0]. predicate (predicate B) ≝
- λB,R. ∀b1. R b1 → ∀b2. R b2 → b1 = b2.
+definition is_mono (B:Type[0]): predicate (predicate B) ≝
+ λR. ∀b1. R b1 → ∀b2. R b2 → b1 = b2.
-definition is_inj2: ∀A,B:Type[0]. predicate (relation2 A B) ≝
- λA,B,R. ∀a1,b. R a1 b → ∀a2. R a2 b → a1 = a2.
+definition is_inj2 (A,B:Type[0]): predicate (relation2 A B) ≝
+ λR. ∀a1,b. R a1 b → ∀a2. R a2 b → a1 = a2.
+
+(* Main properties of equality **********************************************)
+
+theorem canc_sn_eq (A): left_cancellable A (eq …).
+// qed-.
+
+theorem canc_dx_eq (A): right_cancellable A (eq …).
+// qed-.
(* Normal form and strong normalization *************************************)
-definition NF: ∀A. relation A → relation A → predicate A ≝
- λA,R,S,a1. ∀a2. R a1 a2 → S a1 a2.
+definition NF (A): relation A → relation A → predicate A ≝
+ λR,S,a1. ∀a2. R a1 a2 → S a1 a2.
-definition NF_dec: ∀A. relation A → relation A → Prop ≝
- λA,R,S. ∀a1. NF A R S a1 ∨
- ∃∃a2. R … a1 a2 & (S a1 a2 → ⊥).
+definition NF_dec (A): relation A → relation A → Prop ≝
+ λR,S. ∀a1. NF A R S a1 ∨
+ ∃∃a2. R … a1 a2 & (S a1 a2 → ⊥).
inductive SN (A) (R,S:relation A): predicate A ≝
| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a1 a2 → ⊥) → SN A R S a2) → SN A R S a1
.
-lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
+lemma NF_to_SN (A) (R) (S): ∀a. NF A R S a → SN A R S a.
#A #R #S #a1 #Ha1
@SN_intro #a2 #HRa12 #HSa12
elim HSa12 -HSa12 /2 width=1 by/
qed.
-definition NF_sn: ∀A. relation A → relation A → predicate A ≝
- λA,R,S,a2. ∀a1. R a1 a2 → S a1 a2.
+definition NF_sn (A): relation A → relation A → predicate A ≝
+ λR,S,a2. ∀a1. R a1 a2 → S a1 a2.
inductive SN_sn (A) (R,S:relation A): predicate A ≝
| SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a1 a2 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
.
-lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
+lemma NF_to_SN_sn (A) (R) (S): ∀a. NF_sn A R S a → SN_sn A R S a.
#A #R #S #a2 #Ha2
@SN_sn_intro #a1 #HRa12 #HSa12
elim HSa12 -HSa12 /2 width=1 by/
(* Relations on unboxed triples *********************************************)
-definition tri_RC: ∀A,B,C. tri_relation A B C → tri_relation A B C ≝
- λA,B,C,R,a1,b1,c1,a2,b2,c2. R … a1 b1 c1 a2 b2 c2 ∨
- ∧∧ a1 = a2 & b1 = b2 & c1 = c2.
+definition tri_RC (A,B,C): tri_relation A B C → tri_relation A B C ≝
+ λR,a1,b1,c1,a2,b2,c2.
+ ∨∨ R … a1 b1 c1 a2 b2 c2
+ | ∧∧ a1 = a2 & b1 = b2 & c1 = c2.
-lemma tri_RC_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_RC … R).
+lemma tri_RC_reflexive (A) (B) (C): ∀R. tri_reflexive A B C (tri_RC … R).
/3 width=1 by and3_intro, or_intror/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/notation/functions/oplusright_3.ma".
+include "ground_2/lib/relations.ma".
+
+(* STREAMS ******************************************************************)
+
+coinductive stream (A:Type[0]): Type[0] ≝
+| seq: A → stream A → stream A
+.
+
+interpretation "cons (stream)" 'OPlusRight A a u = (seq A a u).
+
+(* Basic properties *********************************************************)
+
+lemma stream_rew (A) (t:stream A): match t with [ seq a u ⇒ a ⨮ u ] = t.
+#A * //
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/notation/relations/ringeq_3.ma".
+include "ground_2/lib/stream.ma".
+
+(* STREAMS ******************************************************************)
+
+coinductive eq_stream (A): relation (stream A) ≝
+| eq_seq: ∀t1,t2,b1,b2. b1 = b2 → eq_stream A t1 t2 → eq_stream A (b1⨮t1) (b2⨮t2)
+.
+
+interpretation "extensional equivalence (stream)"
+ 'RingEq A t1 t2 = (eq_stream A t1 t2).
+
+definition eq_stream_repl (A) (R:relation …) ≝
+ ∀t1,t2. t1 ≗{A} t2 → R t1 t2.
+
+definition eq_stream_repl_back (A) (R:predicate …) ≝
+ ∀t1. R t1 → ∀t2. t1 ≗{A} t2 → R t2.
+
+definition eq_stream_repl_fwd (A) (R:predicate …) ≝
+ ∀t1. R t1 → ∀t2. t2 ≗{A} t1 → R t2.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma eq_stream_inv_seq: ∀A,t1,t2. t1 ≗{A} t2 →
+ ∀u1,u2,a1,a2. a1⨮u1 = t1 → a2⨮u2 = t2 →
+ u1 ≗ u2 ∧ a1 = a2.
+#A #t1 #t2 * -t1 -t2
+#t1 #t2 #b1 #b2 #Hb #Ht #u1 #u2 #a1 #a2 #H1 #H2 destruct /2 width=1 by conj/
+qed-.
+
+(* Basic properties *********************************************************)
+
+corec lemma eq_stream_refl: ∀A. reflexive … (eq_stream A).
+#A * #b #t @eq_seq //
+qed.
+
+corec lemma eq_stream_sym: ∀A. symmetric … (eq_stream A).
+#A #t1 #t2 * -t1 -t2
+#t1 #t2 #b1 #b2 #Hb #Ht @eq_seq /2 width=1 by/
+qed-.
+
+lemma eq_stream_repl_sym: ∀A,R. eq_stream_repl_back A R → eq_stream_repl_fwd A R.
+/3 width=3 by eq_stream_sym/ qed-.
+
+(* Main properties **********************************************************)
+
+corec theorem eq_stream_trans: ∀A. Transitive … (eq_stream A).
+#A #t1 #t * -t1 -t
+#t1 #t #b1 #b * #Ht1 * #b2 #t2 #H cases (eq_stream_inv_seq A … H) -H -b
+/3 width=7 by eq_seq/
+qed-.
+
+theorem eq_stream_canc_sn: ∀A,t,t1,t2. t ≗ t1 → t ≗ t2 → t1 ≗{A} t2.
+/3 width=3 by eq_stream_trans, eq_stream_sym/ qed-.
+
+theorem eq_stream_canc_dx: ∀A,t,t1,t2. t1 ≗ t → t2 ≗ t → t1 ≗{A} t2.
+/3 width=3 by eq_stream_trans, eq_stream_sym/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/notation/functions/downspoon_2.ma".
+include "ground_2/lib/stream_eq.ma".
+include "ground_2/lib/arith.ma".
+
+(* STREAMS ******************************************************************)
+
+definition hd (A:Type[0]): stream A → A ≝
+ λt. match t with [ seq a _ ⇒ a ].
+
+definition tl (A:Type[0]): stream A → stream A ≝
+ λt. match t with [ seq _ t ⇒ t ].
+
+interpretation "tail (stream)" 'DownSpoon A t = (tl A t).
+
+(* basic properties *********************************************************)
+
+lemma hd_rew (A) (a) (t): a = hd A (a⨮t).
+// qed.
+
+lemma tl_rew (A) (a) (t): t = tl A (a⨮t).
+// qed.
+
+lemma eq_stream_split (A) (t): (hd … t) ⨮ ⫰t ≗{A} t.
+#A * //
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/notation/functions/downspoonstar_3.ma".
+include "ground_2/lib/stream_hdtl.ma".
+
+(* STREAMS ******************************************************************)
+
+rec definition tls (A:Type[0]) (n:nat) on n: stream A → stream A ≝ ?.
+cases n -n [ #t @t | #n #t @tl @(tls … n t) ]
+defined.
+
+interpretation "iterated tail (stram)" 'DownSpoonStar A n f = (tls A n f).
+
+(* basic properties *********************************************************)
+
+lemma tls_rew_O (A) (t): t = tls A 0 t.
+// qed.
+
+lemma tls_rew_S (A) (n) (t): ⫰⫰*[n]t = tls A (↑n) t.
+// qed.
+
+lemma tls_S1 (A) (n) (t): ⫰*[n]⫰t = tls A (↑n) t.
+#A #n elim n -n //
+qed.
+
+lemma tls_eq_repl (A) (n): eq_stream_repl A (λt1,t2. ⫰*[n] t1 ≗ ⫰*[n] t2).
+#A #n elim n -n //
+#n #IH * #n1 #t1 * #n2 #t2 #H elim (eq_stream_inv_seq … H) /2 width=7 by/
+qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/notation/functions/oplusright_3.ma".
-include "ground_2/lib/relations.ma".
-
-(* STREAMS ******************************************************************)
-
-coinductive stream (A:Type[0]): Type[0] ≝
-| seq: A → stream A → stream A
-.
-
-interpretation "cons (nstream)" 'OPlusRight A a u = (seq A a u).
-
-(* Basic properties *********************************************************)
-
-lemma stream_rew (A) (t:stream A): match t with [ seq a u ⇒ a ⨮ u ] = t.
-#A * //
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/notation/relations/ringeq_3.ma".
-include "ground_2/lib/streams.ma".
-
-(* STREAMS ******************************************************************)
-
-coinductive eq_stream (A): relation (stream A) ≝
-| eq_seq: ∀t1,t2,b1,b2. b1 = b2 → eq_stream A t1 t2 → eq_stream A (b1⨮t1) (b2⨮t2)
-.
-
-interpretation "extensional equivalence (nstream)"
- 'RingEq A t1 t2 = (eq_stream A t1 t2).
-
-definition eq_stream_repl (A) (R:relation …) ≝
- ∀t1,t2. t1 ≗{A} t2 → R t1 t2.
-
-definition eq_stream_repl_back (A) (R:predicate …) ≝
- ∀t1. R t1 → ∀t2. t1 ≗{A} t2 → R t2.
-
-definition eq_stream_repl_fwd (A) (R:predicate …) ≝
- ∀t1. R t1 → ∀t2. t2 ≗{A} t1 → R t2.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma eq_stream_inv_seq: ∀A,t1,t2. t1 ≗{A} t2 →
- ∀u1,u2,a1,a2. a1⨮u1 = t1 → a2⨮u2 = t2 →
- u1 ≗ u2 ∧ a1 = a2.
-#A #t1 #t2 * -t1 -t2
-#t1 #t2 #b1 #b2 #Hb #Ht #u1 #u2 #a1 #a2 #H1 #H2 destruct /2 width=1 by conj/
-qed-.
-
-(* Basic properties *********************************************************)
-
-corec lemma eq_stream_refl: ∀A. reflexive … (eq_stream A).
-#A * #b #t @eq_seq //
-qed.
-
-corec lemma eq_stream_sym: ∀A. symmetric … (eq_stream A).
-#A #t1 #t2 * -t1 -t2
-#t1 #t2 #b1 #b2 #Hb #Ht @eq_seq /2 width=1 by/
-qed-.
-
-lemma eq_stream_repl_sym: ∀A,R. eq_stream_repl_back A R → eq_stream_repl_fwd A R.
-/3 width=3 by eq_stream_sym/ qed-.
-
-(* Main properties **********************************************************)
-
-corec theorem eq_stream_trans: ∀A. Transitive … (eq_stream A).
-#A #t1 #t * -t1 -t
-#t1 #t #b1 #b * #Ht1 * #b2 #t2 #H cases (eq_stream_inv_seq A … H) -H -b
-/3 width=7 by eq_seq/
-qed-.
-
-theorem eq_stream_canc_sn: ∀A,t,t1,t2. t ≗ t1 → t ≗ t2 → t1 ≗{A} t2.
-/3 width=3 by eq_stream_trans, eq_stream_sym/ qed-.
-
-theorem eq_stream_canc_dx: ∀A,t,t1,t2. t1 ≗ t → t2 ≗ t → t1 ≗{A} t2.
-/3 width=3 by eq_stream_trans, eq_stream_sym/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/notation/functions/downspoon_2.ma".
-include "ground_2/lib/streams_eq.ma".
-include "ground_2/lib/arith.ma".
-
-(* STREAMS ******************************************************************)
-
-definition hd (A:Type[0]): stream A → A ≝
- λt. match t with [ seq a _ ⇒ a ].
-
-definition tl (A:Type[0]): stream A → stream A ≝
- λt. match t with [ seq _ t ⇒ t ].
-
-interpretation "tail (streams)" 'DownSpoon A t = (tl A t).
-
-(* basic properties *********************************************************)
-
-lemma hd_rew (A) (a) (t): a = hd A (a⨮t).
-// qed.
-
-lemma tl_rew (A) (a) (t): t = tl A (a⨮t).
-// qed.
-
-lemma eq_stream_split (A) (t): (hd … t) ⨮ ⫰t ≗{A} t.
-#A * //
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/notation/functions/downspoonstar_3.ma".
-include "ground_2/lib/streams_hdtl.ma".
-
-(* STREAMS ******************************************************************)
-
-rec definition tls (A:Type[0]) (n:nat) on n: stream A → stream A ≝ ?.
-cases n -n [ #t @t | #n #t @tl @(tls … n t) ]
-defined.
-
-interpretation "iterated tail (strams)" 'DownSpoonStar A n f = (tls A n f).
-
-(* basic properties *********************************************************)
-
-lemma tls_rew_O (A) (t): t = tls A 0 t.
-// qed.
-
-lemma tls_rew_S (A) (n) (t): ⫰⫰*[n]t = tls A (↑n) t.
-// qed.
-
-lemma tls_S1 (A) (n) (t): ⫰*[n]⫰t = tls A (↑n) t.
-#A #n elim n -n //
-qed.
-
-lemma tls_eq_repl (A) (n): eq_stream_repl A (λt1,t2. ⫰*[n] t1 ≗ ⫰*[n] t2).
-#A #n elim n -n //
-#n #IH * #n1 #t1 * #n2 #t2 #H elim (eq_stream_inv_seq … H) /2 width=7 by/
-qed.
(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-notation "hvbox( 𝐁❴ break term 46 l, break term 46 h ❵ )"
+notation "hvbox( 𝐁❴ term 46 l, break term 46 h ❵ )"
non associative with precedence 90
for @{ 'Basic $l $h }.
(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-notation "hvbox(f2 ~ \circ break f1)"
+notation "hvbox(f2 ~ \circ break f1)" (**)
right associative with precedence 60
for @{ 'CoCompose $f2 $f1 }.
(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-notation "hvbox( ⫱ * [ term 46 n ] term 46 T )"
+notation "hvbox( ⫱ * [ term 46 n ] break term 46 T )"
non associative with precedence 46
for @{ 'DropPreds $n $T }.
(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-notation "hvbox( ↑ * [ term 46 n ] term 70 T )"
+notation "hvbox( ↑ * [ term 46 n ] break term 70 T )"
non associative with precedence 70
for @{ 'UpArrowStar $n $T }.
(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-notation "hvbox( ⫯ * [ term 46 n ] term 46 T )"
+notation "hvbox( ⫯ * [ term 46 n ] break term 46 T )"
non associative with precedence 46
for @{ 'UpSpoonStar $n $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+(* multiple existental quantifier (1, 4) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0)"
+ non associative with precedence 20
+ for @{ 'Ex4 (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0)"
+ non associative with precedence 20
+ for @{ 'Ex4 (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) }.
+
rec definition pluss (cs:mr2) (i:nat) on cs ≝ match cs with
[ nil2 ⇒ ◊
-| cons2 l m cs ⇒ {l + i, m};pluss cs i
+| cons2 l m cs ⇒ {l + i,m};pluss cs i
].
interpretation "plus (multiple relocation with pairs)"
(* Basic properties *********************************************************)
-lemma pluss_SO2: ∀l,m,cs. ({l, m};cs) + 1 = {↑l, m};cs + 1.
+lemma pluss_SO2: ∀l,m,cs. ({l,m};cs) + 1 = {↑l,m};cs + 1.
normalize // qed.
(* Basic inversion lemmas ***************************************************)
#l #m #cs #H destruct
qed.
-lemma pluss_inv_cons2: ∀i,l,m,cs2,cs. cs + i = {l, m};cs2 →
- ∃∃cs1. cs1 + i = cs2 & cs = {l - i, m};cs1.
+lemma pluss_inv_cons2: ∀i,l,m,cs2,cs. cs + i = {l,m};cs2 →
+ ∃∃cs1. cs1 + i = cs2 & cs = {l - i,m};cs1.
#i #l #m #cs2 *
[ normalize #H destruct
| #l1 #m1 #cs1 whd in ⊢ (??%?→?); #H destruct
(**************************************************************************)
include "ground_2/notation/functions/upspoon_1.ma".
-include "ground_2/lib/streams_tls.ma".
+include "ground_2/lib/stream_tls.ma".
(* RELOCATION N-STREAM ******************************************************)
'RAt i1 f i2 = (at f i1 i2).
definition H_at_div: relation4 rtmap rtmap rtmap rtmap ≝ λf2,g2,f1,g1.
- ∀jf,jg,j. @⦃jf, f2⦄ ≘ j → @⦃jg, g2⦄ ≘ j →
- ∃∃j0. @⦃j0, f1⦄ ≘ jf & @⦃j0, g1⦄ ≘ jg.
+ ∀jf,jg,j. @⦃jf,f2⦄ ≘ j → @⦃jg,g2⦄ ≘ j →
+ ∃∃j0. @⦃j0,f1⦄ ≘ jf & @⦃j0,g1⦄ ≘ jg.
(* Basic inversion lemmas ***************************************************)
-lemma at_inv_ppx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. 0 = i1 → ⫯g = f → 0 = i2.
+lemma at_inv_ppx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. 0 = i1 → ⫯g = f → 0 = i2.
#f #i1 #i2 * -f -i1 -i2 //
[ #f #i1 #i2 #_ #g #j1 #j2 #_ * #_ #x #H destruct
| #f #i1 #i2 #_ #g #j2 * #_ #x #_ #H elim (discr_push_next … H)
]
qed-.
-lemma at_inv_npx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g,j1. ↑j1 = i1 → ⫯g = f →
- ∃∃j2. @⦃j1, g⦄ ≘ j2 & ↑j2 = i2.
+lemma at_inv_npx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g,j1. ↑j1 = i1 → ⫯g = f →
+ ∃∃j2. @⦃j1,g⦄ ≘ j2 & ↑j2 = i2.
#f #i1 #i2 * -f -i1 -i2
[ #f #g #j1 #j2 #_ * #_ #x #x1 #H destruct
| #f #i1 #i2 #Hi #g #j1 #j2 * * * #x #x1 #H #Hf >(injective_push … Hf) -g destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma at_inv_xnx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. ↑g = f →
- ∃∃j2. @⦃i1, g⦄ ≘ j2 & ↑j2 = i2.
+lemma at_inv_xnx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ↑g = f →
+ ∃∃j2. @⦃i1,g⦄ ≘ j2 & ↑j2 = i2.
#f #i1 #i2 * -f -i1 -i2
[ #f #g #j1 #j2 * #_ #_ #x #H elim (discr_next_push … H)
| #f #i1 #i2 #_ #g #j1 #j2 * #_ #_ #x #H elim (discr_next_push … H)
(* Advanced inversion lemmas ************************************************)
-lemma at_inv_ppn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
+lemma at_inv_ppn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
∀g,j2. 0 = i1 → ⫯g = f → ↑j2 = i2 → ⊥.
#f #i1 #i2 #Hf #g #j2 #H1 #H <(at_inv_ppx … Hf … H1 H) -f -g -i1 -i2
#H destruct
qed-.
-lemma at_inv_npp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
+lemma at_inv_npp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
∀g,j1. ↑j1 = i1 → ⫯g = f → 0 = i2 → ⊥.
#f #i1 #i2 #Hf #g #j1 #H1 #H elim (at_inv_npx … Hf … H1 H) -f -i1
#x2 #Hg * -i2 #H destruct
qed-.
-lemma at_inv_npn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
- ∀g,j1,j2. ↑j1 = i1 → ⫯g = f → ↑j2 = i2 → @⦃j1, g⦄ ≘ j2.
+lemma at_inv_npn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+ ∀g,j1,j2. ↑j1 = i1 → ⫯g = f → ↑j2 = i2 → @⦃j1,g⦄ ≘ j2.
#f #i1 #i2 #Hf #g #j1 #j2 #H1 #H elim (at_inv_npx … Hf … H1 H) -f -i1
#x2 #Hg * -i2 #H destruct //
qed-.
-lemma at_inv_xnp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
+lemma at_inv_xnp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
∀g. ↑g = f → 0 = i2 → ⊥.
#f #i1 #i2 #Hf #g #H elim (at_inv_xnx … Hf … H) -f
#x2 #Hg * -i2 #H destruct
qed-.
-lemma at_inv_xnn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
- ∀g,j2. ↑g = f → ↑j2 = i2 → @⦃i1, g⦄ ≘ j2.
+lemma at_inv_xnn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+ ∀g,j2. ↑g = f → ↑j2 = i2 → @⦃i1,g⦄ ≘ j2.
#f #i1 #i2 #Hf #g #j2 #H elim (at_inv_xnx … Hf … H) -f
#x2 #Hg * -i2 #H destruct //
qed-.
-lemma at_inv_pxp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → 0 = i1 → 0 = i2 → ∃g. ⫯g = f.
+lemma at_inv_pxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → 0 = i1 → 0 = i2 → ∃g. ⫯g = f.
#f elim (pn_split … f) * /2 width=2 by ex_intro/
#g #H #i1 #i2 #Hf #H1 #H2 cases (at_inv_xnp … Hf … H H2)
qed-.
-lemma at_inv_pxn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀j2. 0 = i1 → ↑j2 = i2 →
- ∃∃g. @⦃i1, g⦄ ≘ j2 & ↑g = f.
+lemma at_inv_pxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j2. 0 = i1 → ↑j2 = i2 →
+ ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f.
#f elim (pn_split … f) *
#g #H #i1 #i2 #Hf #j2 #H1 #H2
[ elim (at_inv_ppn … Hf … H1 H H2)
]
qed-.
-lemma at_inv_nxp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 →
+lemma at_inv_nxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
∀j1. ↑j1 = i1 → 0 = i2 → ⊥.
#f elim (pn_split f) *
#g #H #i1 #i2 #Hf #j1 #H1 #H2
]
qed-.
-lemma at_inv_nxn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀j1,j2. ↑j1 = i1 → ↑j2 = i2 →
- (∃∃g. @⦃j1, g⦄ ≘ j2 & ⫯g = f) ∨
- ∃∃g. @⦃i1, g⦄ ≘ j2 & ↑g = f.
+lemma at_inv_nxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j1,j2. ↑j1 = i1 → ↑j2 = i2 →
+ (∃∃g. @⦃j1,g⦄ ≘ j2 & ⫯g = f) ∨
+ ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f.
#f elim (pn_split f) *
/4 width=7 by at_inv_xnn, at_inv_npn, ex2_intro, or_intror, or_introl/
qed-.
(* Note: the following inversion lemmas must be checked *)
-lemma at_inv_xpx: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. ⫯g = f →
+lemma at_inv_xpx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ⫯g = f →
(0 = i1 ∧ 0 = i2) ∨
- ∃∃j1,j2. @⦃j1, g⦄ ≘ j2 & ↑j1 = i1 & ↑j2 = i2.
+ ∃∃j1,j2. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1 & ↑j2 = i2.
#f * [2: #i1 ] #i2 #Hf #g #H
[ elim (at_inv_npx … Hf … H) -f /3 width=5 by or_intror, ex3_2_intro/
| >(at_inv_ppx … Hf … H) -f /3 width=1 by conj, or_introl/
]
qed-.
-lemma at_inv_xpp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g. ⫯g = f → 0 = i2 → 0 = i1.
+lemma at_inv_xpp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ⫯g = f → 0 = i2 → 0 = i1.
#f #i1 #i2 #Hf #g #H elim (at_inv_xpx … Hf … H) -f * //
#j1 #j2 #_ #_ * -i2 #H destruct
qed-.
-lemma at_inv_xpn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀g,j2. ⫯g = f → ↑j2 = i2 →
- ∃∃j1. @⦃j1, g⦄ ≘ j2 & ↑j1 = i1.
+lemma at_inv_xpn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g,j2. ⫯g = f → ↑j2 = i2 →
+ ∃∃j1. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1.
#f #i1 #i2 #Hf #g #j2 #H elim (at_inv_xpx … Hf … H) -f *
[ #_ * -i2 #H destruct
| #x1 #x2 #Hg #H1 * -i2 #H destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma at_inv_xxp: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → 0 = i2 →
+lemma at_inv_xxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → 0 = i2 →
∃∃g. 0 = i1 & ⫯g = f.
#f elim (pn_split f) *
#g #H #i1 #i2 #Hf #H2
]
qed-.
-lemma at_inv_xxn: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → ∀j2. ↑j2 = i2 →
- (∃∃g,j1. @⦃j1, g⦄ ≘ j2 & ↑j1 = i1 & ⫯g = f) ∨
- ∃∃g. @⦃i1, g⦄ ≘ j2 & ↑g = f.
+lemma at_inv_xxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j2. ↑j2 = i2 →
+ (∃∃g,j1. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1 & ⫯g = f) ∨
+ ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f.
#f elim (pn_split f) *
#g #H #i1 #i2 #Hf #j2 #H2
[ elim (at_inv_xpn … Hf … H H2) -i2 /3 width=5 by or_introl, ex3_2_intro/
(* Basic forward lemmas *****************************************************)
-lemma at_increasing: ∀i2,i1,f. @⦃i1, f⦄ ≘ i2 → i1 ≤ i2.
+lemma at_increasing: ∀i2,i1,f. @⦃i1,f⦄ ≘ i2 → i1 ≤ i2.
#i2 elim i2 -i2
[ #i1 #f #Hf elim (at_inv_xxp … Hf) -Hf //
| #i2 #IH * //
]
qed-.
-lemma at_increasing_strict: ∀g,i1,i2. @⦃i1, g⦄ ≘ i2 → ∀f. ↑f = g →
- i1 < i2 ∧ @⦃i1, f⦄ ≘ ↓i2.
+lemma at_increasing_strict: ∀g,i1,i2. @⦃i1,g⦄ ≘ i2 → ∀f. ↑f = g →
+ i1 < i2 ∧ @⦃i1,f⦄ ≘ ↓i2.
#g #i1 #i2 #Hg #f #H elim (at_inv_xnx … Hg … H) -Hg -H
/4 width=2 by conj, at_increasing, le_S_S/
qed-.
-lemma at_fwd_id_ex: ∀f,i. @⦃i, f⦄ ≘ i → ∃g. ⫯g = f.
+lemma at_fwd_id_ex: ∀f,i. @⦃i,f⦄ ≘ i → ∃g. ⫯g = f.
#f elim (pn_split f) * /2 width=2 by ex_intro/
#g #H #i #Hf elim (at_inv_xnx … Hf … H) -Hf -H
#j2 #Hg #H destruct lapply (at_increasing … Hg) -Hg
(* Basic properties *********************************************************)
-corec lemma at_eq_repl_back: ∀i1,i2. eq_repl_back (λf. @⦃i1, f⦄ ≘ i2).
+corec lemma at_eq_repl_back: ∀i1,i2. eq_repl_back (λf. @⦃i1,f⦄ ≘ i2).
#i1 #i2 #f1 #H1 cases H1 -f1 -i1 -i2
[ #f1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12 cases (eq_inv_px … H12 … H) -g1 /2 width=2 by at_refl/
| #f1 #i1 #i2 #Hf1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12 cases (eq_inv_px … H12 … H) -g1 /3 width=7 by at_push/
]
qed-.
-lemma at_eq_repl_fwd: ∀i1,i2. eq_repl_fwd (λf. @⦃i1, f⦄ ≘ i2).
+lemma at_eq_repl_fwd: ∀i1,i2. eq_repl_fwd (λf. @⦃i1,f⦄ ≘ i2).
#i1 #i2 @eq_repl_sym /2 width=3 by at_eq_repl_back/
qed-.
-lemma at_le_ex: ∀j2,i2,f. @⦃i2, f⦄ ≘ j2 → ∀i1. i1 ≤ i2 →
- ∃∃j1. @⦃i1, f⦄ ≘ j1 & j1 ≤ j2.
+lemma at_le_ex: ∀j2,i2,f. @⦃i2,f⦄ ≘ j2 → ∀i1. i1 ≤ i2 →
+ ∃∃j1. @⦃i1,f⦄ ≘ j1 & j1 ≤ j2.
#j2 elim j2 -j2 [2: #j2 #IH ] #i2 #f #Hf
[ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
#g [ #x2 ] #Hg [ #H2 ] #H0
]
qed-.
-lemma at_id_le: ∀i1,i2. i1 ≤ i2 → ∀f. @⦃i2, f⦄ ≘ i2 → @⦃i1, f⦄ ≘ i1.
+lemma at_id_le: ∀i1,i2. i1 ≤ i2 → ∀f. @⦃i2,f⦄ ≘ i2 → @⦃i1,f⦄ ≘ i1.
#i1 #i2 #H @(le_elim … H) -i1 -i2 [ #i2 | #i1 #i2 #IH ]
#f #Hf elim (at_fwd_id_ex … Hf) /4 width=7 by at_inv_npn, at_push, at_refl/
qed-.
(* Main properties **********************************************************)
-theorem at_monotonic: ∀j2,i2,f. @⦃i2, f⦄ ≘ j2 → ∀j1,i1. @⦃i1, f⦄ ≘ j1 →
+theorem at_monotonic: ∀j2,i2,f. @⦃i2,f⦄ ≘ j2 → ∀j1,i1. @⦃i1,f⦄ ≘ j1 →
i1 < i2 → j1 < j2.
#j2 elim j2 -j2
[ #i2 #f #H2f elim (at_inv_xxp … H2f) -H2f //
]
qed-.
-theorem at_inv_monotonic: ∀j1,i1,f. @⦃i1, f⦄ ≘ j1 → ∀j2,i2. @⦃i2, f⦄ ≘ j2 →
+theorem at_inv_monotonic: ∀j1,i1,f. @⦃i1,f⦄ ≘ j1 → ∀j2,i2. @⦃i2,f⦄ ≘ j2 →
j1 < j2 → i1 < i2.
#j1 elim j1 -j1
[ #i1 #f #H1f elim (at_inv_xxp … H1f) -H1f //
]
qed-.
-theorem at_mono: ∀f,i,i1. @⦃i, f⦄ ≘ i1 → ∀i2. @⦃i, f⦄ ≘ i2 → i2 = i1.
+theorem at_mono: ∀f,i,i1. @⦃i,f⦄ ≘ i1 → ∀i2. @⦃i,f⦄ ≘ i2 → i2 = i1.
#f #i #i1 #H1 #i2 #H2 elim (lt_or_eq_or_gt i2 i1) //
#Hi elim (lt_le_false i i) /3 width=6 by at_inv_monotonic, eq_sym/
qed-.
-theorem at_inj: ∀f,i1,i. @⦃i1, f⦄ ≘ i → ∀i2. @⦃i2, f⦄ ≘ i → i1 = i2.
+theorem at_inj: ∀f,i1,i. @⦃i1,f⦄ ≘ i → ∀i2. @⦃i2,f⦄ ≘ i → i1 = i2.
#f #i1 #i #H1 #i2 #H2 elim (lt_or_eq_or_gt i2 i1) //
#Hi elim (lt_le_false i i) /3 width=6 by at_monotonic, eq_sym/
qed-.
(* Properties on tls ********************************************************)
-lemma at_pxx_tls: ∀n,f. @⦃0, f⦄ ≘ n → @⦃0, ⫱*[n]f⦄ ≘ 0.
+lemma at_pxx_tls: ∀n,f. @⦃0,f⦄ ≘ n → @⦃0,⫱*[n]f⦄ ≘ 0.
#n elim n -n //
#n #IH #f #Hf
cases (at_inv_pxn … Hf) -Hf [ |*: // ] #g #Hg #H0 destruct
<tls_xn /2 width=1 by/
qed.
-lemma at_tls: ∀i2,f. ⫯⫱*[↑i2]f ≡ ⫱*[i2]f → ∃i1. @⦃i1, f⦄ ≘ i2.
+lemma at_tls: ∀i2,f. ⫯⫱*[↑i2]f ≡ ⫱*[i2]f → ∃i1. @⦃i1,f⦄ ≘ i2.
#i2 elim i2 -i2
[ /4 width=4 by at_eq_repl_back, at_refl, ex_intro/
| #i2 #IH #f <tls_xn <tls_xn in ⊢ (??%→?); #H
(* Inversion lemmas with tls ************************************************)
-lemma at_inv_nxx: ∀n,g,i1,j2. @⦃↑i1, g⦄ ≘ j2 → @⦃0, g⦄ ≘ n →
- ∃∃i2. @⦃i1, ⫱*[↑n]g⦄ ≘ i2 & ↑(n+i2) = j2.
+lemma at_inv_nxx: ∀n,g,i1,j2. @⦃↑i1,g⦄ ≘ j2 → @⦃0,g⦄ ≘ n →
+ ∃∃i2. @⦃i1,⫱*[↑n]g⦄ ≘ i2 & ↑(n+i2) = j2.
#n elim n -n
[ #g #i1 #j2 #Hg #H
elim (at_inv_pxp … H) -H [ |*: // ] #f #H0
]
qed-.
-lemma at_inv_tls: ∀i2,i1,f. @⦃i1, f⦄ ≘ i2 → ⫯⫱*[↑i2]f ≡ ⫱*[i2]f.
+lemma at_inv_tls: ∀i2,i1,f. @⦃i1,f⦄ ≘ i2 → ⫯⫱*[↑i2]f ≡ ⫱*[i2]f.
#i2 elim i2 -i2
[ #i1 #f #Hf elim (at_inv_xxp … Hf) -Hf // #g #H1 #H destruct
/2 width=1 by eq_refl/
(* Advanced inversion lemmas on isid ****************************************)
-lemma isid_inv_at: ∀i,f. 𝐈⦃f⦄ → @⦃i, f⦄ ≘ i.
+lemma isid_inv_at: ∀i,f. 𝐈⦃f⦄ → @⦃i,f⦄ ≘ i.
#i elim i -i
[ #f #H elim (isid_inv_gen … H) -H /2 width=2 by at_refl/
| #i #IH #f #H elim (isid_inv_gen … H) -H /3 width=7 by at_push/
]
qed.
-lemma isid_inv_at_mono: ∀f,i1,i2. 𝐈⦃f⦄ → @⦃i1, f⦄ ≘ i2 → i1 = i2.
+lemma isid_inv_at_mono: ∀f,i1,i2. 𝐈⦃f⦄ → @⦃i1,f⦄ ≘ i2 → i1 = i2.
/3 width=6 by isid_inv_at, at_mono/ qed-.
(* Advanced properties on isid **********************************************)
-corec lemma isid_at: ∀f. (∀i. @⦃i, f⦄ ≘ i) → 𝐈⦃f⦄.
+corec lemma isid_at: ∀f. (∀i. @⦃i,f⦄ ≘ i) → 𝐈⦃f⦄.
#f #Hf lapply (Hf 0)
#H cases (at_fwd_id_ex … H) -H
#g #H @(isid_push … H) /3 width=7 by at_inv_npn/
(* Advanced properties on id ************************************************)
-lemma id_inv_at: ∀f. (∀i. @⦃i, f⦄ ≘ i) → 𝐈𝐝 ≡ f.
+lemma id_inv_at: ∀f. (∀i. @⦃i,f⦄ ≘ i) → 𝐈𝐝 ≡ f.
/3 width=1 by isid_at, eq_id_inv_isid/ qed-.
-lemma id_at: ∀i. @⦃i, 𝐈𝐝⦄ ≘ i.
+lemma id_at: ∀i. @⦃i,𝐈𝐝⦄ ≘ i.
/2 width=1 by isid_inv_at/ qed.
(* Advanced forward lemmas on id ********************************************)
-lemma at_id_fwd: ∀i1,i2. @⦃i1, 𝐈𝐝⦄ ≘ i2 → i1 = i2.
+lemma at_id_fwd: ∀i1,i2. @⦃i1,𝐈𝐝⦄ ≘ i2 → i1 = i2.
/2 width=4 by at_mono/ qed.
(* Main properties on id ****************************************************)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/relocation/rtmap_after.ma".
+include "ground_2/relocation/rtmap_basic.ma".
+
+(* RELOCATION MAP ***********************************************************)
+
+(* Properties with composition **********************************************)
+
+lemma after_basic_rc (m2,m1,n2,n1):
+ m1 ≤ m2 → m2 ≤ m1+n1 → 𝐁❴m2,n2❵ ⊚ 𝐁❴m1,n1❵ ≘ 𝐁❴m1,n2+n1❵.
+#m2 elim m2 -m2
+[ #m1 #n2 #n1 #Hm21 #_
+ <(le_n_O_to_eq … Hm21) -m1 //
+| #m2 #IH *
+ [ #n2 #n1 #_ < plus_O_n #H
+ elim (le_inv_S1 … H) -H #x #Hx #H destruct
+ <plus_n_Sm
+ @after_push [4:|*: // ]
+ @(IH 0 … Hx) -IH -n2 -x //
+ | #m1 #n2 #n1 #H1 #H2
+ lapply (le_S_S_to_le … H1) -H1 #H1
+ lapply (le_S_S_to_le … H2) -H2 #H2
+ /3 width=7 by after_refl/
+ ]
+]
+qed.
(* RELOCATION MAP ***********************************************************)
-definition istot: predicate rtmap ≝ λf. ∀i. ∃j. @⦃i, f⦄ ≘ j.
+definition istot: predicate rtmap ≝ λf. ∀i. ∃j. @⦃i,f⦄ ≘ j.
interpretation "test for totality (rtmap)"
'IsTotal f = (istot f).
(* Main forward lemmas on at ************************************************)
corec theorem at_ext: ∀f1,f2. 𝐓⦃f1⦄ → 𝐓⦃f2⦄ →
- (∀i,i1,i2. @⦃i, f1⦄ ≘ i1 → @⦃i, f2⦄ ≘ i2 → i1 = i2) →
+ (∀i,i1,i2. @⦃i,f1⦄ ≘ i1 → @⦃i,f2⦄ ≘ i2 → i1 = i2) →
f1 ≡ f2.
#f1 cases (pn_split f1) * #g1 #H1
#f2 cases (pn_split f2) * #g2 #H2
(* Advanced properties on at ************************************************)
-lemma at_dec: ∀f,i1,i2. 𝐓⦃f⦄ → Decidable (@⦃i1, f⦄ ≘ i2).
+lemma at_dec: ∀f,i1,i2. 𝐓⦃f⦄ → Decidable (@⦃i1,f⦄ ≘ i2).
#f #i1 #i2 #Hf lapply (Hf i1) -Hf *
#j2 #Hf elim (eq_nat_dec i2 j2)
[ #H destruct /2 width=1 by or_introl/
]
qed-.
-lemma is_at_dec_le: ∀f,i2,i. 𝐓⦃f⦄ → (∀i1. i1 + i ≤ i2 → @⦃i1, f⦄ ≘ i2 → ⊥) →
- Decidable (∃i1. @⦃i1, f⦄ ≘ i2).
+lemma is_at_dec_le: ∀f,i2,i. 𝐓⦃f⦄ → (∀i1. i1 + i ≤ i2 → @⦃i1,f⦄ ≘ i2 → ⊥) →
+ Decidable (∃i1. @⦃i1,f⦄ ≘ i2).
#f #i2 #i #Hf elim i -i
[ #Ht @or_intror * /3 width=3 by at_increasing/
| #i #IH #Ht elim (at_dec f (i2-i) i2) /3 width=2 by ex_intro, or_introl/
]
qed-.
-lemma is_at_dec: ∀f,i2. 𝐓⦃f⦄ → Decidable (∃i1. @⦃i1, f⦄ ≘ i2).
+lemma is_at_dec: ∀f,i2. 𝐓⦃f⦄ → Decidable (∃i1. @⦃i1,f⦄ ≘ i2).
#f #i2 #Hf @(is_at_dec_le ?? (↑i2)) /2 width=4 by lt_le_false/
qed-.
(* Advanced properties on isid **********************************************)
-lemma isid_at_total: ∀f. 𝐓⦃f⦄ → (∀i1,i2. @⦃i1, f⦄ ≘ i2 → i1 = i2) → 𝐈⦃f⦄.
+lemma isid_at_total: ∀f. 𝐓⦃f⦄ → (∀i1,i2. @⦃i1,f⦄ ≘ i2 → i1 = i2) → 𝐈⦃f⦄.
#f #H1f #H2f @isid_at
#i lapply (H1f i) -H1f *
#j #Hf >(H2f … Hf) in ⊢ (???%); -H2f //
inductive eq_t: relation rtc ≝
| eq_t_intro: ∀ri1,ri2,rs1,rs2,ti,ts.
- eq_t (〈ri1, rs1, ti, ts〉) (〈ri2, rs2, ti, ts〉)
+ eq_t (〈ri1,rs1,ti,ts〉) (〈ri2,rs2,ti,ts〉)
.
(* Basic properties *********************************************************)
(* RT-TRANSITION COUNTER ****************************************************)
definition isrt: relation2 nat rtc ≝ λts,c.
- ∃∃ri,rs. 〈ri, rs, 0, ts〉 = c.
+ ∃∃ri,rs. 〈ri,rs,0,ts〉 = c.
interpretation "test for costrained rt-transition counter (rtc)"
'IsRedType ts c = (isrt ts c).
(* Basic properties *********************************************************)
-lemma isr_00: 𝐑𝐓⦃0, 𝟘𝟘⦄.
+lemma isr_00: 𝐑𝐓⦃0,𝟘𝟘⦄.
/2 width=3 by ex1_2_intro/ qed.
-lemma isr_10: 𝐑𝐓⦃0, 𝟙𝟘⦄.
+lemma isr_10: 𝐑𝐓⦃0,𝟙𝟘⦄.
/2 width=3 by ex1_2_intro/ qed.
-lemma isrt_01: 𝐑𝐓⦃1, 𝟘𝟙⦄.
+lemma isrt_01: 𝐑𝐓⦃1,𝟘𝟙⦄.
/2 width=3 by ex1_2_intro/ qed.
-lemma isrt_eq_t_trans: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → eq_t c1 c2 → 𝐑𝐓⦃n, c2⦄.
+lemma isrt_eq_t_trans: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → eq_t c1 c2 → 𝐑𝐓⦃n,c2⦄.
#n #c1 #c2 * #ri1 #rs1 #H destruct
#H elim (eq_t_inv_dx … H) -H /2 width=3 by ex1_2_intro/
qed-.
(* Basic inversion properties ***********************************************)
-lemma isrt_inv_00: ∀n. 𝐑𝐓⦃n, 𝟘𝟘⦄ → 0 = n.
+lemma isrt_inv_00: ∀n. 𝐑𝐓⦃n,𝟘𝟘⦄ → 0 = n.
#n * #ri #rs #H destruct //
qed-.
-lemma isrt_inv_10: ∀n. 𝐑𝐓⦃n, 𝟙𝟘⦄ → 0 = n.
+lemma isrt_inv_10: ∀n. 𝐑𝐓⦃n,𝟙𝟘⦄ → 0 = n.
#n * #ri #rs #H destruct //
qed-.
-lemma isrt_inv_01: ∀n. 𝐑𝐓⦃n, 𝟘𝟙⦄ → 1 = n.
+lemma isrt_inv_01: ∀n. 𝐑𝐓⦃n,𝟘𝟙⦄ → 1 = n.
#n * #ri #rs #H destruct //
qed-.
(* Main inversion properties ************************************************)
-theorem isrt_inj: ∀n1,n2,c. 𝐑𝐓⦃n1, c⦄ → 𝐑𝐓⦃n2, c⦄ → n1 = n2.
+theorem isrt_inj: ∀n1,n2,c. 𝐑𝐓⦃n1,c⦄ → 𝐑𝐓⦃n2,c⦄ → n1 = n2.
#n1 #n2 #c * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct //
qed-.
-theorem isrt_mono: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃n, c2⦄ → eq_t c1 c2.
+theorem isrt_mono: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃n,c2⦄ → eq_t c1 c2.
#n #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct //
qed-.
definition max (c1:rtc) (c2:rtc): rtc ≝ match c1 with [
mk_rtc ri1 rs1 ti1 ts1 ⇒ match c2 with [
- mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1∨ri2, rs1∨rs2, ti1∨ti2, ts1∨ts2〉
+ mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1∨ri2,rs1∨rs2,ti1∨ti2,ts1∨ts2〉
]
].
(* Basic properties *********************************************************)
lemma max_rew: ∀ri1,ri2,rs1,rs2,ti1,ti2,ts1,ts2.
- 〈ri1∨ri2, rs1∨rs2, ti1∨ti2, ts1∨ts2〉 =
+ 〈ri1∨ri2,rs1∨rs2,ti1∨ti2,ts1∨ts2〉 =
(〈ri1,rs1,ti1,ts1〉 ∨ 〈ri2,rs2,ti2,ts2〉).
// qed.
(* Properties with test for constrained rt-transition counter ***************)
-lemma isrt_max: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1, c1⦄ → 𝐑𝐓⦃n2, c2⦄ → 𝐑𝐓⦃n1∨n2, c1∨c2⦄.
+lemma isrt_max: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1,c1⦄ → 𝐑𝐓⦃n2,c2⦄ → 𝐑𝐓⦃n1∨n2,c1∨c2⦄.
#n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct
/2 width=3 by ex1_2_intro/
qed.
-lemma isrt_max_O1: ∀n,c1,c2. 𝐑𝐓⦃0, c1⦄ → 𝐑𝐓⦃n, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄.
+lemma isrt_max_O1: ∀n,c1,c2. 𝐑𝐓⦃0,c1⦄ → 𝐑𝐓⦃n,c2⦄ → 𝐑𝐓⦃n,c1∨c2⦄.
/2 width=1 by isrt_max/ qed.
-lemma isrt_max_O2: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄.
+lemma isrt_max_O2: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1∨c2⦄.
#n #c1 #c2 #H1 #H2 >(max_O2 n) /2 width=1 by isrt_max/
qed.
-lemma isrt_max_idem1: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃n, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄.
+lemma isrt_max_idem1: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃n,c2⦄ → 𝐑𝐓⦃n,c1∨c2⦄.
#n #c1 #c2 #H1 #H2 >(idempotent_max n) /2 width=1 by isrt_max/
qed.
(* Inversion properties with test for constrained rt-transition counter *****)
-lemma isrt_inv_max: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ →
- ∃∃n1,n2. 𝐑𝐓⦃n1, c1⦄ & 𝐑𝐓⦃n2, c2⦄ & (n1 ∨ n2) = n.
+lemma isrt_inv_max: ∀n,c1,c2. 𝐑𝐓⦃n,c1 ∨ c2⦄ →
+ ∃∃n1,n2. 𝐑𝐓⦃n1,c1⦄ & 𝐑𝐓⦃n2,c2⦄ & (n1 ∨ n2) = n.
#n #c1 #c2 * #ri #rs #H
elim (max_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
elim (max_inv_O3 … H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
qed-.
-lemma isrt_O_inv_max: ∀c1,c2. 𝐑𝐓⦃0, c1 ∨ c2⦄ → ∧∧ 𝐑𝐓⦃0, c1⦄ & 𝐑𝐓⦃0, c2⦄.
+lemma isrt_O_inv_max: ∀c1,c2. 𝐑𝐓⦃0,c1 ∨ c2⦄ → ∧∧ 𝐑𝐓⦃0,c1⦄ & 𝐑𝐓⦃0,c2⦄.
#c1 #c2 #H
elim (isrt_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H
elim (max_inv_O3 … H) -H #H1 #H2 destruct
/2 width=1 by conj/
qed-.
-lemma isrt_inv_max_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1⦄.
+lemma isrt_inv_max_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n,c1 ∨ c2⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1⦄.
#n #c1 #c2 #H #H2
elim (isrt_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
lapply (isrt_inj … Hn2 H2) -c2 #H destruct //
qed-.
-lemma isrt_inv_max_eq_t: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ → eq_t c1 c2 →
- ∧∧ 𝐑𝐓⦃n, c1⦄ & 𝐑𝐓⦃n, c2⦄.
+lemma isrt_inv_max_eq_t: ∀n,c1,c2. 𝐑𝐓⦃n,c1 ∨ c2⦄ → eq_t c1 c2 →
+ ∧∧ 𝐑𝐓⦃n,c1⦄ & 𝐑𝐓⦃n,c2⦄.
#n #c1 #c2 #H #Hc12
elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
lapply (isrt_eq_t_trans … Hc1 … Hc12) -Hc12 #H
(* Inversion lemmaswith shift ***********************************************)
-lemma isrt_inv_max_shift_sn: ∀n,c1,c2. 𝐑𝐓⦃n, ↕*c1 ∨ c2⦄ →
- ∧∧ 𝐑𝐓⦃0, c1⦄ & 𝐑𝐓⦃n, c2⦄.
+lemma isrt_inv_max_shift_sn: ∀n,c1,c2. 𝐑𝐓⦃n,↕*c1 ∨ c2⦄ →
+ ∧∧ 𝐑𝐓⦃0,c1⦄ & 𝐑𝐓⦃n,c2⦄.
#n #c1 #c2 #H
elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
elim (isrt_inv_shift … Hc1) -Hc1 #Hc1 * -n1
definition plus (c1:rtc) (c2:rtc): rtc ≝ match c1 with [
mk_rtc ri1 rs1 ti1 ts1 ⇒ match c2 with [
- mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1+ri2, rs1+rs2, ti1+ti2, ts1+ts2〉
+ mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1+ri2,rs1+rs2,ti1+ti2,ts1+ts2〉
]
].
(**) (* plus is not disambiguated parentheses *)
lemma plus_rew: ∀ri1,ri2,rs1,rs2,ti1,ti2,ts1,ts2.
- 〈ri1+ri2, rs1+rs2, ti1+ti2, ts1+ts2〉 =
+ 〈ri1+ri2,rs1+rs2,ti1+ti2,ts1+ts2〉 =
(〈ri1,rs1,ti1,ts1〉) + (〈ri2,rs2,ti2,ts2〉).
// qed.
(* Properties with test for constrained rt-transition counter ***************)
-lemma isrt_plus: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1, c1⦄ → 𝐑𝐓⦃n2, c2⦄ → 𝐑𝐓⦃n1+n2, c1+c2⦄.
+lemma isrt_plus: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1,c1⦄ → 𝐑𝐓⦃n2,c2⦄ → 𝐑𝐓⦃n1+n2,c1+c2⦄.
#n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct
/2 width=3 by ex1_2_intro/
qed.
-lemma isrt_plus_O1: ∀n,c1,c2. 𝐑𝐓⦃0, c1⦄ → 𝐑𝐓⦃n, c2⦄ → 𝐑𝐓⦃n, c1+c2⦄.
+lemma isrt_plus_O1: ∀n,c1,c2. 𝐑𝐓⦃0,c1⦄ → 𝐑𝐓⦃n,c2⦄ → 𝐑𝐓⦃n,c1+c2⦄.
/2 width=1 by isrt_plus/ qed.
-lemma isrt_plus_O2: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1+c2⦄.
+lemma isrt_plus_O2: ∀n,c1,c2. 𝐑𝐓⦃n,c1⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1+c2⦄.
#n #c1 #c2 #H1 #H2 >(plus_n_O n) /2 width=1 by isrt_plus/
qed.
-lemma isrt_succ: ∀n,c. 𝐑𝐓⦃n, c⦄ → 𝐑𝐓⦃↑n, c+𝟘𝟙⦄.
+lemma isrt_succ: ∀n,c. 𝐑𝐓⦃n,c⦄ → 𝐑𝐓⦃↑n,c+𝟘𝟙⦄.
/2 width=1 by isrt_plus/ qed.
(* Inversion properties with test for constrained rt-transition counter *****)
-lemma isrt_inv_plus: ∀n,c1,c2. 𝐑𝐓⦃n, c1 + c2⦄ →
- ∃∃n1,n2. 𝐑𝐓⦃n1, c1⦄ & 𝐑𝐓⦃n2, c2⦄ & n1 + n2 = n.
+lemma isrt_inv_plus: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ →
+ ∃∃n1,n2. 𝐑𝐓⦃n1,c1⦄ & 𝐑𝐓⦃n2,c2⦄ & n1 + n2 = n.
#n #c1 #c2 * #ri #rs #H
elim (plus_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
elim (plus_inv_O3 … H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
qed-.
-lemma isrt_inv_plus_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n, c1 + c2⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1⦄.
+lemma isrt_inv_plus_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ → 𝐑𝐓⦃0,c2⦄ → 𝐑𝐓⦃n,c1⦄.
#n #c1 #c2 #H #H2
elim (isrt_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
lapply (isrt_inj … Hn2 H2) -c2 #H destruct //
qed-.
-lemma isrt_inv_plus_SO_dx: ∀n,c1,c2. 𝐑𝐓⦃n, c1 + c2⦄ → 𝐑𝐓⦃1, c2⦄ →
- ∃∃m. 𝐑𝐓⦃m, c1⦄ & n = ↑m.
+lemma isrt_inv_plus_SO_dx: ∀n,c1,c2. 𝐑𝐓⦃n,c1 + c2⦄ → 𝐑𝐓⦃1,c2⦄ →
+ ∃∃m. 𝐑𝐓⦃m,c1⦄ & n = ↑m.
#n #c1 #c2 #H #H2
elim (isrt_inv_plus … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
lapply (isrt_inj … Hn2 H2) -c2 #H destruct
(* RT-TRANSITION COUNTER ****************************************************)
definition shift (c:rtc): rtc ≝ match c with
-[ mk_rtc ri rs ti ts ⇒ 〈ri∨rs, 0, ti∨ts, 0〉 ].
+[ mk_rtc ri rs ti ts ⇒ 〈ri∨rs,0,ti∨ts,0〉 ].
interpretation "shift (rtc)"
'UpDownArrowStar c = (shift c).
(* Basic properties *********************************************************)
-lemma shift_rew: ∀ri,rs,ti,ts. 〈ri∨rs, 0, ti∨ts, 0〉 = ↕*〈ri, rs, ti, ts〉.
+lemma shift_rew: ∀ri,rs,ti,ts. 〈ri∨rs,0,ti∨ts,0〉 = ↕*〈ri,rs,ti,ts〉.
normalize //
qed.
(* Basic inversion properties ***********************************************)
-lemma shift_inv_dx: ∀ri,rs,ti,ts,c. 〈ri, rs, ti, ts〉 = ↕*c →
+lemma shift_inv_dx: ∀ri,rs,ti,ts,c. 〈ri,rs,ti,ts〉 = ↕*c →
∃∃ri0,rs0,ti0,ts0. (ri0∨rs0) = ri & 0 = rs & (ti0∨ts0) = ti & 0 = ts &
- 〈ri0, rs0, ti0, ts0〉 = c.
+ 〈ri0,rs0,ti0,ts0〉 = c.
#ri #rs #ti #ts * #ri0 #rs0 #ti0 #ts0 <shift_rew #H destruct
/2 width=7 by ex5_4_intro/
qed-.
(* Properties with test for costrained rt-transition counter ****************)
-lemma isr_shift: ∀c. 𝐑𝐓⦃0, c⦄ → 𝐑𝐓⦃0, ↕*c⦄.
+lemma isr_shift: ∀c. 𝐑𝐓⦃0,c⦄ → 𝐑𝐓⦃0,↕*c⦄.
#c * #ri #rs #H destruct /2 width=3 by ex1_2_intro/
qed.
(* Inversion properties with test for costrained rt-counter *****************)
-lemma isrt_inv_shift: ∀n,c. 𝐑𝐓⦃n, ↕*c⦄ → 𝐑𝐓⦃0, c⦄ ∧ 0 = n.
+lemma isrt_inv_shift: ∀n,c. 𝐑𝐓⦃n,↕*c⦄ → 𝐑𝐓⦃0,c⦄ ∧ 0 = n.
#n #c * #ri #rs #H
elim (shift_inv_dx … H) -H #rt0 #rs0 #ti0 #ts0 #_ #_ #H1 #H2 #H3
elim (max_inv_O3 … H1) -H1 /3 width=3 by ex1_2_intro, conj/
qed-.
-lemma isr_inv_shift: ∀c. 𝐑𝐓⦃0, ↕*c⦄ → 𝐑𝐓⦃0, c⦄.
+lemma isr_inv_shift: ∀c. 𝐑𝐓⦃0,↕*c⦄ → 𝐑𝐓⦃0,c⦄.
#c #H elim (isrt_inv_shift … H) -H //
qed-.
"rtmap_fcla ( 𝐂⦃?⦄ ≘ ? )" "rtmap_isfin ( 𝐅⦃?⦄ )" "rtmap_isuni ( 𝐔⦃?⦄ )" "rtmap_uni ( 𝐔❴?❵ )"
"rtmap_sle ( ? ⊆ ? )" "rtmap_sdj ( ? ∥ ? )" "rtmap_sand ( ? ⋒ ? ≘ ? )" "rtmap_sor ( ? ⋓ ? ≘ ? )"
"rtmap_at ( @⦃?,?⦄ ≘ ? )" "rtmap_istot ( 𝐓⦃?⦄ )" "rtmap_after ( ? ⊚ ? ≘ ? )" "rtmap_coafter ( ? ~⊚ ? ≘ ? )"
- "rtmap_basic ( 𝐁❴?,?❵ )"
+ "rtmap_basic ( 𝐁❴?,?❵ )" "rtmap_basic_after"
* ]
[ "nstream ( ⫯? ) ( ↑? )" "nstream_eq" "" ""
"" "" "nstream_isid" "nstream_id ( 𝐈𝐝 )" ""
"" "" "" ""
"" "" "" "nstream_sor"
"" "nstream_istot ( ?@❴?❵ )" "nstream_after ( ? ∘ ? )" "nstream_coafter ( ? ~∘ ? )"
- "nstream_basic"
+ "nstream_basic" ""
* ]
(*
[ "trace ( ∥?∥ )" "trace_at ( @⦃?,?⦄ ≘ ? )" "trace_after ( ? ⊚ ? ≘ ? )" "trace_isid ( 𝐈⦃?⦄ )" "trace_isun ( 𝐔⦃?⦄ )"
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was generated by xoa.native: do not edit *********************)
+
+include "basics/pts.ma".
+
+include "ground_2/notation/xoa/ex_1_4.ma".
+
+(* multiple existental quantifier (1, 4) *)
+
+inductive ex1_4 (A0,A1,A2,A3:Type[0]) (P0:A0→A1→A2→A3→Prop) : Prop ≝
+ | ex1_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → ex1_4 ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (1, 4)" 'Ex4 P0 = (ex1_4 ? ? ? ? P0).
+
include "ground_2/xoa/xoa.ma".
-(* Properties with multiple existental quantifier (5, 1) ********************)
+(* Properties with multiple existental quantifier (4, 1) ********************)
lemma ex4_commute (A0) (P0,P1,P2,P3:A0→Prop):
(∃∃x0. P0 x0 & P1 x0 & P2 x0 & P3 x0) → ∃∃x0. P2 x0 & P3 x0 & P0 x0 & P1 x0.
<key name="objects">ground_2/xoa</key>
<key name="notations">ground_2/notation/xoa</key>
<key name="include">basics/pts.ma</key>
+ <key name="ex">1 4</key>
<key name="ex">5 1</key>
<key name="ex">5 7</key>
<key name="ex">9 3</key>
]
qed-.
+lemma le_ylt_trans (x) (y) (z): x ≤ y → yinj y < z → yinj x < z.
+/3 width=3 by yle_ylt_trans, yle_inj/
+qed-.
+
lemma yle_inv_succ1_lt: ∀x,y:ynat. ↑x ≤ y → 0 < y ∧ x ≤ ↓y.
#x #y #H elim (yle_inv_succ1 … H) -H /3 width=1 by ylt_O1, conj/
qed-.
]
qed-.
+lemma lt_ylt_trans (x) (y) (z): x < y → yinj y < z → yinj x < z.
+/3 width=3 by ylt_trans, ylt_inj/
+qed-.
+
(* Elimination principles ***************************************************)
fact ynat_ind_lt_le_aux: ∀R:predicate ynat.
+++ /dev/null
-ground_2
-static_2
-basic_2
-apps_2
#!/bin/sh
-for SRC in `find grond_2 static_2 basic_2 apps_2 -name "*.ma" -or -name "*.tbl"`; do
- sed "s!$1!$2!g" ${SRC} > ${SRC}.new
+for SRC in `find ground_2 static_2 basic_2 apps_2 -name "*.ma" -or -name "*.tbl"`; do
+ sed "/$1/s!$2!$3!g" ${SRC} > ${SRC}.new
if [ ! -s ${SRC}.new ] || diff ${SRC} ${SRC}.new > /dev/null;
then rm -f ${SRC}.new;
else echo ${SRC}; mv -f ${SRC} ${SRC}.old; mv -f ${SRC}.new ${SRC};
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⊐ⓧ [ break term 46 f ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'ClearSn $f $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/notation/relations/clearsn_3.ma".
+include "static_2/syntax/cext2.ma".
+include "static_2/relocation/sex.ma".
+
+(* CLEAR FOR LOCAL ENVIRONMENTS ON SELECTED ENTRIES *************************)
+
+definition ccl: relation3 lenv bind bind ≝ λL,I1,I2. BUnit Void = I2.
+
+definition scl: rtmap → relation lenv ≝ sex ccl (cext2 ceq).
+
+interpretation
+ "clear (local environment)"
+ 'ClearSn f L1 L2 = (scl f L1 L2).
+
+(* Basic eliminators ********************************************************)
+
+lemma scl_ind (Q:rtmap→relation lenv):
+ (∀f. Q f (⋆) (⋆)) →
+ (∀f,I,K1,K2. K1 ⊐ⓧ[f] K2 → Q f K1 K2 → Q (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I})) →
+ (∀f,I,K1,K2. K1 ⊐ⓧ[f] K2 → Q f K1 K2 → Q (↑f) (K1.ⓘ{I}) (K2.ⓧ)) →
+ ∀f,L1,L2. L1 ⊐ⓧ[f] L2 → Q f L1 L2.
+#Q #IH1 #IH2 #IH3 #f #L1 #L2 #H elim H -f -L1 -L2
+[ //
+| #f #I1 #I2 #K1 #K2 #HK #H #IH destruct /2 by/
+| #f #I1 #I2 #K1 #K2 #HK * #I [| #V1 #V2 #H ] #IH destruct /2 by/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma scl_inv_atom_sn: ∀g,L2. ⋆ ⊐ⓧ[g] L2 → L2 = ⋆.
+/2 width=4 by sex_inv_atom1/ qed-.
+
+lemma scl_inv_push_sn: ∀f,I,K1,L2. K1.ⓘ{I} ⊐ⓧ[⫯f] L2 →
+ ∃∃K2. K1 ⊐ⓧ[f] K2 & L2 = K2.ⓘ{I}.
+#f #I #K1 #L2 #H
+elim (sex_inv_push1 … H) -H #J #K2 #HK12 *
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma scl_inv_next_sn: ∀f,I,K1,L2. K1.ⓘ{I} ⊐ⓧ[↑f] L2 →
+ ∃∃K2. K1 ⊐ⓧ[f] K2 & L2 = K2.ⓧ.
+#f #I #K1 #L2 #H
+elim (sex_inv_next1 … H) -H
+/2 width=3 by ex2_intro/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma scl_inv_bind_sn_gen: ∀g,I,K1,L2. K1.ⓘ{I} ⊐ⓧ[g] L2 →
+ ∨∨ ∃∃f,K2. K1 ⊐ⓧ[f] K2 & g = ⫯f & L2 = K2.ⓘ{I}
+ | ∃∃f,K2. K1 ⊐ⓧ[f] K2 & g = ↑f & L2 = K2.ⓧ.
+#g #I #K1 #L2 #H
+elim (pn_split g) * #f #Hf destruct
+[ elim (scl_inv_push_sn … H) -H
+| elim (scl_inv_next_sn … H) -H
+]
+/3 width=5 by ex3_2_intro, or_intror, or_introl/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma scl_fwd_bind_sn: ∀g,I1,K1,L2. K1.ⓘ{I1} ⊐ⓧ[g] L2 →
+ ∃∃I2,K2. K1 ⊐ⓧ[⫱g] K2 & L2 = K2.ⓘ{I2}.
+#g #I1 #K1 #L2
+elim (pn_split g) * #f #Hf destruct #H
+[ elim (scl_inv_push_sn … H) -H
+| elim (scl_inv_next_sn … H) -H
+]
+/2 width=4 by ex2_2_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma scl_atom: ∀f. ⋆ ⊐ⓧ[f] ⋆.
+/by sex_atom/ qed.
+
+lemma scl_push: ∀f,K1,K2. K1 ⊐ⓧ[f] K2 → ∀I. K1.ⓘ{I} ⊐ⓧ[⫯f] K2.ⓘ{I}.
+#f #K1 #K2 #H * /3 width=1 by sex_push, ext2_unit, ext2_pair/
+qed.
+
+lemma scl_next: ∀f,K1,K2. K1 ⊐ⓧ[f] K2 → ∀I. K1.ⓘ{I} ⊐ⓧ[↑f] K2.ⓧ.
+/2 width=1 by sex_next/ qed.
+
+lemma scl_eq_repl_back: ∀L1,L2. eq_repl_back … (λf. L1 ⊐ⓧ[f] L2).
+/2 width=3 by sex_eq_repl_back/ qed-.
+
+lemma scl_eq_repl_fwd: ∀L1,L2. eq_repl_fwd … (λf. L1 ⊐ⓧ[f] L2).
+/2 width=3 by sex_eq_repl_fwd/ qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma scl_refl: ∀f. 𝐈⦃f⦄ → reflexive … (scl f).
+#f #Hf #L elim L -L
+/3 width=3 by scl_eq_repl_back, scl_push, eq_push_inv_isid/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/scl.ma".
+
+(* CLEAR FOR LOCAL ENVIRONMENTS ON SELECTED ENTRIES *************************)
+
+(* Main properties **********************************************************)
+
+theorem scl_fix: ∀f,L1,L. L1 ⊐ⓧ[f] L →
+ ∀L2. L ⊐ⓧ[f] L2 → L = L2.
+#f #L1 #L #H @(scl_ind … H) -f -L1 -L
+[ #f #L2 #H
+ >(scl_inv_atom_sn … H) -L2 //
+| #f #I #K1 #K2 #_ #IH #L2 #H
+ elim (scl_inv_push_sn … H) -H /3 width=1 by eq_f2/
+| #f #I #K1 #K2 #_ #IH #L2 #H
+ elim (scl_inv_next_sn … H) -H /3 width=1 by eq_f2/
+]
+qed-.
+
+theorem scl_trans: ∀f. Transitive … (scl f).
+#f #L1 #L #H1 #L2 #H2
+<(scl_fix … H1 … H2) -L2 //
+qed-.
--- /dev/null
+definition sh_N: sh ≝ mk_sh S ….
+// defined.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tueq.ma".
+include "static_2/relocation/lifts_lifts.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+(* Properties with tail sort-irrelevant equivalence for terms ***************)
+
+lemma tueq_lifts_sn: liftable2_sn tueq.
+#T1 #T2 #H elim H -T1 -T2
+[ #s1 #s2 #f #X #H >(lifts_inv_sort1 … H) -H
+ /3 width=3 by lifts_sort, tueq_sort, ex2_intro/
+| #i #f #X #H elim (lifts_inv_lref1 … H) -H
+ /3 width=3 by lifts_lref, tueq_lref, ex2_intro/
+| #l #f #X #H >(lifts_inv_gref1 … H) -H
+ /2 width=3 by lifts_gref, tueq_gref, ex2_intro/
+| #p #I #V #T1 #T2 #_ #IHT #f #X #H elim (lifts_inv_bind1 … H) -H
+ #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHT … HTU1) -T1
+ /3 width=3 by lifts_bind, tueq_bind, ex2_intro/
+| #V #T1 #T2 #_ #IHT #f #X #H elim (lifts_inv_flat1 … H) -H
+ #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHT … HTU1) -T1
+ /3 width=3 by lifts_flat, tueq_appl, ex2_intro/
+| #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f #X #H elim (lifts_inv_flat1 … H) -H
+ #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV … HVW1) -V1 elim (IHT … HTU1) -T1
+ /3 width=5 by lifts_flat, tueq_cast, ex2_intro/
+]
+qed-.
+
+lemma tueq_lifts_dx: liftable2_dx tueq.
+/3 width=3 by tueq_lifts_sn, liftable2_sn_dx, tueq_sym/ qed-.
+
+lemma tueq_lifts_bi: liftable2_bi tueq.
+/3 width=6 by tueq_lifts_sn, liftable2_sn_bi/ qed-.
+
+(* Inversion lemmas with tail sort-irrelevant equivalence for terms *********)
+
+lemma tueq_inv_lifts_sn: deliftable2_sn tueq.
+#U1 #U2 #H elim H -U1 -U2
+[ #s1 #s2 #f #X #H >(lifts_inv_sort2 … H) -H
+ /3 width=3 by lifts_sort, tueq_sort, ex2_intro/
+| #i #f #X #H elim (lifts_inv_lref2 … H) -H
+ /3 width=3 by lifts_lref, tueq_lref, ex2_intro/
+| #l #f #X #H >(lifts_inv_gref2 … H) -H
+ /2 width=3 by lifts_gref, tueq_gref, ex2_intro/
+| #p #I #W #U1 #U2 #_ #IHU #f #X #H elim (lifts_inv_bind2 … H) -H
+ #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHU … HTU1) -U1
+ /3 width=3 by lifts_bind, tueq_bind, ex2_intro/
+| #W #U1 #U2 #_ #IHU #f #X #H elim (lifts_inv_flat2 … H) -H
+ #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHU … HTU1) -U1
+ /3 width=3 by lifts_flat, tueq_appl, ex2_intro/
+| #W1 #W2 #U1 #U2 #_ #_ #IHW #IHU #f #X #H elim (lifts_inv_flat2 … H) -H
+ #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW … HVW1) -W1 elim (IHU … HTU1) -U1
+ /3 width=5 by lifts_flat, tueq_cast, ex2_intro/
+]
+qed-.
+
+lemma tueq_inv_lifts_dx: deliftable2_dx tueq.
+/3 width=3 by tueq_inv_lifts_sn, deliftable2_sn_dx, tueq_sym/ qed-.
+
+lemma tueq_inv_lifts_bi: deliftable2_bi tueq.
+/3 width=6 by tueq_inv_lifts_sn, deliftable2_sn_bi/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/notation/relations/approxeq_2.ma".
+include "static_2/syntax/term.ma".
+
+(* TAIL SORT-IRRELEVANT EQUIVALENCE ON TERMS ********************************)
+
+inductive tueq: relation term ≝
+| tueq_sort: ∀s1,s2. tueq (⋆s1) (⋆s2)
+| tueq_lref: ∀i. tueq (#i) (#i)
+| tueq_gref: ∀l. tueq (§l) (§l)
+| tueq_bind: ∀p,I,V,T1,T2. tueq T1 T2 → tueq (ⓑ{p,I}V.T1) (ⓑ{p,I}V.T2)
+| tueq_appl: ∀V,T1,T2. tueq T1 T2 → tueq (ⓐV.T1) (ⓐV.T2)
+| tueq_cast: ∀V1,V2,T1,T2. tueq V1 V2 → tueq T1 T2 → tueq (ⓝV1.T1) (ⓝV2.T2)
+.
+
+interpretation
+ "context-free tail sort-irrelevant equivalence (term)"
+ 'ApproxEq T1 T2 = (tueq T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tueq_refl: reflexive … tueq.
+#T elim T -T * [|||| * ]
+/2 width=1 by tueq_sort, tueq_lref, tueq_gref, tueq_bind, tueq_appl, tueq_cast/
+qed.
+
+lemma tueq_sym: symmetric … tueq.
+#T1 #T2 #H elim H -T1 -T2
+/2 width=3 by tueq_sort, tueq_bind, tueq_appl, tueq_cast/
+qed-.
+
+(* Left basic inversion lemmas **********************************************)
+
+fact tueq_inv_sort1_aux: ∀X,Y. X ≅ Y → ∀s1. X = ⋆s1 →
+ ∃s2. Y = ⋆s2.
+#X #Y * -X -Y
+[ #s1 #s2 #s #H destruct /2 width=2 by ex_intro/
+| #i #s #H destruct
+| #l #s #H destruct
+| #p #I #V #T1 #T2 #_ #s #H destruct
+| #V #T1 #T2 #_ #s #H destruct
+| #V1 #V2 #T1 #T2 #_ #_ #s #H destruct
+]
+qed-.
+
+lemma tueq_inv_sort1: ∀Y,s1. ⋆s1 ≅ Y →
+ ∃s2. Y = ⋆s2.
+/2 width=4 by tueq_inv_sort1_aux/ qed-.
+
+fact tueq_inv_lref1_aux: ∀X,Y. X ≅ Y → ∀i. X = #i → Y = #i.
+#X #Y * -X -Y //
+[ #s1 #s2 #j #H destruct
+| #p #I #V #T1 #T2 #_ #j #H destruct
+| #V #T1 #T2 #_ #j #H destruct
+| #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
+]
+qed-.
+
+lemma tueq_inv_lref1: ∀Y,i. #i ≅ Y → Y = #i.
+/2 width=5 by tueq_inv_lref1_aux/ qed-.
+
+fact tueq_inv_gref1_aux: ∀X,Y. X ≅ Y → ∀l. X = §l → Y = §l.
+#X #Y * -X -Y //
+[ #s1 #s2 #k #H destruct
+| #p #I #V #T1 #T2 #_ #k #H destruct
+| #V #T1 #T2 #_ #k #H destruct
+| #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
+]
+qed-.
+
+lemma tueq_inv_gref1: ∀Y,l. §l ≅ Y → Y = §l.
+/2 width=5 by tueq_inv_gref1_aux/ qed-.
+
+fact tueq_inv_bind1_aux: ∀X,Y. X ≅ Y → ∀p,I,V,T1. X = ⓑ{p,I}V.T1 →
+ ∃∃T2. T1 ≅ T2 & Y = ⓑ{p,I}V.T2.
+#X #Y * -X -Y
+[ #s1 #s2 #q #J #W #U1 #H destruct
+| #i #q #J #W #U1 #H destruct
+| #l #q #J #W #U1 #H destruct
+| #p #I #V #T1 #T2 #HT #q #J #W #U1 #H destruct /2 width=3 by ex2_intro/
+| #V #T1 #T2 #_ #q #J #W #U1 #H destruct
+| #V1 #V2 #T1 #T2 #_ #_ #q #J #W #U1 #H destruct
+]
+qed-.
+
+lemma tueq_inv_bind1: ∀p,I,V,T1,Y. ⓑ{p,I}V.T1 ≅ Y →
+ ∃∃T2. T1 ≅ T2 & Y = ⓑ{p,I}V.T2.
+/2 width=3 by tueq_inv_bind1_aux/ qed-.
+
+fact tueq_inv_appl1_aux: ∀X,Y. X ≅ Y → ∀V,T1. X = ⓐV.T1 →
+ ∃∃T2. T1 ≅ T2 & Y = ⓐV.T2.
+#X #Y * -X -Y
+[ #s1 #s2 #W #U1 #H destruct
+| #i #W #U1 #H destruct
+| #l #W #U1 #H destruct
+| #p #I #V #T1 #T2 #_ #W #U1 #H destruct
+| #V #T1 #T2 #HT #W #U1 #H destruct /2 width=3 by ex2_intro/
+| #V1 #V2 #T1 #T2 #_ #_ #W #U1 #H destruct
+]
+qed-.
+
+lemma tueq_inv_appl1: ∀V,T1,Y. ⓐV.T1 ≅ Y →
+ ∃∃T2. T1 ≅ T2 & Y = ⓐV.T2.
+/2 width=3 by tueq_inv_appl1_aux/ qed-.
+
+fact tueq_inv_cast1_aux: ∀X,Y. X ≅ Y → ∀V1,T1. X = ⓝV1.T1 →
+ ∃∃V2,T2. V1 ≅ V2 & T1 ≅ T2 & Y = ⓝV2.T2.
+#X #Y * -X -Y
+[ #s1 #s2 #W1 #U1 #H destruct
+| #i #W1 #U1 #H destruct
+| #l #W1 #U1 #H destruct
+| #p #I #V #T1 #T2 #_ #W1 #U1 #H destruct
+| #V #T1 #T2 #_ #W1 #U1 #H destruct
+| #V1 #V2 #T1 #T2 #HV #HT #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma tueq_inv_cast1: ∀V1,T1,Y. ⓝV1.T1 ≅ Y →
+ ∃∃V2,T2. V1 ≅ V2 & T1 ≅ T2 & Y = ⓝV2.T2.
+/2 width=3 by tueq_inv_cast1_aux/ qed-.
+
+(* Right basic inversion lemmas *********************************************)
+
+lemma tueq_inv_bind2: ∀p,I,V,T2,X1. X1 ≅ ⓑ{p,I}V.T2 →
+ ∃∃T1. T1 ≅ T2 & X1 = ⓑ{p,I}V.T1.
+#p #I #V #T2 #X1 #H
+elim (tueq_inv_bind1 p I V T2 X1)
+[ #T1 #HT #H destruct ]
+/3 width=3 by tueq_sym, ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tueq.ma".
+
+(* TAIL SORT-IRRELEVANT EQUIVALENCE ON TERMS ********************************)
+
+(* Main properties **********************************************************)
+
+theorem tueq_trans: Transitive … tueq.
+#T1 #T #H elim H -T1 -T
+[ #s1 #s #X #H
+ elim (tueq_inv_sort1 … H) -s /2 width=1 by tueq_sort/
+| #i1 #i #H //
+| #l1 #l #H //
+| #p #I #V #T1 #T #_ #IHT #X #H
+ elim (tueq_inv_bind1 … H) -H /3 width=1 by tueq_bind/
+| #V #T1 #T #_ #IHT #X #H
+ elim (tueq_inv_appl1 … H) -H /3 width=1 by tueq_appl/
+| #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H
+ elim (tueq_inv_cast1 … H) -H /3 width=1 by tueq_cast/
+]
+qed-.
+
+theorem tueq_canc_sn: left_cancellable … tueq.
+/3 width=3 by tueq_trans, tueq_sym/ qed-.
+
+theorem tueq_canc_dx: right_cancellable … tueq.
+/3 width=3 by tueq_trans, tueq_sym/ qed-.
+
+theorem tueq_repl: ∀T1,T2. T1 ≅ T2 →
+ ∀U1. T1 ≅ U1 → ∀U2. T2 ≅ U2 → U1 ≅ U2.
+/3 width=3 by tueq_canc_sn, tueq_trans/ qed-.
(* Basic properties *********************************************************)
-lemma rexs_step_dx: ∀R,L1,L,T. L1 ⪤*[R, T] L →
- ∀L2. L ⪤[R, T] L2 → L1 ⪤*[R, T] L2.
+lemma rexs_step_dx: ∀R,L1,L,T. L1 ⪤*[R,T] L →
+ ∀L2. L ⪤[R,T] L2 → L1 ⪤*[R,T] L2.
#R #L1 #L2 #T #HL1 #L2 @step @HL1 (**) (* auto fails *)
qed-.
-lemma rexs_step_sn: ∀R,L1,L,T. L1 ⪤[R, T] L →
- ∀L2. L ⪤*[R, T] L2 → L1 ⪤*[R, T] L2.
+lemma rexs_step_sn: ∀R,L1,L,T. L1 ⪤[R,T] L →
+ ∀L2. L ⪤*[R,T] L2 → L1 ⪤*[R,T] L2.
#R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *)
qed-.
-lemma rexs_atom: ∀R,I. ⋆ ⪤*[R, ⓪{I}] ⋆.
+lemma rexs_atom: ∀R,I. ⋆ ⪤*[R,⓪{I}] ⋆.
/2 width=1 by inj/ qed.
lemma rexs_sort: ∀R,I,L1,L2,V1,V2,s.
- L1 ⪤*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R, ⋆s] L2.ⓑ{I}V2.
+ L1 ⪤*[R,⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R,⋆s] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
/3 width=4 by rex_sort, rexs_step_dx, inj/
qed.
lemma rexs_pair: ∀R. (∀L. reflexive … (R L)) →
- ∀I,L1,L2,V. L1 ⪤*[R, V] L2 →
- L1.ⓑ{I}V ⪤*[R, #0] L2.ⓑ{I}V.
+ ∀I,L1,L2,V. L1 ⪤*[R,V] L2 →
+ L1.ⓑ{I}V ⪤*[R,#0] L2.ⓑ{I}V.
#R #HR #I #L1 #L2 #V #H elim H -L2
/3 width=5 by rex_pair, rexs_step_dx, inj/
qed.
-lemma rexs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 →
- L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
+lemma rexs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R,cfull,f] L2 →
+ L1.ⓤ{I} ⪤*[R,#0] L2.ⓤ{I}.
/3 width=3 by rex_unit, inj/ qed.
lemma rexs_lref: ∀R,I,L1,L2,V1,V2,i.
- L1 ⪤*[R, #i] L2 → L1.ⓑ{I}V1 ⪤*[R, #↑i] L2.ⓑ{I}V2.
+ L1 ⪤*[R,#i] L2 → L1.ⓑ{I}V1 ⪤*[R,#↑i] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
/3 width=4 by rex_lref, rexs_step_dx, inj/
qed.
lemma rexs_gref: ∀R,I,L1,L2,V1,V2,l.
- L1 ⪤*[R, §l] L2 → L1.ⓑ{I}V1 ⪤*[R, §l] L2.ⓑ{I}V2.
+ L1 ⪤*[R,§l] L2 → L1.ⓑ{I}V1 ⪤*[R,§l] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
/3 width=4 by rex_gref, rexs_step_dx, inj/
qed.
lemma rexs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- ∀L1,L2,T. L1 ⪤*[R1, T] L2 → L1 ⪤*[R2, T] L2.
+ ∀L1,L2,T. L1 ⪤*[R1,T] L2 → L1 ⪤*[R2,T] L2.
#R1 #R2 #HR #L1 #L2 #T #H elim H -L2
/4 width=5 by rex_co, rexs_step_dx, inj/
qed-.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *)
-lemma rexs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤*[R, ⓪{I}] Y2 → Y2 = ⋆.
+lemma rexs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤*[R,⓪{I}] Y2 → Y2 = ⋆.
#R #I #Y2 #H elim H -Y2 /3 width=3 by inj, rex_inv_atom_sn/
qed-.
(* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *)
-lemma rexs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤*[R, ⓪{I}] ⋆ → Y1 = ⋆.
+lemma rexs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤*[R,⓪{I}] ⋆ → Y1 = ⋆.
#R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1
/3 width=3 by inj, rex_inv_atom_dx/
qed-.
-lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 →
+lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R,⋆s] Y2 →
∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤*[R, ⋆s] L2 &
+ | ∃∃I1,I2,L1,L2. L1 ⪤*[R,⋆s] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R #Y1 #Y2 #s #H elim H -Y2
[ #Y2 #H elim (rex_inv_sort … H) -H *
]
qed-.
-lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 →
+lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R,§l] Y2 →
∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤*[R, §l] L2 &
+ | ∃∃I1,I2,L1,L2. L1 ⪤*[R,§l] L2 &
Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R #Y1 #Y2 #l #H elim H -Y2
[ #Y2 #H elim (rex_inv_gref … H) -H *
qed-.
lemma rexs_inv_bind: ∀R. (∀L. reflexive … (R L)) →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- ∧∧ L1 ⪤*[R, V] L2 & L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V.
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+ ∧∧ L1 ⪤*[R,V] L2 & L1.ⓑ{I}V ⪤*[R,T] L2.ⓑ{I}V.
#R #HR #p #I #L1 #L2 #V #T #H elim H -L2
[ #L2 #H elim (rex_inv_bind … V ? H) -H /3 width=1 by inj, conj/
| #L #L2 #_ #H * elim (rex_inv_bind … V ? H) -H /3 width=3 by rexs_step_dx, conj/
]
qed-.
-lemma rexs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 →
- ∧∧ L1 ⪤*[R, V] L2 & L1 ⪤*[R, T] L2.
+lemma rexs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R,ⓕ{I}V.T] L2 →
+ ∧∧ L1 ⪤*[R,V] L2 & L1 ⪤*[R,T] L2.
#R #I #L1 #L2 #V #T #H elim H -L2
[ #L2 #H elim (rex_inv_flat … H) -H /3 width=1 by inj, conj/
| #L #L2 #_ #H * elim (rex_inv_flat … H) -H /3 width=3 by rexs_step_dx, conj/
(* Advanced inversion lemmas ************************************************)
-lemma rexs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤*[R, ⋆s] Y2 →
- ∃∃I2,L2. L1 ⪤*[R, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
+lemma rexs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤*[R,⋆s] Y2 →
+ ∃∃I2,L2. L1 ⪤*[R,⋆s] L2 & Y2 = L2.ⓘ{I2}.
#R #I1 #Y2 #L1 #s #H elim (rexs_inv_sort … H) -H *
[ #H destruct
| #Z #I2 #Y1 #L2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rexs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤*[R, ⋆s] L2.ⓘ{I2} →
- ∃∃I1,L1. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
+lemma rexs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤*[R,⋆s] L2.ⓘ{I2} →
+ ∃∃I1,L1. L1 ⪤*[R,⋆s] L2 & Y1 = L1.ⓘ{I1}.
#R #I2 #Y1 #L2 #s #H elim (rexs_inv_sort … H) -H *
[ #_ #H destruct
| #I1 #Z #L1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rexs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤*[R, §l] Y2 →
- ∃∃I2,L2. L1 ⪤*[R, §l] L2 & Y2 = L2.ⓘ{I2}.
+lemma rexs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤*[R,§l] Y2 →
+ ∃∃I2,L2. L1 ⪤*[R,§l] L2 & Y2 = L2.ⓘ{I2}.
#R #I1 #Y2 #L1 #l #H elim (rexs_inv_gref … H) -H *
[ #H destruct
| #Z #I2 #Y1 #L2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rexs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤*[R, §l] L2.ⓘ{I2} →
- ∃∃I1,L1. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓘ{I1}.
+lemma rexs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤*[R,§l] L2.ⓘ{I2} →
+ ∃∃I1,L1. L1 ⪤*[R,§l] L2 & Y1 = L1.ⓘ{I1}.
#R #I2 #Y1 #L2 #l #H elim (rexs_inv_gref … H) -H *
[ #_ #H destruct
| #I1 #Z #L1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
(* Basic forward lemmas *****************************************************)
-lemma rexs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ②{I}V.T] L2 → L1 ⪤*[R, V] L2.
+lemma rexs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R,②{I}V.T] L2 → L1 ⪤*[R,V] L2.
#R #I #L1 #L2 #V #T #H elim H -L2
/3 width=5 by rex_fwd_pair_sn, rexs_step_dx, inj/
qed-.
lemma rexs_fwd_bind_dx: ∀R. (∀L. reflexive … (R L)) →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V.
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+ L1.ⓑ{I}V ⪤*[R,T] L2.ⓑ{I}V.
#R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind … H) -H //
qed-.
-lemma rexs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 → L1 ⪤*[R, T] L2.
+lemma rexs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R,ⓕ{I}V.T] L2 → L1 ⪤*[R,T] L2.
#R #I #L1 #L2 #V #T #H elim (rexs_inv_flat … H) -H //
qed-.
(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
definition tc_f_dedropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
- ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
- ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 →
+ ∀K2,T. K1 ⪤*[R,T] K2 → ∀U. ⬆*[f] T ≘ U →
+ ∃∃L2. L1 ⪤*[R,U] L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2.
definition tc_f_dropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
- ∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
- ∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ ∀L2,U. L1 ⪤*[R,U] L2 → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K2. K1 ⪤*[R,T] K2 & ⬇*[b,f] L2 ≘ K2.
definition tc_f_dropable_dx: predicate (relation3 lenv term term) ≝
- λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[R, T] K2.
+ λR. ∀L1,L2,U. L1 ⪤*[R,U] L2 →
+ ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤*[R,T] K2.
(* Properties with generic slicing for local environments *******************)
(* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
lemma rexs_pair_refl: ∀R. c_reflexive … R →
- ∀L,V1,V2. CTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R, T] L.ⓑ{I}V2.
+ ∀L,V1,V2. CTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R,T] L.ⓑ{I}V2.
#R #HR #L #V1 #V2 #H elim H -V2
/3 width=3 by rexs_step_dx, rex_pair_refl, inj/
qed.
lemma rexs_tc: ∀R,L1,L2,T,f. 𝐈⦃f⦄ → TC … (sex cfull (cext2 R) f) L1 L2 →
- L1 ⪤*[R, T] L2.
+ L1 ⪤*[R,T] L2.
#R #L1 #L2 #T #f #Hf #H elim H -L2
[ elim (frees_total L1 T) | #L elim (frees_total L T) ]
/5 width=7 by sex_sdj, rexs_step_dx, sdj_isid_sn, inj, ex2_intro/
lemma rexs_ind_sn: ∀R. c_reflexive … R →
∀L1,T. ∀Q:predicate …. Q L1 →
- (∀L,L2. L1 ⪤*[R, T] L → L ⪤[R, T] L2 → Q L → Q L2) →
- ∀L2. L1 ⪤*[R, T] L2 → Q L2.
+ (∀L,L2. L1 ⪤*[R,T] L → L ⪤[R,T] L2 → Q L → Q L2) →
+ ∀L2. L1 ⪤*[R,T] L2 → Q L2.
#R #HR #L1 #T #Q #HL1 #IHL1 #L2 #HL12
@(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by rex_refl/
qed-.
lemma rexs_ind_dx: ∀R. c_reflexive … R →
∀L2,T. ∀Q:predicate …. Q L2 →
- (∀L1,L. L1 ⪤[R, T] L → L ⪤*[R, T] L2 → Q L → Q L1) →
- ∀L1. L1 ⪤*[R, T] L2 → Q L1.
+ (∀L1,L. L1 ⪤[R,T] L → L ⪤*[R,T] L2 → Q L → Q L1) →
+ ∀L1. L1 ⪤*[R,T] L2 → Q L1.
#R #HR #L2 #Q #HL2 #IHL2 #L1 #HL12
@(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by rex_refl/
qed-.
(* Advanced inversion lemmas ************************************************)
lemma rexs_inv_bind_void: ∀R. c_reflexive … R →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- ∧∧ L1 ⪤*[R, V] L2 & L1.ⓧ ⪤*[R, T] L2.ⓧ.
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+ ∧∧ L1 ⪤*[R,V] L2 & L1.ⓧ ⪤*[R,T] L2.ⓧ.
#R #HR #p #I #L1 #L2 #V #T #H @(rexs_ind_sn … HR … H) -L2
[ /3 width=1 by rexs_refl, conj/
| #L #L2 #_ #H * elim (rex_inv_bind_void … H) -H /3 width=3 by rexs_step_dx, conj/
(* Advanced forward lemmas **************************************************)
lemma rexs_fwd_bind_dx_void: ∀R. c_reflexive … R →
- ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- L1.ⓧ ⪤*[R, T] L2.ⓧ.
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R,ⓑ{p,I}V.T] L2 →
+ L1.ⓧ ⪤*[R,T] L2.ⓧ.
#R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind_void … H) -H //
qed-.
(* Forward lemmas with length for local environments ************************)
(* Basic_2A1: uses: TC_lpx_sn_fwd_length *)
-lemma rexs_fwd_length: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 → |L1| = |L2|.
+lemma rexs_fwd_length: ∀R,L1,L2,T. L1 ⪤*[R,T] L2 → |L1| = |L2|.
#R #L1 #L2 #T #H elim H -L2
[ #L2 #HL12 >(rex_fwd_length … HL12) -HL12 //
| #L #L2 #_ #HL2 #IHL1
(* Properties with generic extension of a context sensitive relation ********)
lemma rexs_lex: ∀R. c_reflexive … R →
- ∀L1,L2,T. L1 ⪤[CTC … R] L2 → L1 ⪤*[R, T] L2.
+ ∀L1,L2,T. L1 ⪤[CTC … R] L2 → L1 ⪤*[R,T] L2.
#R #HR #L1 #L2 #T *
/5 width=7 by rexs_tc, sex_inv_tc_dx, sex_co, ext2_inv_tc, ext2_refl/
qed.
lemma rexs_lex_req: ∀R. c_reflexive … R →
∀L1,L. L1 ⪤[CTC … R] L → ∀L2,T. L ≡[T] L2 →
- L1 ⪤*[R, T] L2.
+ L1 ⪤*[R,T] L2.
/3 width=3 by rexs_lex, rexs_step_dx, req_fwd_rex/ qed.
(* Inversion lemmas with generic extension of a context sensitive relation **)
rex_fsge_compatible R →
s_rs_transitive … R (λ_.lex R) →
req_transitive R →
- ∀L1,L2,T. L1 ⪤*[R, T] L2 →
+ ∀L1,L2,T. L1 ⪤*[R,T] L2 →
∃∃L. L1 ⪤[CTC … R] L & L ≡[T] L2.
#R #H1R #H2R #H3R #H4R #L1 #L2 #T #H
lapply (s_rs_transitive_lex_inv_isid … H3R) -H3R #H3R
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L. break ⓓ T1 )"
+notation "hvbox( L. ⓓ break T1 )"
left associative with precedence 50
for @{ 'DxAbbr $L $T1 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L. break ⓛ T1 )"
+notation "hvbox( L. ⓛ break T1 )"
left associative with precedence 51
for @{ 'DxAbst $L $T1 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L. break ⓤ { term 46 I } )"
+notation "hvbox( L. ⓤ { break term 46 I } )"
non associative with precedence 47
for @{ 'DxBind1 $L $I }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L. break ⓑ { term 46 I } break term 49 T1 )"
+notation "hvbox( L. ⓑ { break term 46 I } break term 49 T1 )"
non associative with precedence 48
for @{ 'DxBind2 $L $I $T1 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "𝛚"
+ non associative with precedence 46
+ for @{ 'Omega }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "𝟏"
+ non associative with precedence 46
+ for @{ 'One }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "𝟐"
+ non associative with precedence 46
+ for @{ 'Two }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⫯[ term 46 h ] break term 46 s )"
+ non associative with precedence 46
+ for @{ 'UpSpoon $h $s }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( T1 ≅ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'ApproxEq $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L ⊢ 𝐅 + ⦃ break term 46 T ⦄ ≘ break term 46 f )"
+ non associative with precedence 45
+ for @{ 'FreePlus $L $T $f }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L ⊢ 𝐅 * ⦃ break term 46 T ⦄ ≘ break term 46 f )"
- non associative with precedence 45
- for @{ 'FreeStar $L $T $f }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 L1, break term 46 f1 ⦄ ⫃ 𝐅* ⦃ break term 46 L2, break term 46 f2 ⦄ )"
+notation "hvbox( ⦃ term 46 L1, break term 46 f1 ⦄ ⫃ 𝐅+ ⦃ break term 46 L2, break term 46 f2 ⦄ )"
non associative with precedence 45
for @{ 'LRSubEqF $L1 $f1 $L2 $f2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬆[ term 46 m, break term 46 n ] break term 46 T1 ≘ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'RLift $m $n $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( T1 ≛ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'StarEq $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L ⊢ break term 46 T1 ≛ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'StarEq $L $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( T1 ≛ [ break term 46 h, break term 46 o ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'StarEq $h $o $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L ⊢ break term 46 T1 ≛ [ break term 46 h, break term 46 o ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'StarEq $h $o $L $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ≛ [ break term 46 T ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'StarEqSn $T $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 ≛ [ break term 46 h, break term 46 o, break term 46 T ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'StarEqSn $h $o $T $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≛ ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'StarEqSn $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≛ [ break term 46 h, break term 46 o ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'StarEqSn $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( â¦\83 term 46 G1, break term 46 L1, break term 46 T1 â¦\84 â\8a\90 ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( â¦\83 term 46 G1, break term 46 L1, break term 46 T1 â¦\84 â¬\82 ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTerm $G1 $L1 $T1 $G2 $L2 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( â¦\83 term 46 G1, break term 46 L1, break term 46 T1 â¦\84 â\8a\90 [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( â¦\83 term 46 G1, break term 46 L1, break term 46 T1 â¦\84 â¬\82 [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTerm $b $G1 $L1 $T1 $G2 $L2 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( â¦\83 term 46 G1, break term 46 L1, break term 46 T1 â¦\84 â\8a\90⸮ ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( â¦\83 term 46 G1, break term 46 L1, break term 46 T1 â¦\84 â¬\82⸮ ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTermOpt $G1 $L1 $T1 $G2 $L2 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( â¦\83 term 46 G1, break term 46 L1, break term 46 T1 â¦\84 â\8a\90⸮ [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( â¦\83 term 46 G1, break term 46 L1, break term 46 T1 â¦\84 â¬\82⸮ [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTermOpt $b $G1 $L1 $T1 $G2 $L2 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ + ⦃ break term 46 G2, term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⬂ + ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTermPlus $G1 $L1 $T1 $G2 $L2 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ + [ break term 46 b ] ⦃ break term 46 G2, term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⬂ + [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTermPlus $b $G1 $L1 $T1 $G2 $L2 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ * ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⬂ * ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTermStar $G1 $L1 $T1 $G2 $L2 $T2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ * [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⬂ * [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
non associative with precedence 45
for @{ 'SupTermStar $b $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( T1 ⩳ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'TopIso $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( T1 ⩳ [ break term 46 h, break term 46 o ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'TopIso $h $o $T1 $T2 }.
'RDropStar b f L1 L2 = (drops b f L1 L2).
definition d_liftable1: predicate (relation2 lenv term) ≝
- λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≘ K →
+ λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b,f] L ≘ K →
∀U. ⬆*[f] T ≘ U → R L U.
definition d_liftable1_isuni: predicate (relation2 lenv term) ≝
- λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ →
+ λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b,f] L ≘ K → 𝐔⦃f⦄ →
∀U. ⬆*[f] T ≘ U → R L U.
definition d_deliftable1: predicate (relation2 lenv term) ≝
- λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K →
+ λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b,f] L ≘ K →
∀T. ⬆*[f] T ≘ U → R K T.
definition d_deliftable1_isuni: predicate (relation2 lenv term) ≝
- λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ →
+ λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b,f] L ≘ K → 𝐔⦃f⦄ →
∀T. ⬆*[f] T ≘ U → R K T.
definition d_liftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≘ K →
+ λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b,f] L ≘ K →
∀U1. S f T1 U1 →
∃∃U2. S f T2 U2 & R L U1 U2.
definition d_deliftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≘ K →
+ λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b,f] L ≘ K →
∀T1. S f T1 U1 →
∃∃T2. S f T2 U2 & R K T1 T2.
definition d_liftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≘ K →
+ λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b,f] L ≘ K →
∀U1. S f T1 U1 →
∀U2. S f T2 U2 → R L U1 U2.
definition d_deliftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≘ K →
+ λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b,f] L ≘ K →
∀T1. S f T1 U1 →
∀T2. S f T2 U2 → R K T1 T2.
definition co_dropable_sn: predicate (rtmap → relation lenv) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ →
∀f2,L2. R f2 L1 L2 → ∀f1. f ~⊚ f1 ≘ f2 →
- ∃∃K2. R f1 K1 K2 & ⬇*[b, f] L2 ≘ K2.
+ ∃∃K2. R f1 K1 K2 & ⬇*[b,f] L2 ≘ K2.
definition co_dropable_dx: predicate (rtmap → relation lenv) ≝
λR. ∀f2,L1,L2. R f2 L1 L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
+ ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ →
∀f1. f ~⊚ f1 ≘ f2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & R f1 K1 K2.
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & R f1 K1 K2.
definition co_dedropable_sn: predicate (rtmap → relation lenv) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀f1,K2. R f1 K1 K2 →
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → ∀f1,K2. R f1 K1 K2 →
∀f2. f ~⊚ f1 ≘ f2 →
- ∃∃L2. R f2 L1 L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+ ∃∃L2. R f2 L1 L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2.
(* Basic properties *********************************************************)
-lemma drops_atom_F: ∀f. ⬇*[Ⓕ, f] ⋆ ≘ ⋆.
+lemma drops_atom_F: ∀f. ⬇*[Ⓕ,f] ⋆ ≘ ⋆.
#f @drops_atom #H destruct
qed.
-lemma drops_eq_repl_back: ∀b,L1,L2. eq_repl_back … (λf. ⬇*[b, f] L1 ≘ L2).
+lemma drops_eq_repl_back: ∀b,L1,L2. eq_repl_back … (λf. ⬇*[b,f] L1 ≘ L2).
#b #L1 #L2 #f1 #H elim H -f1 -L1 -L2
[ /4 width=3 by drops_atom, isid_eq_repl_back/
| #f1 #I #L1 #L2 #_ #IH #f2 #H elim (eq_inv_nx … H) -H
]
qed-.
-lemma drops_eq_repl_fwd: ∀b,L1,L2. eq_repl_fwd … (λf. ⬇*[b, f] L1 ≘ L2).
+lemma drops_eq_repl_fwd: ∀b,L1,L2. eq_repl_fwd … (λf. ⬇*[b,f] L1 ≘ L2).
#b #L1 #L2 @eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *)
qed-.
(* Basic_2A1: includes: drop_FT *)
-lemma drops_TF: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ L2.
+lemma drops_TF: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → ⬇*[Ⓕ,f] L1 ≘ L2.
#f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by drops_atom, drops_drop, drops_skip/
qed.
(* Basic_2A1: includes: drop_gen *)
-lemma drops_gen: ∀b,f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[b, f] L1 ≘ L2.
+lemma drops_gen: ∀b,f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → ⬇*[b,f] L1 ≘ L2.
* /2 width=1 by drops_TF/
qed-.
(* Basic_2A1: includes: drop_T *)
-lemma drops_F: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ L2.
+lemma drops_F: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → ⬇*[Ⓕ,f] L1 ≘ L2.
* /2 width=1 by drops_TF/
qed-.
(* Basic inversion lemmas ***************************************************)
-fact drops_inv_atom1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → X = ⋆ →
+fact drops_inv_atom1_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → X = ⋆ →
Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
#b #f #X #Y * -f -X -Y
[ /3 width=1 by conj/
(* Basic_1: includes: drop_gen_sort *)
(* Basic_2A1: includes: drop_inv_atom1 *)
-lemma drops_inv_atom1: ∀b,f,Y. ⬇*[b, f] ⋆ ≘ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
+lemma drops_inv_atom1: ∀b,f,Y. ⬇*[b,f] ⋆ ≘ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
/2 width=3 by drops_inv_atom1_aux/ qed-.
-fact drops_inv_drop1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I,K. X = K.ⓘ{I} → f = ↑g →
- ⬇*[b, g] K ≘ Y.
+fact drops_inv_drop1_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → ∀g,I,K. X = K.ⓘ{I} → f = ↑g →
+ ⬇*[b,g] K ≘ Y.
#b #f #X #Y * -f -X -Y
[ #f #Hf #g #J #K #H destruct
| #f #I #L1 #L2 #HL #g #J #K #H1 #H2 <(injective_next … H2) -g destruct //
(* Basic_1: includes: drop_gen_drop *)
(* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
-lemma drops_inv_drop1: ∀b,f,I,K,Y. ⬇*[b, ↑f] K.ⓘ{I} ≘ Y → ⬇*[b, f] K ≘ Y.
+lemma drops_inv_drop1: ∀b,f,I,K,Y. ⬇*[b,↑f] K.ⓘ{I} ≘ Y → ⬇*[b,f] K ≘ Y.
/2 width=6 by drops_inv_drop1_aux/ qed-.
-fact drops_inv_skip1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ{I1} → f = ⫯g →
- ∃∃I2,K2. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & Y = K2.ⓘ{I2}.
+fact drops_inv_skip1_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ{I1} → f = ⫯g →
+ ∃∃I2,K2. ⬇*[b,g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & Y = K2.ⓘ{I2}.
#b #f #X #Y * -f -X -Y
[ #f #Hf #g #J1 #K1 #H destruct
| #f #I #L1 #L2 #_ #g #J1 #K1 #_ #H2 elim (discr_next_push … H2)
(* Basic_1: includes: drop_gen_skip_l *)
(* Basic_2A1: includes: drop_inv_skip1 *)
-lemma drops_inv_skip1: ∀b,f,I1,K1,Y. ⬇*[b, ⫯f] K1.ⓘ{I1} ≘ Y →
- ∃∃I2,K2. ⬇*[b, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & Y = K2.ⓘ{I2}.
+lemma drops_inv_skip1: ∀b,f,I1,K1,Y. ⬇*[b,⫯f] K1.ⓘ{I1} ≘ Y →
+ ∃∃I2,K2. ⬇*[b,f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & Y = K2.ⓘ{I2}.
/2 width=5 by drops_inv_skip1_aux/ qed-.
-fact drops_inv_skip2_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ{I2} → f = ⫯g →
- ∃∃I1,K1. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & X = K1.ⓘ{I1}.
+fact drops_inv_skip2_aux: ∀b,f,X,Y. ⬇*[b,f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ{I2} → f = ⫯g →
+ ∃∃I1,K1. ⬇*[b,g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & X = K1.ⓘ{I1}.
#b #f #X #Y * -f -X -Y
[ #f #Hf #g #J2 #K2 #H destruct
| #f #I #L1 #L2 #_ #g #J2 #K2 #_ #H2 elim (discr_next_push … H2)
(* Basic_1: includes: drop_gen_skip_r *)
(* Basic_2A1: includes: drop_inv_skip2 *)
-lemma drops_inv_skip2: ∀b,f,I2,X,K2. ⬇*[b, ⫯f] X ≘ K2.ⓘ{I2} →
- ∃∃I1,K1. ⬇*[b, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & X = K1.ⓘ{I1}.
+lemma drops_inv_skip2: ∀b,f,I2,X,K2. ⬇*[b,⫯f] X ≘ K2.ⓘ{I2} →
+ ∃∃I1,K1. ⬇*[b,f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & X = K1.ⓘ{I1}.
/2 width=5 by drops_inv_skip2_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⬇*[b, f2] X ≘ Y → ∀I,K. Y = K.ⓘ{I} →
- ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b, f] X ≘ K.
+fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⬇*[b,f2] X ≘ Y → ∀I,K. Y = K.ⓘ{I} →
+ ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b,f] X ≘ K.
#b #f2 #X #Y #H elim H -f2 -X -Y
[ #f2 #Hf2 #J #K #H destruct
| #f2 #I #L1 #L2 #_ #IHL #J #K #H elim (IHL … H) -IHL
]
qed-.
-lemma drops_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b, f2] X ≘ K.ⓘ{I} →
- ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b, f] X ≘ K.
+lemma drops_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b,f2] X ≘ K.ⓘ{I} →
+ ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b,f] X ≘ K.
/2 width=4 by drops_fwd_drop2_aux/ qed-.
(* Properties with test for identity ****************************************)
(* Basic_2A1: includes: drop_refl *)
-lemma drops_refl: ∀b,L,f. 𝐈⦃f⦄ → ⬇*[b, f] L ≘ L.
+lemma drops_refl: ∀b,L,f. 𝐈⦃f⦄ → ⬇*[b,f] L ≘ L.
#b #L elim L -L /2 width=1 by drops_atom/
#L #I #IHL #f #Hf elim (isid_inv_gen … Hf) -Hf
/3 width=1 by drops_skip, liftsb_refl/
(* Basic_1: includes: drop_gen_refl *)
(* Basic_2A1: includes: drop_inv_O2 *)
-lemma drops_fwd_isid: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → 𝐈⦃f⦄ → L1 = L2.
+lemma drops_fwd_isid: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → 𝐈⦃f⦄ → L1 = L2.
#b #f #L1 #L2 #H elim H -f -L1 -L2 //
[ #f #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) //
| /5 width=5 by isid_inv_push, liftsb_fwd_isid, eq_f2, sym_eq/
]
qed-.
-lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b, f2] X ≘ K.ⓘ{I} →
- ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ↑f1 ≘ f → ⬇*[b, f] X ≘ K.
+lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b,f2] X ≘ K.ⓘ{I} →
+ ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ↑f1 ≘ f → ⬇*[b,f] X ≘ K.
#b #f2 #I #X #K #H #f1 #f #Hf1 #Hf elim (drops_fwd_drop2 … H) -H
#g1 #g #Hg1 #Hg #HK lapply (after_mono_eq … Hg … Hf ??) -Hg -Hf
/3 width=5 by drops_eq_repl_back, isid_inv_eq_repl, eq_next/
(* Forward lemmas with test for finite colength *****************************)
-lemma drops_fwd_isfin: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐅⦃f⦄.
+lemma drops_fwd_isfin: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐅⦃f⦄.
#f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by isfin_next, isfin_push, isfin_isid/
qed-.
(* Properties with test for uniformity **************************************)
-lemma drops_isuni_ex: ∀f. 𝐔⦃f⦄ → ∀L. ∃K. ⬇*[Ⓕ, f] L ≘ K.
+lemma drops_isuni_ex: ∀f. 𝐔⦃f⦄ → ∀L. ∃K. ⬇*[Ⓕ,f] L ≘ K.
#f #H elim H -f /4 width=2 by drops_refl, drops_TF, ex_intro/
#f #_ #g #H #IH destruct * /2 width=2 by ex_intro/
#L #I elim (IH L) -IH /3 width=2 by drops_drop, ex_intro/
(* Inversion lemmas with test for uniformity ********************************)
-lemma drops_inv_isuni: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐔⦃f⦄ →
+lemma drops_inv_isuni: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐔⦃f⦄ →
(𝐈⦃f⦄ ∧ L1 = L2) ∨
- ∃∃g,I,K. ⬇*[Ⓣ, g] K ≘ L2 & 𝐔⦃g⦄ & L1 = K.ⓘ{I} & f = ↑g.
+ ∃∃g,I,K. ⬇*[Ⓣ,g] K ≘ L2 & 𝐔⦃g⦄ & L1 = K.ⓘ{I} & f = ↑g.
#f #L1 #L2 * -f -L1 -L2
[ /4 width=1 by or_introl, conj/
| /4 width=7 by isuni_inv_next, ex4_3_intro, or_intror/
qed-.
(* Basic_2A1: was: drop_inv_O1_pair1 *)
-lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⬇*[b, f] K.ⓘ{I} ≘ L2 →
+lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⬇*[b,f] K.ⓘ{I} ≘ L2 →
(𝐈⦃f⦄ ∧ L2 = K.ⓘ{I}) ∨
- ∃∃g. 𝐔⦃g⦄ & ⬇*[b, g] K ≘ L2 & f = ↑g.
+ ∃∃g. 𝐔⦃g⦄ & ⬇*[b,g] K ≘ L2 & f = ↑g.
#b #f #I #K #L2 #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct
[ lapply (drops_inv_skip1 … H) -H * #Z #Y #HY #HZ #H destruct
<(drops_fwd_isid … HY Hg) -Y >(liftsb_fwd_isid … HZ Hg) -Z
qed-.
(* Basic_2A1: was: drop_inv_O1_pair2 *)
-lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K.ⓘ{I} →
+lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b,f] L1 ≘ K.ⓘ{I} →
(𝐈⦃f⦄ ∧ L1 = K.ⓘ{I}) ∨
- ∃∃g,I1,K1. 𝐔⦃g⦄ & ⬇*[b, g] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1} & f = ↑g.
+ ∃∃g,I1,K1. 𝐔⦃g⦄ & ⬇*[b,g] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1} & f = ↑g.
#b #f #I #K *
[ #Hf #H elim (drops_inv_atom1 … H) -H #H destruct
| #L1 #I1 #Hf #H elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
]
qed-.
-lemma drops_inv_bind2_isuni_next: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, ↑f] L1 ≘ K.ⓘ{I} →
- ∃∃I1,K1. ⬇*[b, f] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1}.
+lemma drops_inv_bind2_isuni_next: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b,↑f] L1 ≘ K.ⓘ{I} →
+ ∃∃I1,K1. ⬇*[b,f] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1}.
#b #f #I #K #L1 #Hf #H elim (drops_inv_bind2_isuni … H) -H /2 width=3 by isuni_next/ -Hf *
[ #H elim (isid_inv_next … H) -H //
| /2 width=4 by ex2_2_intro/
]
qed-.
-fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ, f] L1 ≘ L2 → 𝐔⦃f⦄ →
- ∀I,K. L2 = K.ⓘ{I} → ⬇*[Ⓣ, f] L1 ≘ K.ⓘ{I}.
+fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ,f] L1 ≘ L2 → 𝐔⦃f⦄ →
+ ∀I,K. L2 = K.ⓘ{I} → ⬇*[Ⓣ,f] L1 ≘ K.ⓘ{I}.
#f #L1 #L2 #H elim H -f -L1 -L2
[ #f #_ #_ #J #K #H destruct
| #f #I #L1 #L2 #_ #IH #Hf #J #K #H destruct
qed-.
(* Basic_2A1: includes: drop_inv_FT *)
-lemma drops_inv_TF: ∀f,I,L,K. ⬇*[Ⓕ, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≘ K.ⓘ{I}.
+lemma drops_inv_TF: ∀f,I,L,K. ⬇*[Ⓕ,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ,f] L ≘ K.ⓘ{I}.
/2 width=3 by drops_inv_TF_aux/ qed-.
(* Basic_2A1: includes: drop_inv_gen *)
-lemma drops_inv_gen: ∀b,f,I,L,K. ⬇*[b, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≘ K.ⓘ{I}.
+lemma drops_inv_gen: ∀b,f,I,L,K. ⬇*[b,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ,f] L ≘ K.ⓘ{I}.
* /2 width=1 by drops_inv_TF/
qed-.
(* Basic_2A1: includes: drop_inv_T *)
-lemma drops_inv_F: ∀b,f,I,L,K. ⬇*[Ⓕ, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[b, f] L ≘ K.ⓘ{I}.
+lemma drops_inv_F: ∀b,f,I,L,K. ⬇*[Ⓕ,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[b,f] L ≘ K.ⓘ{I}.
* /2 width=1 by drops_inv_TF/
qed-.
(* Basic_1: was: drop_S *)
(* Basic_2A1: was: drop_fwd_drop2 *)
-lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K. 𝐔⦃f⦄ → ⬇*[b, f] X ≘ K.ⓘ{I} → ⬇*[b, ↑f] X ≘ K.
+lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K. 𝐔⦃f⦄ → ⬇*[b,f] X ≘ K.ⓘ{I} → ⬇*[b,↑f] X ≘ K.
/3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-.
(* Inversion lemmas with uniform relocations ********************************)
-lemma drops_inv_atom2: ∀b,L,f. ⬇*[b, f] L ≘ ⋆ →
- ∃∃n,f1. ⬇*[b, 𝐔❴n❵] L ≘ ⋆ & 𝐔❴n❵ ⊚ f1 ≘ f.
+lemma drops_inv_atom2: ∀b,L,f. ⬇*[b,f] L ≘ ⋆ →
+ ∃∃n,f1. ⬇*[b,𝐔❴n❵] L ≘ ⋆ & 𝐔❴n❵ ⊚ f1 ≘ f.
#b #L elim L -L
[ /3 width=4 by drops_atom, after_isid_sn, ex2_2_intro/
| #L #I #IH #f #H elim (pn_split f) * #g #H0 destruct
(* Properties with uniform relocations **************************************)
-lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⬇*[i] L ≘ K.ⓘ{I}.
+lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⬇*[i] L ≘ K.ⓘ{I}.
#L elim L -L /2 width=1 by or_introl/
#L #I #IH * /4 width=3 by drops_refl, ex1_2_intro, or_intror/
#i elim (IH i) -IH /3 width=1 by drops_drop, or_introl/
qed-.
(* Basic_2A1: includes: drop_split *)
-lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ∀f1,f2. f1 ⊚ f2 ≘ f → 𝐔⦃f1⦄ →
- ∃∃L. ⬇*[b, f1] L1 ≘ L & ⬇*[b, f2] L ≘ L2.
+lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → ∀f1,f2. f1 ⊚ f2 ≘ f → 𝐔⦃f1⦄ →
+ ∃∃L. ⬇*[b,f1] L1 ≘ L & ⬇*[b,f2] L ≘ L2.
#b #f #L1 #L2 #H elim H -f -L1 -L2
[ #f #H0f #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom
#H lapply (H0f H) -b
]
qed-.
-lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b, f1] L1 ≘ L → ∀f2,f. f1 ⊚ f2 ≘ f → 𝐔⦃f2⦄ →
- ∃∃L2. ⬇*[Ⓕ, f2] L ≘ L2 & ⬇*[Ⓕ, f] L1 ≘ L2.
+lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b,f1] L1 ≘ L → ∀f2,f. f1 ⊚ f2 ≘ f → 𝐔⦃f2⦄ →
+ ∃∃L2. ⬇*[Ⓕ,f2] L ≘ L2 & ⬇*[Ⓕ,f] L1 ≘ L2.
#b #f1 #L1 #L #H elim H -f1 -L1 -L
[ #f1 #Hf1 #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct
| #f1 #I #L1 #L #HL1 #IH #f2 #f #Hf #Hf2 elim (after_inv_nxx … Hf) -Hf [2,3: // ]
⬇*[b,⫯⫱*[↑i2]f] L1 ≘ L2.
/3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-.
-lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⬇*[b, f] L ≘ K0.ⓘ{I} → ∀i. @⦃O, f⦄ ≘ i →
- ∃∃J,K. ⬇*[i]L ≘ K.ⓘ{J} & ⬇*[b, ⫱*[↑i]f] K ≘ K0 & ⬆*[⫱*[↑i]f] I ≘ J.
+lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⬇*[b,f] L ≘ K0.ⓘ{I} → ∀i. @⦃O,f⦄ ≘ i →
+ ∃∃J,K. ⬇*[i]L ≘ K.ⓘ{J} & ⬇*[b,⫱*[↑i]f] K ≘ K0 & ⬆*[⫱*[↑i]f] I ≘ J.
#b #f #I #L #K0 #H #i #Hf
elim (drops_split_trans … H) -H [ |5: @(after_uni_dx … Hf) |2,3: skip ] /2 width=1 by after_isid_dx/ #Y #HLY #H
lapply (drops_tls_at … Hf … H) -H #H
(* Main properties **********************************************************)
(* Basic_2A1: includes: drop_conf_ge drop_conf_be drop_conf_le *)
-theorem drops_conf: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f,L2. ⬇*[b2, f] L1 ≘ L2 →
- ∀f2. f1 ⊚ f2 ≘ f → ⬇*[b2, f2] L ≘ L2.
+theorem drops_conf: ∀b1,f1,L1,L. ⬇*[b1,f1] L1 ≘ L →
+ ∀b2,f,L2. ⬇*[b2,f] L1 ≘ L2 →
+ ∀f2. f1 ⊚ f2 ≘ f → ⬇*[b2,f2] L ≘ L2.
#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
[ #f1 #_ #b2 #f #L2 #HL2 #f2 #Hf12 elim (drops_inv_atom1 … HL2) -b1 -HL2
#H #Hf destruct @drops_atom
(* Basic_2A1: includes: drop_trans_ge drop_trans_le drop_trans_ge_comm
drops_drop_trans
*)
-theorem drops_trans: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f2,L2. ⬇*[b2, f2] L ≘ L2 →
- ∀f. f1 ⊚ f2 ≘ f → ⬇*[b1∧b2, f] L1 ≘ L2.
+theorem drops_trans: ∀b1,f1,L1,L. ⬇*[b1,f1] L1 ≘ L →
+ ∀b2,f2,L2. ⬇*[b2,f2] L ≘ L2 →
+ ∀f. f1 ⊚ f2 ≘ f → ⬇*[b1∧b2,f] L1 ≘ L2.
#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
[ #f1 #Hf1 #b2 #f2 #L2 #HL2 #f #Hf elim (drops_inv_atom1 … HL2) -HL2
#H #Hf2 destruct @drops_atom #H elim (andb_inv_true_dx … H) -H
(* Advanced properties ******************************************************)
(* Basic_2A1: includes: drop_mono *)
-lemma drops_mono: ∀b1,f,L,L1. ⬇*[b1, f] L ≘ L1 →
- ∀b2,L2. ⬇*[b2, f] L ≘ L2 → L1 = L2.
+lemma drops_mono: ∀b1,f,L,L1. ⬇*[b1,f] L ≘ L1 →
+ ∀b2,L2. ⬇*[b2,f] L ≘ L2 → L1 = L2.
#b1 #f #L #L1 lapply (after_isid_dx 𝐈𝐝 … f)
/3 width=8 by drops_conf, drops_fwd_isid/
qed-.
+lemma drops_inv_uni: ∀L,i. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ → ∀I,K. ⬇*[i] L ≘ K.ⓘ{I} → ⊥.
+#L #i #H1 #I #K #H2
+lapply (drops_F … H2) -H2 #H2
+lapply (drops_mono … H2 … H1) -L -i #H destruct
+qed-.
+
+lemma drops_ldec_dec: ∀L,i. Decidable (∃∃K,W. ⬇*[i] L ≘ K.ⓛW).
+#L #i elim (drops_F_uni L i) [| * * [ #I #K1 | * #W1 #K1 ] ]
+[4: /3 width=3 by ex1_2_intro, or_introl/
+|*: #H1L @or_intror * #K2 #W2 #H2L
+ lapply (drops_mono … H2L … H1L) -L #H destruct
+]
+qed-.
+
(* Basic_2A1: includes: drop_conf_lt *)
-lemma drops_conf_skip1: ∀b2,f,L,L2. ⬇*[b2, f] L ≘ L2 →
- ∀b1,f1,I1,K1. ⬇*[b1, f1] L ≘ K1.ⓘ{I1} →
+lemma drops_conf_skip1: ∀b2,f,L,L2. ⬇*[b2,f] L ≘ L2 →
+ ∀b1,f1,I1,K1. ⬇*[b1,f1] L ≘ K1.ⓘ{I1} →
∀f2. f1 ⊚ ⫯f2 ≘ f →
∃∃I2,K2. L2 = K2.ⓘ{I2} &
- ⬇*[b2, f2] K1 ≘ K2 & ⬆*[f2] I2 ≘ I1.
+ ⬇*[b2,f2] K1 ≘ K2 & ⬆*[f2] I2 ≘ I1.
#b2 #f #L #L2 #H2 #b1 #f1 #I1 #K1 #H1 #f2 #Hf lapply (drops_conf … H1 … H2 … Hf) -L -Hf
#H elim (drops_inv_skip1 … H) -H /2 width=5 by ex3_2_intro/
qed-.
(* Basic_2A1: includes: drop_trans_lt *)
-lemma drops_trans_skip2: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f2,I2,K2. ⬇*[b2, f2] L ≘ K2.ⓘ{I2} →
+lemma drops_trans_skip2: ∀b1,f1,L1,L. ⬇*[b1,f1] L1 ≘ L →
+ ∀b2,f2,I2,K2. ⬇*[b2,f2] L ≘ K2.ⓘ{I2} →
∀f. f1 ⊚ f2 ≘ ⫯f →
∃∃I1,K1. L1 = K1.ⓘ{I1} &
- ⬇*[b1∧b2, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1.
+ ⬇*[b1∧b2,f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1.
#b1 #f1 #L1 #L #H1 #b2 #f2 #I2 #K2 #H2 #f #Hf
lapply (drops_trans … H1 … H2 … Hf) -L -Hf
#H elim (drops_inv_skip2 … H) -H /2 width=5 by ex3_2_intro/
(* Basic_2A1: includes: drops_conf_div *)
lemma drops_conf_div_bind: ∀f1,f2,I1,I2,L,K.
- ⬇*[Ⓣ, f1] L ≘ K.ⓘ{I1} → ⬇*[Ⓣ, f2] L ≘ K.ⓘ{I2} →
+ ⬇*[Ⓣ,f1] L ≘ K.ⓘ{I1} → ⬇*[Ⓣ,f2] L ≘ K.ⓘ{I2} →
𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≡ f2 ∧ I1 = I2.
#f1 #f2 #I1 #I2 #L #K #Hf1 #Hf2 #HU1 #HU2
lapply (drops_isuni_fwd_drop2 … Hf1) // #H1
lapply (drops_mono … H0 … Hf2) -L #H
destruct /2 width=1 by conj/
qed-.
-
-lemma drops_inv_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ → ∀I,K. ⬇*[i] L ≘ K.ⓘ{I} → ⊥.
-#L #i #H1 #I #K #H2
-lapply (drops_F … H2) -H2 #H2
-lapply (drops_mono … H2 … H1) -L -i #H destruct
-qed-.
(* Forward lemmas with length for local environments ************************)
(* Basic_2A1: includes: drop_fwd_length_le4 *)
-lemma drops_fwd_length_le4: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → |L2| ≤ |L1|.
+lemma drops_fwd_length_le4: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → |L2| ≤ |L1|.
#b #f #L1 #L2 #H elim H -f -L1 -L2 /2 width=1 by le_S, le_S_S/
qed-.
(* Basic_2A1: includes: drop_fwd_length_eq1 *)
-theorem drops_fwd_length_eq1: ∀b1,b2,f,L1,K1. ⬇*[b1, f] L1 ≘ K1 →
- ∀L2,K2. ⬇*[b2, f] L2 ≘ K2 →
+theorem drops_fwd_length_eq1: ∀b1,b2,f,L1,K1. ⬇*[b1,f] L1 ≘ K1 →
+ ∀L2,K2. ⬇*[b2,f] L2 ≘ K2 →
|L1| = |L2| → |K1| = |K2|.
#b1 #b2 #f #L1 #K1 #HLK1 elim HLK1 -f -L1 -K1
[ #f #_ #L2 #K2 #HLK2 #H lapply (length_inv_zero_sn … H) -H
(* forward lemmas with finite colength assignment ***************************)
-lemma drops_fwd_fcla: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
+lemma drops_fwd_fcla: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 →
∃∃n. 𝐂⦃f⦄ ≘ n & |L1| = |L2| + n.
#f #L1 #L2 #H elim H -f -L1 -L2
[ /4 width=3 by fcla_isid, ex2_intro/
qed-.
(* Basic_2A1: includes: drop_fwd_length *)
-lemma drops_fcla_fwd: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
+lemma drops_fcla_fwd: ∀f,L1,L2,n. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
|L1| = |L2| + n.
#f #l1 #l2 #n #Hf #Hn elim (drops_fwd_fcla … Hf) -Hf
#k #Hm #H <(fcla_mono … Hm … Hn) -f //
qed-.
-lemma drops_fwd_fcla_le2: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
+lemma drops_fwd_fcla_le2: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 →
∃∃n. 𝐂⦃f⦄ ≘ n & n ≤ |L1|.
#f #L1 #L2 #H elim (drops_fwd_fcla … H) -H /2 width=3 by ex2_intro/
qed-.
(* Basic_2A1: includes: drop_fwd_length_le2 *)
-lemma drops_fcla_fwd_le2: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
+lemma drops_fcla_fwd_le2: ∀f,L1,L2,n. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
n ≤ |L1|.
#f #L1 #L2 #n #H #Hn elim (drops_fwd_fcla_le2 … H) -H
#k #Hm #H <(fcla_mono … Hm … Hn) -f //
qed-.
-lemma drops_fwd_fcla_lt2: ∀f,L1,I2,K2. ⬇*[Ⓣ, f] L1 ≘ K2.ⓘ{I2} →
+lemma drops_fwd_fcla_lt2: ∀f,L1,I2,K2. ⬇*[Ⓣ,f] L1 ≘ K2.ⓘ{I2} →
∃∃n. 𝐂⦃f⦄ ≘ n & n < |L1|.
#f #L1 #I2 #K2 #H elim (drops_fwd_fcla … H) -H
#n #Hf #H >H -L1 /3 width=3 by le_S_S, ex2_intro/
(* Basic_2A1: includes: drop_fwd_length_lt2 *)
lemma drops_fcla_fwd_lt2: ∀f,L1,I2,K2,n.
- ⬇*[Ⓣ, f] L1 ≘ K2.ⓘ{I2} → 𝐂⦃f⦄ ≘ n →
+ ⬇*[Ⓣ,f] L1 ≘ K2.ⓘ{I2} → 𝐂⦃f⦄ ≘ n →
n < |L1|.
#f #L1 #I2 #K2 #n #H #Hn elim (drops_fwd_fcla_lt2 … H) -H
#k #Hm #H <(fcla_mono … Hm … Hn) -f //
qed-.
(* Basic_2A1: includes: drop_fwd_length_lt4 *)
-lemma drops_fcla_fwd_lt4: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → 0 < n →
+lemma drops_fcla_fwd_lt4: ∀f,L1,L2,n. ⬇*[Ⓣ,f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → 0 < n →
|L2| < |L1|.
#f #L1 #L2 #n #H #Hf #Hn lapply (drops_fcla_fwd … H Hf) -f
/2 width=1 by lt_minus_to_plus_r/ qed-.
(* Basic_2A1: includes: drop_inv_length_eq *)
-lemma drops_inv_length_eq: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → |L1| = |L2| → 𝐈⦃f⦄.
+lemma drops_inv_length_eq: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 → |L1| = |L2| → 𝐈⦃f⦄.
#f #L1 #L2 #H #HL12 elim (drops_fwd_fcla … H) -H
#n #Hn <HL12 -L2 #H lapply (discr_plus_x_xy … H) -H
/2 width=3 by fcla_inv_xp/
qed-.
(* Basic_2A1: includes: drop_fwd_length_eq2 *)
-theorem drops_fwd_length_eq2: ∀f,L1,L2,K1,K2. ⬇*[Ⓣ, f] L1 ≘ K1 → ⬇*[Ⓣ, f] L2 ≘ K2 →
+theorem drops_fwd_length_eq2: ∀f,L1,L2,K1,K2. ⬇*[Ⓣ,f] L1 ≘ K1 → ⬇*[Ⓣ,f] L2 ≘ K2 →
|K1| = |K2| → |L1| = |L2|.
#f #L1 #L2 #K1 #K2 #HLK1 #HLK2 #HL12
elim (drops_fwd_fcla … HLK1) -HLK1 #n1 #Hn1 #H1 >H1 -L1
<(fcla_mono … Hn2 … Hn1) -f //
qed-.
-theorem drops_conf_div: ∀f1,f2,L1,L2. ⬇*[Ⓣ, f1] L1 ≘ L2 → ⬇*[Ⓣ, f2] L1 ≘ L2 →
+theorem drops_conf_div: ∀f1,f2,L1,L2. ⬇*[Ⓣ,f1] L1 ≘ L2 → ⬇*[Ⓣ,f2] L1 ≘ L2 →
∃∃n. 𝐂⦃f1⦄ ≘ n & 𝐂⦃f2⦄ ≘ n.
#f1 #f2 #L1 #L2 #H1 #H2
elim (drops_fwd_fcla … H1) -H1 #n1 #Hf1 #H1
qed-.
theorem drops_conf_div_fcla: ∀f1,f2,L1,L2,n1,n2.
- ⬇*[Ⓣ, f1] L1 ≘ L2 → ⬇*[Ⓣ, f2] L1 ≘ L2 → 𝐂⦃f1⦄ ≘ n1 → 𝐂⦃f2⦄ ≘ n2 →
+ ⬇*[Ⓣ,f1] L1 ≘ L2 → ⬇*[Ⓣ,f2] L1 ≘ L2 → 𝐂⦃f1⦄ ≘ n1 → 𝐂⦃f2⦄ ≘ n2 →
n1 = n2.
#f1 #f2 #L1 #L2 #n1 #n2 #Hf1 #Hf2 #Hn1 #Hn2
lapply (drops_fcla_fwd … Hf1 Hn1) -f1 #H1
(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
definition dedropable_sn: predicate … ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀K2. K1 ⪤[R] K2 →
- ∃∃L2. L1 ⪤[R] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → ∀K2. K1 ⪤[R] K2 →
+ ∃∃L2. L1 ⪤[R] L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2.
definition dropable_sn: predicate … ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ → ∀L2. L1 ⪤[R] L2 →
- ∃∃K2. K1 ⪤[R] K2 & ⬇*[b, f] L2 ≘ K2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ → ∀L2. L1 ⪤[R] L2 →
+ ∃∃K2. K1 ⪤[R] K2 & ⬇*[b,f] L2 ≘ K2.
definition dropable_dx: predicate … ≝
- λR. ∀L1,L2. L1 ⪤[R] L2 → ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[R] K2.
+ λR. ∀L1,L2. L1 ⪤[R] L2 → ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ →
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤[R] K2.
(* Properties with generic extension ****************************************)
(* Basic_2A1: includes: lpx_sn_drop_conf *)
lemma lex_drops_conf_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
- ∀b,f,I,K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
- ∃∃K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R] K2 & R K1 V1 V2.
+ ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≘ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
+ ∃∃K2,V2. ⬇*[b,f] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R] K2 & R K1 V1 V2.
#R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf
elim (sex_drops_conf_push … HL12 … HLK1 Hf f2) -L1 -Hf
[ #Z2 #K2 #HLK2 #HK12 #H
(* Basic_2A1: includes: lpx_sn_drop_trans *)
lemma lex_drops_trans_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
- ∀b,f,I,K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
- ∃∃K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R] K2 & R K1 V1 V2.
+ ∀b,f,I,K2,V2. ⬇*[b,f] L2 ≘ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
+ ∃∃K1,V1. ⬇*[b,f] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R] K2 & R K1 V1 V2.
#R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K2 #V2 #HLK2 #Hf
elim (sex_drops_trans_push … HL12 … HLK2 Hf f2) -L2 -Hf
[ #Z1 #K1 #HLK1 #HK12 #H
(* Basic_2A1: includes: lreq_drop_trans_be *)
lemma seq_drops_trans_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
- ∀b,f,I,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} → 𝐔⦃f⦄ →
+ ∀b,f,I,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} & K1 ≡[f1] K2.
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I} & K1 ≡[f1] K2.
#f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
elim (sex_drops_trans_next … HL12 … HLK2 Hf … Hf2) -f2 -L2 -Hf
#I1 #K1 #HLK1 #HK12 #H <(ceq_ext_inv_eq … H) -I2
(* Basic_2A1: includes: lreq_drop_conf_be *)
lemma seq_drops_conf_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
- ∀b,f,I,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} → 𝐔⦃f⦄ →
+ ∀b,f,I,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} & K1 ≡[f1] K2.
+ ∃∃K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I} & K1 ≡[f1] K2.
#f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
elim (seq_drops_trans_next … (seq_sym … HL12) … HLK1 … Hf2) // -f2 -L1 -Hf
/3 width=3 by seq_sym, ex2_intro/
qed-.
lemma drops_seq_trans_next: ∀f1,K1,K2. K1 ≡[f1] K2 →
- ∀b,f,I,L1. ⬇*[b, f] L1.ⓘ{I} ≘ K1 →
+ ∀b,f,I,L1. ⬇*[b,f] L1.ⓘ{I} ≘ K1 →
∀f2. f ~⊚ f1 ≘ ↑f2 →
- ∃∃L2. ⬇*[b, f] L2.ⓘ{I} ≘ K2 & L1 ≡[f2] L2 & L1.ⓘ{I} ≡[f] L2.ⓘ{I}.
+ ∃∃L2. ⬇*[b,f] L2.ⓘ{I} ≘ K2 & L1 ≡[f2] L2 & L1.ⓘ{I} ≡[f] L2.ⓘ{I}.
#f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (drops_sex_trans_next … HK12 … HLK1 … Hf2) -f1 -K1
/2 width=6 by cfull_lift_sn, ceq_lift_sn/
qed-.
lemma sex_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- ∀f2,L1,L2. L1 ⪤[cfull, RP, f2] L2 → ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 →
- ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- f ~⊚ f1 ≘ f2 → L1 ⪤[RN, RP, f2] L2.
+ ∀f2,L1,L2. L1 ⪤[cfull,RP,f2] L2 → ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+ ∀b,f. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
+ f ~⊚ f1 ≘ f2 → L1 ⪤[RN,RP,f2] L2.
#RN #RP #HRN #HRP #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
#g2 #I1 #I2 #L1 #L2 #HL12 #HI12 #IH #f1 #Y1 #Y2 #HK12 #b #f #HY1 #HY2 #H
[ elim (coafter_inv_xxn … H) [ |*: // ] -H #g #g1 #Hg2 #H1 #H2 destruct
]
qed-.
-fact sex_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
- ∀f2,L1. L1 ⪤[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[RN, RP, f1] K2.
+fact sex_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ →
+ ∀f2,L1. L1 ⪤[RN,RP,f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤[RN,RP,f1] K2.
#RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2
[ #f #Hf #_ #f2 #X #H #f1 #Hf2 lapply (sex_inv_atom2 … H) -H
#H destruct /4 width=3 by sex_atom, drops_atom, ex2_intro/
/2 width=5 by sex_dropable_dx_aux/ qed-.
lemma sex_drops_conf_next: ∀RN,RP.
- ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+ ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ ∀b,f,I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN, RP, f1] K2 & RN K1 I1 I2.
+ ∃∃I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (sex_inv_next1 … HX) -HX
qed-.
lemma sex_drops_conf_push: ∀RN,RP.
- ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+ ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ ∀b,f,I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ⫯f1 ≘ f2 →
- ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN, RP, f1] K2 & RP K1 I1 I2.
+ ∃∃I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (sex_inv_push1 … HX) -HX
#I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ ∀b,f,I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN, RP, f1] K2 & RN K1 I1 I2.
+ ∃∃I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN,RP,f1] K2 & RN K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (sex_inv_next2 … HX) -HX
#I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ ∀b,f,I2,K2. ⬇*[b,f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
∀f1. f ~⊚ ⫯f1 ≘ f2 →
- ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN, RP, f1] K2 & RP K1 I1 I2.
+ ∃∃I1,K1. ⬇*[b,f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN,RP,f1] K2 & RP K1 I1 I2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (sex_inv_push2 … HX) -HX
lemma drops_sex_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 →
- ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
+ ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+ ∀b,f,I1,L1. ⬇*[b,f] L1.ⓘ{I1} ≘ K1 →
∀f2. f ~⊚ f1 ≘ ↑f2 →
- ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN, RP, f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+ ∃∃I2,L2. ⬇*[b,f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (sex_inv_next1 … HX) -HX
lemma drops_sex_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 →
- ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
+ ∀f1,K1,K2. K1 ⪤[RN,RP,f1] K2 →
+ ∀b,f,I1,L1. ⬇*[b,f] L1.ⓘ{I1} ≘ K1 →
∀f2. f ~⊚ f1 ≘ ⫯f2 →
- ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN, RP, f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+ ∃∃I2,L2. ⬇*[b,f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN,RP,f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (sex_inv_push1 … HX) -HX
#I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
qed-.
-lemma drops_atom2_sex_conf: ∀RN,RP,b,f1,L1. ⬇*[b, f1] L1 ≘ ⋆ → 𝐔⦃f1⦄ →
- ∀f,L2. L1 ⪤[RN, RP, f] L2 →
- ∀f2. f1 ~⊚ f2 ≘f → ⬇*[b, f1] L2 ≘ ⋆.
+lemma drops_atom2_sex_conf: ∀RN,RP,b,f1,L1. ⬇*[b,f1] L1 ≘ ⋆ → 𝐔⦃f1⦄ →
+ ∀f,L2. L1 ⪤[RN,RP,f] L2 →
+ ∀f2. f1 ~⊚ f2 ≘f → ⬇*[b,f1] L2 ≘ ⋆.
#RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3
elim (sex_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1
#L #H #HL2 lapply (sex_inv_atom1 … H) -H //
definition d_liftable1_all: predicate (relation2 lenv term) ≝
λR. ∀K,Ts. all … (R K) Ts →
- ∀b,f,L. ⬇*[b, f] L ≘ K →
+ ∀b,f,L. ⬇*[b,f] L ≘ K →
∀Us. ⬆*[f] Ts ≘ Us → all … (R L) Us.
(* Properties with generic relocation for term vectors **********************)
(* Forward lemmas with weight for local environments ************************)
(* Basic_2A1: includes: drop_fwd_lw *)
-lemma drops_fwd_lw: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ♯{L2} ≤ ♯{L1}.
+lemma drops_fwd_lw: ∀b,f,L1,L2. ⬇*[b,f] L1 ≘ L2 → ♯{L2} ≤ ♯{L1}.
#b #f #L1 #L2 #H elim H -f -L1 -L2 //
[ /2 width=3 by transitive_le/
| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 normalize
qed-.
(* Basic_2A1: includes: drop_fwd_lw_lt *)
-lemma drops_fwd_lw_lt: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
+lemma drops_fwd_lw_lt: ∀f,L1,L2. ⬇*[Ⓣ,f] L1 ≘ L2 →
(𝐈⦃f⦄ → ⊥) → ♯{L2} < ♯{L1}.
#f #L1 #L2 #H elim H -f -L1 -L2
[ #f #Hf #Hnf elim Hnf -Hnf /2 width=1 by/
(* Forward lemmas with restricted weight for closures ***********************)
(* Basic_2A1: includes: drop_fwd_rfw *)
-lemma drops_bind2_fwd_rfw: ∀b,f,I,L,K,V. ⬇*[b, f] L ≘ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}.
+lemma drops_bind2_fwd_rfw: ∀b,f,I,L,K,V. ⬇*[b,f] L ≘ K.ⓑ{I}V → ∀T. ♯{K,V} < ♯{L,T}.
#b #f #I #L #K #V #HLK lapply (drops_fwd_lw … HLK) -HLK
normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt, monotonic_lt_plus_r/
qed-.
(* Advanced inversion lemma *************************************************)
-lemma drops_inv_x_bind_xy: ∀b,f,I,L. ⬇*[b, f] L ≘ L.ⓘ{I} → ⊥.
+lemma drops_inv_x_bind_xy: ∀b,f,I,L. ⬇*[b,f] L ≘ L.ⓘ{I} → ⊥.
#b #f #I #L #H lapply (drops_fwd_lw … H) -b -f
/2 width=4 by lt_le_false/ (**) (* full auto is a bit slow: 19s *)
qed-.
(* GENERIC EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **************)
definition lex (R): relation lenv ≝
- λL1,L2. ∃∃f. 𝐈⦃f⦄ & L1 ⪤[cfull, cext2 R, f] L2.
+ λL1,L2. ∃∃f. 𝐈⦃f⦄ & L1 ⪤[cfull,cext2 R,f] L2.
interpretation "generic extension (local environment)"
'Relation R L1 L2 = (lex R L1 L2).
*)
inductive lifts: rtmap → relation term ≝
| lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
-| lifts_lref: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → lifts f (#i1) (#i2)
+| lifts_lref: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → lifts f (#i1) (#i2)
| lifts_gref: ∀f,l. lifts f (§l) (§l)
| lifts_bind: ∀f,p,I,V1,V2,T1,T2.
lifts f V1 V2 → lifts (⫯f) T1 T2 →
λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≘ U1 →
∀T2. ⬆*[f] T2 ≘ U2 → R T1 T2.
+definition liftable2_dx: predicate (relation term) ≝
+ λR. ∀T1,T2. R T1 T2 → ∀f,U2. ⬆*[f] T2 ≘ U2 →
+ ∃∃U1. ⬆*[f] T1 ≘ U1 & R U1 U2.
+
definition deliftable2_dx: predicate (relation term) ≝
λR. ∀U1,U2. R U1 U2 → ∀f,T2. ⬆*[f] T2 ≘ U2 →
∃∃T1. ⬆*[f] T1 ≘ U1 & R T1 T2.
/2 width=4 by lifts_inv_sort1_aux/ qed-.
fact lifts_inv_lref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i1. X = #i1 →
- ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2.
+ ∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2.
#f #X #Y * -f -X -Y
[ #f #s #x #H destruct
| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
(* Basic_1: was: lift1_lref *)
(* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
lemma lifts_inv_lref1: ∀f,Y,i1. ⬆*[f] #i1 ≘ Y →
- ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2.
+ ∃∃i2. @⦃i1,f⦄ ≘ i2 & Y = #i2.
/2 width=3 by lifts_inv_lref1_aux/ qed-.
fact lifts_inv_gref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. X = §l → Y = §l.
/2 width=4 by lifts_inv_sort2_aux/ qed-.
fact lifts_inv_lref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i2. Y = #i2 →
- ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1.
+ ∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1.
#f #X #Y * -f -X -Y
[ #f #s #x #H destruct
| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
(* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *)
(* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *)
lemma lifts_inv_lref2: ∀f,X,i2. ⬆*[f] X ≘ #i2 →
- ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1.
+ ∃∃i1. @⦃i1,f⦄ ≘ i2 & X = #i1.
/2 width=3 by lifts_inv_lref2_aux/ qed-.
fact lifts_inv_gref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. Y = §l → X = §l.
lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≘ Y →
∨∨ ∃∃s. I = Sort s & Y = ⋆s
- | ∃∃i,j. @⦃i, f⦄ ≘ j & I = LRef i & Y = #j
+ | ∃∃i,j. @⦃i,f⦄ ≘ j & I = LRef i & Y = #j
| ∃∃l. I = GRef l & Y = §l.
#f * #n #Y #H
[ lapply (lifts_inv_sort1 … H)
lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≘ ⓪{I} →
∨∨ ∃∃s. X = ⋆s & I = Sort s
- | ∃∃i,j. @⦃i, f⦄ ≘ j & X = #i & I = LRef j
+ | ∃∃i,j. @⦃i,f⦄ ≘ j & X = #i & I = LRef j
| ∃∃l. X = §l & I = GRef l.
#f * #n #X #H
[ lapply (lifts_inv_sort2 … H)
(* Basic properties *********************************************************)
+lemma liftable2_sn_dx (R): symmetric … R → liftable2_sn R → liftable2_dx R.
+#R #H2R #H1R #T1 #T2 #HT12 #f #U2 #HTU2
+elim (H1R … T1 … HTU2) -H1R /3 width=3 by ex2_intro/
+qed-.
+
lemma deliftable2_sn_dx (R): symmetric … R → deliftable2_sn R → deliftable2_dx R.
#R #H2R #H1R #U1 #U2 #HU12 #f #T2 #HTU2
elim (H1R … U1 … HTU2) -H1R /3 width=3 by ex2_intro/
]
qed-.
-lemma lifts_push_zero (f): ⬆*[⫯f]#O ≘ #0.
+lemma lifts_push_zero (f): ⬆*[⫯f]#0 ≘ #0.
/2 width=1 by lifts_lref/ qed.
lemma lifts_push_lref (f) (i1) (i2): ⬆*[f]#i1 ≘ #i2 → ⬆*[⫯f]#(↑i1) ≘ #(↑i2).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/relocation/rtmap_basic_after.ma".
+include "static_2/notation/relations/rlift_4.ma".
+include "static_2/relocation/lifts.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+interpretation "basic relocation (term)"
+ 'RLift m n T1 T2 = (lifts (basic m n) T1 T2).
+
+(* Properties with basic relocation *****************************************)
+
+lemma lifts_lref_lt (m,n,i): i < m → ⬆[m,n] #i ≘ #i.
+/3 width=1 by lifts_lref, at_basic_lt/ qed.
+
+lemma lifts_lref_ge (m,n,i): m ≤ i → ⬆[m,n] #i ≘ #(n+i).
+/3 width=1 by lifts_lref, at_basic_ge/ qed.
+
+lemma lifts_lref_ge_minus (m,n,i): n+m ≤ i → ⬆[m,n] #(i-n) ≘ #i.
+#m #n #i #Hi
+>(plus_minus_m_m i n) in ⊢ (???%);
+/3 width=2 by lifts_lref_ge, le_plus_to_minus_l, le_plus_b/
+qed.
(* GENERIC RELOCATION FOR TERMS *********************************************)
-(* Properties with degree-based equivalence for terms ***********************)
+(* Properties with sort-irrelevant equivalence for terms ********************)
-lemma tdeq_lifts_sn: ∀h,o. liftable2_sn (tdeq h o).
-#h #o #T1 #T2 #H elim H -T1 -T2 [||| * ]
-[ #s1 #s2 #d #Hs1 #Hs2 #f #X #H >(lifts_inv_sort1 … H) -H
- /3 width=5 by lifts_sort, tdeq_sort, ex2_intro/
+lemma tdeq_lifts_sn: liftable2_sn tdeq.
+#T1 #T2 #H elim H -T1 -T2 [||| * ]
+[ #s1 #s2 #f #X #H >(lifts_inv_sort1 … H) -H
+ /3 width=3 by lifts_sort, tdeq_sort, ex2_intro/
| #i #f #X #H elim (lifts_inv_lref1 … H) -H
/3 width=3 by lifts_lref, tdeq_lref, ex2_intro/
| #l #f #X #H >(lifts_inv_gref1 … H) -H
]
qed-.
-lemma tdeq_lifts_bi: ∀h,o. liftable2_bi (tdeq h o).
+lemma tdeq_lifts_dx: liftable2_dx tdeq.
+/3 width=3 by tdeq_lifts_sn, liftable2_sn_dx, tdeq_sym/ qed-.
+
+lemma tdeq_lifts_bi: liftable2_bi tdeq.
/3 width=6 by tdeq_lifts_sn, liftable2_sn_bi/ qed-.
-(* Inversion lemmas with degree-based equivalence for terms *****************)
+(* Inversion lemmas with sort-irrelevant equivalence for terms **************)
-lemma tdeq_inv_lifts_sn: ∀h,o. deliftable2_sn (tdeq h o).
-#h #o #U1 #U2 #H elim H -U1 -U2 [||| * ]
-[ #s1 #s2 #d #Hs1 #Hs2 #f #X #H >(lifts_inv_sort2 … H) -H
- /3 width=5 by lifts_sort, tdeq_sort, ex2_intro/
+lemma tdeq_inv_lifts_sn: deliftable2_sn tdeq.
+#U1 #U2 #H elim H -U1 -U2 [||| * ]
+[ #s1 #s2 #f #X #H >(lifts_inv_sort2 … H) -H
+ /3 width=3 by lifts_sort, tdeq_sort, ex2_intro/
| #i #f #X #H elim (lifts_inv_lref2 … H) -H
/3 width=3 by lifts_lref, tdeq_lref, ex2_intro/
| #l #f #X #H >(lifts_inv_gref2 … H) -H
]
qed-.
-lemma tdeq_inv_lifts_dx (h) (o): deliftable2_dx (tdeq h o).
+lemma tdeq_inv_lifts_dx: deliftable2_dx tdeq.
/3 width=3 by tdeq_inv_lifts_sn, deliftable2_sn_dx, tdeq_sym/ qed-.
-lemma tdeq_inv_lifts_bi: ∀h,o. deliftable2_bi (tdeq h o).
+lemma tdeq_inv_lifts_bi: deliftable2_bi tdeq.
/3 width=6 by tdeq_inv_lifts_sn, deliftable2_sn_bi/ qed-.
-lemma tdeq_lifts_inv_pair_sn (h) (o) (I) (f:rtmap):
- ∀X,T. ⬆*[f]X ≘ T → ∀V. ②{I}V.T ≛[h,o] X → ⊥.
-#h #o #I #f #X #T #H elim H -f -X -T
+lemma tdeq_lifts_inv_pair_sn (I) (f:rtmap):
+ ∀X,T. ⬆*[f]X ≘ T → ∀V. ②{I}V.T ≛ X → ⊥.
+#I #f #X #T #H elim H -f -X -T
[ #f #s #V #H
elim (tdeq_inv_pair1 … H) -H #X1 #X2 #_ #_ #H destruct
| #f #i #j #_ #V #H
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/toeq.ma".
+include "static_2/relocation/lifts_lifts.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+(* Properties with sort-irrelevant outer equivalence for terms **************)
+
+lemma toeq_lifts_sn: liftable2_sn toeq.
+#T1 #T2 #H elim H -T1 -T2 [||| * ]
+[ #s1 #s2 #f #X #H
+ >(lifts_inv_sort1 … H) -H
+ /2 width=3 by toeq_sort, ex2_intro/
+| #i #f #X #H
+ elim (lifts_inv_lref1 … H) -H #j #Hj #H destruct
+ /3 width=3 by toeq_lref, lifts_lref, ex2_intro/
+| #l #f #X #H
+ >(lifts_inv_gref1 … H) -H
+ /2 width=3 by toeq_gref, ex2_intro/
+| #p #I #V1 #V2 #T1 #T2 #f #X #H
+ elim (lifts_inv_bind1 … H) -H #W1 #U1 #_ #_ #H destruct -V1 -T1
+ elim (lifts_total V2 f) #W2 #HVW2
+ elim (lifts_total T2 (⫯f)) #U2 #HTU2
+ /3 width=3 by toeq_pair, lifts_bind, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #f #X #H
+ elim (lifts_inv_flat1 … H) -H #W1 #U1 #_ #_ #H destruct -V1 -T1
+ elim (lifts_total V2 f) #W2 #HVW2
+ elim (lifts_total T2 f) #U2 #HTU2
+ /3 width=3 by toeq_pair, lifts_flat, ex2_intro/
+]
+qed-.
+
+lemma toeq_lifts_dx: liftable2_dx toeq.
+/3 width=3 by toeq_lifts_sn, liftable2_sn_dx, toeq_sym/ qed-.
+
+lemma toeq_lifts_bi: liftable2_bi toeq.
+/3 width=6 by toeq_lifts_sn, liftable2_sn_bi/ qed-.
+
+(* Inversion lemmas with sort-irrelevant outer equivalence for terms ********)
+
+lemma toeq_inv_lifts_bi: deliftable2_bi toeq.
+#U1 #U2 #H elim H -U1 -U2 [||| * ]
+[ #s1 #s2 #f #X1 #H1 #X2 #H2
+ >(lifts_inv_sort2 … H1) -H1 >(lifts_inv_sort2 … H2) -H2
+ /1 width=1 by toeq_sort/
+| #j #f #X1 #H1 #X2 #H2
+ elim (lifts_inv_lref2 … H1) -H1 #i1 #Hj1 #H destruct
+ elim (lifts_inv_lref2 … H2) -H2 #i2 #Hj2 #H destruct
+ <(at_inj … Hj2 … Hj1) -j -f -i1
+ /1 width=1 by toeq_lref/
+| #l #f #X1 #H1 #X2 #H2
+ >(lifts_inv_gref2 … H1) -H1 >(lifts_inv_gref2 … H2) -H2
+ /1 width=1 by toeq_gref/
+| #p #I #W1 #W2 #U1 #U2 #f #X1 #H1 #X2 #H2
+ elim (lifts_inv_bind2 … H1) -H1 #V1 #T1 #_ #_ #H destruct -W1 -U1
+ elim (lifts_inv_bind2 … H2) -H2 #V2 #T2 #_ #_ #H destruct -W2 -U2
+ /1 width=1 by toeq_pair/
+| #I #W1 #W2 #U1 #U2 #f #X1 #H1 #X2 #H2
+ elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #_ #_ #H destruct -W1 -U1
+ elim (lifts_inv_flat2 … H2) -H2 #V2 #T2 #_ #_ #H destruct -W2 -U2
+ /1 width=1 by toeq_pair/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tweq.ma".
+include "static_2/relocation/lifts_lifts.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+(* Properties with sort-irrelevant whd equivalence for terms ****************)
+
+lemma tweq_lifts_sn: liftable2_sn tweq.
+#T1 #T2 #H elim H -T1 -T2
+[ #s1 #s2 #f #X #H >(lifts_inv_sort1 … H) -H
+ /3 width=3 by lifts_sort, tweq_sort, ex2_intro/
+| #i #f #X #H elim (lifts_inv_lref1 … H) -H
+ /3 width=3 by lifts_lref, tweq_lref, ex2_intro/
+| #l #f #X #H >(lifts_inv_gref1 … H) -H
+ /2 width=3 by lifts_gref, tweq_gref, ex2_intro/
+| #p #V1 #V2 #T1 #T2 #_ #IHT #f #X #H
+ elim (lifts_inv_bind1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (lifts_total V2 f) #W2 #HVW2
+ elim (true_or_false p) #H destruct
+ [ elim (IHT … HTU1) -T1 [| // ] #U2 #HTU2 #HU12
+ | elim (lifts_total T2 (⫯f)) #U2 #HTU2
+ ]
+ /3 width=4 by tweq_abbr_pos, lifts_bind, ex2_intro/
+| #p #V1 #V2 #T1 #T2 #f #X #H
+ elim (lifts_inv_bind1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (lifts_total V2 f) #W2 #HVW2
+ elim (lifts_total T2 (⫯f)) #U2 #HTU2
+ /3 width=3 by lifts_bind, ex2_intro/
+| #V1 #V2 #T1 #T2 #_ #IHT #f #X #H
+ elim (lifts_inv_flat1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (lifts_total V2 f) #W2 #HVW2
+ elim (IHT … HTU1) -T1 #U2 #HTU2 #HU12
+ /3 width=4 by lifts_flat, tweq_appl, ex2_intro/
+| #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f #X #H
+ elim (lifts_inv_flat1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV … HVW1) -V1 #W2 #HVW2 #HW12
+ elim (IHT … HTU1) -T1 #U2 #HTU2 #HU12
+ /3 width=5 by lifts_flat, tweq_cast, ex2_intro/
+]
+qed-.
+
+lemma tweq_lifts_dx: liftable2_dx tweq.
+/3 width=3 by tweq_lifts_sn, liftable2_sn_dx, tweq_sym/ qed-.
+
+lemma tweq_lifts_bi: liftable2_bi tweq.
+/3 width=6 by tweq_lifts_sn, liftable2_sn_bi/ qed-.
+
+(* Inversion lemmas with sort-irrelevant whd equivalence for terms **********)
+
+lemma tweq_inv_lifts_bi: deliftable2_bi tweq.
+#U1 #U2 #H elim H -U1 -U2
+[ #s1 #s2 #f #X1 #H1 #X2 #H2
+ >(lifts_inv_sort2 … H1) -H1 >(lifts_inv_sort2 … H2) -H2
+ /1 width=1 by tweq_sort/
+| #j #f #X1 #H1 #X2 #H2
+ elim (lifts_inv_lref2 … H1) -H1 #i1 #Hj1 #H destruct
+ elim (lifts_inv_lref2 … H2) -H2 #i2 #Hj2 #H destruct
+ <(at_inj … Hj2 … Hj1) -j -f -i1
+ /1 width=1 by tweq_lref/
+| #l #f #X1 #H1 #X2 #H2
+ >(lifts_inv_gref2 … H1) -H1 >(lifts_inv_gref2 … H2) -H2
+ /1 width=1 by tweq_gref/
+| #p #W1 #W2 #U1 #U2 #_ #IH #f #X1 #H1 #X2 #H2
+ elim (lifts_inv_bind2 … H1) -H1 #V1 #T1 #_ #HTU1 #H destruct -W1
+ elim (lifts_inv_bind2 … H2) -H2 #V2 #T2 #_ #HTU2 #H destruct -W2
+ elim (true_or_false p) #H destruct
+ [ /3 width=3 by tweq_abbr_pos/
+ | /1 width=1 by tweq_abbr_neg/
+ ]
+| #p #W1 #W2 #U1 #U2 #f #X1 #H1 #X2 #H2
+ elim (lifts_inv_bind2 … H1) -H1 #V1 #T1 #_ #_ #H destruct -W1 -U1
+ elim (lifts_inv_bind2 … H2) -H2 #V2 #T2 #_ #_ #H destruct -W2 -U2
+ /1 width=1 by tweq_abst/
+| #W1 #W2 #U1 #U2 #_ #IH #f #X1 #H1 #X2 #H2
+ elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #_ #HTU1 #H destruct -W1
+ elim (lifts_inv_flat2 … H2) -H2 #V2 #T2 #_ #HTU2 #H destruct -W2
+ /3 width=3 by tweq_appl/
+| #W1 #W2 #U1 #U2 #_ #_ #IHW #IHU #f #X1 #H1 #X2 #H2
+ elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (lifts_inv_flat2 … H2) -H2 #V2 #T2 #HVW2 #HTU2 #H destruct
+ /3 width=3 by tweq_cast/
+]
+qed-.
+
+lemma tweq_inv_abbr_pos_x_lifts_y_y (T) (f:rtmap):
+ ∀V,U. +ⓓV.U ≅ T → ⬆*[f]T ≘ U → ⊥.
+@(f_ind … tw) #n #IH #T #Hn #f #V #U #H1 #H2 destruct
+elim (tweq_inv_abbr_pos_sn … H1) -H1 #X1 #X2 #HX2 #H destruct -V
+elim (lifts_inv_bind1 … H2) -H2 #Y1 #Y2 #_ #HXY2 #H destruct
+/2 width=7 by/
+qed-.
relation3 rtmap lenv bind ≝
λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
- ∀L1. L0 ⪤[RN1, RP1, f] L1 → ∀L2. L0 ⪤[RN2, RP2, f] L2 →
+ ∀L1. L0 ⪤[RN1,RP1,f] L1 → ∀L2. L0 ⪤[RN2,RP2,f] L2 →
∃∃I. R2 L1 I1 I & R1 L2 I2 I.
definition sex_transitive: relation3 lenv bind bind → relation3 lenv bind bind →
relation3 lenv bind bind → relation3 lenv bind bind →
relation3 rtmap lenv bind ≝
λR1,R2,R3,RN,RP,f,L1,I1.
- ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN, RP, f] L2 →
+ ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN,RP,f] L2 →
∀I2. R2 L2 I I2 → R3 L1 I1 I2.
(* Basic inversion lemmas ***************************************************)
-fact sex_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → X = ⋆ → Y = ⋆.
+fact sex_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → X = ⋆ → Y = ⋆.
#RN #RP #f #X #Y * -f -X -Y //
#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_atom1 *)
-lemma sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN, RP, f] Y → Y = ⋆.
+lemma sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN,RP,f] Y → Y = ⋆.
/2 width=6 by sex_inv_atom1_aux/ qed-.
-fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g →
- ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
+fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g →
+ ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J1 #K1 #H destruct
| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_next … H2) -g destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma sex_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN, RP, ↑g] Y →
- ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
+lemma sex_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN,RP,↑g] Y →
+ ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
/2 width=7 by sex_inv_next1_aux/ qed-.
-fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g →
- ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
+fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g →
+ ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J1 #K1 #H destruct
| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_next_push … H)
]
qed-.
-lemma sex_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN, RP, ⫯g] Y →
- ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
+lemma sex_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN,RP,⫯g] Y →
+ ∃∃J2,K2. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
/2 width=7 by sex_inv_push1_aux/ qed-.
-fact sex_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → Y = ⋆ → X = ⋆.
+fact sex_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → Y = ⋆ → X = ⋆.
#RN #RP #f #X #Y * -f -X -Y //
#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_atom2 *)
-lemma sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN, RP, f] ⋆ → X = ⋆.
+lemma sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN,RP,f] ⋆ → X = ⋆.
/2 width=6 by sex_inv_atom2_aux/ qed-.
-fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g →
- ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
+fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g →
+ ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J2 #K2 #H destruct
| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_next … H2) -g destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma sex_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤[RN, RP, ↑g] K2.ⓘ{J2} →
- ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
+lemma sex_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤[RN,RP,↑g] K2.ⓘ{J2} →
+ ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
/2 width=7 by sex_inv_next2_aux/ qed-.
-fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g →
- ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
+fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN,RP,f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g →
+ ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
#RN #RP #f #X #Y * -f -X -Y
[ #f #J2 #K2 #g #H destruct
| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_next_push … H)
]
qed-.
-lemma sex_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤[RN, RP, ⫯g] K2.ⓘ{J2} →
- ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
+lemma sex_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤[RN,RP,⫯g] K2.ⓘ{J2} →
+ ∃∃J1,K1. K1 ⪤[RN,RP,g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
/2 width=7 by sex_inv_push2_aux/ qed-.
(* Basic_2A1: includes lpx_sn_inv_pair *)
lemma sex_inv_next: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, ↑f] L2.ⓘ{I2} →
- L1 ⪤[RN, RP, f] L2 ∧ RN L1 I1 I2.
+ L1.ⓘ{I1} ⪤[RN,RP,↑f] L2.ⓘ{I2} →
+ L1 ⪤[RN,RP,f] L2 ∧ RN L1 I1 I2.
#RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_next1 … H) -H
#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
qed-.
lemma sex_inv_push: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, ⫯f] L2.ⓘ{I2} →
- L1 ⪤[RN, RP, f] L2 ∧ RP L1 I1 I2.
+ L1.ⓘ{I1} ⪤[RN,RP,⫯f] L2.ⓘ{I2} →
+ L1 ⪤[RN,RP,f] L2 ∧ RP L1 I1 I2.
#RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_push1 … H) -H
#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
qed-.
-lemma sex_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤[RN, RP, ⫱f] L2 →
+lemma sex_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤[RN,RP,⫱f] L2 →
RN L1 I1 I2 → RP L1 I1 I2 →
- L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2}.
+ L1.ⓘ{I1} ⪤[RN,RP,f] L2.ⓘ{I2}.
#RN #RP #f #I1 #I2 #L2 #L2 elim (pn_split f) *
/2 width=1 by sex_next, sex_push/
qed-.
(* Basic forward lemmas *****************************************************)
lemma sex_fwd_bind: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} →
- L1 ⪤[RN, RP, ⫱f] L2.
+ L1.ⓘ{I1} ⪤[RN,RP,f] L2.ⓘ{I2} →
+ L1 ⪤[RN,RP,⫱f] L2.
#RN #RP #f #I1 #I2 #L1 #L2 #Hf
elim (pn_split f) * #g #H destruct
[ elim (sex_inv_push … Hf) | elim (sex_inv_next … Hf) ] -Hf //
(* Basic properties *********************************************************)
-lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN, RP, f] L2).
+lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN,RP,f] L2).
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H
[ elim (eq_inv_nx … H) -H /3 width=3 by sex_next/
]
qed-.
-lemma sex_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN, RP, f] L2).
+lemma sex_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN,RP,f] L2).
#RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by sex_eq_repl_back/ (**) (* full auto fails *)
qed-.
qed-.
lemma sex_pair_repl: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} →
+ L1.ⓘ{I1} ⪤[RN,RP,f] L2.ⓘ{I2} →
∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 →
- L1.ⓘ{J1} ⪤[RN, RP, f] L2.ⓘ{J2}.
+ L1.ⓘ{J1} ⪤[RN,RP,f] L2.ⓘ{J2}.
/3 width=3 by sex_inv_tl, sex_fwd_bind/ qed-.
lemma sex_co: ∀RN1,RP1,RN2,RP2. RN1 ⊆ RN2 → RP1 ⊆ RP2 →
- ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → L1 ⪤[RN2, RP2, f] L2.
+ ∀f,L1,L2. L1 ⪤[RN1,RP1,f] L2 → L1 ⪤[RN2,RP2,f] L2.
#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by sex_atom, sex_next, sex_push/
qed-.
lemma sex_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 →
- ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
- L1 ⪤[RN2, RP2, f] L2.
+ ∀f,L1,L2. L1 ⪤[RN1,RP1,f] L2 → 𝐈⦃f⦄ →
+ L1 ⪤[RN2,RP2,f] L2.
#RN1 #RP1 #RN2 #RP2 #HR #f #L1 #L2 #H elim H -f -L1 -L2 //
#f #I1 #I2 #K1 #K2 #_ #HI12 #IH #H
[ elim (isid_inv_next … H) -H //
qed-.
lemma sex_sdj: ∀RN,RP. RP ⊆ RN →
- ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 →
- ∀f2. f1 ∥ f2 → L1 ⪤[RP, RN, f2] L2.
+ ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 →
+ ∀f2. f1 ∥ f2 → L1 ⪤[RP,RN,f2] L2.
#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
[ elim (sdj_inv_nx … H12) -H12 [2,3: // ]
qed-.
lemma sle_sex_trans: ∀RN,RP. RN ⊆ RP →
- ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
- ∀f1. f1 ⊆ f2 → L1 ⪤[RN, RP, f1] L2.
+ ∀f2,L1,L2. L1 ⪤[RN,RP,f2] L2 →
+ ∀f1. f1 ⊆ f2 → L1 ⪤[RN,RP,f1] L2.
#RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
#f2 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12
[ elim (pn_split f1) * ]
qed-.
lemma sle_sex_conf: ∀RN,RP. RP ⊆ RN →
- ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 →
- ∀f2. f1 ⊆ f2 → L1 ⪤[RN, RP, f2] L2.
+ ∀f1,L1,L2. L1 ⪤[RN,RP,f1] L2 →
+ ∀f2. f1 ⊆ f2 → L1 ⪤[RN,RP,f2] L2.
#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
[2: elim (pn_split f2) * ]
qed-.
lemma sex_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
- ∀f,L1,L2. L1 ⪤[R1, RP, f] L2 → ∀g. f ⊆ g →
- ∃∃L. L1 ⪤[R1, RP, g] L & L ⪤[R2, cfull, f] L2.
+ ∀f,L1,L2. L1 ⪤[R1,RP,f] L2 → ∀g. f ⊆ g →
+ ∃∃L. L1 ⪤[R1,RP,g] L & L ⪤[R2,cfull,f] L2.
#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
[ /2 width=3 by sex_atom, ex2_intro/ ]
#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
qed-.
lemma sex_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
- ∀f,L1,L2. L1 ⪤[R1, RP, f] L2 → ∀g. f ∥ g →
- ∃∃L. L1 ⪤[RP, R1, g] L & L ⪤[R2, cfull, f] L2.
+ ∀f,L1,L2. L1 ⪤[R1,RP,f] L2 → ∀g. f ∥ g →
+ ∃∃L. L1 ⪤[RP,R1,g] L & L ⪤[R2,cfull,f] L2.
#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
[ /2 width=3 by sex_atom, ex2_intro/ ]
#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
lemma sex_dec: ∀RN,RP.
(∀L,I1,I2. Decidable (RN L I1 I2)) →
(∀L,I1,I2. Decidable (RP L I1 I2)) →
- ∀L1,L2,f. Decidable (L1 ⪤[RN, RP, f] L2).
+ ∀L1,L2,f. Decidable (L1 ⪤[RN,RP,f] L2).
#RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #IH * ]
[ /2 width=1 by sex_atom, or_introl/
| #L2 #I2 #f @or_intror #H
(* Forward lemmas with length for local environments ************************)
(* Note: "#f #I1 #I2 #L1 #L2 >length_bind >length_bind //" was needed to conclude *)
-lemma sex_fwd_length: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → |L1| = |L2|.
+lemma sex_fwd_length: ∀RN,RP,f,L1,L2. L1 ⪤[RN,RP,f] L2 → |L1| = |L2|.
#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2 //
qed-.
(* Properties with length for local environments ****************************)
-lemma sex_length_cfull: ∀L1,L2. |L1| = |L2| → ∀f. L1 ⪤[cfull, cfull, f] L2.
+lemma sex_length_cfull: ∀L1,L2. |L1| = |L2| → ∀f. L1 ⪤[cfull,cfull,f] L2.
#L1 elim L1 -L1
[ #Y2 #H >(length_inv_zero_sn … H) -Y2 //
| #L1 #I1 #IH #Y2 #H #f
qed.
lemma sex_length_isid: ∀R,L1,L2. |L1| = |L2| →
- ∀f. 𝐈⦃f⦄ → L1 ⪤[R, cfull, f] L2.
+ ∀f. 𝐈⦃f⦄ → L1 ⪤[R,cfull,f] L2.
#R #L1 elim L1 -L1
[ #Y2 #H >(length_inv_zero_sn … H) -Y2 //
| #L1 #I1 #IH #Y2 #H #f #Hf
∀L1,f.
(∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → sex_transitive RN1 RN2 RN RN1 RP1 g K I) →
(∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → sex_transitive RP1 RP2 RP RN1 RP1 g K I) →
- ∀L0. L1 ⪤[RN1, RP1, f] L0 →
- ∀L2. L0 ⪤[RN2, RP2, f] L2 →
- L1 ⪤[RN, RP, f] L2.
+ ∀L0. L1 ⪤[RN1,RP1,f] L0 →
+ ∀L2. L0 ⪤[RN2,RP2,f] L2 →
+ L1 ⪤[RN,RP,f] L2.
#RN1 #RP1 #RN2 #RP2 #RN #RP #L1 elim L1 -L1
[ #f #_ #_ #L0 #H1 #L2 #H2
lapply (sex_inv_atom1 … H1) -H1 #H destruct
Transitive … (sex RN RP f).
/2 width=9 by sex_trans_gen/ qed-.
-theorem sex_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤[R1, cfull, f] L → 𝐈⦃f⦄ →
- ∀L2. L ⪤[R2, cfull, f] L2 → L1 ⪤[R3, cfull, f] L2.
+theorem sex_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤[R1,cfull,f] L → 𝐈⦃f⦄ →
+ ∀L2. L ⪤[R2,cfull,f] L2 → L1 ⪤[R3,cfull,f] L2.
#R1 #R2 #R3 #L1 #L #f #H elim H -L1 -L -f
[ #f #Hf #L2 #H >(sex_inv_atom1 … H) -L2 // ]
#f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H
/3 width=3 by/ qed-.
lemma sex_meet: ∀RN,RP,L1,L2.
- ∀f1. L1 ⪤[RN, RP, f1] L2 →
- ∀f2. L1 ⪤[RN, RP, f2] L2 →
- ∀f. f1 ⋒ f2 ≘ f → L1 ⪤[RN, RP, f] L2.
+ ∀f1. L1 ⪤[RN,RP,f1] L2 →
+ ∀f2. L1 ⪤[RN,RP,f2] L2 →
+ ∀f. f1 ⋒ f2 ≘ f → L1 ⪤[RN,RP,f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct
qed-.
lemma sex_join: ∀RN,RP,L1,L2.
- ∀f1. L1 ⪤[RN, RP, f1] L2 →
- ∀f2. L1 ⪤[RN, RP, f2] L2 →
- ∀f. f1 ⋓ f2 ≘ f → L1 ⪤[RN, RP, f] L2.
+ ∀f1. L1 ⪤[RN,RP,f1] L2 →
+ ∀f2. L1 ⪤[RN,RP,f2] L2 →
+ ∀f. f1 ⋓ f2 ≘ f → L1 ⪤[RN,RP,f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct
qed.
lemma sex_tc_next_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. L1 ⪤[RN, RP, f] L2 →
+ ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. L1 ⪤[RN,RP,f] L2 →
TC … (sex RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
/4 width=5 by sex_refl, sex_next, step, inj/
qed.
lemma sex_tc_push_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. L1 ⪤[RN, RP, f] L2 →
+ ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. L1 ⪤[RN,RP,f] L2 →
TC … (sex RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
/4 width=5 by sex_refl, sex_push, step, inj/
qed.
-lemma sex_tc_inj_sn: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → L1 ⪤[CTC … RN, RP, f] L2.
+lemma sex_tc_inj_sn: ∀RN,RP,f,L1,L2. L1 ⪤[RN,RP,f] L2 → L1 ⪤[CTC … RN,RP,f] L2.
#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by sex_push, sex_next, inj/
qed.
-lemma sex_tc_inj_dx: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → L1 ⪤[RN, CTC … RP, f] L2.
+lemma sex_tc_inj_dx: ∀RN,RP,f,L1,L2. L1 ⪤[RN,RP,f] L2 → L1 ⪤[RN,CTC … RP,f] L2.
#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by sex_push, sex_next, inj/
qed.
(* Basic_2A1: uses: TC_lpx_sn_ind *)
theorem sex_tc_step_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
- ∀f,L1,L. L1 ⪤[RN, RP, f] L → 𝐈⦃f⦄ →
- ∀L2. L ⪤[RN, CTC … RP, f] L2 → L1⪤ [RN, CTC … RP, f] L2.
+ ∀f,L1,L. L1 ⪤[RN,RP,f] L → 𝐈⦃f⦄ →
+ ∀L2. L ⪤[RN,CTC … RP,f] L2 → L1⪤ [RN,CTC … RP,f] L2.
#RN #RP #HRP #f #L1 #L #H elim H -f -L1 -L
[ #f #_ #Y #H -HRP >(sex_inv_atom1 … H) -Y // ]
#f #I1 #I #L1 #L #HL1 #HI1 #IH #Hf #Y #H
(* Advanced properties ******************************************************)
lemma sex_tc_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
- ∀f. 𝐈⦃f⦄ → ∀L1,L2. TC … (sex RN RP f) L1 L2 → L1 ⪤[RN, CTC … RP, f] L2.
+ ∀f. 𝐈⦃f⦄ → ∀L1,L2. TC … (sex RN RP f) L1 L2 → L1 ⪤[RN,CTC … RP,f] L2.
#RN #RP #HRP #f #Hf #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
/3 width=3 by sex_tc_step_dx, sex_tc_inj_dx/
qed.
(* Advanced inversion lemmas ************************************************)
lemma sex_inv_tc_sn: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,L1,L2. L1 ⪤[CTC … RN, RP, f] L2 → TC … (sex RN RP f) L1 L2.
+ ∀f,L1,L2. L1 ⪤[CTC … RN,RP,f] L2 → TC … (sex RN RP f) L1 L2.
#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
/2 width=1 by sex_tc_next, sex_tc_push_sn, sex_atom, inj/
qed-.
lemma sex_inv_tc_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,L1,L2. L1 ⪤[RN, CTC … RP, f] L2 → TC … (sex RN RP f) L1 L2.
+ ∀f,L1,L2. L1 ⪤[RN,CTC … RP,f] L2 → TC … (sex RN RP f) L1 L2.
#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
/2 width=1 by sex_tc_push, sex_tc_next_sn, sex_atom, inj/
qed-.
(* Basic properties *********************************************************)
-lemma fqu_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+lemma fqu_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄.
/2 width=1 by tri_inj/ qed.
lemma fqup_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+ ⦃G1,L1,T1⦄ ⬂+[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄.
/2 width=5 by tri_step/ qed.
lemma fqup_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+ ⦃G1,L1,T1⦄ ⬂[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⬂+[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄.
/2 width=5 by tri_TC_strap/ qed.
-lemma fqup_pair_sn: ∀b,I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+[b] ⦃G, L, V⦄.
+lemma fqup_pair_sn: ∀b,I,G,L,V,T. ⦃G,L,②{I}V.T⦄ ⬂+[b] ⦃G,L,V⦄.
/2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
-lemma fqup_bind_dx: ∀b,p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[b] ⦃G, L.ⓑ{I}V, T⦄.
-/2 width=1 by fqu_bind_dx, fqu_fqup/ qed.
+lemma fqup_bind_dx: ∀p,I,G,L,V,T. ⦃G,L,ⓑ{p,I}V.T⦄ ⬂+[Ⓣ] ⦃G,L.ⓑ{I}V,T⦄.
+/3 width=1 by fqu_bind_dx, fqu_fqup/ qed.
-lemma fqup_clear: ∀p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[Ⓕ] ⦃G, L.ⓧ, T⦄.
+lemma fqup_clear: ∀p,I,G,L,V,T. ⦃G,L,ⓑ{p,I}V.T⦄ ⬂+[Ⓕ] ⦃G,L.ⓧ,T⦄.
/3 width=1 by fqu_clear, fqu_fqup/ qed.
-lemma fqup_flat_dx: ∀b,I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+[b] ⦃G, L, T⦄.
+lemma fqup_flat_dx: ∀b,I,G,L,V,T. ⦃G,L,ⓕ{I}V.T⦄ ⬂+[b] ⦃G,L,T⦄.
/2 width=1 by fqu_flat_dx, fqu_fqup/ qed.
-lemma fqup_flat_dx_pair_sn: ∀b,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+[b] ⦃G, L, V2⦄.
+lemma fqup_flat_dx_pair_sn: ∀b,I1,I2,G,L,V1,V2,T. ⦃G,L,ⓕ{I1}V1.②{I2}V2.T⦄ ⬂+[b] ⦃G,L,V2⦄.
/2 width=5 by fqu_pair_sn, fqup_strap1/ qed.
-lemma fqup_bind_dx_flat_dx: ∀b,p,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I1}V1, T⦄.
+lemma fqup_bind_dx_flat_dx: ∀p,G,I1,I2,L,V1,V2,T. ⦃G,L,ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⬂+[Ⓣ] ⦃G,L.ⓑ{I1}V1,T⦄.
/2 width=5 by fqu_flat_dx, fqup_strap1/ qed.
-lemma fqup_flat_dx_bind_dx: ∀b,p,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I2}V2, T⦄.
-/2 width=5 by fqu_bind_dx, fqup_strap1/ qed.
+lemma fqup_flat_dx_bind_dx: ∀p,I1,I2,G,L,V1,V2,T. ⦃G,L,ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⬂+[Ⓣ] ⦃G,L.ⓑ{I2}V2,T⦄.
+/3 width=5 by fqu_bind_dx, fqup_strap1/ qed.
(* Basic eliminators ********************************************************)
lemma fqup_ind: ∀b,G1,L1,T1. ∀Q:relation3 ….
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
+ (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
+ (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⬂[b] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2.
#b #G1 #L1 #T1 #Q #IH1 #IH2 #G2 #L2 #T2 #H
@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
qed-.
lemma fqup_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 ….
- (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1) →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1.
+ (∀G1,L1,T1. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1) →
+ (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ⬂[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⬂+[b] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1.
#b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H
@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
qed-.
(* Advanced properties ******************************************************)
lemma fqup_zeta (b) (p) (I) (G) (K) (V):
- â\88\80T1,T2. â¬\86*[1]T2 â\89\98 T1 â\86\92 â¦\83G,K,â\93\91{p,I}V.T1â¦\84 â\8a\90+[b] ⦃G,K,T2⦄.
-/4 width=5 by fqup_strap2, fqu_fqup, fqu_drop/ qed.
+ â\88\80T1,T2. â¬\86*[1]T2 â\89\98 T1 â\86\92 â¦\83G,K,â\93\91{p,I}V.T1â¦\84 â¬\82+[b] ⦃G,K,T2⦄.
+* /4 width=5 by fqup_strap2, fqu_fqup, fqu_drop, fqu_clear, fqu_bind_dx/ qed.
(* Basic_2A1: removed theorems 1: fqup_drop *)
(* Properties with generic slicing for local environments *******************)
lemma fqup_drops_succ: ∀b,G,K,T,i,L,U. ⬇*[↑i] L ≘ K → ⬆*[↑i] T ≘ U →
- ⦃G, L, U⦄ ⊐+[b] ⦃G, K, T⦄.
+ ⦃G,L,U⦄ ⬂+[b] ⦃G,K,T⦄.
#b #G #K #T #i elim i -i
[ #L #U #HLK #HTU elim (drops_inv_succ … HLK) -HLK
#I #Y #HY #H destruct <(drops_fwd_isid … HY) -K //
qed.
lemma fqup_drops_strap1: ∀b,G1,G2,L1,K1,K2,T1,T2,U1,i. ⬇*[i] L1 ≘ K1 → ⬆*[i] T1 ≘ U1 →
- ⦃G1, K1, T1⦄ ⊐[b] ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, K2, T2⦄.
+ ⦃G1,K1,T1⦄ ⬂[b] ⦃G2,K2,T2⦄ → ⦃G1,L1,U1⦄ ⬂+[b] ⦃G2,K2,T2⦄.
#b #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 *
[ #HLK1 #HTU1 #HT12
>(drops_fwd_isid … HLK1) -L1 //
]
qed-.
-lemma fqup_lref: ∀b,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+[b] ⦃G, K, V⦄.
+lemma fqup_lref: ∀b,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,L,#i⦄ ⬂+[b] ⦃G,K,V⦄.
/2 width=6 by fqup_drops_strap1/ qed.
(* Forward lemmas with weight for closures **********************************)
-lemma fqup_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+lemma fqup_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ →
+ ♯{G2,L2,T2} < ♯{G1,L1,T1}.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
/3 width=3 by fqu_fwd_fw, transitive_lt/
qed-.
(* Advanced eliminators *****************************************************)
lemma fqup_wf_ind: ∀b. ∀Q:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) → ∀G1,L1,T1. Q G1 L1 T1.
#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct
qed-.
lemma fqup_wf_ind_eq: ∀b. ∀Q:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → Q G2 L2 T2
) → ∀G1,L1,T1. Q G1 L1 T1.
#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct
(* Basic eliminators ********************************************************)
lemma fqus_ind: ∀b,G1,L1,T1. ∀Q:relation3 …. Q G1 L1 T1 →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
+ (∀G,G2,L,L2,T,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ → Q G L T → Q G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2.
#b #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
qed-.
lemma fqus_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 …. Q G2 L2 T2 →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1.
+ (∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⬂*[b] ⦃G2,L2,T2⦄ → Q G L T → Q G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ → Q G1 L1 T1.
#b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H
@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
qed-.
lemma fqus_refl: ∀b. tri_reflexive … (fqus b).
/2 width=1 by tri_inj/ qed.
-lemma fquq_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fquq_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄.
/2 width=1 by tri_inj/ qed.
-lemma fqus_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G,L,T⦄ →
+ ⦃G,L,T⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄.
/2 width=5 by tri_step/ qed-.
-lemma fqus_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G,L,T⦄ →
+ ⦃G,L,T⦄ ⬂*[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄.
/2 width=5 by tri_TC_strap/ qed-.
(* Basic inversion lemmas ***************************************************)
-lemma fqus_inv_fqu_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_fqu_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
(∧∧ G1 = G2 & L1 = L2 & T1 = T2) ∨
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+ ∃∃G,L,T. ⦃G1,L1,T1⦄ ⬂[b] ⦃G,L,T⦄ & ⦃G,L,T⦄ ⬂*[b] ⦃G2,L2,T2⦄.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H12 @(fqus_ind_dx … H12) -G1 -L1 -T1 /3 width=1 by and3_intro, or_introl/
#G1 #G #L1 #L #T1 #T * /3 width=5 by ex2_3_intro, or_intror/
* #HG #HL #HT #_ destruct //
qed-.
-lemma fqus_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1,L1,⋆s⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
(∧∧ G1 = G2 & L1 = L2 & ⋆s = T2) ∨
- ∃∃J,L. ⦃G1, L, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}.
+ ∃∃J,L. ⦃G1,L,⋆s⦄ ⬂*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J}.
#b #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_sort1 … H) -H /3 width=4 by ex2_2_intro, or_intror/
qed-.
-lemma fqus_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & #i = T2
- | ∃∃J,L,V. ⦃G1, L, V⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V & i = 0
- | ∃∃J,L,j. ⦃G1, L, #j⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J} & i = ↑j.
+ | ∃∃J,L,V. ⦃G1,L,V⦄ ⬂*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓑ{J}V & i = 0
+ | ∃∃J,L,j. ⦃G1,L,#j⦄ ⬂*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J} & i = ↑j.
#b #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/
#G #L #T #H elim (fqu_inv_lref1 … H) -H * /3 width=7 by or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro/
qed-.
-lemma fqus_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1,L1,§l⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
(∧∧ G1 = G2 & L1 = L2 & §l = T2) ∨
- ∃∃J,L. ⦃G1, L, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}.
+ ∃∃J,L. ⦃G1,L,§l⦄ ⬂*[b] ⦃G2,L2,T2⦄ & L1 = L.ⓘ{J}.
#b #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_gref1 … H) -H /3 width=4 by ex2_2_intro, or_intror/
qed-.
-lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓑ{p,I}V1.T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
- | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄
- | â¦\83G1, L1.â\93\91{I}V1, T1â¦\84 â\8a\90*[b] â¦\83G2, L2, T2â¦\84
- | â¦\83G1, L1.â\93§, T1â¦\84 â\8a\90*[b] â¦\83G2, L2, T2â¦\84 â\88§ b = Ⓕ
- | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
+ | ⦃G1,L1,V1⦄ ⬂*[b] ⦃G2,L2,T2⦄
+ | â\88§â\88§ â¦\83G1,L1.â\93\91{I}V1,T1â¦\84 â¬\82*[b] â¦\83G2,L2,T2â¦\84 & b = â\93\89
+ | â\88§â\88§ â¦\83G1,L1.â\93§,T1â¦\84 â¬\82*[b] â¦\83G2,L2,T2â¦\84 & b = Ⓕ
+ | ∃∃J,L,T. ⦃G1,L,T⦄ ⬂*[b] ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
#b #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or5_intro0/
#G #L #T #H elim (fqu_inv_bind1 … H) -H *
-[4: #J ] #H1 #H2 #H3 [4: #Hb ] #H destruct
+[4: #J ] #H1 #H2 #H3 [3,4: #Hb ] #H destruct
/3 width=6 by or5_intro1, or5_intro2, or5_intro3, or5_intro4, ex3_3_intro, conj/
qed-.
-lemma fqus_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+lemma fqus_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓑ{p,I}V1.T1⦄ ⬂* ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
- | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
- | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐* ⦃G2, L2, T2⦄
- | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐* ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
-#p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_bind1 … H) -H [1,4: * ]
-/3 width=1 by and3_intro, or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex3_3_intro/
+ | ⦃G1,L1,V1⦄ ⬂* ⦃G2,L2,T2⦄
+ | ⦃G1,L1.ⓑ{I}V1,T1⦄ ⬂* ⦃G2,L2,T2⦄
+ | ∃∃J,L,T. ⦃G1,L,T⦄ ⬂* ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
+#p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_bind1 … H) -H [1,3,4: * ]
+/3 width=1 by and3_intro, or4_intro0, or4_intro1, or4_intro2, or4_intro3/
#_ #H destruct
qed-.
-lemma fqus_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓕ{I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1,L1,ⓕ{I}V1.T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓕ{I}V1.T1 = T2
- | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄
- | ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄
- | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓕ{I}V1.T1 & L1 = L.ⓘ{J}.
+ | ⦃G1,L1,V1⦄ ⬂*[b] ⦃G2,L2,T2⦄
+ | ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄
+ | ∃∃J,L,T. ⦃G1,L,T⦄ ⬂*[b] ⦃G2,L2,T2⦄ & ⬆*[1] T ≘ ⓕ{I}V1.T1 & L1 = L.ⓘ{J}.
#b #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or4_intro0/
#G #L #T #H elim (fqu_inv_flat1 … H) -H *
[3: #J ] #H1 #H2 #H3 #H destruct
(* Advanced inversion lemmas ************************************************)
-lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1,⋆,⓪{I}⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & ⋆ = L2 & ⓪{I} = T2.
#b #I #G1 #G2 #L2 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /2 width=1 by and3_intro/
#G #L #T #H elim (fqu_inv_atom1 … H)
qed-.
-lemma fqus_inv_sort1_bind: ∀b,I,G1,G2,L1,L2,T2,s. ⦃G1, L1.ⓘ{I}, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & ⋆s = T2) ∨ ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_inv_sort1_bind: ∀b,I,G1,G2,L1,L2,T2,s. ⦃G1,L1.ⓘ{I},⋆s⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & ⋆s = T2) ∨ ⦃G1,L1,⋆s⦄ ⬂*[b] ⦃G2,L2,T2⦄.
#b #I #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_sort1_bind … H) -H
#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
qed-.
-lemma fqus_inv_zero1_pair: ∀b,I,G1,G2,L1,L2,V1,T2. ⦃G1, L1.ⓑ{I}V1, #0⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_inv_zero1_pair: ∀b,I,G1,G2,L1,L2,V1,T2. ⦃G1,L1.ⓑ{I}V1,#0⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1,L1,V1⦄ ⬂*[b] ⦃G2,L2,T2⦄.
#b #I #G1 #G2 #L1 #L2 #V1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_zero1_pair … H) -H
#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
qed-.
-lemma fqus_inv_lref1_bind: ∀b,I,G1,G2,L1,L2,T2,i. ⦃G1, L1.ⓘ{I}, #↑i⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & #(↑i) = T2) ∨ ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_inv_lref1_bind: ∀b,I,G1,G2,L1,L2,T2,i. ⦃G1,L1.ⓘ{I},#↑i⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & #(↑i) = T2) ∨ ⦃G1,L1,#i⦄ ⬂*[b] ⦃G2,L2,T2⦄.
#b #I #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_lref1_bind … H) -H
#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
qed-.
-lemma fqus_inv_gref1_bind: ∀b,I,G1,G2,L1,L2,T2,l. ⦃G1, L1.ⓘ{I}, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & §l = T2) ∨ ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqus_inv_gref1_bind: ∀b,I,G1,G2,L1,L2,T2,l. ⦃G1,L1.ⓘ{I},§l⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & §l = T2) ∨ ⦃G1,L1,§l⦄ ⬂*[b] ⦃G2,L2,T2⦄.
#b #I #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
#G #L #T #H elim (fqu_inv_gref1_bind … H) -H
#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
(* Properties with generic slicing for local environments *******************)
lemma fqus_drops: ∀b,G,L,K,T,U,i. ⬇*[i] L ≘ K → ⬆*[i] T ≘ U →
- ⦃G, L, U⦄ ⊐*[b] ⦃G, K, T⦄.
+ ⦃G,L,U⦄ ⬂*[b] ⦃G,K,T⦄.
#b #G #L #K #T #U * /3 width=3 by fqup_drops_succ, fqup_fqus/
#HLK #HTU <(lifts_fwd_isid … HTU) -U // <(drops_fwd_isid … HLK) -K //
qed.
(* Alternative definition with plus-iterated supclosure *********************)
-lemma fqup_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqup_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/
qed.
(* Basic_2A1: was: fqus_inv_gen *)
-lemma fqus_inv_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
+lemma fqus_inv_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 //
#G #G2 #L #L2 #T #T2 #_ *
[ #H2 * /3 width=5 by fqup_strap1, or_introl/
(* Advanced properties ******************************************************)
-lemma fqus_strap1_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+lemma fqus_strap1_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄.
#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H1) -H1
[ /2 width=5 by fqup_strap1/
| * /2 width=1 by fqu_fqup/
]
qed-.
-lemma fqus_strap2_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+lemma fqus_strap2_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄.
#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H2) -H2
[ /2 width=5 by fqup_strap2/
| * /2 width=1 by fqu_fqup/
]
qed-.
-lemma fqus_fqup_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+lemma fqus_fqup_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G,L,T⦄ → ⦃G,L,T⦄ ⬂+[b] ⦃G2,L2,T2⦄ →
+ ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄.
#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2
/2 width=5 by fqus_strap1_fqu, fqup_strap1/
qed-.
-lemma fqup_fqus_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+lemma fqup_fqus_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G,L,T⦄ →
+ ⦃G,L,T⦄ ⬂*[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄.
#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1
/3 width=5 by fqus_strap2_fqu, fqup_strap2/
qed-.
(* Advanced inversion lemmas for plus-iterated supclosure *******************)
-lemma fqup_inv_step_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+lemma fqup_inv_step_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+[b] ⦃G2,L2,T2⦄ →
+ ∃∃G,L,T. ⦃G1,L1,T1⦄ ⬂[b] ⦃G,L,T⦄ & ⦃G,L,T⦄ ⬂*[b] ⦃G2,L2,T2⦄.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_intro/
qed-.
(* Forward lemmas with weight for closures **********************************)
-lemma fqus_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+lemma fqus_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ ♯{G2,L2,T2} ≤ ♯{G1,L1,T1}.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2
/3 width=3 by fquq_fwd_fw, transitive_le/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma fqus_inv_refl_atom3: ∀b,I,G,L,X. ⦃G, L, ⓪{I}⦄ ⊐*[b] ⦃G, L, X⦄ → ⓪{I} = X.
+lemma fqus_inv_refl_atom3: ∀b,I,G,L,X. ⦃G,L,⓪{I}⦄ ⬂*[b] ⦃G,L,X⦄ → ⓪{I} = X.
#b #I #G #L #X #H elim (fqus_inv_fqu_sn … H) -H * //
#G0 #L0 #T0 #H1 #H2 lapply (fqu_fwd_fw … H1) lapply (fqus_fwd_fw … H2) -H2 -H1
#H2 #H1 lapply (le_to_lt_to_lt … H2 H1) -G0 -L0 -T0
inductive fqu (b:bool): tri_relation genv lenv term ≝
| fqu_lref_O : ∀I,G,L,V. fqu b G (L.ⓑ{I}V) (#0) G L V
| fqu_pair_sn: ∀I,G,L,V,T. fqu b G L (②{I}V.T) G L V
-| fqu_bind_dx: ∀p,I,G,L,V,T. fqu b G L (ⓑ{p,I}V.T) G (L.ⓑ{I}V) T
+| fqu_bind_dx: ∀p,I,G,L,V,T. b = Ⓣ → fqu b G L (ⓑ{p,I}V.T) G (L.ⓑ{I}V) T
| fqu_clear : ∀p,I,G,L,V,T. b = Ⓕ → fqu b G L (ⓑ{p,I}V.T) G (L.ⓧ) T
| fqu_flat_dx: ∀I,G,L,V,T. fqu b G L (ⓕ{I}V.T) G L T
| fqu_drop : ∀I,G,L,T,U. ⬆*[1] T ≘ U → fqu b G (L.ⓘ{I}) U G L T
(* Basic properties *********************************************************)
-lemma fqu_sort: ∀b,I,G,L,s. ⦃G, L.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G, L, ⋆s⦄.
+lemma fqu_sort: ∀b,I,G,L,s. ⦃G,L.ⓘ{I},⋆s⦄ ⬂[b] ⦃G,L,⋆s⦄.
/2 width=1 by fqu_drop/ qed.
-lemma fqu_lref_S: ∀b,I,G,L,i. ⦃G, L.ⓘ{I}, #↑i⦄ ⊐[b] ⦃G, L, #i⦄.
+lemma fqu_lref_S: ∀b,I,G,L,i. ⦃G,L.ⓘ{I},#↑i⦄ ⬂[b] ⦃G,L,#i⦄.
/2 width=1 by fqu_drop/ qed.
-lemma fqu_gref: ∀b,I,G,L,l. ⦃G, L.ⓘ{I}, §l⦄ ⊐[b] ⦃G, L, §l⦄.
+lemma fqu_gref: ∀b,I,G,L,l. ⦃G,L.ⓘ{I},§l⦄ ⬂[b] ⦃G,L,§l⦄.
/2 width=1 by fqu_drop/ qed.
(* Basic inversion lemmas ***************************************************)
-fact fqu_inv_sort1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_inv_sort1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∀s. T1 = ⋆s →
∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s.
#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #T #s #H destruct
| #I #G #L #V #T #s #H destruct
-| #p #I #G #L #V #T #s #H destruct
+| #p #I #G #L #V #T #_ #s #H destruct
| #p #I #G #L #V #T #_ #s #H destruct
| #I #G #L #V #T #s #H destruct
| #I #G #L #T #U #HI12 #s #H destruct
]
qed-.
-lemma fqu_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1,L1,⋆s⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s.
/2 width=4 by fqu_inv_sort1_aux/ qed-.
-fact fqu_inv_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_inv_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∀i. T1 = #i →
(∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j.
#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #T #i #H destruct /3 width=4 by ex4_2_intro, or_introl/
| #I #G #L #V #T #i #H destruct
-| #p #I #G #L #V #T #i #H destruct
+| #p #I #G #L #V #T #_ #i #H destruct
| #p #I #G #L #V #T #_ #i #H destruct
| #I #G #L #V #T #i #H destruct
| #I #G #L #T #U #HI12 #i #H destruct
]
qed-.
-lemma fqu_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⬂[b] ⦃G2,L2,T2⦄ →
(∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j.
/2 width=4 by fqu_inv_lref1_aux/ qed-.
-fact fqu_inv_gref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_inv_gref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∀l. T1 = §l →
∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l.
#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #T #l #H destruct
| #I #G #L #V #T #l #H destruct
-| #p #I #G #L #V #T #l #H destruct
+| #p #I #G #L #V #T #_ #l #H destruct
| #p #I #G #L #V #T #_ #l #H destruct
| #I #G #L #V #T #s #H destruct
| #I #G #L #T #U #HI12 #l #H destruct
]
qed-.
-lemma fqu_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1,L1,§l⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l.
/2 width=4 by fqu_inv_gref1_aux/ qed-.
-fact fqu_inv_bind1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_inv_bind1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∀p,I,V1,U1. T1 = ⓑ{p,I}V1.U1 →
∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
- | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
+ | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2 & b = Ⓣ
| ∧∧ G1 = G2 & L1.ⓧ = L2 & U1 = T2 & b = Ⓕ
| ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1.
#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #T #q #J #V0 #U0 #H destruct
| #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or4_intro0/
-| #p #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or4_intro1/
+| #p #I #G #L #V #T #Hb #q #J #V0 #U0 #H destruct /3 width=1 by and4_intro, or4_intro1/
| #p #I #G #L #V #T #Hb #q #J #V0 #U0 #H destruct /3 width=1 by and4_intro, or4_intro2/
| #I #G #L #V #T #q #J #V0 #U0 #H destruct
| #I #G #L #T #U #HTU #q #J #V0 #U0 #H destruct /3 width=2 by or4_intro3, ex3_intro/
]
qed-.
-lemma fqu_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1,L1,ⓑ{p,I}V1.U1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
- | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
+ | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2 & b = Ⓣ
| ∧∧ G1 = G2 & L1.ⓧ = L2 & U1 = T2 & b = Ⓕ
| ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1.
/2 width=4 by fqu_inv_bind1_aux/ qed-.
-lemma fqu_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐ ⦃G2, L2, T2⦄ →
+lemma fqu_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1,L1,ⓑ{p,I}V1.U1⦄ ⬂ ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
| ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
| ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1.
#p #I #G1 #G2 #L1 #L2 #V1 #U1 #T2 #H elim (fqu_inv_bind1 … H) -H
-/3 width=1 by or3_intro0, or3_intro1, or3_intro2/
-* #_ #_ #_ #H destruct
+/3 width=1 by or3_intro0, or3_intro2/
+* #HG #HL #HU #H destruct
+/3 width=1 by and3_intro, or3_intro1/
qed-.
-fact fqu_inv_flat1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_inv_flat1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∀I,V1,U1. T1 = ⓕ{I}V1.U1 →
∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
| ∧∧ G1 = G2 & L1 = L2 & U1 = T2
#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #T #J #V0 #U0 #H destruct
| #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro0/
-| #p #I #G #L #V #T #J #V0 #U0 #H destruct
+| #p #I #G #L #V #T #_ #J #V0 #U0 #H destruct
| #p #I #G #L #V #T #_ #J #V0 #U0 #H destruct
| #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro1/
| #I #G #L #T #U #HTU #J #V0 #U0 #H destruct /3 width=2 by or3_intro2, ex3_intro/
]
qed-.
-lemma fqu_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓕ{I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1,L1,ⓕ{I}V1.U1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
| ∧∧ G1 = G2 & L1 = L2 & U1 = T2
| ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓕ{I}V1.U1.
(* Advanced inversion lemmas ************************************************)
-lemma fqu_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐[b] ⦃G2, L2, T2⦄ → ⊥.
+lemma fqu_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1,⋆,⓪{I}⦄ ⬂[b] ⦃G2,L2,T2⦄ → ⊥.
#b * #x #G1 #G2 #L2 #T2 #H
[ elim (fqu_inv_sort1 … H) | elim (fqu_inv_lref1 … H) * | elim (fqu_inv_gref1 … H) ] -H
#I [2: #V |3: #i ] #_ #H destruct
qed-.
-lemma fqu_inv_sort1_bind: ∀b,I,G1,G2,K,L2,T2,s. ⦃G1, K.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_sort1_bind: ∀b,I,G1,G2,K,L2,T2,s. ⦃G1,K.ⓘ{I},⋆s⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & L2 = K & T2 = ⋆s.
#b #I #G1 #G2 #K #L2 #T2 #s #H elim (fqu_inv_sort1 … H) -H
#Z #X #H1 #H2 destruct /2 width=1 by and3_intro/
qed-.
-lemma fqu_inv_zero1_pair: ∀b,I,G1,G2,K,L2,V,T2. ⦃G1, K.ⓑ{I}V, #0⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_zero1_pair: ∀b,I,G1,G2,K,L2,V,T2. ⦃G1,K.ⓑ{I}V,#0⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & L2 = K & T2 = V.
#b #I #G1 #G2 #K #L2 #V #T2 #H elim (fqu_inv_lref1 … H) -H *
#Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
qed-.
-lemma fqu_inv_lref1_bind: ∀b,I,G1,G2,K,L2,T2,i. ⦃G1, K.ⓘ{I}, #(↑i)⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_lref1_bind: ∀b,I,G1,G2,K,L2,T2,i. ⦃G1,K.ⓘ{I},#(↑i)⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & L2 = K & T2 = #i.
#b #I #G1 #G2 #K #L2 #T2 #i #H elim (fqu_inv_lref1 … H) -H *
#Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
qed-.
-lemma fqu_inv_gref1_bind: ∀b,I,G1,G2,K,L2,T2,l. ⦃G1, K.ⓘ{I}, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_inv_gref1_bind: ∀b,I,G1,G2,K,L2,T2,l. ⦃G1,K.ⓘ{I},§l⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∧∧ G1 = G2 & L2 = K & T2 = §l.
#b #I #G1 #G2 #K #L2 #T2 #l #H elim (fqu_inv_gref1 … H) -H
#Z #H1 #H2 #H3 destruct /2 width=1 by and3_intro/
(* Forward lemmas with length for local environments ************************)
-fact fqu_fwd_length_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+fact fqu_fwd_length_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
∀i. T1 = #i → |L2| < |L1|.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 // [2,3: #p]
-#I #G #L #V #T [2: #_ ] #j #H destruct
+#I #G #L #V #T [1,2: #_ ] #j #H destruct
qed-.
-lemma fqu_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+lemma fqu_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⬂[b] ⦃G2,L2,T2⦄ →
|L2| < |L1|.
/2 width=8 by fqu_fwd_length_lref1_aux/
qed-.
(* SUPCLOSURE ***************************************************************)
-(* Inversion lemmas with context-free degree-based equivalence for terms ****)
+(* Inversion lemmas with context-free sort-irrelevant equivalence for terms *)
-fact fqu_inv_tdeq_aux: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- G1 = G2 → |L1| = |L2| → T1 ≛[h, o] T2 → ⊥.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
+fact fqu_inv_tdeq_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ G1 = G2 → |L1| = |L2| → T1 ≛ T2 → ⊥.
+#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
[1: #I #G #L #V #_ #H elim (succ_inv_refl_sn … H)
|6: #I #G #L #T #U #_ #_ #H elim (succ_inv_refl_sn … H)
]
qed-.
(* Basic_2A1: uses: fqu_inv_eq *)
-lemma fqu_inv_tdeq: ∀h,o,b,G,L1,L2,T1,T2. ⦃G, L1, T1⦄ ⊐[b] ⦃G, L2, T2⦄ →
- |L1| = |L2| → T1 ≛[h, o] T2 → ⊥.
-#h #o #b #G #L1 #L2 #T1 #T2 #H
+lemma fqu_inv_tdeq: ∀b,G,L1,L2,T1,T2. ⦃G,L1,T1⦄ ⬂[b] ⦃G,L2,T2⦄ →
+ |L1| = |L2| → T1 ≛ T2 → ⊥.
+#b #G #L1 #L2 #T1 #T2 #H
@(fqu_inv_tdeq_aux … H) // (**) (* full auto fails *)
qed-.
(* Forward lemmas with weight for closures **********************************)
-lemma fqu_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+lemma fqu_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ →
+ ♯{G2,L2,T2} < ♯{G1,L1,T1}.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
#I #I1 #I2 #G #L #HI12 normalize in ⊢ (?%%); -I1
<(lifts_fwd_tw … HI12) /3 width=1 by monotonic_lt_plus_r, monotonic_lt_plus_l/
(* Advanced eliminators *****************************************************)
lemma fqu_wf_ind: ∀b. ∀Q:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) → ∀G1,L1,T1. Q G1 L1 T1.
#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=2 by fqu_fwd_fw/
lemma fquq_refl: ∀b. tri_reflexive … (fquq b).
// qed.
-lemma fqu_fquq: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄.
+lemma fqu_fquq: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂[b] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄.
/2 width=1 by or_introl/ qed.
(* Basic_2A1: removed theorems 8:
(* Forward lemmas with length for local environments ************************)
-lemma fquq_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+lemma fquq_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1,L1,#i⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
|L2| ≤ |L1|.
#b #G1 #G2 #L1 #L2 #T2 #i #H elim H -H [2: * ]
/3 width=6 by fqu_fwd_length_lref1, lt_to_le/
(* Forward lemmas with weight for closures **********************************)
-lemma fquq_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+lemma fquq_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮[b] ⦃G2,L2,T2⦄ →
+ ♯{G2,L2,T2} ≤ ♯{G1,L1,T1}.
#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [2: * ]
/3 width=2 by fqu_fwd_fw, lt_to_le/
qed-.
(* Basic inversion lemmas ***************************************************)
-fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀s. T = ⋆s → A = ⓪.
+fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀s. T = ⋆s → A = ⓪.
#G #L #T #A * -G -L -T -A //
[ #I #G #L #V #B #_ #s #H destruct
| #I #G #L #A #i #_ #s #H destruct
]
qed-.
-lemma aaa_inv_sort: ∀G,L,A,s. ⦃G, L⦄ ⊢ ⋆s ⁝ A → A = ⓪.
+lemma aaa_inv_sort: ∀G,L,A,s. ⦃G,L⦄ ⊢ ⋆s ⁝ A → A = ⓪.
/2 width=6 by aaa_inv_sort_aux/ qed-.
-fact aaa_inv_zero_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → T = #0 →
- ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+fact aaa_inv_zero_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → T = #0 →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ⁝ A.
#G #L #T #A * -G -L -T -A /2 width=5 by ex2_3_intro/
[ #G #L #s #H destruct
| #I #G #L #A #i #_ #H destruct
]
qed-.
-lemma aaa_inv_zero: ∀G,L,A. ⦃G, L⦄ ⊢ #0 ⁝ A →
- ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+lemma aaa_inv_zero: ∀G,L,A. ⦃G,L⦄ ⊢ #0 ⁝ A →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ⁝ A.
/2 width=3 by aaa_inv_zero_aux/ qed-.
-fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #(↑i) →
- ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A.
+fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀i. T = #(↑i) →
+ ∃∃I,K. L = K.ⓘ{I} & ⦃G,K⦄ ⊢ #i ⁝ A.
#G #L #T #A * -G -L -T -A
[ #G #L #s #j #H destruct
| #I #G #L #V #B #_ #j #H destruct
]
qed-.
-lemma aaa_inv_lref: ∀G,L,A,i. ⦃G, L⦄ ⊢ #↑i ⁝ A →
- ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A.
+lemma aaa_inv_lref: ∀G,L,A,i. ⦃G,L⦄ ⊢ #↑i ⁝ A →
+ ∃∃I,K. L = K.ⓘ{I} & ⦃G,K⦄ ⊢ #i ⁝ A.
/2 width=3 by aaa_inv_lref_aux/ qed-.
-fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀l. T = §l → ⊥.
+fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀l. T = §l → ⊥.
#G #L #T #A * -G -L -T -A
[ #G #L #s #k #H destruct
| #I #G #L #V #B #_ #k #H destruct
]
qed-.
-lemma aaa_inv_gref: ∀G,L,A,l. ⦃G, L⦄ ⊢ §l ⁝ A → ⊥.
+lemma aaa_inv_gref: ∀G,L,A,l. ⦃G,L⦄ ⊢ §l ⁝ A → ⊥.
/2 width=7 by aaa_inv_gref_aux/ qed-.
-fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓓ{p}W.U →
- ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L.ⓓW⦄ ⊢ U ⁝ A.
+fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓓ{p}W.U →
+ ∃∃B. ⦃G,L⦄ ⊢ W ⁝ B & ⦃G,L.ⓓW⦄ ⊢ U ⁝ A.
#G #L #T #A * -G -L -T -A
[ #G #L #s #q #W #U #H destruct
| #I #G #L #V #B #_ #q #W #U #H destruct
]
qed-.
-lemma aaa_inv_abbr: ∀p,G,L,V,T,A. ⦃G, L⦄ ⊢ ⓓ{p}V.T ⁝ A →
- ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L.ⓓV⦄ ⊢ T ⁝ A.
+lemma aaa_inv_abbr: ∀p,G,L,V,T,A. ⦃G,L⦄ ⊢ ⓓ{p}V.T ⁝ A →
+ ∃∃B. ⦃G,L⦄ ⊢ V ⁝ B & ⦃G,L.ⓓV⦄ ⊢ T ⁝ A.
/2 width=4 by aaa_inv_abbr_aux/ qed-.
-fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓛ{p}W.U →
- ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ U ⁝ B2 & A = ②B1.B2.
+fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓛ{p}W.U →
+ ∃∃B1,B2. ⦃G,L⦄ ⊢ W ⁝ B1 & ⦃G,L.ⓛW⦄ ⊢ U ⁝ B2 & A = ②B1.B2.
#G #L #T #A * -G -L -T -A
[ #G #L #s #q #W #U #H destruct
| #I #G #L #V #B #_ #q #W #U #H destruct
]
qed-.
-lemma aaa_inv_abst: ∀p,G,L,W,T,A. ⦃G, L⦄ ⊢ ⓛ{p}W.T ⁝ A →
- ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ T ⁝ B2 & A = ②B1.B2.
+lemma aaa_inv_abst: ∀p,G,L,W,T,A. ⦃G,L⦄ ⊢ ⓛ{p}W.T ⁝ A →
+ ∃∃B1,B2. ⦃G,L⦄ ⊢ W ⁝ B1 & ⦃G,L.ⓛW⦄ ⊢ T ⁝ B2 & A = ②B1.B2.
/2 width=4 by aaa_inv_abst_aux/ qed-.
-fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U →
- ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L⦄ ⊢ U ⁝ ②B.A.
+fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U →
+ ∃∃B. ⦃G,L⦄ ⊢ W ⁝ B & ⦃G,L⦄ ⊢ U ⁝ ②B.A.
#G #L #T #A * -G -L -T -A
[ #G #L #s #W #U #H destruct
| #I #G #L #V #B #_ #W #U #H destruct
]
qed-.
-lemma aaa_inv_appl: ∀G,L,V,T,A. ⦃G, L⦄ ⊢ ⓐV.T ⁝ A →
- ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L⦄ ⊢ T ⁝ ②B.A.
+lemma aaa_inv_appl: ∀G,L,V,T,A. ⦃G,L⦄ ⊢ ⓐV.T ⁝ A →
+ ∃∃B. ⦃G,L⦄ ⊢ V ⁝ B & ⦃G,L⦄ ⊢ T ⁝ ②B.A.
/2 width=3 by aaa_inv_appl_aux/ qed-.
-fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW.U →
- ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ U ⁝ A.
+fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW.U →
+ ⦃G,L⦄ ⊢ W ⁝ A ∧ ⦃G,L⦄ ⊢ U ⁝ A.
#G #L #T #A * -G -L -T -A
[ #G #L #s #W #U #H destruct
| #I #G #L #V #B #_ #W #U #H destruct
]
qed-.
-lemma aaa_inv_cast: ∀G,L,W,T,A. ⦃G, L⦄ ⊢ ⓝW.T ⁝ A →
- ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ T ⁝ A.
+lemma aaa_inv_cast: ∀G,L,W,T,A. ⦃G,L⦄ ⊢ ⓝW.T ⁝ A →
+ ⦃G,L⦄ ⊢ W ⁝ A ∧ ⦃G,L⦄ ⊢ T ⁝ A.
/2 width=3 by aaa_inv_cast_aux/ qed-.
(* Main inversion lemmas ****************************************************)
-theorem aaa_mono: ∀G,L,T,A1. ⦃G, L⦄ ⊢ T ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T ⁝ A2 → A1 = A2.
+theorem aaa_mono: ∀G,L,T,A1. ⦃G,L⦄ ⊢ T ⁝ A1 → ∀A2. ⦃G,L⦄ ⊢ T ⁝ A2 → A1 = A2.
#G #L #T #A1 #H elim H -G -L -T -A1
[ #G #L #s #A2 #H >(aaa_inv_sort … H) -H //
| #I1 #G #L #V1 #B #_ #IH #A2 #H
elim (aaa_inv_cast … H) -H /2 width=1 by/
]
qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma aaa_aaa_inv_appl (G) (L) (V) (T) (B) (X):
+ ∀A. ⦃G,L⦄ ⊢ ⓐV.T ⁝ A → ⦃G,L⦄ ⊢ V ⁝ B → ⦃G,L⦄⊢ T ⁝ X → ②B.A = X.
+#G #L #V #T #B #X #A #H #H1V #H1T
+elim (aaa_inv_appl … H) -H #B0 #H2V #H2T
+lapply (aaa_mono … H2V … H1V) -V #H destruct
+lapply (aaa_mono … H2T … H1T) -G -L -T //
+qed-.
+
+lemma aaa_aaa_inv_cast (G) (L) (U) (T) (B) (A):
+ ∀X. ⦃G,L⦄ ⊢ ⓝU.T ⁝ X → ⦃G,L⦄ ⊢ U ⁝ B → ⦃G,L⦄⊢ T ⁝ A → ∧∧ B = X & A = X.
+#G #L #U #T #B #A #X #H #H1U #H1T
+elim (aaa_inv_cast … H) -H #H2U #H2T
+lapply (aaa_mono … H1U … H2U) -U #HB
+lapply (aaa_mono … H1T … H2T) -G -L -T #HA
+/2 width=1 by conj/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/static/aaa_drops.ma".
+include "static_2/static/aaa_aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Main properties **********************************************************)
+
+theorem aaa_dec (G) (L) (T): Decidable (∃A. ⦃G,L⦄ ⊢ T ⁝ A).
+#G #L #T @(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T
+#G0 #L0 #T0 #IH #G #L * * [||| #p * | * ]
+[ #s #HG #HL #HT destruct -IH
+ /3 width=2 by aaa_sort, ex_intro, or_introl/
+| #i #HG #HL #HT destruct
+ elim (drops_F_uni L i) [| * * #I [| #V ] #K ] #HLK
+ [1,2: -IH
+ @or_intror * #A #H
+ elim (aaa_inv_lref_drops … H) -H #J #Y #X #HLY #_ -G -A
+ lapply (drops_mono … HLY … HLK) -L -i #H destruct
+ | elim (IH G K V) -IH [3: /2 width=2 by fqup_lref/ ]
+ [ * /4 width=6 by aaa_lref_drops, ex_intro, or_introl/
+ | #H0 @or_intror * #A #H
+ lapply (aaa_pair_inv_lref … H … HLK) -I -L -i
+ /3 width=2 by ex_intro/
+ ]
+ ]
+| #l #HG #HL #HT destruct -IH
+ @or_intror * #A #H
+ @(aaa_inv_gref … H)
+| #V #T #HG #HL #HT destruct
+ elim (IH G L V) [ * #B #HB | #HnB | // ]
+ [ elim (IH G (L.ⓓV) T) [ * #A #HA | #HnA | // ] ] -IH
+ [ /4 width=2 by aaa_abbr, ex_intro, or_introl/ ]
+ @or_intror * #A #H
+ elim (aaa_inv_abbr … H) -H #B0 #HB0 #HA0
+ /3 width=2 by ex_intro/
+| #W #T #HG #HL #HT destruct
+ elim (IH G L W) [ * #B #HB | #HnB | // ]
+ [ elim (IH G (L.ⓛW) T) [ * #A #HA | #HnA | // ] ] -IH
+ [ /4 width=2 by aaa_abst, ex_intro, or_introl/ ]
+ @or_intror * #A #H
+ elim (aaa_inv_abst … H) -H #B0 #A0 #HB0 #HA0 #H destruct
+ /3 width=2 by ex_intro/
+| #V #T #HG #HL #HT destruct
+ elim (IH G L V) [ * #B #HB | #HnB | // ]
+ [ elim (IH G L T) [ * #X #HX | #HnX | // ] ] -IH
+ [ elim (is_apear_dec B X) [ * #A #H destruct | #HnX ]
+ [ /4 width=4 by aaa_appl, ex_intro, or_introl/ ]
+ ]
+ @or_intror * #A #H
+ [ lapply (aaa_aaa_inv_appl … H HB HX) -G -L -V -T
+ |*: elim (aaa_inv_appl … H) -H #B0 #HB0 #HA0
+ ]
+ /3 width=2 by ex_intro/
+| #U #T #HG #HL #HT destruct
+ elim (IH G L U) [ * #B #HB | #HnB | // ]
+ [ elim (IH G L T) [ * #A #HA | #HnA | // ] ] -IH
+ [ elim (eq_aarity_dec B A) [ #H destruct | #HnA ]
+ [ /4 width=3 by aaa_cast, ex_intro, or_introl/ ]
+ ]
+ @or_intror * #X #H
+ [ elim (aaa_aaa_inv_cast … H HB HA) -G -L -U -T
+ |*: elim (aaa_inv_cast … H) -H #HU #HT
+ ]
+ /3 width=2 by ex_intro/
+]
+qed-.
(* Advanced properties ******************************************************)
(* Basic_2A1: was: aaa_lref *)
-lemma aaa_lref_drops: ∀I,G,K,V,B,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B.
+lemma aaa_lref_drops: ∀I,G,K,V,B,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ⁝ B → ⦃G,L⦄ ⊢ #i ⁝ B.
#I #G #K #V #B #i elim i -i
[ #L #H lapply (drops_fwd_isid … H ?) -H //
#H destruct /2 width=1 by aaa_zero/
(* Advanced inversion lemmas ************************************************)
(* Basic_2A1: was: aaa_inv_lref *)
-lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A →
- ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G,L⦄ ⊢ #i ⁝ A →
+ ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G,K⦄ ⊢ V ⁝ A.
#G #A #i elim i -i
[ #L #H elim (aaa_inv_zero … H) -H /3 width=5 by drops_refl, ex2_3_intro/
| #i #IH #L #H elim (aaa_inv_lref … H) -H
]
qed-.
+lemma aaa_pair_inv_lref (G) (L) (i):
+ ∀A. ⦃G,L⦄ ⊢ #i ⁝ A → ∀I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G,K⦄ ⊢ V ⁝ A.
+#G #L #i #A #H #I #K #V #HLK
+elim (aaa_inv_lref_drops … H) -H #J #Y #X #HLY #HX
+lapply (drops_mono … HLY … HLK) -L -i #H destruct //
+qed-.
+
(* Properties with generic slicing for local environments *******************)
(* Basic_2A1: includes: aaa_lift *)
(* Note: it should use drops_split_trans_pair2 *)
-lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b, f] L2 ≘ L1 →
- ∀T2. ⬆*[f] T1 ≘ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+lemma aaa_lifts: ∀G,L1,T1,A. ⦃G,L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b,f] L2 ≘ L1 →
+ ∀T2. ⬆*[f] T1 ≘ T2 → ⦃G,L2⦄ ⊢ T2 ⁝ A.
@(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L1 * *
[ #s #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -IH
lapply (aaa_inv_sort … H) -H #H destruct
(* Inversion lemmas with generic slicing for local environments *************)
(* Basic_2A1: includes: aaa_inv_lift *)
-lemma aaa_inv_lifts: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀b,f,L1. ⬇*[b, f] L2 ≘ L1 →
- ∀T1. ⬆*[f] T1 ≘ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
+lemma aaa_inv_lifts: ∀G,L2,T2,A. ⦃G,L2⦄ ⊢ T2 ⁝ A → ∀b,f,L1. ⬇*[b,f] L2 ≘ L1 →
+ ∀T1. ⬆*[f] T1 ≘ T2 → ⦃G,L1⦄ ⊢ T1 ⁝ A.
@(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L2 * *
[ #s #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -b -IH
lapply (aaa_inv_sort … H) -H #H destruct
(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-(* Properties with degree-based equivalence on referred entries *************)
+(* Properties with sort-irrelevant equivalence on referred entries **********)
-lemma aaa_fdeq_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
- ∀A. ⦃G1, L1⦄ ⊢ T1 ⁝ A → ⦃G2, L2⦄ ⊢ T2 ⁝ A.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/2 width=7 by aaa_tdeq_conf_rdeq/ qed-.
+lemma aaa_fdeq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ →
+ ∀A. ⦃G1,L1⦄ ⊢ T1 ⁝ A → ⦃G2,L2⦄ ⊢ T2 ⁝ A.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/2 width=5 by aaa_tdeq_conf_rdeq/ qed-.
(* Properties on supclosure *************************************************)
-lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂ ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #T #A #H elim (aaa_inv_zero … H) -H
#J #K #V #H #HA destruct /2 width=2 by ex_intro/
| elim (aaa_inv_appl … H)
| elim (aaa_inv_cast … H)
] -H /2 width=2 by ex_intro/
-| #p * #G #L #V #T #X #H
+| #p * #G #L #V #T #_ #X #H
[ elim (aaa_inv_abbr … H)
| elim (aaa_inv_abst … H)
] -H /2 width=2 by ex_intro/
]
qed-.
-lemma aaa_fquq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+lemma aaa_fquq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂⸮ ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=6 by aaa_fqu_conf/
* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
qed-.
-lemma aaa_fqup_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+lemma aaa_fqup_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂+ ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A ]
/2 width=6 by aaa_fqu_conf/
qed-.
-lemma aaa_fqus_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+lemma aaa_fqus_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ⬂* ⦃G2,L2,T2⦄ →
+ ∀A1. ⦃G1,L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2,L2⦄ ⊢ T2 ⁝ A2.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim(fqus_inv_fqup … H) -H /2 width=6 by aaa_fqup_conf/
* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
qed-.
(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-(* Properties with degree-based equivalence on referred entries *************)
+(* Properties with sort-irrelevant equivalence on referred entries **********)
-lemma aaa_tdeq_conf_rdeq: ∀h,o,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀T2. T1 ≛[h, o] T2 →
- ∀L2. L1 ≛[h, o, T1] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-#h #o #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
+lemma aaa_tdeq_conf_rdeq: ∀G,L1,T1,A. ⦃G,L1⦄ ⊢ T1 ⁝ A → ∀T2. T1 ≛ T2 →
+ ∀L2. L1 ≛[T1] L2 → ⦃G,L2⦄ ⊢ T2 ⁝ A.
+#G #L1 #T1 #A #H elim H -G -L1 -T1 -A
[ #G #L1 #s1 #X #H1 elim (tdeq_inv_sort1 … H1) -H1 //
| #I #G #L1 #V1 #B #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1
#Y #H2 elim (rdeq_inv_zero_pair_sn … H2) -H2
(* *)
(**************************************************************************)
-include "static_2/notation/relations/stareqsn_8.ma".
+include "static_2/notation/relations/stareqsn_6.ma".
include "static_2/syntax/genv.ma".
include "static_2/static/rdeq.ma".
-(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+(* SORT-IRRELEVANT EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *************)
-inductive fdeq (h) (o) (G) (L1) (T1): relation3 genv lenv term ≝
-| fdeq_intro_sn: ∀L2,T2. L1 ≛[h, o, T1] L2 → T1 ≛[h, o] T2 →
- fdeq h o G L1 T1 G L2 T2
+inductive fdeq (G) (L1) (T1): relation3 genv lenv term ≝
+| fdeq_intro_sn: ∀L2,T2. L1 ≛[T1] L2 → T1 ≛ T2 →
+ fdeq G L1 T1 G L2 T2
.
interpretation
- "degree-based equivalence on referred entries (closure)"
- 'StarEqSn h o G1 L1 T1 G2 L2 T2 = (fdeq h o G1 L1 T1 G2 L2 T2).
+ "sort-irrelevant equivalence on referred entries (closure)"
+ 'StarEqSn G1 L1 T1 G2 L2 T2 = (fdeq G1 L1 T1 G2 L2 T2).
(* Basic_properties *********************************************************)
-lemma fdeq_intro_dx (h) (o) (G): ∀L1,L2,T2. L1 ≛[h, o, T2] L2 →
- ∀T1. T1 ≛[h, o] T2 → ⦃G, L1, T1⦄ ≛[h, o] ⦃G, L2, T2⦄.
+lemma fdeq_intro_dx (G): ∀L1,L2,T2. L1 ≛[T2] L2 →
+ ∀T1. T1 ≛ T2 → ⦃G,L1,T1⦄ ≛ ⦃G,L2,T2⦄.
/3 width=3 by fdeq_intro_sn, tdeq_rdeq_div/ qed.
(* Basic inversion lemmas ***************************************************)
-lemma fdeq_inv_gen_sn: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & L1 ≛[h, o, T1] L2 & T1 ≛[h, o] T2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and3_intro/
+lemma fdeq_inv_gen_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ →
+ ∧∧ G1 = G2 & L1 ≛[T1] L2 & T1 ≛ T2.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and3_intro/
qed-.
-lemma fdeq_inv_gen_dx: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & L1 ≛[h, o, T2] L2 & T1 ≛[h, o] T2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+lemma fdeq_inv_gen_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ →
+ ∧∧ G1 = G2 & L1 ≛[T2] L2 & T1 ≛ T2.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
/3 width=3 by tdeq_rdeq_conf, and3_intro/
qed-.
include "static_2/static/rdeq_rdeq.ma".
include "static_2/static/fdeq.ma".
-(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+(* SORT-IRRELEVANT EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *************)
(* Advanced properties ******************************************************)
-lemma fdeq_sym: ∀h,o. tri_symmetric … (fdeq h o).
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1
+lemma fdeq_sym: tri_symmetric … fdeq.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1
/3 width=1 by fdeq_intro_dx, rdeq_sym, tdeq_sym/
qed-.
(* Main properties **********************************************************)
-theorem fdeq_trans: ∀h,o. tri_transitive … (fdeq h o).
-#h #o #G1 #G #L1 #L #T1 #T * -G -L -T
+theorem fdeq_trans: tri_transitive … fdeq.
+#G1 #G #L1 #L #T1 #T * -G -L -T
#L #T #HL1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
/4 width=5 by fdeq_intro_sn, rdeq_trans, tdeq_rdeq_div, tdeq_trans/
qed-.
-theorem fdeq_canc_sn: ∀h,o,G,G1,G2,L,L1,L2,T,T1,T2.
- â¦\83G, L, Tâ¦\84 â\89\9b[h, o] â¦\83G1, L1, T1â¦\84â\86\92 â¦\83G, L, Tâ¦\84 â\89\9b[h, o] â¦\83G2, L2, T2â¦\84 â\86\92 â¦\83G1, L1, T1â¦\84 â\89\9b[h, o] â¦\83G2, L2, T2⦄.
+theorem fdeq_canc_sn: ∀G,G1,L,L1,T,T1. ⦃G,L,T⦄ ≛ ⦃G1,L1,T1⦄→
+ â\88\80G2,L2,T2. â¦\83G,L,Tâ¦\84 â\89\9b â¦\83G2,L2,T2â¦\84 â\86\92 â¦\83G1,L1,T1â¦\84 â\89\9b â¦\83G2,L2,T2⦄.
/3 width=5 by fdeq_trans, fdeq_sym/ qed-.
-theorem fdeq_canc_dx: ∀h,o,G1,G2,G,L1,L2,L,T1,T2,T.
- â¦\83G1, L1, T1â¦\84 â\89\9b[h, o] â¦\83G, L, Tâ¦\84 â\86\92 â¦\83G2, L2, T2â¦\84 â\89\9b[h, o] â¦\83G, L, Tâ¦\84 â\86\92 â¦\83G1, L1, T1â¦\84 â\89\9b[h, o] â¦\83G2, L2, T2⦄.
+theorem fdeq_canc_dx: ∀G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ →
+ â\88\80G2,L2,T2. â¦\83G2,L2,T2â¦\84 â\89\9b â¦\83G,L,Tâ¦\84 â\86\92 â¦\83G1,L1,T1â¦\84 â\89\9b â¦\83G2,L2,T2⦄.
/3 width=5 by fdeq_trans, fdeq_sym/ qed-.
(* Main inversion lemmas with degree-based equivalence on terms *************)
-theorem fdeq_tdneq_repl_dx: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
- ∀U1,U2. ⦃G1, L1, U1⦄ ≛[h, o] ⦃G2, L2, U2⦄ →
- (T2 ≛[h, o] U2 → ⊥) → (T1 ≛[h, o] U1 → ⊥).
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #HT #U1 #U2 #HU #HnTU2 #HTU1
+theorem fdeq_tdneq_repl_dx: ∀G1,G2,L1,L2,T1,T2. ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄ →
+ ∀U1,U2. ⦃G1,L1,U1⦄ ≛ ⦃G2,L2,U2⦄ →
+ (T2 ≛ U2 → ⊥) → (T1 ≛ U1 → ⊥).
+#G1 #G2 #L1 #L2 #T1 #T2 #HT #U1 #U2 #HU #HnTU2 #HTU1
elim (fdeq_inv_gen_sn … HT) -HT #_ #_ #HT
elim (fdeq_inv_gen_sn … HU) -HU #_ #_ #HU
/3 width=5 by tdeq_repl/
include "static_2/static/rdeq_fqup.ma".
include "static_2/static/fdeq.ma".
-(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+(* SORT-IRRELEVANT EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *************)
-(* Properties with degree-based equivalence for terms ***********************)
+(* Properties with sort-irrelevant equivalence for terms ********************)
-lemma tdeq_fdeq: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
- ∀G,L. ⦃G, L, T1⦄ ≛[h, o] ⦃G, L, T2⦄.
+lemma tdeq_fdeq: ∀T1,T2. T1 ≛ T2 →
+ ∀G,L. ⦃G,L,T1⦄ ≛ ⦃G,L,T2⦄.
/2 width=1 by fdeq_intro_sn/ qed.
(* Advanced properties ******************************************************)
-lemma fdeq_refl: ∀h,o. tri_reflexive … (fdeq h o).
+lemma fdeq_refl: tri_reflexive … fdeq.
/2 width=1 by fdeq_intro_sn/ qed.
include "static_2/static/rdeq_fqus.ma".
include "static_2/static/fdeq.ma".
-(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+(* SORT-IRRELEVANT EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *************)
(* Properties with star-iterated structural successor for closures **********)
-lemma fdeq_fqus_trans: ∀h,o,b,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∃∃G,L0,T0. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L0, T0⦄ & ⦃G, L0, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #b #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H2
+lemma fdeq_fqus_trans: ∀b,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≛ ⦃G,L,T⦄ →
+ ∀G2,L2,T2. ⦃G,L,T⦄ ⬂*[b] ⦃G2,L2,T2⦄ →
+ ∃∃G,L0,T0. ⦃G1,L1,T1⦄ ⬂*[b] ⦃G,L0,T0⦄ & ⦃G,L0,T0⦄ ≛ ⦃G2,L2,T2⦄.
+#b #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H2
elim(fdeq_inv_gen_dx … H1) -H1 #HG #HL1 #HT1 destruct
elim (rdeq_fqus_trans … H2 … HL1) -L #L #T0 #H2 #HT02 #HL2
elim (tdeq_fqus_trans … H2 … HT1) -T #L0 #T #H2 #HT0 #HL0
include "static_2/static/rdeq_req.ma".
include "static_2/static/fdeq.ma".
-(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+(* SORT-IRRELEVANT EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *************)
(* Properties with syntactic equivalence on referred entries ****************)
-lemma req_rdeq_trans: ∀h,o,L1,L,T1. L1 ≡[T1] L →
- ∀G1,G2,L2,T2. ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #L1 #L #T1 #HL1 #G1 #G2 #L2 #T2 #H
+lemma req_rdeq_trans: ∀L1,L,T1. L1 ≡[T1] L →
+ ∀G1,G2,L2,T2. ⦃G1,L,T1⦄ ≛ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≛ ⦃G2,L2,T2⦄.
+#L1 #L #T1 #HL1 #G1 #G2 #L2 #T2 #H
elim (fdeq_inv_gen_sn … H) -H #H #HL2 #T12 destruct
/3 width=3 by fdeq_intro_sn, req_rdeq_trans/
qed-.
(**************************************************************************)
include "ground_2/relocation/rtmap_sor.ma".
-include "static_2/notation/relations/freestar_3.ma".
+include "static_2/notation/relations/freeplus_3.ma".
include "static_2/syntax/lenv.ma".
(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
interpretation
"context-sensitive free variables (term)"
- 'FreeStar L T f = (frees L T f).
+ 'FreePlus L T f = (frees L T f).
(* Basic inversion lemmas ***************************************************)
-fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = ⋆x → 𝐈⦃f⦄.
+fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀x. X = ⋆x → 𝐈⦃f⦄.
#L #X #f #H elim H -f -L -X //
[ #f #i #_ #x #H destruct
| #f #_ #L #V #_ #_ #x #H destruct
]
qed-.
-lemma frees_inv_sort: ∀f,L,s. L ⊢ 𝐅*⦃⋆s⦄ ≘ f → 𝐈⦃f⦄.
+lemma frees_inv_sort: ∀f,L,s. L ⊢ 𝐅+⦃⋆s⦄ ≘ f → 𝐈⦃f⦄.
/2 width=5 by frees_inv_sort_aux/ qed-.
-fact frees_inv_atom_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀i. L = ⋆ → X = #i →
- ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+fact frees_inv_atom_aux:
+ ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀i. L = ⋆ → X = #i →
+ ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
#f #L #X #H elim H -f -L -X
[ #f #L #s #_ #j #_ #H destruct
| #f #i #Hf #j #_ #H destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma frees_inv_atom: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+lemma frees_inv_atom: ∀f,i. ⋆ ⊢ 𝐅+⦃#i⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
/2 width=5 by frees_inv_atom_aux/ qed-.
-fact frees_inv_pair_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,V. L = K.ⓑ{I}V → X = #0 →
- ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
+fact frees_inv_pair_aux:
+ ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀I,K,V. L = K.ⓑ{I}V → X = #0 →
+ ∃∃g. K ⊢ 𝐅+⦃V⦄ ≘ g & f = ↑g.
#f #L #X * -f -L -X
[ #f #L #s #_ #Z #Y #X #_ #H destruct
| #f #i #_ #Z #Y #X #H destruct
]
qed-.
-lemma frees_inv_pair: ∀f,I,K,V. K.ⓑ{I}V ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
+lemma frees_inv_pair: ∀f,I,K,V. K.ⓑ{I}V ⊢ 𝐅+⦃#0⦄ ≘ f → ∃∃g. K ⊢ 𝐅+⦃V⦄ ≘ g & f = ↑g.
/2 width=6 by frees_inv_pair_aux/ qed-.
-fact frees_inv_unit_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K. L = K.ⓤ{I} → X = #0 →
- ∃∃g. 𝐈⦃g⦄ & f = ↑g.
+fact frees_inv_unit_aux:
+ ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀I,K. L = K.ⓤ{I} → X = #0 →
+ ∃∃g. 𝐈⦃g⦄ & f = ↑g.
#f #L #X * -f -L -X
[ #f #L #s #_ #Z #Y #_ #H destruct
| #f #i #_ #Z #Y #H destruct
]
qed-.
-lemma frees_inv_unit: ∀f,I,K. K.ⓤ{I} ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ↑g.
+lemma frees_inv_unit: ∀f,I,K. K.ⓤ{I} ⊢ 𝐅+⦃#0⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ↑g.
/2 width=7 by frees_inv_unit_aux/ qed-.
-fact frees_inv_lref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,j. L = K.ⓘ{I} → X = #(↑j) →
- ∃∃g. K ⊢ 𝐅*⦃#j⦄ ≘ g & f = ⫯g.
+fact frees_inv_lref_aux:
+ ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀I,K,j. L = K.ⓘ{I} → X = #(↑j) →
+ ∃∃g. K ⊢ 𝐅+⦃#j⦄ ≘ g & f = ⫯g.
#f #L #X * -f -L -X
[ #f #L #s #_ #Z #Y #j #_ #H destruct
| #f #i #_ #Z #Y #j #H destruct
]
qed-.
-lemma frees_inv_lref: ∀f,I,K,i. K.ⓘ{I} ⊢ 𝐅*⦃#(↑i)⦄ ≘ f →
- ∃∃g. K ⊢ 𝐅*⦃#i⦄ ≘ g & f = ⫯g.
+lemma frees_inv_lref:
+ ∀f,I,K,i. K.ⓘ{I} ⊢ 𝐅+⦃#(↑i)⦄ ≘ f →
+ ∃∃g. K ⊢ 𝐅+⦃#i⦄ ≘ g & f = ⫯g.
/2 width=6 by frees_inv_lref_aux/ qed-.
-fact frees_inv_gref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = §x → 𝐈⦃f⦄.
+fact frees_inv_gref_aux: ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀x. X = §x → 𝐈⦃f⦄.
#f #L #X #H elim H -f -L -X //
[ #f #i #_ #x #H destruct
| #f #_ #L #V #_ #_ #x #H destruct
]
qed-.
-lemma frees_inv_gref: ∀f,L,l. L ⊢ 𝐅*⦃§l⦄ ≘ f → 𝐈⦃f⦄.
+lemma frees_inv_gref: ∀f,L,l. L ⊢ 𝐅+⦃§l⦄ ≘ f → 𝐈⦃f⦄.
/2 width=5 by frees_inv_gref_aux/ qed-.
-fact frees_inv_bind_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀p,I,V,T. X = ⓑ{p,I}V.T →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+fact frees_inv_bind_aux:
+ ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀p,I,V,T. X = ⓑ{p,I}V.T →
+ ∃∃f1,f2. L ⊢ 𝐅+⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅+⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
#f #L #X * -f -L -X
[ #f #L #s #_ #q #J #W #U #H destruct
| #f #i #_ #q #J #W #U #H destruct
]
qed-.
-lemma frees_inv_bind: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+lemma frees_inv_bind:
+ ∀f,p,I,L,V,T. L ⊢ 𝐅+⦃ⓑ{p,I}V.T⦄ ≘ f →
+ ∃∃f1,f2. L ⊢ 𝐅+⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅+⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
/2 width=4 by frees_inv_bind_aux/ qed-.
-fact frees_inv_flat_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,V,T. X = ⓕ{I}V.T →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
+fact frees_inv_flat_aux: ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀I,V,T. X = ⓕ{I}V.T →
+ ∃∃f1,f2. L ⊢ 𝐅+⦃V⦄ ≘ f1 & L ⊢ 𝐅+⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
#f #L #X * -f -L -X
[ #f #L #s #_ #J #W #U #H destruct
| #f #i #_ #J #W #U #H destruct
]
qed-.
-lemma frees_inv_flat: ∀f,I,L,V,T. L ⊢ 𝐅*⦃ⓕ{I}V.T⦄ ≘ f →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
+lemma frees_inv_flat:
+ ∀f,I,L,V,T. L ⊢ 𝐅+⦃ⓕ{I}V.T⦄ ≘ f →
+ ∃∃f1,f2. L ⊢ 𝐅+⦃V⦄ ≘ f1 & L ⊢ 𝐅+⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
/2 width=4 by frees_inv_flat_aux/ qed-.
(* Basic properties ********************************************************)
-lemma frees_eq_repl_back: ∀L,T. eq_repl_back … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
+lemma frees_eq_repl_back: ∀L,T. eq_repl_back … (λf. L ⊢ 𝐅+⦃T⦄ ≘ f).
#L #T #f1 #H elim H -f1 -L -T
[ /3 width=3 by frees_sort, isid_eq_repl_back/
| #f1 #i #Hf1 #g2 #H
]
qed-.
-lemma frees_eq_repl_fwd: ∀L,T. eq_repl_fwd … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
+lemma frees_eq_repl_fwd: ∀L,T. eq_repl_fwd … (λf. L ⊢ 𝐅+⦃T⦄ ≘ f).
#L #T @eq_repl_sym /2 width=3 by frees_eq_repl_back/
qed-.
-lemma frees_lref_push: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ⋆ ⊢ 𝐅*⦃#↑i⦄ ≘ ⫯f.
+lemma frees_lref_push: ∀f,i. ⋆ ⊢ 𝐅+⦃#i⦄ ≘ f → ⋆ ⊢ 𝐅+⦃#↑i⦄ ≘ ⫯f.
#f #i #H
elim (frees_inv_atom … H) -H #g #Hg #H destruct
/2 width=1 by frees_atom/
(* Forward lemmas with test for finite colength *****************************)
-lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f → 𝐅⦃f⦄.
+lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅+⦃T⦄ ≘ f → 𝐅⦃f⦄.
#f #L #T #H elim H -f -L -T
/4 width=5 by sor_isfin, isfin_isid, isfin_tl, isfin_pushs, isfin_push, isfin_next/
qed-.
(* Properties with append for local environments ****************************)
-lemma frees_append_void: ∀f,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f → ⓧ.K ⊢ 𝐅*⦃T⦄ ≘ f.
+lemma frees_append_void: ∀f,K,T. K ⊢ 𝐅+⦃T⦄ ≘ f → ⓧ.K ⊢ 𝐅+⦃T⦄ ≘ f.
#f #K #T #H elim H -f -K -T
[ /2 width=1 by frees_sort/
| #f * /3 width=1 by frees_atom, frees_unit, frees_lref/
(* Inversion lemmas with append for local environments **********************)
-fact frees_inv_append_void_aux: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
- ∀K. L = ⓧ.K → K ⊢ 𝐅*⦃T⦄ ≘ f.
+fact frees_inv_append_void_aux:
+ ∀f,L,T. L ⊢ 𝐅+⦃T⦄ ≘ f →
+ ∀K. L = ⓧ.K → K ⊢ 𝐅+⦃T⦄ ≘ f.
#f #L #T #H elim H -f -L -T
[ /2 width=1 by frees_sort/
| #f #i #_ #K #H
]
qed-.
-lemma frees_inv_append_void: ∀f,K,T. ⓧ.K ⊢ 𝐅*⦃T⦄ ≘ f → K ⊢ 𝐅*⦃T⦄ ≘ f.
+lemma frees_inv_append_void: ∀f,K,T. ⓧ.K ⊢ 𝐅+⦃T⦄ ≘ f → K ⊢ 𝐅+⦃T⦄ ≘ f.
/2 width=3 by frees_inv_append_void_aux/ qed-.
(* Advanced properties ******************************************************)
-lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≘ ⋆ →
- ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i]↑f.
+lemma frees_atom_drops:
+ ∀b,L,i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ →
+ ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅+⦃#i⦄ ≘ ⫯*[i]↑f.
#b #L elim L -L /2 width=1 by frees_atom/
#L #I #IH *
[ #H lapply (drops_fwd_isid … H ?) -H // #H destruct
]
qed.
-lemma frees_pair_drops: ∀f,K,V. K ⊢ 𝐅*⦃V⦄ ≘ f →
- ∀i,I,L. ⬇*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
+lemma frees_pair_drops:
+ ∀f,K,V. K ⊢ 𝐅+⦃V⦄ ≘ f →
+ ∀i,I,L. ⬇*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅+⦃#i⦄ ≘ ⫯*[i] ↑f.
#f #K #V #Hf #i elim i -i
[ #I #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_pair/
| #i #IH #I #L #H elim (drops_inv_succ … H) -H /3 width=2 by frees_lref/
]
qed.
-lemma frees_unit_drops: ∀f. 𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≘ K.ⓤ{I} →
- L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
+lemma frees_unit_drops:
+ ∀f. 𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≘ K.ⓤ{I} →
+ L ⊢ 𝐅+⦃#i⦄ ≘ ⫯*[i] ↑f.
#f #Hf #I #K #i elim i -i
[ #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_unit/
| #i #IH #Y #H elim (drops_inv_succ … H) -H
]
qed.
(*
-lemma frees_sort_pushs: ∀f,K,s. K ⊢ 𝐅*⦃⋆s⦄ ≘ f →
- ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃⋆s⦄ ≘ ⫯*[i] f.
+lemma frees_sort_pushs:
+ ∀f,K,s. K ⊢ 𝐅+⦃⋆s⦄ ≘ f →
+ ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅+⦃⋆s⦄ ≘ ⫯*[i] f.
#f #K #s #Hf #i elim i -i
[ #L #H lapply (drops_fwd_isid … H ?) -H //
| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_sort/
]
qed.
*)
-lemma frees_lref_pushs: ∀f,K,j. K ⊢ 𝐅*⦃#j⦄ ≘ f →
- ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃#(i+j)⦄ ≘ ⫯*[i] f.
+lemma frees_lref_pushs:
+ ∀f,K,j. K ⊢ 𝐅+⦃#j⦄ ≘ f →
+ ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅+⦃#(i+j)⦄ ≘ ⫯*[i] f.
#f #K #j #Hf #i elim i -i
[ #L #H lapply (drops_fwd_isid … H ?) -H //
| #i #IH #L #H elim (drops_inv_succ … H) -H
]
qed.
(*
-lemma frees_gref_pushs: ∀f,K,l. K ⊢ 𝐅*⦃§l⦄ ≘ f →
- ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃§l⦄ ≘ ⫯*[i] f.
+lemma frees_gref_pushs:
+ ∀f,K,l. K ⊢ 𝐅+⦃§l⦄ ≘ f →
+ ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅+⦃§l⦄ ≘ ⫯*[i] f.
#f #K #l #Hf #i elim i -i
[ #L #H lapply (drops_fwd_isid … H ?) -H //
| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_gref/
*)
(* Advanced inversion lemmas ************************************************)
-lemma frees_inv_lref_drops: ∀L,i,f. L ⊢ 𝐅*⦃#i⦄ ≘ f →
- ∨∨ ∃∃g. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
- | ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≘ g &
- ⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
- | ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
+lemma frees_inv_lref_drops:
+ ∀L,i,f. L ⊢ 𝐅+⦃#i⦄ ≘ f →
+ ∨∨ ∃∃g. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
+ | ∃∃g,I,K,V. K ⊢ 𝐅+⦃V⦄ ≘ g & ⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
+ | ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
#L elim L -L
[ #i #g | #L #I #IH * [ #g cases I -I [ #I | #I #V ] -IH | #i #g ] ] #H
[ elim (frees_inv_atom … H) -H #f #Hf #H destruct
(* Properties with generic slicing for local environments *******************)
-lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 →
- ∀f,L. ⬇*[b, f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
- ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅*⦃U⦄ ≘ f2.
+lemma frees_lifts:
+ ∀b,f1,K,T. K ⊢ 𝐅+⦃T⦄ ≘ f1 →
+ ∀f,L. ⬇*[b,f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
+ ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅+⦃U⦄ ≘ f2.
#b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
[ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
]
qed-.
-lemma frees_lifts_SO: ∀b,L,K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
- ∀f. K ⊢ 𝐅*⦃T⦄ ≘ f → L ⊢ 𝐅*⦃U⦄ ≘ ⫯f.
+lemma frees_lifts_SO:
+ ∀b,L,K. ⬇*[b,𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
+ ∀f. K ⊢ 𝐅+⦃T⦄ ≘ f → L ⊢ 𝐅+⦃U⦄ ≘ ⫯f.
#b #L #K #HLK #T #U #HTU #f #Hf
@(frees_lifts b … Hf … HTU) // (**) (* auto fails *)
qed.
(* Forward lemmas with generic slicing for local environments ***************)
-lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
- ∀f1. K ⊢ 𝐅*⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
+lemma frees_fwd_coafter:
+ ∀b,f2,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f2 →
+ ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∀f1. K ⊢ 𝐅+⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
/4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
(* Inversion lemmas with generic slicing for local environments *************)
-lemma frees_inv_lifts_ex: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
- ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅*⦃T⦄ ≘ f1.
+lemma frees_inv_lifts_ex:
+ ∀b,f2,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f2 →
+ ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅+⦃T⦄ ≘ f1.
#b #f2 #L #U #Hf2 #f #K #HLK #T elim (frees_total K T)
/3 width=9 by frees_fwd_coafter, ex2_intro/
qed-.
-lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f →
- ∀K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U →
- K ⊢ 𝐅*⦃T⦄ ≘ ⫱f.
+lemma frees_inv_lifts_SO:
+ ∀b,f,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f →
+ ∀K. ⬇*[b,𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U →
+ K ⊢ 𝐅+⦃T⦄ ≘ ⫱f.
#b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
#f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf
/3 width=5 by frees_eq_repl_back, coafter_isid_inv_sn/
qed-.
-lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
- ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅*⦃T⦄ ≘ f1.
+lemma frees_inv_lifts:
+ ∀b,f2,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f2 →
+ ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅+⦃T⦄ ≘ f1.
#b #f2 #L #U #H #f #K #HLK #T #HTU #f1 #Hf2 elim (frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
/3 width=7 by frees_eq_repl_back, coafter_inj/
qed-.
(* Note: this is used by rex_conf and might be modified *)
-lemma frees_inv_drops_next: ∀f1,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 →
- ∀I2,L2,V2,n. ⬇*[n] L1 ≘ L2.ⓑ{I2}V2 →
- ∀g1. ↑g1 = ⫱*[n] f1 →
- ∃∃g2. L2 ⊢ 𝐅*⦃V2⦄ ≘ g2 & g2 ⊆ g1.
+lemma frees_inv_drops_next:
+ ∀f1,L1,T1. L1 ⊢ 𝐅+⦃T1⦄ ≘ f1 →
+ ∀I2,L2,V2,n. ⬇*[n] L1 ≘ L2.ⓑ{I2}V2 →
+ ∀g1. ↑g1 = ⫱*[n] f1 →
+ ∃∃g2. L2 ⊢ 𝐅+⦃V2⦄ ≘ g2 & g2 ⊆ g1.
#f1 #L1 #T1 #H elim H -f1 -L1 -T1
[ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -s
lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
(* Advanced properties ******************************************************)
(* Note: this replaces lemma 1400 concluding the "big tree" theorem *)
-lemma frees_total: ∀L,T. ∃f. L ⊢ 𝐅*⦃T⦄ ≘ f.
+lemma frees_total: ∀L,T. ∃f. L ⊢ 𝐅+⦃T⦄ ≘ f.
#L #T @(fqup_wf_ind_eq (Ⓣ) … (⋆) L T) -L -T
#G0 #L0 #T0 #IH #G #L * *
[ /3 width=2 by frees_sort, ex_intro/
(* Advanced main properties *************************************************)
-theorem frees_bind_void: ∀f1,L,V. L ⊢ 𝐅*⦃V⦄ ≘ f1 → ∀f2,T. L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 →
- ∀f. f1 ⋓ ⫱f2 ≘ f → ∀p,I. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f.
+theorem frees_bind_void:
+ ∀f1,L,V. L ⊢ 𝐅+⦃V⦄ ≘ f1 → ∀f2,T. L.ⓧ ⊢ 𝐅+⦃T⦄ ≘ f2 →
+ ∀f. f1 ⋓ ⫱f2 ≘ f → ∀p,I. L ⊢ 𝐅+⦃ⓑ{p,I}V.T⦄ ≘ f.
#f1 #L #V #Hf1 #f2 #T #Hf2 #f #Hf #p #I
elim (frees_total (L.ⓑ{I}V) T) #f0 #Hf0
lapply (lsubr_lsubf … Hf2 … Hf0) -Hf2 /2 width=5 by lsubr_unit/ #H02
(* Advanced inversion lemmas ************************************************)
-lemma frees_inv_bind_void: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+lemma frees_inv_bind_void:
+ ∀f,p,I,L,V,T. L ⊢ 𝐅+⦃ⓑ{p,I}V.T⦄ ≘ f →
+ ∃∃f1,f2. L ⊢ 𝐅+⦃V⦄ ≘ f1 & L.ⓧ ⊢ 𝐅+⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
#f #p #I #L #V #T #H
elim (frees_inv_bind … H) -H #f1 #f2 #Hf1 #Hf2 #Hf
elim (frees_total (L.ⓧ) T) #f0 #Hf0
]
qed-.
-lemma frees_ind_void: ∀Q:relation3 ….
- (
- ∀f,L,s. 𝐈⦃f⦄ → Q L (⋆s) f
- ) → (
- ∀f,i. 𝐈⦃f⦄ → Q (⋆) (#i) (⫯*[i]↑f)
- ) → (
- ∀f,I,L,V.
- L ⊢ 𝐅*⦃V⦄ ≘ f → Q L V f→ Q (L.ⓑ{I}V) (#O) (↑f)
- ) → (
- ∀f,I,L. 𝐈⦃f⦄ → Q (L.ⓤ{I}) (#O) (↑f)
- ) → (
- ∀f,I,L,i.
- L ⊢ 𝐅*⦃#i⦄ ≘ f → Q L (#i) f → Q (L.ⓘ{I}) (#(↑i)) (⫯f)
- ) → (
- ∀f,L,l. 𝐈⦃f⦄ → Q L (§l) f
- ) → (
- ∀f1,f2,f,p,I,L,V,T.
- L ⊢ 𝐅*⦃V⦄ ≘ f1 → L.ⓧ ⊢𝐅*⦃T⦄≘ f2 → f1 ⋓ ⫱f2 ≘ f →
- Q L V f1 → Q (L.ⓧ) T f2 → Q L (ⓑ{p,I}V.T) f
- ) → (
- ∀f1,f2,f,I,L,V,T.
- L ⊢ 𝐅*⦃V⦄ ≘ f1 → L ⊢𝐅*⦃T⦄ ≘ f2 → f1 ⋓ f2 ≘ f →
- Q L V f1 → Q L T f2 → Q L (ⓕ{I}V.T) f
- ) →
- ∀L,T,f. L ⊢ 𝐅*⦃T⦄ ≘ f → Q L T f.
+lemma frees_ind_void (Q:relation3 …):
+ (
+ ∀f,L,s. 𝐈⦃f⦄ → Q L (⋆s) f
+ ) → (
+ ∀f,i. 𝐈⦃f⦄ → Q (⋆) (#i) (⫯*[i]↑f)
+ ) → (
+ ∀f,I,L,V.
+ L ⊢ 𝐅+⦃V⦄ ≘ f → Q L V f→ Q (L.ⓑ{I}V) (#O) (↑f)
+ ) → (
+ ∀f,I,L. 𝐈⦃f⦄ → Q (L.ⓤ{I}) (#O) (↑f)
+ ) → (
+ ∀f,I,L,i.
+ L ⊢ 𝐅+⦃#i⦄ ≘ f → Q L (#i) f → Q (L.ⓘ{I}) (#(↑i)) (⫯f)
+ ) → (
+ ∀f,L,l. 𝐈⦃f⦄ → Q L (§l) f
+ ) → (
+ ∀f1,f2,f,p,I,L,V,T.
+ L ⊢ 𝐅+⦃V⦄ ≘ f1 → L.ⓧ ⊢𝐅+⦃T⦄≘ f2 → f1 ⋓ ⫱f2 ≘ f →
+ Q L V f1 → Q (L.ⓧ) T f2 → Q L (ⓑ{p,I}V.T) f
+ ) → (
+ ∀f1,f2,f,I,L,V,T.
+ L ⊢ 𝐅+⦃V⦄ ≘ f1 → L ⊢𝐅+⦃T⦄ ≘ f2 → f1 ⋓ f2 ≘ f →
+ Q L V f1 → Q L T f2 → Q L (ⓕ{I}V.T) f
+ ) →
+ ∀L,T,f. L ⊢ 𝐅+⦃T⦄ ≘ f → Q L T f.
#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #L #T
@(fqup_wf_ind_eq (Ⓕ) … (⋆) L T) -L -T #G0 #L0 #T0 #IH #G #L * *
[ #s #HG #HL #HT #f #H destruct -IH
(* Main inversion lemmas ****************************************************)
-theorem frees_mono: ∀f1,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f1 → ∀f2. L ⊢ 𝐅*⦃T⦄ ≘ f2 → f1 ≡ f2.
+theorem frees_mono: ∀f1,L,T. L ⊢ 𝐅+⦃T⦄ ≘ f1 → ∀f2. L ⊢ 𝐅+⦃T⦄ ≘ f2 → f1 ≡ f2.
#f1 #L #T #H elim H -f1 -L -T
[ /3 width=3 by frees_inv_sort, isid_inv_eq_repl/
| #f1 #i #Hf1 #g2 #H
(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
definition fsle: bi_relation lenv term ≝ λL1,T1,L2,T2.
- ∃∃n1,n2,f1,f2. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 &
- L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
+ ∃∃n1,n2,f1,f2. L1 ⊢ 𝐅+⦃T1⦄ ≘ f1 & L2 ⊢ 𝐅+⦃T2⦄ ≘ f2 &
+ L1 ≋ⓧ*[n1,n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
interpretation "free variables inclusion (restricted closure)"
'SubSetEq L1 T1 L2 T2 = (fsle L1 T1 L2 T2).
(* Basic properties *********************************************************)
-lemma fsle_sort: ∀L,s1,s2. ⦃L, ⋆s1⦄ ⊆ ⦃L, ⋆s2⦄.
+lemma fsle_sort: ∀L,s1,s2. ⦃L,⋆s1⦄ ⊆ ⦃L,⋆s2⦄.
/3 width=8 by frees_sort, sle_refl, ex4_4_intro/ qed.
-lemma fsle_gref: ∀L,l1,l2. ⦃L, §l1⦄ ⊆ ⦃L, §l2⦄.
+lemma fsle_gref: ∀L,l1,l2. ⦃L,§l1⦄ ⊆ ⦃L,§l2⦄.
/3 width=8 by frees_gref, sle_refl, ex4_4_intro/ qed.
(* Advanced properties ******************************************************)
lemma fsle_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≘ U1 → ∀L1,L2. |L2| ≤ |L1| →
- ∀T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ⦃L1.ⓧ, U1⦄ ⊆ ⦃L2, T2⦄.
+ ∀T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ → ⦃L1.ⓧ,U1⦄ ⊆ ⦃L2,T2⦄.
#T1 #U1 #HTU1 #L1 #L2 #H1L #T2
* #n #m #f #g #Hf #Hg #H2L #Hfg
lapply (lveq_length_fwd_dx … H2L ?) // -H1L #H destruct
@(ex4_4_intro … Hf Hg) /2 width=4 by lveq_void_dx/ (**) (* explict constructor *)
qed-.
-lemma fsle_lifts_SO_sn: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1, V1⦄ ⊆ ⦃K2, V2⦄ →
- ∀W1. ⬆*[1] V1 ≘ W1 → ∀I1,I2. ⦃K1.ⓘ{I1}, W1⦄ ⊆ ⦃K2.ⓑ{I2}V2, #O⦄.
+lemma fsle_lifts_SO_sn: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1,V1⦄ ⊆ ⦃K2,V2⦄ →
+ ∀W1. ⬆*[1] V1 ≘ W1 → ∀I1,I2. ⦃K1.ⓘ{I1},W1⦄ ⊆ ⦃K2.ⓑ{I2}V2,#O⦄.
#K1 #K2 #HK #V1 #V2
* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
#W1 #HVW1 #I1 #I2
/5 width=12 by frees_lifts_SO, frees_pair, drops_refl, drops_drop, lveq_bind, sle_weak, ex4_4_intro/
qed.
-lemma fsle_lifts_SO: ∀K1,K2. |K1| = |K2| → ∀T1,T2. ⦃K1, T1⦄ ⊆ ⦃K2, T2⦄ →
+lemma fsle_lifts_SO: ∀K1,K2. |K1| = |K2| → ∀T1,T2. ⦃K1,T1⦄ ⊆ ⦃K2,T2⦄ →
∀U1,U2. ⬆*[1] T1 ≘ U1 → ⬆*[1] T2 ≘ U2 →
- ∀I1,I2. ⦃K1.ⓘ{I1}, U1⦄ ⊆ ⦃K2.ⓘ{I2}, U2⦄.
+ ∀I1,I2. ⦃K1.ⓘ{I1},U1⦄ ⊆ ⦃K2.ⓘ{I2},U2⦄.
#K1 #K2 #HK #T1 #T2
* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
#U1 #U2 #HTU1 #HTU2 #I1 #I2
(* Advanced inversion lemmas ************************************************)
lemma fsle_inv_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≘ U1 →
- ∀I1,I2,L1,L2,V1,V2,U2. ⦃L1.ⓑ{I1}V1,U1⦄ ⊆ ⦃L2.ⓑ{I2}V2, U2⦄ →
- ∀p. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.U2⦄.
+ ∀I1,I2,L1,L2,V1,V2,U2. ⦃L1.ⓑ{I1}V1,U1⦄ ⊆ ⦃L2.ⓑ{I2}V2,U2⦄ →
+ ∀p. ⦃L1,T1⦄ ⊆ ⦃L2,ⓑ{p,I2}V2.U2⦄.
#T1 #U1 #HTU1 #I1 #I2 #L1 #L2 #V1 #V2 #U2
* #n #m #f2 #g2 #Hf2 #Hg2 #HL #Hfg2 #p
elim (lveq_inv_pair_pair … HL) -HL #HL #H1 #H2 destruct
qed.
lemma fsle_shift: ∀L1,L2. |L1| = |L2| →
- ∀I,T1,T2,V. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I}V, T2⦄ →
- ∀p. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V.T2⦄.
+ ∀I,T1,T2,V. ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2.ⓑ{I}V,T2⦄ →
+ ∀p. ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2,ⓑ{p,I}V.T2⦄.
#L1 #L2 #H1L #I #T1 #T2 #V
* #n #m #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
elim (lveq_inj_length … H2L) // -H1L #H1 #H2 destruct
/4 width=10 by frees_bind, lveq_void_sn, sle_tl, sle_trans, ex4_4_intro/
qed.
-lemma fsle_bind_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀p,I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
+lemma fsle_bind_dx_sn: ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀p,I,T2. ⦃L1,V1⦄ ⊆ ⦃L2,ⓑ{p,I}V2.T2⦄.
#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #p #I #T2
elim (frees_total (L2.ⓧ) T2) #g2 #Hg2
elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
/4 width=5 by frees_bind_void, sor_inv_sle_sn, sor_tls, sle_trans/
qed.
-lemma fsle_bind_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2.ⓧ, T2⦄ → |L1| ≤ |L2| →
- ∀p,I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
+lemma fsle_bind_dx_dx: ∀L1,L2,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2.ⓧ,T2⦄ → |L1| ≤ |L2| →
+ ∀p,I,V2. ⦃L1,T1⦄ ⊆ ⦃L2,ⓑ{p,I}V2.T2⦄.
#L1 #L2 #T1 #T2 * #n1 #x1 #f2 #g2 #Hf2 #Hg2 #H #Hfg2 #HL12 #p #I #V2
elim (lveq_inv_void_dx_length … H HL12) -H -HL12 #m1 #HL12 #H1 #H2 destruct
<tls_xn in Hfg2; #Hfg2
/4 width=5 by frees_bind_void, sor_inv_sle_dx, sor_tls, sle_trans/
qed.
-lemma fsle_flat_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
+lemma fsle_flat_dx_sn: ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀I,T2. ⦃L1,V1⦄ ⊆ ⦃L2,ⓕ{I}V2.T2⦄.
#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #I #T2
elim (frees_total L2 T2) #g2 #Hg2
elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
/4 width=5 by frees_flat, sor_inv_sle_sn, sor_tls, sle_trans/
qed.
-lemma fsle_flat_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
+lemma fsle_flat_dx_dx: ∀L1,L2,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
+ ∀I,V2. ⦃L1,T1⦄ ⊆ ⦃L2,ⓕ{I}V2.T2⦄.
#L1 #L2 #T1 #T2 * #n1 #m1 #f2 #g2 #Hf2 #Hg2 #HL12 #Hfg2 #I #V2
elim (frees_total L2 V2) #g1 #Hg1
elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
(* Advanced forward lemmas ***************************************************)
-lemma fsle_fwd_pair_sn: ∀I1,I2,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄.
+lemma fsle_fwd_pair_sn: ∀I1,I2,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I1}V1,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄ →
+ ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄.
#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 *
#n1 #n2 #f1 #f2 #Hf1 #Hf2 #HL12 #Hf12
elim (lveq_inv_pair_pair … HL12) -HL12 #HL12 #H1 #H2 destruct
(* Advanced inversion lemmas ************************************************)
-lemma fsle_frees_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
- ∃∃n1,n2,f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 &
- L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
+lemma fsle_frees_trans:
+ ∀L1,L2,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
+ ∀f2. L2 ⊢ 𝐅+⦃T2⦄ ≘ f2 →
+ ∃∃n1,n2,f1. L1 ⊢ 𝐅+⦃T1⦄ ≘ f1 & L1 ≋ⓧ*[n1,n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
#L1 #L2 #T1 #T2 * #n1 #n2 #f1 #g2 #Hf1 #Hg2 #HL #Hn #f2 #Hf2
lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hgf2
lapply (tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
/2 width=6 by ex3_3_intro/
qed-.
-lemma fsle_frees_trans_eq: ∀L1,L2. |L1| = |L2| →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
- ∃∃f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 & f1 ⊆ f2.
+lemma fsle_frees_trans_eq:
+ ∀L1,L2. |L1| = |L2| →
+ ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ → ∀f2. L2 ⊢ 𝐅+⦃T2⦄ ≘ f2 →
+ ∃∃f1. L1 ⊢ 𝐅+⦃T1⦄ ≘ f1 & f1 ⊆ f2.
#L1 #L2 #H1L #T1 #T2 #H2L #f2 #Hf2
elim (fsle_frees_trans … H2L … Hf2) -T2 #n1 #n2 #f1 #Hf1 #H2L #Hf12
elim (lveq_inj_length … H2L) // -L2 #H1 #H2 destruct
/2 width=3 by ex2_intro/
qed-.
-lemma fsle_inv_frees_eq: ∀L1,L2. |L1| = |L2| →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
- f1 ⊆ f2.
+lemma fsle_inv_frees_eq:
+ ∀L1,L2. |L1| = |L2| →
+ ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
+ ∀f1. L1 ⊢ 𝐅+⦃T1⦄ ≘ f1 → ∀f2. L2 ⊢ 𝐅+⦃T2⦄ ≘ f2 →
+ f1 ⊆ f2.
#L1 #L2 #H1L #T1 #T2 #H2L #f1 #Hf1 #f2 #Hf2
elim (fsle_frees_trans_eq … H2L … Hf2) // -L2 -T2
/3 width=6 by frees_mono, sle_eq_repl_back1/
(* Main properties **********************************************************)
-theorem fsle_trans_sn: ∀L1,L2,T1,T. ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀T2. ⦃L2, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+theorem fsle_trans_sn:
+ ∀L1,L2,T1,T. ⦃L1,T1⦄ ⊆ ⦃L2,T⦄ →
+ ∀T2. ⦃L2,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
#L1 #L2 #T1 #T
* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
#T2
/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
qed-.
-theorem fsle_trans_dx: ∀L1,T1,T. ⦃L1, T1⦄ ⊆ ⦃L1, T⦄ →
- ∀L2,T2. ⦃L1, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+theorem fsle_trans_dx:
+ ∀L1,T1,T. ⦃L1,T1⦄ ⊆ ⦃L1,T⦄ →
+ ∀L2,T2. ⦃L1,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
#L1 #T1 #T
* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
#L2 #T2
/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
qed-.
-theorem fsle_trans_rc: ∀L1,L,T1,T. |L1| = |L| → ⦃L1, T1⦄ ⊆ ⦃L, T⦄ →
- ∀L2,T2. |L| = |L2| → ⦃L, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+theorem fsle_trans_rc:
+ ∀L1,L,T1,T. |L1| = |L| → ⦃L1,T1⦄ ⊆ ⦃L,T⦄ →
+ ∀L2,T2. |L| = |L2| → ⦃L,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
#L1 #L #T1 #T #HL1
* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
#L2 #T2 #HL2
/3 width=10 by lveq_length_eq, sle_trans, ex4_4_intro/
qed-.
-theorem fsle_bind_sn_ge: ∀L1,L2. |L2| ≤ |L1| →
- ∀V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀p,I. ⦃L1, ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2, T⦄.
+theorem fsle_bind_sn_ge:
+ ∀L1,L2. |L2| ≤ |L1| →
+ ∀V1,T1,T. ⦃L1,V1⦄ ⊆ ⦃L2,T⦄ → ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2,T⦄ →
+ ∀p,I. ⦃L1,ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2,T⦄.
#L1 #L2 #HL #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #p #I
elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
elim (lveq_inj_void_sn_ge … H1n1 … H1n2) -H1n2 // #H1 #H2 #H3 destruct
/4 width=12 by frees_bind_void, sor_inv_sle, sor_tls, ex4_4_intro/
qed.
-theorem fsle_flat_sn: ∀L1,L2,V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⊆ ⦃L2, T⦄.
+theorem fsle_flat_sn:
+ ∀L1,L2,V1,T1,T. ⦃L1,V1⦄ ⊆ ⦃L2,T⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T⦄ →
+ ∀I. ⦃L1,ⓕ{I}V1.T1⦄ ⊆ ⦃L2,T⦄.
#L1 #L2 #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #I
elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
elim (lveq_inj … H1n1 … H1n2) -H1n2 #H1 #H2 destruct
/4 width=12 by frees_flat, sor_inv_sle, sor_tls, ex4_4_intro/
qed.
-theorem fsle_bind_eq: ∀L1,L2. |L1| = |L2| → ∀V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀I2,T1,T2. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ∀p,I1. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
+theorem fsle_bind_eq:
+ ∀L1,L2. |L1| = |L2| → ∀V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀I2,T1,T2. ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄ →
+ ∀p,I1. ⦃L1,ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2,ⓑ{p,I2}V2.T2⦄.
#L1 #L2 #HL #V1 #V2
* #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I2 #T1 #T2
* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p #I1
/4 width=15 by frees_bind_void, frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
qed.
-theorem fsle_bind: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ∀p. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
+theorem fsle_bind:
+ ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄ →
+ ∀p. ⦃L1,ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2,ⓑ{p,I2}V2.T2⦄.
#L1 #L2 #V1 #V2
* #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I1 #I2 #T1 #T2
* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
/4 width=15 by frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
qed.
-theorem fsle_flat: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀I1,I2. ⦃L1, ⓕ{I1}V1.T1⦄ ⊆ ⦃L2, ⓕ{I2}V2.T2⦄.
+theorem fsle_flat:
+ ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
+ ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
+ ∀I1,I2. ⦃L1,ⓕ{I1}V1.T1⦄ ⊆ ⦃L2,ⓕ{I2}V2.T2⦄.
/3 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn/ qed-.
(* Properties with length for local environments ****************************)
-lemma fsle_sort_bi: ∀L1,L2,s1,s2. |L1| = |L2| → ⦃L1, ⋆s1⦄ ⊆ ⦃L2, ⋆s2⦄.
+lemma fsle_sort_bi: ∀L1,L2,s1,s2. |L1| = |L2| → ⦃L1,⋆s1⦄ ⊆ ⦃L2,⋆s2⦄.
/3 width=8 by lveq_length_eq, frees_sort, sle_refl, ex4_4_intro/ qed.
-lemma fsle_gref_bi: ∀L1,L2,l1,l2. |L1| = |L2| → ⦃L1, §l1⦄ ⊆ ⦃L2, §l2⦄.
+lemma fsle_gref_bi: ∀L1,L2,l1,l2. |L1| = |L2| → ⦃L1,§l1⦄ ⊆ ⦃L2,§l2⦄.
/3 width=8 by lveq_length_eq, frees_gref, sle_refl, ex4_4_intro/ qed.
-lemma fsle_pair_bi: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1, V1⦄ ⊆ ⦃K2, V2⦄ →
- ∀I1,I2. ⦃K1.ⓑ{I1}V1, #O⦄ ⊆ ⦃K2.ⓑ{I2}V2, #O⦄.
+lemma fsle_pair_bi: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1,V1⦄ ⊆ ⦃K2,V2⦄ →
+ ∀I1,I2. ⦃K1.ⓑ{I1}V1,#O⦄ ⊆ ⦃K2.ⓑ{I2}V2,#O⦄.
#K1 #K2 #HK #V1 #V2
* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
#I1 #I2
qed.
lemma fsle_unit_bi: ∀K1,K2. |K1| = |K2| →
- ∀I1,I2. ⦃K1.ⓤ{I1}, #O⦄ ⊆ ⦃K2.ⓤ{I2}, #O⦄.
+ ∀I1,I2. ⦃K1.ⓤ{I1},#O⦄ ⊆ ⦃K2.ⓤ{I2},#O⦄.
/3 width=8 by frees_unit, lveq_length_eq, sle_refl, ex4_4_intro/
qed.
(* GENERIC COMPUTATION PROPERTIES *******************************************)
-definition nf ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- λG,L,T. NF … (RR G L) RS T.
+definition nf (RR:relation4 genv lenv term term) (RS:relation term) ≝
+ λG,L,T. NF … (RR G L) RS T.
definition candidate: Type[0] ≝ relation3 genv lenv term.
-definition CP0 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- ∀G. d_liftable1 (nf RR RS G).
+definition CP0 (RR) (RS) ≝ ∀G. d_liftable1 (nf RR RS G).
-definition CP1 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- ∀G,L. ∃s. NF … (RR G L) RS (⋆s).
+definition CP1 (RR) (RS) ≝ ∀G,L,s. nf RR RS G L (⋆s).
-definition CP2 ≝ λRP:candidate. ∀G. d_liftable1 (RP G).
+definition CP2 (RP:candidate) ≝ ∀G. d_liftable1 (RP G).
-definition CP3 ≝ λRP:candidate.
- ∀G,L,T,s. RP G L (ⓐ⋆s.T) → RP G L T.
+definition CP3 (RP:candidate) ≝ ∀G,L,T,s. RP G L (ⓐ⋆s.T) → RP G L T.
(* requirements for generic computation properties *)
(* Basic_1: includes: nf2_lift1 *)
(* Basic_1: was: sc3_arity_csubc *)
theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀b,f,L0. ⬇*[b, f] L0 ≘ L1 →
+ ∀G,L1,T,A. ⦃G,L1⦄ ⊢ T ⁝ A → ∀b,f,L0. ⬇*[b,f] L0 ≘ L1 →
∀T0. ⬆*[f] T ≘ T0 → ∀L2. G ⊢ L2 ⫃[RP] L0 →
- ⦃G, L2, T0⦄ ϵ[RP] 〚A〛.
+ ⦃G,L2,T0⦄ ϵ[RP] 〚A〛.
#RR #RS #RP #H1RP #H2RP #G #L1 #T @(fqup_wf_ind_eq (Ⓣ) … G L1 T) -G -L1 -T
#Z #Y #X #IH #G #L1 * [ * | * [ #p ] * ]
[ #s #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct -IH
lapply (aaa_inv_sort … HA) -HA #H destruct
>(lifts_inv_sort1 … H0) -H0
lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
- lapply (s4 … HAtom G L2 (Ⓔ)) /2 width=1 by/
+ lapply (s2 … HAtom G L2 (Ⓔ)) /3 width=7 by cp1, simple_atom/
| #i #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
elim (aaa_inv_lref_drops … HA) -HA #I #K1 #V1 #HLK1 #HKV1
elim (lifts_inv_lref1 … H0) -H0 #j #Hf #H destruct
(* Basic_1: was: sc3_arity *)
lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
+ ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → ⦃G,L,T⦄ ϵ[RP] 〚A〛.
/3 width=9 by drops_refl, lifts_refl, acr_aaa_csubc_lifts/ qed.
lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
+ ∀G,L,T,A. ⦃G,L⦄ ⊢ T ⁝ A → RP G L T.
#RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
lapply (acr_gcr … H1RP H2RP A) #HA
@(s1 … HA) /2 width=4 by acr_aaa/
(* Note: this is Tait's iii, or Girard's CR4 *)
definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
∀G,L,Vs. all … (RP G L) Vs →
- ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
+ ∀T. 𝐒⦃T⦄ → nf RR RS G L T → C G L (ⒶVs.T).
(* Note: this generalizes Tait's ii *)
definition S3 ≝ λC:candidate.
∀a,G,L,Vs,V,T,W.
C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
-definition S4 ≝ λRP,C:candidate.
- ∀G,L,Vs. all … (RP G L) Vs → ∀s. C G L (ⒶVs.⋆s).
-
definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
C G L (ⒶVs.V2) → ⬆*[↑i] V1 ≘ V2 →
⬇*[i] L ≘ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
{ s1: S1 RP C;
s2: S2 RR RS RP C;
s3: S3 C;
- s4: S4 RP C;
s5: S5 C;
s6: S6 RP C;
s7: S7 C
(* the functional construction for candidates *)
definition cfun: candidate → candidate → candidate ≝
λC1,C2,G,K,T. ∀f,L,W,U.
- ⬇*[Ⓕ, f] L ≘ K → ⬆*[f] T ≘ U → C1 G L W → C2 G L (ⓐW.U).
+ ⬇*[Ⓕ,f] L ≘ K → ⬆*[f] T ≘ U → C1 G L W → C2 G L (ⓐW.U).
(* the reducibility candidate associated to an atomic arity *)
rec definition acr (RP:candidate) (A:aarity) on A: candidate ≝
(* Basic_1: was:
sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast
*)
+(* Note: one sort must exist *)
lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
∀A. gcr RR RS RP (acr RP A).
#RR #RS #RP #H1RP #H2RP #A elim A -A //
#B #A #IHB #IHA @mk_gcr
[ #G #L #T #H
- elim (cp1 … H1RP G L) #s #HK
+ letin s ≝ 0 (* one sort must exist *)
+ lapply (cp1 … H1RP G L s) #HK
lapply (s2 … IHB G L (Ⓔ) … HK) // #HB
lapply (H (𝐈𝐝) L (⋆s) T ? ? ?) -H
/3 width=6 by s1, cp3, drops_refl, lifts_refl/
elim (lifts_inv_flat1 … H0) -H0 #U0 #X #HU0 #HX #H destruct
elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
@(s3 … IHA … (V0⨮V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
-| #G #L #Vs #HVs #s #f #L0 #V0 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
- >(lifts_inv_sort1 … H0) -X0
- lapply (s1 … IHB … HB) #HV0
- @(s4 … IHA … (V0⨮V0s)) /3 width=7 by gcp2_all, conj/
| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #f #L0 #V0 #X #HL0 #H #HB
elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
elim (lifts_inv_lref1 … H0) -H0 #j #Hf #H destruct
qed.
lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀p,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
- ∀b,f,L0,V0,W0,T0. ⬇*[b, f] L0 ≘ L → ⬆*[f] W ≘ W0 → ⬆*[⫯f] T ≘ T0 →
- ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
+ ∀p,G,L,W,T,A,B. ⦃G,L,W⦄ ϵ[RP] 〚B〛 → (
+ ∀b,f,L0,V0,W0,T0. ⬇*[b,f] L0 ≘ L → ⬆*[f] W ≘ W0 → ⬆*[⫯f] T ≘ T0 →
+ ⦃G,L0,V0⦄ ϵ[RP] 〚B〛 → ⦃G,L0,W0⦄ ϵ[RP] 〚B〛 → ⦃G,L0.ⓓⓝW0.V0,T0⦄ ϵ[RP] 〚A〛
) →
- ⦃G, L, ⓛ{p}W.T⦄ ϵ[RP] 〚②B.A〛.
+ ⦃G,L,ⓛ{p}W.T⦄ ϵ[RP] 〚②B.A〛.
#RR #RS #RP #H1RP #H2RP #p #G #L #W #T #A #B #HW #HA #f #L0 #V0 #X #HL0 #H #HB
lapply (acr_gcr … H1RP H2RP A) #HCA
lapply (acr_gcr … H1RP H2RP B) #HCB
inductive lsuba (G:genv): relation lenv ≝
| lsuba_atom: lsuba G (⋆) (⋆)
| lsuba_bind: ∀I,L1,L2. lsuba G L1 L2 → lsuba G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsuba_beta: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
+| lsuba_beta: ∀L1,L2,W,V,A. ⦃G,L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G,L2⦄ ⊢ W ⁝ A →
lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
.
fact lsuba_inv_bind1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
(∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K2,W,V,A. ⦃G,K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃⁝ K2 & I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
#G #L1 #L2 * -L1 -L2
[ #J #K1 #H destruct
lemma lsuba_inv_bind1: ∀I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃⁝ L2 →
(∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+ ∃∃K2,W,V,A. ⦃G,K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G,K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
/2 width=3 by lsuba_inv_bind1_aux/ qed-.
fact lsuba_inv_bind2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
(∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓘ{I}) ∨
- ∃∃K1,V,W, A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K1,V,W,A. ⦃G,K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃⁝ K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
#G #L1 #L2 * -L1 -L2
[ #J #K2 #H destruct
lemma lsuba_inv_bind2: ∀I,G,L1,K2. G ⊢ L1 ⫃⁝ K2.ⓘ{I} →
(∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓘ{I}) ∨
- ∃∃K1,V,W,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+ ∃∃K1,V,W,A. ⦃G,K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G,K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
I = BPair Abst W & L1 = K1.ⓓⓝW.V.
/2 width=3 by lsuba_inv_bind2_aux/ qed-.
(* Properties with atomic arity assignment **********************************)
-lemma lsuba_aaa_conf: ∀G,L1,V,A. ⦃G, L1⦄ ⊢ V ⁝ A →
- ∀L2. G ⊢ L1 ⫃⁝ L2 → ⦃G, L2⦄ ⊢ V ⁝ A.
+lemma lsuba_aaa_conf: ∀G,L1,V,A. ⦃G,L1⦄ ⊢ V ⁝ A →
+ ∀L2. G ⊢ L1 ⫃⁝ L2 → ⦃G,L2⦄ ⊢ V ⁝ A.
#G #L1 #V #A #H elim H -G -L1 -V -A
[ //
| #I #G #L1 #V #A #HA #IH #L2 #H
]
qed-.
-lemma lsuba_aaa_trans: ∀G,L2,V,A. ⦃G, L2⦄ ⊢ V ⁝ A →
- ∀L1. G ⊢ L1 ⫃⁝ L2 → ⦃G, L1⦄ ⊢ V ⁝ A.
+lemma lsuba_aaa_trans: ∀G,L2,V,A. ⦃G,L2⦄ ⊢ V ⁝ A →
+ ∀L1. G ⊢ L1 ⫃⁝ L2 → ⦃G,L1⦄ ⊢ V ⁝ A.
#G #L2 #V #A #H elim H -G -L2 -V -A
[ //
| #I #G #L2 #V #A #HA #IH #L1 #H
(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
(* Basic_2A1: includes: lsuba_drop_O1_conf *)
lemma lsuba_drops_conf_isuni: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 →
- ∀b,f,K1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K1 →
- ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇*[b, f] L2 ≘ K2.
+ ∀b,f,K1. 𝐔⦃f⦄ → ⬇*[b,f] L1 ≘ K1 →
+ ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇*[b,f] L2 ≘ K2.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #HL12 #IH #b #f #K1 #Hf #H
(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
(* Basic_2A1: includes: lsuba_drop_O1_trans *)
lemma lsuba_drops_trans_isuni: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 →
- ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2 →
- ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇*[b, f] L1 ≘ K1.
+ ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b,f] L2 ≘ K2 →
+ ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇*[b,f] L1 ≘ K1.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #HL12 #IH #b #f #K2 #Hf #H
inductive lsubc (RP) (G): relation lenv ≝
| lsubc_atom: lsubc RP G (⋆) (⋆)
| lsubc_bind: ∀I,L1,L2. lsubc RP G L1 L2 → lsubc RP G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsubc_beta: ∀L1,L2,V,W,A. ⦃G, L1, V⦄ ϵ[RP] 〚A〛 → ⦃G, L1, W⦄ ϵ[RP] 〚A〛 → ⦃G, L2⦄ ⊢ W ⁝ A →
+| lsubc_beta: ∀L1,L2,V,W,A. ⦃G,L1,V⦄ ϵ[RP] 〚A〛 → ⦃G,L1,W⦄ ϵ[RP] 〚A〛 → ⦃G,L2⦄ ⊢ W ⁝ A →
lsubc RP G L1 L2 → lsubc RP G (L1. ⓓⓝW.V) (L2.ⓛW)
.
fact lsubc_inv_bind1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K1. L1 = K1.ⓘ{I} →
(∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K2,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L2 = K2. ⓛW & I = BPair Abbr (ⓝW.V).
#RP #G #L1 #L2 * -L1 -L2
(* Basic_1: was: csubc_gen_head_r *)
lemma lsubc_inv_bind1: ∀RP,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃[RP] L2 →
(∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K2,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V).
/2 width=3 by lsubc_inv_bind1_aux/ qed-.
fact lsubc_inv_bind2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K2. L2 = K2.ⓘ{I} →
(∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1. ⓘ{I}) ∨
- ∃∃K1,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K1,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L1 = K1.ⓓⓝW.V & I = BPair Abst W.
#RP #G #L1 #L2 * -L1 -L2
(* Basic_1: was just: csubc_gen_head_l *)
lemma lsubc_inv_bind2: ∀RP,I,G,L1,K2. G ⊢ L1 ⫃[RP] K2.ⓘ{I} →
(∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1.ⓘ{I}) ∨
- ∃∃K1,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ ∃∃K1,V,W,A. ⦃G,K1,V⦄ ϵ[RP] 〚A〛 & ⦃G,K1,W⦄ ϵ[RP] 〚A〛 & ⦃G,K2⦄ ⊢ W ⁝ A &
G ⊢ K1 ⫃[RP] K2 &
L1 = K1.ⓓⓝW.V & I = BPair Abst W.
/2 width=3 by lsubc_inv_bind2_aux/ qed-.
(* Basic_1: includes: csubc_drop_conf_O *)
(* Basic_2A1: includes: lsubc_drop_O1_trans *)
lemma lsubc_drops_trans_isuni: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 →
- ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & G ⊢ K1 ⫃[RP] K2.
+ ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b,f] L2 ≘ K2 →
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & G ⊢ K1 ⫃[RP] K2.
#RP #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
| #I #L1 #L2 #HL12 #IH #b #f #K2 #Hf #H
(* Basic_1: includes: csubc_drop_conf_rev *)
(* Basic_2A1: includes: drop_lsubc_trans *)
lemma drops_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
- ∀b,f,G,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
- ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇*[b, f] L2 ≘ K2.
+ ∀b,f,G,L1,K1. ⬇*[b,f] L1 ≘ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
+ ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇*[b,f] L2 ≘ K2.
#RR #RS #RP #HR #b #f #G #L1 #K1 #H elim H -f -L1 -K1
[ #f #Hf #Y #H lapply (lsubc_inv_atom1 … H) -H
#H destruct /4 width=3 by lsubc_atom, drops_atom, ex2_intro/
lsubf (L1.ⓘ{I1}) (⫯f1) (L2.ⓘ{I2}) (⫯f2)
| lsubf_bind: ∀f1,f2,I,L1,L2. lsubf L1 f1 L2 f2 →
lsubf (L1.ⓘ{I}) (↑f1) (L2.ⓘ{I}) (↑f2)
-| lsubf_beta: ∀f,f0,f1,f2,L1,L2,W,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
+| lsubf_beta: ∀f,f0,f1,f2,L1,L2,W,V. L1 ⊢ 𝐅+⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
lsubf L1 f0 L2 f2 → lsubf (L1.ⓓⓝW.V) (↑f1) (L2.ⓛW) (↑f2)
-| lsubf_unit: ∀f,f0,f1,f2,I1,I2,L1,L2,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
+| lsubf_unit: ∀f,f0,f1,f2,I1,I2,L1,L2,V. L1 ⊢ 𝐅+⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
lsubf L1 f0 L2 f2 → lsubf (L1.ⓑ{I1}V) (↑f1) (L2.ⓤ{I2}) (↑f2)
.
(* Basic inversion lemmas ***************************************************)
-fact lsubf_inv_atom1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 = ⋆ →
- f1 ≡ f2 ∧ L2 = ⋆.
+fact lsubf_inv_atom1_aux:
+ ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → L1 = ⋆ →
+ ∧∧ f1 ≡ f2 & L2 = ⋆.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ /2 width=1 by conj/
| #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
]
qed-.
-lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f1 ≡ f2 ∧ L2 = ⋆.
+lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → ∧∧ f1 ≡ f2 & L2 = ⋆.
/2 width=3 by lsubf_inv_atom1_aux/ qed-.
-fact lsubf_inv_push1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ{I1} →
- ∃∃g2,I2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
+fact lsubf_inv_push1_aux:
+ ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+ ∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ{I1} →
+ ∃∃g2,I2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ #f1 #f2 #_ #g1 #J1 #K1 #_ #H destruct
| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J1 #K1 #H1 #H2 destruct
]
qed-.
-lemma lsubf_inv_push1: ∀g1,f2,I1,K1,L2. ⦃K1.ⓘ{I1}, ⫯g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∃∃g2,I2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
+lemma lsubf_inv_push1:
+ ∀g1,f2,I1,K1,L2. ⦃K1.ⓘ{I1},⫯g1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+ ∃∃g2,I2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
/2 width=6 by lsubf_inv_push1_aux/ qed-.
-fact lsubf_inv_pair1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g1,I,K1,X. f1 = ↑g1 → L1 = K1.ⓑ{I}X →
- ∨∨ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
- | ∃∃g,g0,g2,K2,W,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
- I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
- | ∃∃g,g0,g2,J,K2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
- L2 = K2.ⓤ{J}.
+fact lsubf_inv_pair1_aux:
+ ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+ ∀g1,I,K1,X. f1 = ↑g1 → L1 = K1.ⓑ{I}X →
+ ∨∨ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
+ | ∃∃g,g0,g2,K2,W,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
+ K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
+ I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
+ | ∃∃g,g0,g2,J,K2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
+ K1 ⊢ 𝐅+⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 & L2 = K2.ⓤ{J}.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ #f1 #f2 #_ #g1 #J #K1 #X #_ #H destruct
| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #X #H elim (discr_push_next … H)
]
qed-.
-lemma lsubf_inv_pair1: ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X, ↑g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∨∨ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
- | ∃∃g,g0,g2,K2,W,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
- I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
- | ∃∃g,g0,g2,J,K2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
- L2 = K2.ⓤ{J}.
+lemma lsubf_inv_pair1:
+ ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X,↑g1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+ ∨∨ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
+ | ∃∃g,g0,g2,K2,W,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
+ K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
+ I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
+ | ∃∃g,g0,g2,J,K2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
+ K1 ⊢ 𝐅+⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 & L2 = K2.ⓤ{J}.
/2 width=5 by lsubf_inv_pair1_aux/ qed-.
-fact lsubf_inv_unit1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ{I} →
- ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
+fact lsubf_inv_unit1_aux:
+ ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+ ∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ{I} →
+ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ #f1 #f2 #_ #g1 #J #K1 #_ #H destruct
| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #H elim (discr_push_next … H)
]
qed-.
-lemma lsubf_inv_unit1: ∀g1,f2,I,K1,L2. ⦃K1.ⓤ{I}, ↑g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
+lemma lsubf_inv_unit1:
+ ∀g1,f2,I,K1,L2. ⦃K1.ⓤ{I},↑g1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+ ∃∃g2,K2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
/2 width=5 by lsubf_inv_unit1_aux/ qed-.
-fact lsubf_inv_atom2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L2 = ⋆ →
- f1 ≡ f2 ∧ L1 = ⋆.
+fact lsubf_inv_atom2_aux:
+ ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → L2 = ⋆ →
+ ∧∧ f1 ≡ f2 & L1 = ⋆.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ /2 width=1 by conj/
| #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
]
qed-.
-lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → f1 ≡ f2 ∧ L1 = ⋆.
+lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1,f1⦄ ⫃𝐅+ ⦃⋆,f2⦄ → ∧∧f1 ≡ f2 & L1 = ⋆.
/2 width=3 by lsubf_inv_atom2_aux/ qed-.
-fact lsubf_inv_push2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ{I2} →
- ∃∃g1,I1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
+fact lsubf_inv_push2_aux:
+ ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+ ∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ{I2} →
+ ∃∃g1,I1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ #f1 #f2 #_ #g2 #J2 #K2 #_ #H destruct
| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J2 #K2 #H1 #H2 destruct
]
qed-.
-lemma lsubf_inv_push2: ∀f1,g2,I2,L1,K2. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓘ{I2}, ⫯g2⦄ →
- ∃∃g1,I1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
+lemma lsubf_inv_push2:
+ ∀f1,g2,I2,L1,K2. ⦃L1,f1⦄ ⫃𝐅+ ⦃K2.ⓘ{I2},⫯g2⦄ →
+ ∃∃g1,I1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
/2 width=6 by lsubf_inv_push2_aux/ qed-.
-fact lsubf_inv_pair2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g2,I,K2,W. f2 = ↑g2 → L2 = K2.ⓑ{I}W →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
- | ∃∃g,g0,g1,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
- I = Abst & L1 = K1.ⓓⓝW.V.
+fact lsubf_inv_pair2_aux:
+ ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+ ∀g2,I,K2,W. f2 = ↑g2 → L2 = K2.ⓑ{I}W →
+ ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
+ | ∃∃g,g0,g1,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
+ K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+ I = Abst & L1 = K1.ⓓⓝW.V.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ #f1 #f2 #_ #g2 #J #K2 #X #_ #H destruct
| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #X #H elim (discr_push_next … H)
]
qed-.
-lemma lsubf_inv_pair2: ∀f1,g2,I,L1,K2,W. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓑ{I}W, ↑g2⦄ →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
- | ∃∃g,g0,g1,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
- I = Abst & L1 = K1.ⓓⓝW.V.
+lemma lsubf_inv_pair2:
+ ∀f1,g2,I,L1,K2,W. ⦃L1,f1⦄ ⫃𝐅+ ⦃K2.ⓑ{I}W,↑g2⦄ →
+ ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
+ | ∃∃g,g0,g1,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
+ K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+ I = Abst & L1 = K1.ⓓⓝW.V.
/2 width=5 by lsubf_inv_pair2_aux/ qed-.
-fact lsubf_inv_unit2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g2,I,K2. f2 = ↑g2 → L2 = K2.ⓤ{I} →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
- | ∃∃g,g0,g1,J,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
- L1 = K1.ⓑ{J}V.
+fact lsubf_inv_unit2_aux:
+ ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+ ∀g2,I,K2. f2 = ↑g2 → L2 = K2.ⓤ{I} →
+ ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
+ | ∃∃g,g0,g1,J,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
+ K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 & L1 = K1.ⓑ{J}V.
#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
[ #f1 #f2 #_ #g2 #J #K2 #_ #H destruct
| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #H elim (discr_push_next … H)
]
qed-.
-lemma lsubf_inv_unit2: ∀f1,g2,I,L1,K2. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓤ{I}, ↑g2⦄ →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
- | ∃∃g,g0,g1,J,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
- L1 = K1.ⓑ{J}V.
+lemma lsubf_inv_unit2:
+ ∀f1,g2,I,L1,K2. ⦃L1,f1⦄ ⫃𝐅+ ⦃K2.ⓤ{I},↑g2⦄ →
+ ∨∨ ∃∃g1,K1. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
+ | ∃∃g,g0,g1,J,K1,V. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ &
+ K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 & L1 = K1.ⓑ{J}V.
/2 width=5 by lsubf_inv_unit2_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lsubf_inv_atom: ∀f1,f2. ⦃⋆, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → f1 ≡ f2.
+lemma lsubf_inv_atom: ∀f1,f2. ⦃⋆,f1⦄ ⫃𝐅+ ⦃⋆,f2⦄ → f1 ≡ f2.
#f1 #f2 #H elim (lsubf_inv_atom1 … H) -H //
qed-.
-lemma lsubf_inv_push_sn: ∀g1,f2,I1,I2,K1,K2. ⦃K1.ⓘ{I1}, ⫯g1⦄ ⫃𝐅* ⦃K2.ⓘ{I2}, f2⦄ →
- ∃∃g2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2.
+lemma lsubf_inv_push_sn:
+ ∀g1,f2,I1,I2,K1,K2. ⦃K1.ⓘ{I1},⫯g1⦄ ⫃𝐅+ ⦃K2.ⓘ{I2},f2⦄ →
+ ∃∃g2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ⫯g2.
#g1 #f2 #I #K1 #K2 #X #H elim (lsubf_inv_push1 … H) -H
#g2 #I #Y #H0 #H2 #H destruct /2 width=3 by ex2_intro/
qed-.
-lemma lsubf_inv_bind_sn: ∀g1,f2,I,K1,K2. ⦃K1.ⓘ{I}, ↑g1⦄ ⫃𝐅* ⦃K2.ⓘ{I}, f2⦄ →
- ∃∃g2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2.
+lemma lsubf_inv_bind_sn:
+ ∀g1,f2,I,K1,K2. ⦃K1.ⓘ{I},↑g1⦄ ⫃𝐅+ ⦃K2.ⓘ{I},f2⦄ →
+ ∃∃g2. ⦃K1,g1⦄ ⫃𝐅+ ⦃K2,g2⦄ & f2 = ↑g2.
#g1 #f2 * #I [2: #X ] #K1 #K2 #H
[ elim (lsubf_inv_pair1 … H) -H *
[ #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma lsubf_inv_beta_sn: ∀g1,f2,K1,K2,V,W. ⦃K1.ⓓⓝW.V, ↑g1⦄ ⫃𝐅* ⦃K2.ⓛW, f2⦄ →
- ∃∃g,g0,g2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
+lemma lsubf_inv_beta_sn:
+ ∀g1,f2,K1,K2,V,W. ⦃K1.ⓓⓝW.V,↑g1⦄ ⫃𝐅+ ⦃K2.ⓛW,f2⦄ →
+ ∃∃g,g0,g2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
#g1 #f2 #K1 #K2 #V #W #H elim (lsubf_inv_pair1 … H) -H *
[ #z2 #Y2 #_ #_ #H destruct
| #z #z0 #z2 #Y2 #X0 #X #H02 #Hz #Hg1 #H #_ #H0 #H1 destruct
]
qed-.
-lemma lsubf_inv_unit_sn: ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V, ↑g1⦄ ⫃𝐅* ⦃K2.ⓤ{J}, f2⦄ →
- ∃∃g,g0,g2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
+lemma lsubf_inv_unit_sn:
+ ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V,↑g1⦄ ⫃𝐅+ ⦃K2.ⓤ{J},f2⦄ →
+ ∃∃g,g0,g2. ⦃K1,g0⦄ ⫃𝐅+ ⦃K2,g2⦄ & K1 ⊢ 𝐅+⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
#g1 #f2 #I #J #K1 #K2 #V #H elim (lsubf_inv_pair1 … H) -H *
[ #z2 #Y2 #_ #_ #H destruct
| #z #z0 #z2 #Y2 #X0 #X #_ #_ #_ #_ #_ #_ #H destruct
]
qed-.
-lemma lsubf_inv_refl: ∀L,f1,f2. ⦃L,f1⦄ ⫃𝐅* ⦃L,f2⦄ → f1 ≡ f2.
+lemma lsubf_inv_refl: ∀L,f1,f2. ⦃L,f1⦄ ⫃𝐅+ ⦃L,f2⦄ → f1 ≡ f2.
#L elim L -L /2 width=1 by lsubf_inv_atom/
#L #I #IH #f1 #f2 #H12
elim (pn_split f1) * #g1 #H destruct
(* Basic forward lemmas *****************************************************)
-lemma lsubf_fwd_bind_tl: ∀f1,f2,I,L1,L2.
- ⦃L1.ⓘ{I}, f1⦄ ⫃𝐅* ⦃L2.ⓘ{I}, f2⦄ → ⦃L1, ⫱f1⦄ ⫃𝐅* ⦃L2, ⫱f2⦄.
+lemma lsubf_fwd_bind_tl:
+ ∀f1,f2,I,L1,L2. ⦃L1.ⓘ{I},f1⦄ ⫃𝐅+ ⦃L2.ⓘ{I},f2⦄ → ⦃L1,⫱f1⦄ ⫃𝐅+ ⦃L2,⫱f2⦄.
#f1 #f2 #I #L1 #L2 #H
elim (pn_split f1) * #g1 #H0 destruct
[ elim (lsubf_inv_push_sn … H) | elim (lsubf_inv_bind_sn … H) ] -H
#g2 #H12 #H destruct //
qed-.
-lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → 𝐈⦃f2⦄ → 𝐈⦃f1⦄.
+lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → 𝐈⦃f2⦄ → 𝐈⦃f1⦄.
#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
[ /2 width=3 by isid_eq_repl_fwd/
| /4 width=3 by isid_inv_push, isid_push/
]
qed-.
-lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → 𝐈⦃f1⦄ → 𝐈⦃f2⦄.
+lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → 𝐈⦃f1⦄ → 𝐈⦃f2⦄.
#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
[ /2 width=3 by isid_eq_repl_back/
| /4 width=3 by isid_inv_push, isid_push/
]
qed-.
-lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f2 ⊆ f1.
+lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → f2 ⊆ f1.
#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
/3 width=5 by sor_inv_sle_sn_trans, sle_next, sle_push, sle_refl_eq, eq_sym/
qed-.
(* Basic properties *********************************************************)
-axiom lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+axiom lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
-lemma lsubf_eq_repl_fwd1: ∀f2,L1,L2. eq_repl_fwd … (λf1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+lemma lsubf_eq_repl_fwd1: ∀f2,L1,L2. eq_repl_fwd … (λf1. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
#f2 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back1/
qed-.
-axiom lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+axiom lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
-lemma lsubf_eq_repl_fwd2: ∀f1,L1,L2. eq_repl_fwd … (λf2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+lemma lsubf_eq_repl_fwd2: ∀f1,L1,L2. eq_repl_fwd … (λf2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄).
#f1 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back2/
qed-.
/2 width=1 by lsubf_push, lsubf_bind/
qed.
-lemma lsubf_refl_eq: ∀f1,f2,L. f1 ≡ f2 → ⦃L, f1⦄ ⫃𝐅* ⦃L, f2⦄.
+lemma lsubf_refl_eq: ∀f1,f2,L. f1 ≡ f2 → ⦃L,f1⦄ ⫃𝐅+ ⦃L,f2⦄.
/2 width=3 by lsubf_eq_repl_back2/ qed.
-lemma lsubf_bind_tl_dx: ∀g1,f2,I,L1,L2. ⦃L1, g1⦄ ⫃𝐅* ⦃L2, ⫱f2⦄ →
- ∃∃f1. ⦃L1.ⓘ{I}, f1⦄ ⫃𝐅* ⦃L2.ⓘ{I}, f2⦄ & g1 = ⫱f1.
+lemma lsubf_bind_tl_dx:
+ ∀g1,f2,I,L1,L2. ⦃L1,g1⦄ ⫃𝐅+ ⦃L2,⫱f2⦄ →
+ ∃∃f1. ⦃L1.ⓘ{I},f1⦄ ⫃𝐅+ ⦃L2.ⓘ{I},f2⦄ & g1 = ⫱f1.
#g1 #f2 #I #L1 #L2 #H
elim (pn_split f2) * #g2 #H2 destruct
@ex2_intro [1,2,4,5: /2 width=2 by lsubf_push, lsubf_bind/ ] // (**) (* constructor needed *)
qed-.
-lemma lsubf_beta_tl_dx: ∀f,f0,g1,L1,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ g1 →
- ∀f2,L2,W. ⦃L1, f0⦄ ⫃𝐅* ⦃L2, ⫱f2⦄ →
- ∃∃f1. ⦃L1.ⓓⓝW.V, f1⦄ ⫃𝐅* ⦃L2.ⓛW, f2⦄ & ⫱f1 ⊆ g1.
+lemma lsubf_beta_tl_dx:
+ ∀f,f0,g1,L1,V. L1 ⊢ 𝐅+⦃V⦄ ≘ f → f0 ⋓ f ≘ g1 →
+ ∀f2,L2,W. ⦃L1,f0⦄ ⫃𝐅+ ⦃L2,⫱f2⦄ →
+ ∃∃f1. ⦃L1.ⓓⓝW.V,f1⦄ ⫃𝐅+ ⦃L2.ⓛW,f2⦄ & ⫱f1 ⊆ g1.
#f #f0 #g1 #L1 #V #Hf #Hg1 #f2
elim (pn_split f2) * #x2 #H2 #L2 #W #HL12 destruct
[ /3 width=4 by lsubf_push, sor_inv_sle_sn, ex2_intro/
qed-.
(* Note: this might be moved *)
-lemma lsubf_inv_sor_dx: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀f2l,f2r. f2l⋓f2r ≘ f2 →
- ∃∃f1l,f1r. ⦃L1, f1l⦄ ⫃𝐅* ⦃L2, f2l⦄ & ⦃L1, f1r⦄ ⫃𝐅* ⦃L2, f2r⦄ & f1l⋓f1r ≘ f1.
+lemma lsubf_inv_sor_dx:
+ ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ →
+ ∀f2l,f2r. f2l⋓f2r ≘ f2 →
+ ∃∃f1l,f1r. ⦃L1,f1l⦄ ⫃𝐅+ ⦃L2,f2l⦄ & ⦃L1,f1r⦄ ⫃𝐅+ ⦃L2,f2r⦄ & f1l⋓f1r ≘ f1.
#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
[ /3 width=7 by sor_eq_repl_fwd3, ex3_2_intro/
| #g1 #g2 #I1 #I2 #L1 #L2 #_ #IH #f2l #f2r #H
(* Properties with context-sensitive free variables *************************)
-lemma lsubf_frees_trans: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 →
- ∀f1,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 ⊢ 𝐅*⦃T⦄ ≘ f1.
+lemma lsubf_frees_trans:
+ ∀f2,L2,T. L2 ⊢ 𝐅+⦃T⦄ ≘ f2 →
+ ∀f1,L1. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → L1 ⊢ 𝐅+⦃T⦄ ≘ f1.
#f2 #L2 #T #H elim H -f2 -L2 -T
[ /3 width=5 by lsubf_fwd_isid_dx, frees_sort/
| #f2 #i #Hf2 #g1 #Y1 #H
(* Main properties **********************************************************)
-theorem lsubf_sor: ∀K,L,g1,f1. ⦃K, g1⦄ ⫃𝐅* ⦃L, f1⦄ →
- ∀g2,f2. ⦃K, g2⦄ ⫃𝐅* ⦃L, f2⦄ →
- ∀g. g1 ⋓ g2 ≘ g → ∀f. f1 ⋓ f2 ≘ f → ⦃K, g⦄ ⫃𝐅* ⦃L, f⦄.
+theorem lsubf_sor:
+ ∀K,L,g1,f1. ⦃K,g1⦄ ⫃𝐅+ ⦃L,f1⦄ →
+ ∀g2,f2. ⦃K,g2⦄ ⫃𝐅+ ⦃L,f2⦄ →
+ ∀g. g1 ⋓ g2 ≘ g → ∀f. f1 ⋓ f2 ≘ f → ⦃K,g⦄ ⫃𝐅+ ⦃L,f⦄.
#K elim K -K
[ #L #g1 #f1 #H1 #g2 #f2 #H2 #g #Hg #f #Hf
elim (lsubf_inv_atom1 … H1) -H1 #H1 #H destruct
(* Forward lemmas with restricted refinement for local environments *********)
-lemma lsubf_fwd_lsubr_isdiv: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- 𝛀⦃f1⦄ → 𝛀⦃f2⦄ → L1 ⫃ L2.
+lemma lsubf_fwd_lsubr_isdiv:
+ ∀f1,f2,L1,L2. ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄ → 𝛀⦃f1⦄ → 𝛀⦃f2⦄ → L1 ⫃ L2.
#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
/4 width=3 by lsubr_bind, isdiv_inv_next/
[ #f1 #f2 #I1 #I2 #L1 #L2 #_ #_ #H
(* Properties with restricted refinement for local environments *************)
-lemma lsubr_lsubf_isid: ∀L1,L2. L1 ⫃ L2 →
- ∀f1,f2. 𝐈⦃f1⦄ → 𝐈⦃f2⦄ → ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄.
+lemma lsubr_lsubf_isid:
+ ∀L1,L2. L1 ⫃ L2 → ∀f1,f2. 𝐈⦃f1⦄ → 𝐈⦃f2⦄ → ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄.
#L1 #L2 #H elim H -L1 -L2
[ /3 width=1 by lsubf_atom, isid_inv_eq_repl/
| #I #L1 #L2 | #L1 #L2 #V #W | #I1 #I2 #L1 #L2 #V
/3 width=1 by lsubf_push/
qed.
-lemma lsubr_lsubf: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 → ∀L1. L1 ⫃ L2 →
- ∀f1. L1 ⊢ 𝐅*⦃T⦄ ≘ f1 → ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄.
+lemma lsubr_lsubf:
+ ∀f2,L2,T. L2 ⊢ 𝐅+⦃T⦄ ≘ f2 → ∀L1. L1 ⫃ L2 →
+ ∀f1. L1 ⊢ 𝐅+⦃T⦄ ≘ f1 → ⦃L1,f1⦄ ⫃𝐅+ ⦃L2,f2⦄.
#f2 #L2 #T #H elim H -f2 -L2 -T
[ #f2 #L2 #s #Hf2 #L1 #HL12 #f1 #Hf1
lapply (frees_inv_sort … Hf1) -Hf1 /2 width=1 by lsubr_lsubf_isid/
lemma lsubr_inv_atom1: ∀L2. ⋆ ⫃ L2 → L2 = ⋆.
/2 width=3 by lsubr_inv_atom1_aux/ qed-.
-fact lsubr_inv_bind1_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
- I = BPair Abbr (ⓝW.V)
- | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
- I = BPair J1 V.
+fact lsubr_inv_bind1_aux:
+ ∀L1,L2. L1 ⫃ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V)
+ | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} & I = BPair J1 V.
#L1 #L2 * -L1 -L2
[ #J #K1 #H destruct
| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by or3_intro0, ex2_intro/
qed-.
(* Basic_2A1: uses: lsubr_inv_pair1 *)
-lemma lsubr_inv_bind1: ∀I,K1,L2. K1.ⓘ{I} ⫃ L2 →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
- I = BPair Abbr (ⓝW.V)
- | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
- I = BPair J1 V.
+lemma lsubr_inv_bind1:
+ ∀I,K1,L2. K1.ⓘ{I} ⫃ L2 →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V)
+ | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} & I = BPair J1 V.
/2 width=3 by lsubr_inv_bind1_aux/ qed-.
fact lsubr_inv_atom2_aux: ∀L1,L2. L1 ⫃ L2 → L2 = ⋆ → L1 = ⋆.
lemma lsubr_inv_atom2: ∀L1. L1 ⫃ ⋆ → L1 = ⋆.
/2 width=3 by lsubr_inv_atom2_aux/ qed-.
-fact lsubr_inv_bind2_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
+fact lsubr_inv_bind2_aux:
+ ∀L1,L2. L1 ⫃ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
#L1 #L2 * -L1 -L2
[ #J #K2 #H destruct
| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or3_intro0/
]
qed-.
-lemma lsubr_inv_bind2: ∀I,L1,K2. L1 ⫃ K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
+lemma lsubr_inv_bind2:
+ ∀I,L1,K2. L1 ⫃ K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
/2 width=3 by lsubr_inv_bind2_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lsubr_inv_abst1: ∀K1,L2,W. K1.ⓛW ⫃ L2 →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW
- | ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓤ{I2}.
+lemma lsubr_inv_abst1:
+ ∀K1,L2,W. K1.ⓛW ⫃ L2 →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW
+ | ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓤ{I2}.
#K1 #L2 #W #H elim (lsubr_inv_bind1 … H) -H *
/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
#K2 #V2 #W2 #_ #_ #H destruct
qed-.
-lemma lsubr_inv_unit1: ∀I,K1,L2. K1.ⓤ{I} ⫃ L2 →
- ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓤ{I}.
+lemma lsubr_inv_unit1:
+ ∀I,K1,L2. K1.ⓤ{I} ⫃ L2 →
+ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓤ{I}.
#I #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
[ #K2 #HK12 #H destruct /2 width=3 by ex2_intro/
| #K2 #V #W #_ #_ #H destruct
-| #I1 #I2 #K2 #V #_ #_ #H destruct
+| #J1 #J2 #K2 #V #_ #_ #H destruct
]
qed-.
-lemma lsubr_inv_pair2: ∀I,L1,K2,W. L1 ⫃ K2.ⓑ{I}W →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓑ{I}W
- | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = Abst.
+lemma lsubr_inv_pair2:
+ ∀I,L1,K2,W. L1 ⫃ K2.ⓑ{I}W →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓑ{I}W
+ | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = Abst.
#I #L1 #K2 #W #H elim (lsubr_inv_bind2 … H) -H *
[ /3 width=3 by ex2_intro, or_introl/
-| #K2 #X #V #HK12 #H1 #H2 destruct /3 width=4 by ex3_2_intro, or_intror/
-| #I1 #I1 #K2 #V #_ #_ #H destruct
+| #K1 #X #V #HK12 #H1 #H2 destruct /3 width=4 by ex3_2_intro, or_intror/
+| #J1 #J1 #K1 #V #_ #_ #H destruct
]
qed-.
-lemma lsubr_inv_abbr2: ∀L1,K2,V. L1 ⫃ K2.ⓓV →
- ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓV.
+lemma lsubr_inv_abbr2:
+ ∀L1,K2,V. L1 ⫃ K2.ⓓV →
+ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓV.
#L1 #K2 #V #H elim (lsubr_inv_pair2 … H) -H *
[ /2 width=3 by ex2_intro/
| #K1 #X #_ #_ #H destruct
]
qed-.
-lemma lsubr_inv_abst2: ∀L1,K2,W. L1 ⫃ K2.ⓛW →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓛW
- | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V.
+lemma lsubr_inv_abst2:
+ ∀L1,K2,W. L1 ⫃ K2.ⓛW →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓛW
+ | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V.
#L1 #K2 #W #H elim (lsubr_inv_pair2 … H) -H *
/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
qed-.
-lemma lsubr_inv_unit2: ∀I,L1,K2. L1 ⫃ K2.ⓤ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓤ{I}
- | ∃∃J,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J}V.
+lemma lsubr_inv_unit2:
+ ∀I,L1,K2. L1 ⫃ K2.ⓤ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓤ{I}
+ | ∃∃J,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J}V.
#I #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
[ /3 width=3 by ex2_intro, or_introl/
| #K1 #W #V #_ #_ #H destruct
-| #I1 #I2 #K1 #V #HK12 #H1 #H2 destruct /3 width=5 by ex2_3_intro, or_intror/
+| #J1 #J2 #K1 #V #HK12 #H1 #H2 destruct /3 width=5 by ex2_3_intro, or_intror/
]
qed-.
(* Basic forward lemmas *****************************************************)
-lemma lsubr_fwd_bind1: ∀I1,K1,L2. K1.ⓘ{I1} ⫃ L2 →
- ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓘ{I2}.
+lemma lsubr_fwd_bind1:
+ ∀I1,K1,L2. K1.ⓘ{I1} ⫃ L2 →
+ ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓘ{I2}.
#I1 #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
[ #K2 #HK12 #H destruct /3 width=4 by ex2_2_intro/
| #K2 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-| #I1 #I2 #K2 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+| #J1 #J2 #K2 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
]
qed-.
-lemma lsubr_fwd_bind2: ∀I2,L1,K2. L1 ⫃ K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⫃ K2 & L1 = K1.ⓘ{I1}.
+lemma lsubr_fwd_bind2:
+ ∀I2,L1,K2. L1 ⫃ K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⫃ K2 & L1 = K1.ⓘ{I1}.
#I2 #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
[ #K1 #HK12 #H destruct /3 width=4 by ex2_2_intro/
| #K1 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-| #I1 #I2 #K1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+| #J1 #J2 #K1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
]
qed-.
(* Forward lemmas with generic slicing for local environments ***************)
(* Basic_2A1: includes: lsubr_fwd_drop2_pair *)
-lemma lsubr_fwd_drops2_bind: ∀L1,L2. L1 ⫃ L2 →
- ∀b,f,I,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓘ{I}
- | ∃∃K1,W,V. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓑ{J1}V & I = BUnit J2.
+lemma lsubr_fwd_drops2_bind:
+ ∀L1,L2. L1 ⫃ L2 →
+ ∀b,f,I,K2. 𝐔⦃f⦄ → ⬇*[b,f] L2 ≘ K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & ⬇*[b,f] L1 ≘ K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & ⬇*[b,f] L1 ≘ K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & ⬇*[b,f] L1 ≘ K1.ⓑ{J1}V & I = BUnit J2.
#L1 #L2 #H elim H -L1 -L2
[ #b #f #I #K2 #_ #H
elim (drops_inv_atom1 … H) -H #H destruct
qed-.
(* Basic_2A1: includes: lsubr_fwd_drop2_abbr *)
-lemma lsubr_fwd_drops2_abbr: ∀L1,L2. L1 ⫃ L2 →
- ∀b,f,K2,V. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2.ⓓV →
- ∃∃K1. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓓV.
+lemma lsubr_fwd_drops2_abbr:
+ ∀L1,L2. L1 ⫃ L2 →
+ ∀b,f,K2,V. 𝐔⦃f⦄ → ⬇*[b,f] L2 ≘ K2.ⓓV →
+ ∃∃K1. K1 ⫃ K2 & ⬇*[b,f] L1 ≘ K1.ⓓV.
#L1 #L2 #HL12 #b #f #K2 #V #Hf #HLK2
elim (lsubr_fwd_drops2_bind … HL12 … Hf HLK2) -L2 -Hf // *
[ #K1 #W #V #_ #_ #H destruct
theorem lsubr_trans: Transitive … lsubr.
#L1 #L #H elim H -L1 -L //
[ #I #L1 #L #_ #IH #X #H elim (lsubr_inv_bind1 … H) -H *
- [ #L2 #HL2 #H | #L2 #V #W #HL2 #H1 #H2 | #I1 #I2 #L2 #V #Hl2 #H1 #H2 ]
+ [ #L2 #HL2 #H | #L2 #V #W #HL2 #H1 #H2 | #I1 #I2 #L2 #V #HL2 #H1 #H2 ]
destruct /3 width=1 by lsubr_bind, lsubr_beta, lsubr_unit/
| #L1 #L #V #W #_ #IH #X #H elim (lsubr_inv_abst1 … H) -H *
[ #L2 #HL2 #H | #I #L2 #HL2 #H ]
destruct /3 width=1 by lsubr_beta, lsubr_unit/
| #I1 #I2 #L1 #L #V #_ #IH #X #H elim (lsubr_inv_unit1 … H) -H
- /4 width=1 by lsubr_unit/
+ #L2 #HL2 #H destruct /4 width=1 by lsubr_unit/
]
qed-.
(* *)
(**************************************************************************)
-include "static_2/notation/relations/stareqsn_5.ma".
+include "static_2/notation/relations/stareqsn_3.ma".
include "static_2/syntax/tdeq_ext.ma".
include "static_2/static/rex.ma".
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+(* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
-definition rdeq (h) (o): relation3 term lenv lenv ≝
- rex (cdeq h o).
+definition rdeq: relation3 term lenv lenv ≝
+ rex cdeq.
interpretation
- "degree-based equivalence on referred entries (local environment)"
- 'StarEqSn h o T L1 L2 = (rdeq h o T L1 L2).
+ "sort-irrelevant equivalence on referred entries (local environment)"
+ 'StarEqSn T L1 L2 = (rdeq T L1 L2).
interpretation
- "degree-based ranged equivalence (local environment)"
- 'StarEqSn h o f L1 L2 = (sex (cdeq_ext h o) cfull f L1 L2).
+ "sort-irrelevant ranged equivalence (local environment)"
+ 'StarEqSn f L1 L2 = (sex cdeq_ext cfull f L1 L2).
(* Basic properties ***********************************************************)
-lemma frees_tdeq_conf_rdeq (h) (o): ∀f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f → ∀T2. T1 ≛[h, o] T2 →
- ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≘ f.
-#h #o #f #L1 #T1 #H elim H -f -L1 -T1
+lemma frees_tdeq_conf_rdeq: ∀f,L1,T1. L1 ⊢ 𝐅+⦃T1⦄ ≘ f → ∀T2. T1 ≛ T2 →
+ ∀L2. L1 ≛[f] L2 → L2 ⊢ 𝐅+⦃T2⦄ ≘ f.
+#f #L1 #T1 #H elim H -f -L1 -T1
[ #f #L1 #s1 #Hf #X #H1 #L2 #_
- elim (tdeq_inv_sort1 … H1) -H1 #s2 #d #_ #_ #H destruct
+ elim (tdeq_inv_sort1 … H1) -H1 #s2 #H destruct
/2 width=3 by frees_sort/
| #f #i #Hf #X #H1
>(tdeq_inv_lref1 … H1) -X #Y #H2
]
qed-.
-lemma frees_tdeq_conf (h) (o): ∀f,L,T1. L ⊢ 𝐅*⦃T1⦄ ≘ f →
- ∀T2. T1 ≛[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≘ f.
+lemma frees_tdeq_conf: ∀f,L,T1. L ⊢ 𝐅+⦃T1⦄ ≘ f →
+ ∀T2. T1 ≛ T2 → L ⊢ 𝐅+⦃T2⦄ ≘ f.
/4 width=7 by frees_tdeq_conf_rdeq, sex_refl, ext2_refl/ qed-.
-lemma frees_rdeq_conf (h) (o): ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≘ f →
- ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≘ f.
+lemma frees_rdeq_conf: ∀f,L1,T. L1 ⊢ 𝐅+⦃T⦄ ≘ f →
+ ∀L2. L1 ≛[f] L2 → L2 ⊢ 𝐅+⦃T⦄ ≘ f.
/2 width=7 by frees_tdeq_conf_rdeq, tdeq_refl/ qed-.
-lemma tdeq_rex_conf (R) (h) (o): s_r_confluent1 … (cdeq h o) (rex R).
-#R #h #o #L1 #T1 #T2 #HT12 #L2 *
+lemma tdeq_rex_conf (R): s_r_confluent1 … cdeq (rex R).
+#R #L1 #T1 #T2 #HT12 #L2 *
/3 width=5 by frees_tdeq_conf, ex2_intro/
qed-.
-lemma tdeq_rex_div (R) (h) (o): ∀T1,T2. T1 ≛[h, o] T2 →
- ∀L1,L2. L1 ⪤[R, T2] L2 → L1 ⪤[R, T1] L2.
+lemma tdeq_rex_div (R): ∀T1,T2. T1 ≛ T2 →
+ ∀L1,L2. L1 ⪤[R,T2] L2 → L1 ⪤[R,T1] L2.
/3 width=5 by tdeq_rex_conf, tdeq_sym/ qed-.
-lemma tdeq_rdeq_conf (h) (o): s_r_confluent1 … (cdeq h o) (rdeq h o).
+lemma tdeq_rdeq_conf: s_r_confluent1 … cdeq rdeq.
/2 width=5 by tdeq_rex_conf/ qed-.
-lemma tdeq_rdeq_div (h) (o): ∀T1,T2. T1 ≛[h, o] T2 →
- ∀L1,L2. L1 ≛[h, o, T2] L2 → L1 ≛[h, o, T1] L2.
+lemma tdeq_rdeq_div: ∀T1,T2. T1 ≛ T2 →
+ ∀L1,L2. L1 ≛[T2] L2 → L1 ≛[T1] L2.
/2 width=5 by tdeq_rex_div/ qed-.
-lemma rdeq_atom (h) (o): ∀I. ⋆ ≛[h, o, ⓪{I}] ⋆.
+lemma rdeq_atom: ∀I. ⋆ ≛[⓪{I}] ⋆.
/2 width=1 by rex_atom/ qed.
-lemma rdeq_sort (h) (o): ∀I1,I2,L1,L2,s.
- L1 ≛[h, o, ⋆s] L2 → L1.ⓘ{I1} ≛[h, o, ⋆s] L2.ⓘ{I2}.
+lemma rdeq_sort: ∀I1,I2,L1,L2,s.
+ L1 ≛[⋆s] L2 → L1.ⓘ{I1} ≛[⋆s] L2.ⓘ{I2}.
/2 width=1 by rex_sort/ qed.
-lemma rdeq_pair (h) (o): ∀I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 → V1 ≛[h, o] V2 →
- L1.ⓑ{I}V1 ≛[h, o, #0] L2.ⓑ{I}V2.
+lemma rdeq_pair: ∀I,L1,L2,V1,V2.
+ L1 ≛[V1] L2 → V1 ≛ V2 → L1.ⓑ{I}V1 ≛[#0] L2.ⓑ{I}V2.
/2 width=1 by rex_pair/ qed.
(*
-lemma rdeq_unit (h) (o): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cdeq_ext h o, cfull, f] L2 →
- L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}.
+lemma rdeq_unit: ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cdeq_ext,cfull,f] L2 →
+ L1.ⓤ{I} ≛[#0] L2.ⓤ{I}.
/2 width=3 by rex_unit/ qed.
*)
-lemma rdeq_lref (h) (o): ∀I1,I2,L1,L2,i.
- L1 ≛[h, o, #i] L2 → L1.ⓘ{I1} ≛[h, o, #↑i] L2.ⓘ{I2}.
+lemma rdeq_lref: ∀I1,I2,L1,L2,i.
+ L1 ≛[#i] L2 → L1.ⓘ{I1} ≛[#↑i] L2.ⓘ{I2}.
/2 width=1 by rex_lref/ qed.
-lemma rdeq_gref (h) (o): ∀I1,I2,L1,L2,l.
- L1 ≛[h, o, §l] L2 → L1.ⓘ{I1} ≛[h, o, §l] L2.ⓘ{I2}.
+lemma rdeq_gref: ∀I1,I2,L1,L2,l.
+ L1 ≛[§l] L2 → L1.ⓘ{I1} ≛[§l] L2.ⓘ{I2}.
/2 width=1 by rex_gref/ qed.
-lemma rdeq_bind_repl_dx (h) (o): ∀I,I1,L1,L2.∀T:term.
- L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I1} →
- ∀I2. I ≛[h, o] I2 →
- L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I2}.
+lemma rdeq_bind_repl_dx: ∀I,I1,L1,L2.∀T:term.
+ L1.ⓘ{I} ≛[T] L2.ⓘ{I1} →
+ ∀I2. I ≛ I2 →
+ L1.ⓘ{I} ≛[T] L2.ⓘ{I2}.
/2 width=2 by rex_bind_repl_dx/ qed-.
(* Basic inversion lemmas ***************************************************)
-lemma rdeq_inv_atom_sn (h) (o): ∀Y2. ∀T:term. ⋆ ≛[h, o, T] Y2 → Y2 = ⋆.
+lemma rdeq_inv_atom_sn: ∀Y2. ∀T:term. ⋆ ≛[T] Y2 → Y2 = ⋆.
/2 width=3 by rex_inv_atom_sn/ qed-.
-lemma rdeq_inv_atom_dx (h) (o): ∀Y1. ∀T:term. Y1 ≛[h, o, T] ⋆ → Y1 = ⋆.
+lemma rdeq_inv_atom_dx: ∀Y1. ∀T:term. Y1 ≛[T] ⋆ → Y1 = ⋆.
/2 width=3 by rex_inv_atom_dx/ qed-.
(*
-lemma rdeq_inv_zero (h) (o): ∀Y1,Y2. Y1 ≛[h, o, #0] Y2 →
- ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
- | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cdeq_ext h o, cfull, f] L2 &
- Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
-#h #o #Y1 #Y2 #H elim (rex_inv_zero … H) -H *
+lemma rdeq_inv_zero: ∀Y1,Y2. Y1 ≛[#0] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I,L1,L2,V1,V2. L1 ≛[V1] L2 & V1 ≛ V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
+ | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cdeq_ext h o,cfull,f] L2 &
+ Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
+#Y1 #Y2 #H elim (rex_inv_zero … H) -H *
/3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/
qed-.
*)
-lemma rdeq_inv_lref (h) (o): ∀Y1,Y2,i. Y1 ≛[h, o, #↑i] Y2 →
- ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ≛[h, o, #i] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+lemma rdeq_inv_lref: ∀Y1,Y2,i. Y1 ≛[#↑i] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ≛[#i] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
/2 width=1 by rex_inv_lref/ qed-.
(* Basic_2A1: uses: lleq_inv_bind lleq_inv_bind_O *)
-lemma rdeq_inv_bind (h) (o): ∀p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 →
- ∧∧ L1 ≛[h, o, V] L2 & L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
+lemma rdeq_inv_bind: ∀p,I,L1,L2,V,T. L1 ≛[ⓑ{p,I}V.T] L2 →
+ ∧∧ L1 ≛[V] L2 & L1.ⓑ{I}V ≛[T] L2.ⓑ{I}V.
/2 width=2 by rex_inv_bind/ qed-.
(* Basic_2A1: uses: lleq_inv_flat *)
-lemma rdeq_inv_flat (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 →
- ∧∧ L1 ≛[h, o, V] L2 & L1 ≛[h, o, T] L2.
+lemma rdeq_inv_flat: ∀I,L1,L2,V,T. L1 ≛[ⓕ{I}V.T] L2 →
+ ∧∧ L1 ≛[V] L2 & L1 ≛[T] L2.
/2 width=2 by rex_inv_flat/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma rdeq_inv_zero_pair_sn (h) (o): ∀I,Y2,L1,V1. L1.ⓑ{I}V1 ≛[h, o, #0] Y2 →
- ∃∃L2,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y2 = L2.ⓑ{I}V2.
+lemma rdeq_inv_zero_pair_sn: ∀I,Y2,L1,V1. L1.ⓑ{I}V1 ≛[#0] Y2 →
+ ∃∃L2,V2. L1 ≛[V1] L2 & V1 ≛ V2 & Y2 = L2.ⓑ{I}V2.
/2 width=1 by rex_inv_zero_pair_sn/ qed-.
-lemma rdeq_inv_zero_pair_dx (h) (o): ∀I,Y1,L2,V2. Y1 ≛[h, o, #0] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y1 = L1.ⓑ{I}V1.
+lemma rdeq_inv_zero_pair_dx: ∀I,Y1,L2,V2. Y1 ≛[#0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ≛[V1] L2 & V1 ≛ V2 & Y1 = L1.ⓑ{I}V1.
/2 width=1 by rex_inv_zero_pair_dx/ qed-.
-lemma rdeq_inv_lref_bind_sn (h) (o): ∀I1,Y2,L1,i. L1.ⓘ{I1} ≛[h, o, #↑i] Y2 →
- ∃∃I2,L2. L1 ≛[h, o, #i] L2 & Y2 = L2.ⓘ{I2}.
+lemma rdeq_inv_lref_bind_sn: ∀I1,Y2,L1,i. L1.ⓘ{I1} ≛[#↑i] Y2 →
+ ∃∃I2,L2. L1 ≛[#i] L2 & Y2 = L2.ⓘ{I2}.
/2 width=2 by rex_inv_lref_bind_sn/ qed-.
-lemma rdeq_inv_lref_bind_dx (h) (o): ∀I2,Y1,L2,i. Y1 ≛[h, o, #↑i] L2.ⓘ{I2} →
- ∃∃I1,L1. L1 ≛[h, o, #i] L2 & Y1 = L1.ⓘ{I1}.
+lemma rdeq_inv_lref_bind_dx: ∀I2,Y1,L2,i. Y1 ≛[#↑i] L2.ⓘ{I2} →
+ ∃∃I1,L1. L1 ≛[#i] L2 & Y1 = L1.ⓘ{I1}.
/2 width=2 by rex_inv_lref_bind_dx/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma rdeq_fwd_zero_pair (h) (o): ∀I,K1,K2,V1,V2.
- K1.ⓑ{I}V1 ≛[h, o, #0] K2.ⓑ{I}V2 → K1 ≛[h, o, V1] K2.
+lemma rdeq_fwd_zero_pair: ∀I,K1,K2,V1,V2.
+ K1.ⓑ{I}V1 ≛[#0] K2.ⓑ{I}V2 → K1 ≛[V1] K2.
/2 width=3 by rex_fwd_zero_pair/ qed-.
(* Basic_2A1: uses: lleq_fwd_bind_sn lleq_fwd_flat_sn *)
-lemma rdeq_fwd_pair_sn (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ②{I}V.T] L2 → L1 ≛[h, o, V] L2.
+lemma rdeq_fwd_pair_sn: ∀I,L1,L2,V,T. L1 ≛[②{I}V.T] L2 → L1 ≛[V] L2.
/2 width=3 by rex_fwd_pair_sn/ qed-.
(* Basic_2A1: uses: lleq_fwd_bind_dx lleq_fwd_bind_O_dx *)
-lemma rdeq_fwd_bind_dx (h) (o): ∀p,I,L1,L2,V,T.
- L1 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
+lemma rdeq_fwd_bind_dx: ∀p,I,L1,L2,V,T.
+ L1 ≛[ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≛[T] L2.ⓑ{I}V.
/2 width=2 by rex_fwd_bind_dx/ qed-.
(* Basic_2A1: uses: lleq_fwd_flat_dx *)
-lemma rdeq_fwd_flat_dx (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 → L1 ≛[h, o, T] L2.
+lemma rdeq_fwd_flat_dx: ∀I,L1,L2,V,T. L1 ≛[ⓕ{I}V.T] L2 → L1 ≛[T] L2.
/2 width=3 by rex_fwd_flat_dx/ qed-.
-lemma rdeq_fwd_dx (h) (o): ∀I2,L1,K2. ∀T:term. L1 ≛[h, o, T] K2.ⓘ{I2} →
- ∃∃I1,K1. L1 = K1.ⓘ{I1}.
+lemma rdeq_fwd_dx: ∀I2,L1,K2. ∀T:term. L1 ≛[T] K2.ⓘ{I2} →
+ ∃∃I1,K1. L1 = K1.ⓘ{I1}.
/2 width=5 by rex_fwd_dx/ qed-.
include "static_2/static/rex_drops.ma".
include "static_2/static/rdeq.ma".
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+(* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
(* Properties with generic slicing for local environments *******************)
-lemma rdeq_lifts_sn: ∀h,o. f_dedropable_sn (cdeq h o).
+lemma rdeq_lifts_sn: f_dedropable_sn cdeq.
/3 width=5 by rex_liftable_dedropable_sn, tdeq_lifts_sn/ qed-.
(* Inversion lemmas with generic slicing for local environments *************)
-lemma rdeq_inv_lifts_sn: ∀h,o. f_dropable_sn (cdeq h o).
+lemma rdeq_inv_lifts_sn: f_dropable_sn cdeq.
/2 width=5 by rex_dropable_sn/ qed-.
-lemma rdeq_inv_lifts_dx: ∀h,o. f_dropable_dx (cdeq h o).
+lemma rdeq_inv_lifts_dx: f_dropable_dx cdeq.
/2 width=5 by rex_dropable_dx/ qed-.
-lemma rdeq_inv_lifts_bi: ∀h,o,L1,L2,U. L1 ≛[h, o, U] L2 → ∀b,f. 𝐔⦃f⦄ →
- ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- ∀T. ⬆*[f] T ≘ U → K1 ≛[h, o, T] K2.
+lemma rdeq_inv_lifts_bi: ∀L1,L2,U. L1 ≛[U] L2 → ∀b,f. 𝐔⦃f⦄ →
+ ∀K1,K2. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
+ ∀T. ⬆*[f] T ≘ U → K1 ≛[T] K2.
/2 width=10 by rex_inv_lifts_bi/ qed-.
-lemma rdeq_inv_lref_pair_sn: ∀h,o,L1,L2,i. L1 ≛[h, o, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ≛[h, o, V1] K2 & V1 ≛[h, o] V2.
+lemma rdeq_inv_lref_pair_sn: ∀L1,L2,i. L1 ≛[#i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ≛[V1] K2 & V1 ≛ V2.
/2 width=3 by rex_inv_lref_pair_sn/ qed-.
-lemma rdeq_inv_lref_pair_dx: ∀h,o,L1,L2,i. L1 ≛[h, o, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ≛[h, o, V1] K2 & V1 ≛[h, o] V2.
+lemma rdeq_inv_lref_pair_dx: ∀L1,L2,i. L1 ≛[#i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ≛[V1] K2 & V1 ≛ V2.
/2 width=3 by rex_inv_lref_pair_dx/ qed-.
-lemma rdeq_inv_lref_pair_bi (h) (o) (L1) (L2) (i):
- L1 ≛[h,o,#i] L2 →
+lemma rdeq_inv_lref_pair_bi (L1) (L2) (i):
+ L1 ≛[#i] L2 →
∀I1,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I1}V1 →
∀I2,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I2}V2 →
- ∧∧ K1 ≛[h,o,V1] K2 & V1 ≛[h,o] V2 & I1 = I2.
+ ∧∧ K1 ≛[V1] K2 & V1 ≛ V2 & I1 = I2.
/2 width=6 by rex_inv_lref_pair_bi/ qed-.
include "static_2/static/rex_fqup.ma".
include "static_2/static/rdeq.ma".
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+(* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
(* Advanced properties ******************************************************)
-lemma rdeq_refl: ∀h,o,T. reflexive … (rdeq h o T).
+lemma rdeq_refl: ∀T. reflexive … (rdeq T).
/2 width=1 by rex_refl/ qed.
-lemma rdeq_pair_refl: ∀h,o,V1,V2. V1 ≛[h, o] V2 →
- ∀I,L. ∀T:term. L.ⓑ{I}V1 ≛[h, o, T] L.ⓑ{I}V2.
+lemma rdeq_pair_refl: ∀V1,V2. V1 ≛ V2 →
+ ∀I,L. ∀T:term. L.ⓑ{I}V1 ≛[T] L.ⓑ{I}V2.
/2 width=1 by rex_pair_refl/ qed.
(* Advanced inversion lemmas ************************************************)
-lemma rdeq_inv_bind_void: ∀h,o,p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 →
- L1 ≛[h, o, V] L2 ∧ L1.ⓧ ≛[h, o, T] L2.ⓧ.
+lemma rdeq_inv_bind_void: ∀p,I,L1,L2,V,T. L1 ≛[ⓑ{p,I}V.T] L2 →
+ L1 ≛[V] L2 ∧ L1.ⓧ ≛[T] L2.ⓧ.
/2 width=3 by rex_inv_bind_void/ qed-.
(* Advanced forward lemmas **************************************************)
-lemma rdeq_fwd_bind_dx_void: ∀h,o,p,I,L1,L2,V,T.
- L1 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓧ ≛[h, o, T] L2.ⓧ.
+lemma rdeq_fwd_bind_dx_void: ∀p,I,L1,L2,V,T.
+ L1 ≛[ⓑ{p,I}V.T] L2 → L1.ⓧ ≛[T] L2.ⓧ.
/2 width=4 by rex_fwd_bind_dx_void/ qed-.
include "static_2/static/rdeq_fqup.ma".
include "static_2/static/rdeq_rdeq.ma".
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+(* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
(* Properties with extended structural successor for closures ***************)
-lemma fqu_tdeq_conf: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
- ∀U2. U1 ≛[h, o] U2 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & L2 ≛[h, o, T1] L & T1 ≛[h, o] T2.
-#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -G1 -G2 -L1 -L2 -U1 -T1
+lemma fqu_tdeq_conf: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⬂[b] ⦃G2,L2,T1⦄ →
+ ∀U2. U1 ≛ U2 →
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⬂[b] ⦃G2,L,T2⦄ & L2 ≛[T1] L & T1 ≛ T2.
+#b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -G1 -G2 -L1 -L2 -U1 -T1
[ #I #G #L #W #X #H >(tdeq_inv_lref1 … H) -X
/2 width=5 by fqu_lref_O, ex3_2_intro/
| #I #G #L #W1 #U1 #X #H
elim (tdeq_inv_pair1 … H) -H #W2 #U2 #HW12 #_ #H destruct
/2 width=5 by fqu_pair_sn, ex3_2_intro/
-| #p #I #G #L #W1 #U1 #X #H
+| #p #I #G #L #W1 #U1 #Hb #X #H
elim (tdeq_inv_pair1 … H) -H #W2 #U2 #HW12 #HU12 #H destruct
/3 width=5 by rdeq_pair_refl, fqu_bind_dx, ex3_2_intro/
| #p #I #G #L #W1 #U1 #Hb #X #H
]
qed-.
-lemma tdeq_fqu_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
- ∀U2. U2 ≛[h, o] U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
-#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H12 #U2 #HU21
-elim (fqu_tdeq_conf … o … H12 U2) -H12
+lemma tdeq_fqu_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⬂[b] ⦃G2,L2,T1⦄ →
+ ∀U2. U2 ≛ U1 →
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⬂[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+#b #G1 #G2 #L1 #L2 #U1 #T1 #H12 #U2 #HU21
+elim (fqu_tdeq_conf … H12 U2) -H12
/3 width=5 by rdeq_sym, tdeq_sym, ex3_2_intro/
qed-.
(* Basic_2A1: uses: lleq_fqu_trans *)
-lemma rdeq_fqu_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐[b] ⦃G2, K2, U⦄ →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
-#h #o #b #G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
+lemma rdeq_fqu_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⬂[b] ⦃G2,K2,U⦄ →
+ ∀L1. L1 ≛[T] L2 →
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⬂[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+#b #G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
[ #I #G #L2 #V2 #L1 #H elim (rdeq_inv_zero_pair_dx … H) -H
#K1 #V1 #HV1 #HV12 #H destruct
/3 width=7 by tdeq_rdeq_conf, fqu_lref_O, ex3_2_intro/
| elim (rdeq_inv_flat … H)
] -H
/2 width=5 by fqu_pair_sn, ex3_2_intro/
-| #p #I #G #L2 #V #T #L1 #H elim (rdeq_inv_bind … H) -H
- /2 width=5 by fqu_bind_dx, ex3_2_intro/
+| #p #I #G #L2 #V #T #Hb #L1 #H elim (rdeq_inv_bind … H) -H
+ /3 width=5 by fqu_bind_dx, ex3_2_intro/
| #p #I #G #L2 #V #T #Hb #L1 #H elim (rdeq_inv_bind_void … H) -H
/3 width=5 by fqu_clear, ex3_2_intro/
| #I #G #L2 #V #T #L1 #H elim (rdeq_inv_flat … H) -H
(* Properties with optional structural successor for closures ***************)
-lemma tdeq_fquq_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, T1⦄ →
- ∀U2. U2 ≛[h, o] U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐⸮[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
-#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -H
+lemma tdeq_fquq_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⬂⸮[b] ⦃G2,L2,T1⦄ →
+ ∀U2. U2 ≛ U1 →
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⬂⸮[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+#b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -H
[ #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
/3 width=5 by fqu_fquq, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
qed-.
(* Basic_2A1: was just: lleq_fquq_trans *)
-lemma rdeq_fquq_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮[b] ⦃G2, K2, U⦄ →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐⸮[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
-#h #o #b #G1 #G2 #L2 #K2 #T #U #H elim H -H
+lemma rdeq_fquq_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⬂⸮[b] ⦃G2,K2,U⦄ →
+ ∀L1. L1 ≛[T] L2 →
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⬂⸮[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+#b #G1 #G2 #L2 #K2 #T #U #H elim H -H
[ #H #L1 #HL12 elim (rdeq_fqu_trans … H … HL12) -L2 /3 width=5 by fqu_fquq, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
]
(* Properties with plus-iterated structural successor for closures **********)
(* Basic_2A1: was just: lleq_fqup_trans *)
-lemma rdeq_fqup_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+[b] ⦃G2, K2, U⦄ →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐+[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
-#h #o #b #G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
+lemma rdeq_fqup_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⬂+[b] ⦃G2,K2,U⦄ →
+ ∀L1. L1 ≛[T] L2 →
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⬂+[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+#b #G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
[ #G2 #K2 #U #HTU #L1 #HL12 elim (rdeq_fqu_trans … HTU … HL12) -L2
/3 width=5 by fqu_fqup, ex3_2_intro/
| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12
]
qed-.
-lemma tdeq_fqup_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, T1⦄ →
- ∀U2. U2 ≛[h, o] U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐+[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
-#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -U1
+lemma tdeq_fqup_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⬂+[b] ⦃G2,L2,T1⦄ →
+ ∀U2. U2 ≛ U1 →
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⬂+[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+#b #G1 #G2 #L1 #L2 #U1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -U1
[ #G1 #L1 #U1 #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
/3 width=5 by fqu_fqup, ex3_2_intro/
| #G1 #G #L1 #L #U1 #U #H #_ #IH #U2 #HU21
(* Properties with star-iterated structural successor for closures **********)
-lemma tdeq_fqus_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, T1⦄ →
- ∀U2. U2 ≛[h, o] U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐*[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
-#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H #U2 #HU21 elim(fqus_inv_fqup … H) -H
+lemma tdeq_fqus_trans: ∀b,G1,G2,L1,L2,U1,T1. ⦃G1,L1,U1⦄ ⬂*[b] ⦃G2,L2,T1⦄ →
+ ∀U2. U2 ≛ U1 →
+ ∃∃L,T2. ⦃G1,L1,U2⦄ ⬂*[b] ⦃G2,L,T2⦄ & T2 ≛ T1 & L ≛[T1] L2.
+#b #G1 #G2 #L1 #L2 #U1 #T1 #H #U2 #HU21 elim(fqus_inv_fqup … H) -H
[ #H elim (tdeq_fqup_trans … H … HU21) -U1 /3 width=5 by fqup_fqus, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_2A1: was just: lleq_fqus_trans *)
-lemma rdeq_fqus_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐*[b] ⦃G2, K2, U⦄ →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐*[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
-#h #o #b #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_fqup … H) -H
+lemma rdeq_fqus_trans: ∀b,G1,G2,L2,K2,T,U. ⦃G1,L2,T⦄ ⬂*[b] ⦃G2,K2,U⦄ →
+ ∀L1. L1 ≛[T] L2 →
+ ∃∃K1,U0. ⦃G1,L1,T⦄ ⬂*[b] ⦃G2,K1,U0⦄ & U0 ≛ U & K1 ≛[U] K2.
+#b #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_fqup … H) -H
[ #H elim (rdeq_fqup_trans … H … HL12) -L2 /3 width=5 by fqup_fqus, ex3_2_intro/
| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
]
include "static_2/static/rex_fsle.ma".
include "static_2/static/rdeq.ma".
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+(* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
(* Advanved properties with free variables inclusion ************************)
-lemma rdeq_fsge_comp (h) (o): rex_fsge_compatible (cdeq h o).
-#h #o #L1 #L2 #T * #f1 #Hf1 #HL12
-lapply (frees_rdeq_conf h o … Hf1 … HL12)
+lemma rdeq_fsge_comp: rex_fsge_compatible cdeq.
+#L1 #L2 #T * #f1 #Hf1 #HL12
+lapply (frees_rdeq_conf … Hf1 … HL12)
lapply (sex_fwd_length … HL12)
/3 width=8 by lveq_length_eq, ex4_4_intro/ (**) (* full auto fails *)
qed-.
(* Properties with length for local environments ****************************)
(* Basic_2A1: uses: lleq_sort *)
-lemma rdeq_sort_length (h) (o): ∀L1,L2. |L1| = |L2| → ∀s. L1 ≛[h, o, ⋆s] L2.
+lemma rdeq_sort_length: ∀L1,L2. |L1| = |L2| → ∀s. L1 ≛[⋆s] L2.
/2 width=1 by rex_sort_length/ qed.
(* Basic_2A1: uses: lleq_gref *)
-lemma rdeq_gref_length (h) (o): ∀L1,L2. |L1| = |L2| → ∀l. L1 ≛[h, o, §l] L2.
+lemma rdeq_gref_length: ∀L1,L2. |L1| = |L2| → ∀l. L1 ≛[§l] L2.
/2 width=1 by rex_gref_length/ qed.
-lemma rdeq_unit_length (h) (o): ∀L1,L2. |L1| = |L2| →
- ∀I. L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}.
+lemma rdeq_unit_length: ∀L1,L2. |L1| = |L2| →
+ ∀I. L1.ⓤ{I} ≛[#0] L2.ⓤ{I}.
/2 width=1 by rex_unit_length/ qed.
(* Basic_2A1: uses: lleq_lift_le lleq_lift_ge *)
-lemma rdeq_lifts_bi (h) (o): ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ≛[h, o, T] K2 →
- ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- ∀U. ⬆*[f] T ≘ U → L1 ≛[h, o, U] L2.
+lemma rdeq_lifts_bi: ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ≛[T] K2 →
+ ∀b,f. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
+ ∀U. ⬆*[f] T ≘ U → L1 ≛[U] L2.
/3 width=9 by rex_lifts_bi, tdeq_lifts_sn/ qed-.
(* Forward lemmas with length for local environments ************************)
(* Basic_2A1: lleq_fwd_length *)
-lemma rdeq_fwd_length (h) (o): ∀L1,L2. ∀T:term. L1 ≛[h, o, T] L2 → |L1| = |L2|.
+lemma rdeq_fwd_length: ∀L1,L2. ∀T:term. L1 ≛[T] L2 → |L1| = |L2|.
/2 width=3 by rex_fwd_length/ qed-.
include "static_2/syntax/tdeq_tdeq.ma".
include "static_2/static/rdeq_length.ma".
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+(* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: lleq_sym *)
-lemma rdeq_sym: ∀h,o,T. symmetric … (rdeq h o T).
+lemma rdeq_sym: ∀T. symmetric … (rdeq T).
/3 width=3 by rdeq_fsge_comp, rex_sym, tdeq_sym/ qed-.
(* Basic_2A1: uses: lleq_dec *)
-lemma rdeq_dec: ∀h,o,L1,L2. ∀T:term. Decidable (L1 ≛[h, o, T] L2).
+lemma rdeq_dec: ∀L1,L2. ∀T:term. Decidable (L1 ≛[T] L2).
/3 width=1 by rex_dec, tdeq_dec/ qed-.
(* Main properties **********************************************************)
(* Basic_2A1: uses: lleq_bind lleq_bind_O *)
-theorem rdeq_bind: ∀h,o,p,I,L1,L2,V1,V2,T.
- L1 ≛[h, o, V1] L2 → L1.ⓑ{I}V1 ≛[h, o, T] L2.ⓑ{I}V2 →
- L1 ≛[h, o, ⓑ{p,I}V1.T] L2.
+theorem rdeq_bind: ∀p,I,L1,L2,V1,V2,T.
+ L1 ≛[V1] L2 → L1.ⓑ{I}V1 ≛[T] L2.ⓑ{I}V2 →
+ L1 ≛[ⓑ{p,I}V1.T] L2.
/2 width=2 by rex_bind/ qed.
(* Basic_2A1: uses: lleq_flat *)
-theorem rdeq_flat: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, V] L2 → L1 ≛[h, o, T] L2 →
- L1 ≛[h, o, ⓕ{I}V.T] L2.
+theorem rdeq_flat: ∀I,L1,L2,V,T.
+ L1 ≛[V] L2 → L1 ≛[T] L2 → L1 ≛[ⓕ{I}V.T] L2.
/2 width=1 by rex_flat/ qed.
-theorem rdeq_bind_void: ∀h,o,p,I,L1,L2,V,T.
- L1 ≛[h, o, V] L2 → L1.ⓧ ≛[h, o, T] L2.ⓧ →
- L1 ≛[h, o, ⓑ{p,I}V.T] L2.
+theorem rdeq_bind_void: ∀p,I,L1,L2,V,T.
+ L1 ≛[V] L2 → L1.ⓧ ≛[T] L2.ⓧ → L1 ≛[ⓑ{p,I}V.T] L2.
/2 width=1 by rex_bind_void/ qed.
(* Basic_2A1: uses: lleq_trans *)
-theorem rdeq_trans: ∀h,o,T. Transitive … (rdeq h o T).
-#h #o #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
+theorem rdeq_trans: ∀T. Transitive … (rdeq T).
+#T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
lapply (frees_tdeq_conf_rdeq … Hf1 T … HL1) // #H0
lapply (frees_mono … Hf2 … H0) -Hf2 -H0
/5 width=7 by sex_trans, sex_eq_repl_back, tdeq_trans, ext2_trans, ex2_intro/
qed-.
(* Basic_2A1: uses: lleq_canc_sn *)
-theorem rdeq_canc_sn: ∀h,o,T. left_cancellable … (rdeq h o T).
+theorem rdeq_canc_sn: ∀T. left_cancellable … (rdeq T).
/3 width=3 by rdeq_trans, rdeq_sym/ qed-.
(* Basic_2A1: uses: lleq_canc_dx *)
-theorem rdeq_canc_dx: ∀h,o,T. right_cancellable … (rdeq h o T).
+theorem rdeq_canc_dx: ∀T. right_cancellable … (rdeq T).
/3 width=3 by rdeq_trans, rdeq_sym/ qed-.
-theorem rdeq_repl: ∀h,o,L1,L2. ∀T:term. L1 ≛[h, o, T] L2 →
- ∀K1. L1 ≛[h, o, T] K1 → ∀K2. L2 ≛[h, o, T] K2 → K1 ≛[h, o, T] K2.
+theorem rdeq_repl: ∀L1,L2. ∀T:term. L1 ≛[T] L2 →
+ ∀K1. L1 ≛[T] K1 → ∀K2. L2 ≛[T] K2 → K1 ≛[T] K2.
/3 width=3 by rdeq_canc_sn, rdeq_trans/ qed-.
(* Negated properties *******************************************************)
(* Note: auto works with /4 width=8/ so rdeq_canc_sn is preferred **********)
(* Basic_2A1: uses: lleq_nlleq_trans *)
-lemma rdeq_rdneq_trans: ∀h,o.∀T:term.∀L1,L. L1 ≛[h, o, T] L →
- ∀L2. (L ≛[h, o, T] L2 → ⊥) → (L1 ≛[h, o, T] L2 → ⊥).
+lemma rdeq_rdneq_trans: ∀T:term.∀L1,L. L1 ≛[T] L →
+ ∀L2. (L ≛[T] L2 → ⊥) → (L1 ≛[T] L2 → ⊥).
/3 width=3 by rdeq_canc_sn/ qed-.
(* Basic_2A1: uses: nlleq_lleq_div *)
-lemma rdneq_rdeq_div: ∀h,o.∀T:term.∀L2,L. L2 ≛[h, o, T] L →
- ∀L1. (L1 ≛[h, o, T] L → ⊥) → (L1 ≛[h, o, T] L2 → ⊥).
+lemma rdneq_rdeq_div: ∀T:term.∀L2,L. L2 ≛[T] L →
+ ∀L1. (L1 ≛[T] L → ⊥) → (L1 ≛[T] L2 → ⊥).
/3 width=3 by rdeq_trans/ qed-.
-theorem rdneq_rdeq_canc_dx: ∀h,o,L1,L. ∀T:term. (L1 ≛[h, o, T] L → ⊥) →
- ∀L2. L2 ≛[h, o, T] L → L1 ≛[h, o, T] L2 → ⊥.
+theorem rdneq_rdeq_canc_dx: ∀L1,L. ∀T:term. (L1 ≛[T] L → ⊥) →
+ ∀L2. L2 ≛[T] L → L1 ≛[T] L2 → ⊥.
/3 width=3 by rdeq_trans/ qed-.
(* Negated inversion lemmas *************************************************)
(* Basic_2A1: uses: nlleq_inv_bind nlleq_inv_bind_O *)
-lemma rdneq_inv_bind: ∀h,o,p,I,L1,L2,V,T. (L1 ≛[h, o, ⓑ{p,I}V.T] L2 → ⊥) →
- (L1 ≛[h, o, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V → ⊥).
+lemma rdneq_inv_bind: ∀p,I,L1,L2,V,T. (L1 ≛[ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ≛[V] L2 → ⊥) ∨ (L1.ⓑ{I}V ≛[T] L2.ⓑ{I}V → ⊥).
/3 width=2 by rnex_inv_bind, tdeq_dec/ qed-.
(* Basic_2A1: uses: nlleq_inv_flat *)
-lemma rdneq_inv_flat: ∀h,o,I,L1,L2,V,T. (L1 ≛[h, o, ⓕ{I}V.T] L2 → ⊥) →
- (L1 ≛[h, o, V] L2 → ⊥) ∨ (L1 ≛[h, o, T] L2 → ⊥).
+lemma rdneq_inv_flat: ∀I,L1,L2,V,T. (L1 ≛[ⓕ{I}V.T] L2 → ⊥) →
+ (L1 ≛[V] L2 → ⊥) ∨ (L1 ≛[T] L2 → ⊥).
/3 width=2 by rnex_inv_flat, tdeq_dec/ qed-.
-lemma rdneq_inv_bind_void: ∀h,o,p,I,L1,L2,V,T. (L1 ≛[h, o, ⓑ{p,I}V.T] L2 → ⊥) →
- (L1 ≛[h, o, V] L2 → ⊥) ∨ (L1.ⓧ ≛[h, o, T] L2.ⓧ → ⊥).
+lemma rdneq_inv_bind_void: ∀p,I,L1,L2,V,T. (L1 ≛[ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ≛[V] L2 → ⊥) ∨ (L1.ⓧ ≛[T] L2.ⓧ → ⊥).
/3 width=3 by rnex_inv_bind_void, tdeq_dec/ qed-.
include "static_2/static/req_fsle.ma".
include "static_2/static/rdeq.ma".
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+(* SORT-IRRELEVANT EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ***)
(* Properties with syntactic equivalence on referred entries ****************)
-lemma req_rdeq: ∀h,o,L1,L2. ∀T:term. L1 ≡[T] L2 → L1 ≛[h, o, T] L2.
+lemma req_rdeq: ∀L1,L2. ∀T:term. L1 ≡[T] L2 → L1 ≛[T] L2.
/2 width=3 by rex_co/ qed.
-lemma req_rdeq_trans: ∀h,o,L1,L. ∀T:term. L1 ≡[T] L →
- ∀L2. L ≛[h, o, T] L2 → L1 ≛[h, o, T] L2.
+lemma req_rdeq_trans: ∀L1,L. ∀T:term. L1 ≡[T] L →
+ ∀L2. L ≛[T] L2 → L1 ≛[T] L2.
/2 width=3 by req_rex_trans/ qed-.
(* Basic_2A1: was: llpx_sn_lrefl *)
(* Basic_2A1: this should have been lleq_fwd_llpx_sn *)
lemma req_fwd_rex: ∀R. c_reflexive … R →
- ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤[R, T] L2.
+ ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤[R,T] L2.
#R #HR #L1 #L2 #T * #f #Hf #HL12
/4 width=7 by sex_co, cext2_co, ex2_intro/
qed-.
(* Basic_properties *********************************************************)
-lemma frees_req_conf: ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≘ f →
- ∀L2. L1 ≡[T] L2 → L2 ⊢ 𝐅*⦃T⦄ ≘ f.
+lemma frees_req_conf: ∀f,L1,T. L1 ⊢ 𝐅+⦃T⦄ ≘ f →
+ ∀L2. L1 ≡[T] L2 → L2 ⊢ 𝐅+⦃T⦄ ≘ f.
#f #L1 #T #H elim H -f -L1 -T
[ /2 width=3 by frees_sort/
| #f #i #Hf #L2 #H2
(* Basic_2A1: uses: lleq_inv_lift_le lleq_inv_lift_be lleq_inv_lift_ge *)
lemma req_inv_lifts_bi: ∀L1,L2,U. L1 ≡[U] L2 → ∀b,f. 𝐔⦃f⦄ →
- ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+ ∀K1,K2. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
∀T. ⬆*[f] T ≘ U → K1 ≡[T] K2.
/2 width=10 by rex_inv_lifts_bi/ qed-.
(* Forward lemmas with free variables inclusion for restricted closures *****)
lemma req_rex_trans: ∀R. req_transitive R →
- ∀L1,L,T. L1 ≡[T] L → ∀L2. L ⪤[R, T] L2 → L1 ⪤[R, T] L2.
+ ∀L1,L,T. L1 ≡[T] L → ∀L2. L ⪤[R,T] L2 → L1 ⪤[R,T] L2.
/4 width=16 by req_fsle_comp, rex_trans_fsle, rex_trans_next/ qed-.
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
definition rex (R) (T): relation lenv ≝
- λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≘ f & L1 ⪤[cext2 R, cfull, f] L2.
+ λL1,L2. ∃∃f. L1 ⊢ 𝐅+⦃T⦄ ≘ f & L1 ⪤[cext2 R,cfull,f] L2.
interpretation "generic extension on referred entries (local environment)"
'Relation R T L1 L2 = (rex R T L1 L2).
(relation3 lenv term term) … ≝
λR1,R2,RP1,RP2.
∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. L0 ⪤[RP1, T0] L1 → ∀L2. L0 ⪤[RP2, T0] L2 →
+ ∀L1. L0 ⪤[RP1,T0] L1 → ∀L2. L0 ⪤[RP2,T0] L2 →
∃∃T. R2 L1 T1 T & R1 L2 T2 T.
definition rex_confluent: relation … ≝
λR1,R2.
- ∀K1,K,V1. K1 ⪤[R1, V1] K → ∀V. R1 K1 V1 V →
- ∀K2. K ⪤[R2, V] K2 → K ⪤[R2, V1] K2.
+ ∀K1,K,V1. K1 ⪤[R1,V1] K → ∀V. R1 K1 V1 V →
+ ∀K2. K ⪤[R2,V] K2 → K ⪤[R2,V1] K2.
definition rex_transitive: relation3 ? (relation3 ?? term) … ≝
λR1,R2,R3.
- ∀K1,K,V1. K1 ⪤[R1, V1] K →
+ ∀K1,K,V1. K1 ⪤[R1,V1] K →
∀V. R1 K1 V1 V → ∀V2. R2 K V V2 → R3 K1 V1 V2.
(* Basic inversion lemmas ***************************************************)
-lemma rex_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤[R, T] Y2 → Y2 = ⋆.
+lemma rex_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤[R,T] Y2 → Y2 = ⋆.
#R #Y2 #T * /2 width=4 by sex_inv_atom1/
qed-.
-lemma rex_inv_atom_dx (R): ∀Y1,T. Y1 ⪤[R, T] ⋆ → Y1 = ⋆.
+lemma rex_inv_atom_dx (R): ∀Y1,T. Y1 ⪤[R,T] ⋆ → Y1 = ⋆.
#R #I #Y1 * /2 width=4 by sex_inv_atom2/
qed-.
-lemma rex_inv_sort (R): ∀Y1,Y2,s. Y1 ⪤[R, ⋆s] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤[R, ⋆s] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+lemma rex_inv_sort (R):
+ ∀Y1,Y2,s. Y1 ⪤[R,⋆s] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤[R,⋆s] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R * [ | #Y1 #I1 ] #Y2 #s * #f #H1 #H2
[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
| lapply (frees_inv_sort … H1) -H1 #Hf
]
qed-.
-lemma rex_inv_zero (R): ∀Y1,Y2. Y1 ⪤[R, #0] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 & R L1 V1 V2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
- | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cext2 R, cfull, f] L2 &
- Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
+lemma rex_inv_zero (R):
+ ∀Y1,Y2. Y1 ⪤[R,#0] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
+ | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cext2 R,cfull,f] L2 &
+ Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
#R * [ | #Y1 * #I1 [ | #X ] ] #Y2 * #f #H1 #H2
[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or3_intro0, conj/
| elim (frees_inv_unit … H1) -H1 #g #HX #H destruct
]
qed-.
-lemma rex_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤[R, #↑i] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤[R, #i] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+lemma rex_inv_lref (R):
+ ∀Y1,Y2,i. Y1 ⪤[R,#↑i] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤[R,#i] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R * [ | #Y1 #I1 ] #Y2 #i * #f #H1 #H2
[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
| elim (frees_inv_lref … H1) -H1 #g #Hg #H destruct
]
qed-.
-lemma rex_inv_gref (R): ∀Y1,Y2,l. Y1 ⪤[R, §l] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤[R, §l] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+lemma rex_inv_gref (R):
+ ∀Y1,Y2,l. Y1 ⪤[R,§l] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤[R,§l] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R * [ | #Y1 #I1 ] #Y2 #l * #f #H1 #H2
[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
| lapply (frees_inv_gref … H1) -H1 #Hf
qed-.
(* Basic_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *)
-lemma rex_inv_bind (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
- ∧∧ L1 ⪤[R, V1] L2 & L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2.
+lemma rex_inv_bind (R):
+ ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R,ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
+ ∧∧ L1 ⪤[R,V1] L2 & L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2.
#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
/6 width=6 by sle_sex_trans, sex_inv_tl, ext2_pair, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
qed-.
(* Basic_2A1: uses: llpx_sn_inv_flat *)
-lemma rex_inv_flat (R): ∀I,L1,L2,V,T. L1 ⪤[R, ⓕ{I}V.T] L2 →
- ∧∧ L1 ⪤[R, V] L2 & L1 ⪤[R, T] L2.
+lemma rex_inv_flat (R):
+ ∀I,L1,L2,V,T. L1 ⪤[R,ⓕ{I}V.T] L2 →
+ ∧∧ L1 ⪤[R,V] L2 & L1 ⪤[R,T] L2.
#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
/5 width=6 by sle_sex_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma rex_inv_sort_bind_sn (R): ∀I1,K1,L2,s. K1.ⓘ{I1} ⪤[R, ⋆s] L2 →
- ∃∃I2,K2. K1 ⪤[R, ⋆s] K2 & L2 = K2.ⓘ{I2}.
+lemma rex_inv_sort_bind_sn (R):
+ ∀I1,K1,L2,s. K1.ⓘ{I1} ⪤[R,⋆s] L2 →
+ ∃∃I2,K2. K1 ⪤[R,⋆s] K2 & L2 = K2.ⓘ{I2}.
#R #I1 #K1 #L2 #s #H elim (rex_inv_sort … H) -H *
[ #H destruct
| #Z1 #I2 #Y1 #K2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rex_inv_sort_bind_dx (R): ∀I2,K2,L1,s. L1 ⪤[R, ⋆s] K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⪤[R, ⋆s] K2 & L1 = K1.ⓘ{I1}.
+lemma rex_inv_sort_bind_dx (R):
+ ∀I2,K2,L1,s. L1 ⪤[R,⋆s] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤[R,⋆s] K2 & L1 = K1.ⓘ{I1}.
#R #I2 #K2 #L1 #s #H elim (rex_inv_sort … H) -H *
[ #_ #H destruct
| #I1 #Z2 #K1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rex_inv_zero_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R, #0] L2 →
- ∃∃K2,V2. K1 ⪤[R, V1] K2 & R K1 V1 V2 &
- L2 = K2.ⓑ{I}V2.
+lemma rex_inv_zero_pair_sn (R):
+ ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R,#0] L2 →
+ ∃∃K2,V2. K1 ⪤[R,V1] K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
#R #I #L2 #K1 #V1 #H elim (rex_inv_zero … H) -H *
[ #H destruct
| #Z #Y1 #K2 #X1 #V2 #HK12 #HV12 #H1 #H2 destruct
]
qed-.
-lemma rex_inv_zero_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R, #0] K2.ⓑ{I}V2 →
- ∃∃K1,V1. K1 ⪤[R, V1] K2 & R K1 V1 V2 &
- L1 = K1.ⓑ{I}V1.
+lemma rex_inv_zero_pair_dx (R):
+ ∀I,L1,K2,V2. L1 ⪤[R,#0] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. K1 ⪤[R,V1] K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
#R #I #L1 #K2 #V2 #H elim (rex_inv_zero … H) -H *
[ #_ #H destruct
| #Z #K1 #Y2 #V1 #X2 #HK12 #HV12 #H1 #H2 destruct
]
qed-.
-lemma rex_inv_zero_unit_sn (R): ∀I,K1,L2. K1.ⓤ{I} ⪤[R, #0] L2 →
- ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤[cext2 R, cfull, f] K2 &
- L2 = K2.ⓤ{I}.
+lemma rex_inv_zero_unit_sn (R):
+ ∀I,K1,L2. K1.ⓤ{I} ⪤[R,#0] L2 →
+ ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤[cext2 R,cfull,f] K2 & L2 = K2.ⓤ{I}.
#R #I #K1 #L2 #H elim (rex_inv_zero … H) -H *
[ #H destruct
| #Z #Y1 #Y2 #X1 #X2 #_ #_ #H destruct
]
qed-.
-lemma rex_inv_zero_unit_dx (R): ∀I,L1,K2. L1 ⪤[R, #0] K2.ⓤ{I} →
- ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤[cext2 R, cfull, f] K2 &
- L1 = K1.ⓤ{I}.
+lemma rex_inv_zero_unit_dx (R):
+ ∀I,L1,K2. L1 ⪤[R,#0] K2.ⓤ{I} →
+ ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤[cext2 R,cfull,f] K2 & L1 = K1.ⓤ{I}.
#R #I #L1 #K2 #H elim (rex_inv_zero … H) -H *
[ #_ #H destruct
| #Z #Y1 #Y2 #X1 #X2 #_ #_ #_ #H destruct
]
qed-.
-lemma rex_inv_lref_bind_sn (R): ∀I1,K1,L2,i. K1.ⓘ{I1} ⪤[R, #↑i] L2 →
- ∃∃I2,K2. K1 ⪤[R, #i] K2 & L2 = K2.ⓘ{I2}.
+lemma rex_inv_lref_bind_sn (R):
+ ∀I1,K1,L2,i. K1.ⓘ{I1} ⪤[R,#↑i] L2 →
+ ∃∃I2,K2. K1 ⪤[R,#i] K2 & L2 = K2.ⓘ{I2}.
#R #I1 #K1 #L2 #i #H elim (rex_inv_lref … H) -H *
[ #H destruct
| #Z1 #I2 #Y1 #K2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rex_inv_lref_bind_dx (R): ∀I2,K2,L1,i. L1 ⪤[R, #↑i] K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⪤[R, #i] K2 & L1 = K1.ⓘ{I1}.
+lemma rex_inv_lref_bind_dx (R):
+ ∀I2,K2,L1,i. L1 ⪤[R,#↑i] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤[R,#i] K2 & L1 = K1.ⓘ{I1}.
#R #I2 #K2 #L1 #i #H elim (rex_inv_lref … H) -H *
[ #_ #H destruct
| #I1 #Z2 #K1 #Y2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rex_inv_gref_bind_sn (R): ∀I1,K1,L2,l. K1.ⓘ{I1} ⪤[R, §l] L2 →
- ∃∃I2,K2. K1 ⪤[R, §l] K2 & L2 = K2.ⓘ{I2}.
+lemma rex_inv_gref_bind_sn (R):
+ ∀I1,K1,L2,l. K1.ⓘ{I1} ⪤[R,§l] L2 →
+ ∃∃I2,K2. K1 ⪤[R,§l] K2 & L2 = K2.ⓘ{I2}.
#R #I1 #K1 #L2 #l #H elim (rex_inv_gref … H) -H *
[ #H destruct
| #Z1 #I2 #Y1 #K2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma rex_inv_gref_bind_dx (R): ∀I2,K2,L1,l. L1 ⪤[R, §l] K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⪤[R, §l] K2 & L1 = K1.ⓘ{I1}.
+lemma rex_inv_gref_bind_dx (R):
+ ∀I2,K2,L1,l. L1 ⪤[R,§l] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤[R,§l] K2 & L1 = K1.ⓘ{I1}.
#R #I2 #K2 #L1 #l #H elim (rex_inv_gref … H) -H *
[ #_ #H destruct
| #I1 #Z2 #K1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
(* Basic forward lemmas *****************************************************)
-lemma rex_fwd_zero_pair (R): ∀I,K1,K2,V1,V2.
- K1.ⓑ{I}V1 ⪤[R, #0] K2.ⓑ{I}V2 → K1 ⪤[R, V1] K2.
+lemma rex_fwd_zero_pair (R):
+ ∀I,K1,K2,V1,V2. K1.ⓑ{I}V1 ⪤[R,#0] K2.ⓑ{I}V2 → K1 ⪤[R,V1] K2.
#R #I #K1 #K2 #V1 #V2 #H
elim (rex_inv_zero_pair_sn … H) -H #Y #X #HK12 #_ #H destruct //
qed-.
(* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *)
-lemma rex_fwd_pair_sn (R): ∀I,L1,L2,V,T. L1 ⪤[R, ②{I}V.T] L2 → L1 ⪤[R, V] L2.
+lemma rex_fwd_pair_sn (R): ∀I,L1,L2,V,T. L1 ⪤[R,②{I}V.T] L2 → L1 ⪤[R,V] L2.
#R * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL
[ elim (frees_inv_bind … Hf) | elim (frees_inv_flat … Hf) ] -Hf
/4 width=6 by sle_sex_trans, sor_inv_sle_sn, ex2_intro/
qed-.
(* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *)
-lemma rex_fwd_bind_dx (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R, ⓑ{p,I}V1.T] L2 →
- R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2.
+lemma rex_fwd_bind_dx (R):
+ ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R,ⓑ{p,I}V1.T] L2 →
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2.
#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (rex_inv_bind … H HV) -H -HV //
qed-.
(* Basic_2A1: uses: llpx_sn_fwd_flat_dx *)
-lemma rex_fwd_flat_dx (R): ∀I,L1,L2,V,T. L1 ⪤[R, ⓕ{I}V.T] L2 → L1 ⪤[R, T] L2.
+lemma rex_fwd_flat_dx (R): ∀I,L1,L2,V,T. L1 ⪤[R,ⓕ{I}V.T] L2 → L1 ⪤[R,T] L2.
#R #I #L1 #L2 #V #T #H elim (rex_inv_flat … H) -H //
qed-.
-lemma rex_fwd_dx (R): ∀I2,L1,K2,T. L1 ⪤[R, T] K2.ⓘ{I2} →
- ∃∃I1,K1. L1 = K1.ⓘ{I1}.
+lemma rex_fwd_dx (R):
+ ∀I2,L1,K2,T. L1 ⪤[R,T] K2.ⓘ{I2} →
+ ∃∃I1,K1. L1 = K1.ⓘ{I1}.
#R #I2 #L1 #K2 #T * #f elim (pn_split f) * #g #Hg #_ #Hf destruct
[ elim (sex_inv_push2 … Hf) | elim (sex_inv_next2 … Hf) ] -Hf #I1 #K1 #_ #_ #H destruct
/2 width=3 by ex1_2_intro/
(* Basic properties *********************************************************)
-lemma rex_atom (R): ∀I. ⋆ ⪤[R, ⓪{I}] ⋆.
+lemma rex_atom (R): ∀I. ⋆ ⪤[R,⓪{I}] ⋆.
#R * /3 width=3 by frees_sort, frees_atom, frees_gref, sex_atom, ex2_intro/
qed.
-lemma rex_sort (R): ∀I1,I2,L1,L2,s.
- L1 ⪤[R, ⋆s] L2 → L1.ⓘ{I1} ⪤[R, ⋆s] L2.ⓘ{I2}.
+lemma rex_sort (R):
+ ∀I1,I2,L1,L2,s. L1 ⪤[R,⋆s] L2 → L1.ⓘ{I1} ⪤[R,⋆s] L2.ⓘ{I2}.
#R #I1 #I2 #L1 #L2 #s * #f #Hf #H12
lapply (frees_inv_sort … Hf) -Hf
/4 width=3 by frees_sort, sex_push, isid_push, ex2_intro/
qed.
-lemma rex_pair (R): ∀I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 →
- R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R, #0] L2.ⓑ{I}V2.
+lemma rex_pair (R):
+ ∀I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 →
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R,#0] L2.ⓑ{I}V2.
#R #I1 #I2 #L1 #L2 #V1 *
/4 width=3 by ext2_pair, frees_pair, sex_next, ex2_intro/
qed.
-lemma rex_unit (R): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 →
- L1.ⓤ{I} ⪤[R, #0] L2.ⓤ{I}.
+lemma rex_unit (R):
+ ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R,cfull,f] L2 →
+ L1.ⓤ{I} ⪤[R,#0] L2.ⓤ{I}.
/4 width=3 by frees_unit, sex_next, ext2_unit, ex2_intro/ qed.
-lemma rex_lref (R): ∀I1,I2,L1,L2,i.
- L1 ⪤[R, #i] L2 → L1.ⓘ{I1} ⪤[R, #↑i] L2.ⓘ{I2}.
+lemma rex_lref (R):
+ ∀I1,I2,L1,L2,i. L1 ⪤[R,#i] L2 → L1.ⓘ{I1} ⪤[R,#↑i] L2.ⓘ{I2}.
#R #I1 #I2 #L1 #L2 #i * /3 width=3 by sex_push, frees_lref, ex2_intro/
qed.
-lemma rex_gref (R): ∀I1,I2,L1,L2,l.
- L1 ⪤[R, §l] L2 → L1.ⓘ{I1} ⪤[R, §l] L2.ⓘ{I2}.
+lemma rex_gref (R):
+ ∀I1,I2,L1,L2,l. L1 ⪤[R,§l] L2 → L1.ⓘ{I1} ⪤[R,§l] L2.ⓘ{I2}.
#R #I1 #I2 #L1 #L2 #l * #f #Hf #H12
lapply (frees_inv_gref … Hf) -Hf
/4 width=3 by frees_gref, sex_push, isid_push, ex2_intro/
qed.
-lemma rex_bind_repl_dx (R): ∀I,I1,L1,L2,T.
- L1.ⓘ{I} ⪤[R, T] L2.ⓘ{I1} →
- ∀I2. cext2 R L1 I I2 →
- L1.ⓘ{I} ⪤[R, T] L2.ⓘ{I2}.
+lemma rex_bind_repl_dx (R):
+ ∀I,I1,L1,L2,T. L1.ⓘ{I} ⪤[R,T] L2.ⓘ{I1} →
+ ∀I2. cext2 R L1 I I2 → L1.ⓘ{I} ⪤[R,T] L2.ⓘ{I2}.
#R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #HR
/3 width=5 by sex_pair_repl, ex2_intro/
qed-.
(* Basic_2A1: uses: llpx_sn_co *)
-lemma rex_co (R1) (R2): (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- ∀L1,L2,T. L1 ⪤[R1, T] L2 → L1 ⪤[R2, T] L2.
+lemma rex_co (R1) (R2):
+ (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
+ ∀L1,L2,T. L1 ⪤[R1,T] L2 → L1 ⪤[R2,T] L2.
#R1 #R2 #HR #L1 #L2 #T * /5 width=7 by sex_co, cext2_co, ex2_intro/
qed-.
-lemma rex_isid (R1) (R2): ∀L1,L2,T1,T2.
- (∀f. L1 ⊢ 𝐅*⦃T1⦄ ≘ f → 𝐈⦃f⦄) →
- (∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅*⦃T2⦄ ≘ f) →
- L1 ⪤[R1, T1] L2 → L1 ⪤[R2, T2] L2.
+lemma rex_isid (R1) (R2):
+ ∀L1,L2,T1,T2.
+ (∀f. L1 ⊢ 𝐅+⦃T1⦄ ≘ f → 𝐈⦃f⦄) →
+ (∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅+⦃T2⦄ ≘ f) →
+ L1 ⪤[R1,T1] L2 → L1 ⪤[R2,T2] L2.
#R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 *
/4 width=7 by sex_co_isid, ex2_intro/
qed-.
lemma rex_unit_sn (R1) (R2):
- ∀I,K1,L2. K1.ⓤ{I} ⪤[R1, #0] L2 → K1.ⓤ{I} ⪤[R2, #0] L2.
+ ∀I,K1,L2. K1.ⓤ{I} ⪤[R1,#0] L2 → K1.ⓤ{I} ⪤[R2,#0] L2.
#R1 #R2 #I #K1 #L2 #H
elim (rex_inv_zero_unit_sn … H) -H #f #K2 #Hf #HK12 #H destruct
/3 width=7 by rex_unit, sex_co_isid/
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
definition f_dedropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
- ∀K2,T. K1 ⪤[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
- ∃∃L2. L1 ⪤[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 →
+ ∀K2,T. K1 ⪤[R,T] K2 → ∀U. ⬆*[f] T ≘ U →
+ ∃∃L2. L1 ⪤[R,U] L2 & ⬇*[b,f] L2 ≘ K2 & L1 ≡[f] L2.
definition f_dropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
- ∀L2,U. L1 ⪤[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
- ∃∃K2. K1 ⪤[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
+ λR. ∀b,f,L1,K1. ⬇*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ ∀L2,U. L1 ⪤[R,U] L2 → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K2. K1 ⪤[R,T] K2 & ⬇*[b,f] L2 ≘ K2.
definition f_dropable_dx: predicate (relation3 lenv term term) ≝
- λR. ∀L1,L2,U. L1 ⪤[R, U] L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[R, T] K2.
+ λR. ∀L1,L2,U. L1 ⪤[R,U] L2 →
+ ∀b,f,K2. ⬇*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K1. ⬇*[b,f] L1 ≘ K1 & K1 ⪤[R,T] K2.
definition f_transitive_next: relation3 … ≝ λR1,R2,R3.
- ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
+ ∀f,L,T. L ⊢ 𝐅+⦃T⦄ ≘ f →
∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f →
sex_transitive (cext2 R1) (cext2 R2) (cext2 R3) (cext2 R1) cfull g K I.
(* Properties with generic slicing for local environments *******************)
-lemma rex_liftable_dedropable_sn: ∀R. (∀L. reflexive ? (R L)) →
- d_liftable2_sn … lifts R → f_dedropable_sn R.
+lemma rex_liftable_dedropable_sn (R):
+ (∀L. reflexive ? (R L)) →
+ d_liftable2_sn … lifts R → f_dedropable_sn R.
#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 #T * #f1 #Hf1 #HK12 #U #HTU
elim (frees_total L1 U) #f2 #Hf2
lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
/3 width=6 by cext2_d_liftable2_sn, cfull_lift_sn, ext2_refl, ex3_intro, ex2_intro/
qed-.
-lemma rex_trans_next: ∀R1,R2,R3. rex_transitive R1 R2 R3 → f_transitive_next R1 R2 R3.
+lemma rex_trans_next (R1) (R2) (R3):
+ rex_transitive R1 R2 R3 → f_transitive_next R1 R2 R3.
#R1 #R2 #R3 #HR #f #L1 #T #Hf #g #I1 #K1 #n #HLK #Hgf #I #H
generalize in match HLK; -HLK elim H -I1 -I
[ #I #_ #L2 #_ #I2 #H
(* Basic_2A1: uses: llpx_sn_inv_lift_le llpx_sn_inv_lift_be llpx_sn_inv_lift_ge *)
(* Basic_2A1: was: llpx_sn_drop_conf_O *)
-lemma rex_dropable_sn: ∀R. f_dropable_sn R.
+lemma rex_dropable_sn (R): f_dropable_sn R.
#R #b #f #L1 #K1 #HLK1 #H1f #L2 #U * #f2 #Hf2 #HL12 #T #HTU
elim (frees_total K1 T) #f1 #Hf1
lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #H2f
(* Basic_2A1: was: llpx_sn_drop_trans_O *)
(* Note: the proof might be simplified *)
-lemma rex_dropable_dx: ∀R. f_dropable_dx R.
+lemma rex_dropable_dx (R): f_dropable_dx R.
#R #L1 #L2 #U * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #H1f #T #HTU
elim (drops_isuni_ex … H1f L1) #K1 #HLK1
elim (frees_total K1 T) #f1 #Hf1
qed-.
(* Basic_2A1: uses: llpx_sn_inv_lift_O *)
-lemma rex_inv_lifts_bi: ∀R,L1,L2,U. L1 ⪤[R, U] L2 → ∀b,f. 𝐔⦃f⦄ →
- ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- ∀T. ⬆*[f] T ≘ U → K1 ⪤[R, T] K2.
+lemma rex_inv_lifts_bi (R):
+ ∀L1,L2,U. L1 ⪤[R,U] L2 → ∀b,f. 𝐔⦃f⦄ →
+ ∀K1,K2. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
+ ∀T. ⬆*[f] T ≘ U → K1 ⪤[R,T] K2.
#R #L1 #L2 #U #HL12 #b #f #Hf #K1 #K2 #HLK1 #HLK2 #T #HTU
elim (rex_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY
lapply (drops_mono … HY … HLK2) -b -f -L2 #H destruct //
qed-.
-lemma rex_inv_lref_pair_sn: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R, V1] K2 & R K1 V1 V2.
+lemma rex_inv_lref_pair_sn (R):
+ ∀L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R,V1] K2 & R K1 V1 V2.
#R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (rex_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
#Y #HY #HLK2 elim (rex_inv_zero_pair_sn … HY) -HY
#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma rex_inv_lref_pair_dx: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R, V1] K2 & R K1 V1 V2.
+lemma rex_inv_lref_pair_dx (R):
+ ∀L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R,V1] K2 & R K1 V1 V2.
#R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (rex_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
#Y #HLK1 #HY elim (rex_inv_zero_pair_dx … HY) -HY
#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
lemma rex_inv_lref_pair_bi (R) (L1) (L2) (i):
- L1 ⪤[R, #i] L2 →
- ∀I1,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I1}V1 →
- ∀I2,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I2}V2 →
- ∧∧ K1 ⪤[R, V1] K2 & R K1 V1 V2 & I1 = I2.
+ L1 ⪤[R,#i] L2 →
+ ∀I1,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I1}V1 →
+ ∀I2,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I2}V2 →
+ ∧∧ K1 ⪤[R,V1] K2 & R K1 V1 V2 & I1 = I2.
#R #L1 #L2 #i #H12 #I1 #K1 #V1 #H1 #I2 #K2 #V2 #H2
elim (rex_inv_lref_pair_sn … H12 … H1) -L1 #Y2 #X2 #HLY2 #HK12 #HV12
lapply (drops_mono … HLY2 … H2) -HLY2 -H2 #H destruct
/2 width=1 by and3_intro/
qed-.
-lemma rex_inv_lref_unit_sn: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} →
- ∃∃f,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} & K1 ⪤[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
+lemma rex_inv_lref_unit_sn (R):
+ ∀L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} →
+ ∃∃f,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} & K1 ⪤[cext2 R,cfull,f] K2 & 𝐈⦃f⦄.
#R #L1 #L2 #i #HL12 #I #K1 #HLK1 elim (rex_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
#Y #HY #HLK2 elim (rex_inv_zero_unit_sn … HY) -HY
#f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma rex_inv_lref_unit_dx: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} →
- ∃∃f,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} & K1 ⪤[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
+lemma rex_inv_lref_unit_dx (R):
+ ∀L1,L2,i. L1 ⪤[R,#i] L2 → ∀I,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} →
+ ∃∃f,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} & K1 ⪤[cext2 R,cfull,f] K2 & 𝐈⦃f⦄.
#R #L1 #L2 #i #HL12 #I #K2 #HLK2 elim (rex_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
#Y #HLK1 #HY elim (rex_inv_zero_unit_dx … HY) -HY
#f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: llpx_sn_refl *)
-lemma rex_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⪤[R, T] L.
+lemma rex_refl (R): (∀L. reflexive … (R L)) → ∀L,T. L ⪤[R,T] L.
#R #HR #L #T elim (frees_total L T)
/4 width=3 by sex_refl, ext2_refl, ex2_intro/
qed.
-lemma rex_pair_refl: ∀R. (∀L. reflexive … (R L)) →
- ∀L,V1,V2. R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤[R, T] L.ⓑ{I}V2.
+lemma rex_pair_refl (R):
+ (∀L. reflexive … (R L)) →
+ ∀L,V1,V2. R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤[R,T] L.ⓑ{I}V2.
#R #HR #L #V1 #V2 #HV12 #I #T
elim (frees_total (L.ⓑ{I}V1) T) #f #Hf
elim (pn_split f) * #g #H destruct
(* Advanced inversion lemmas ************************************************)
-lemma rex_inv_bind_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R, ⓑ{p,I}V.T] L2 →
- L1 ⪤[R, V] L2 ∧ L1.ⓧ ⪤[R, T] L2.ⓧ.
+lemma rex_inv_bind_void (R):
+ ∀p,I,L1,L2,V,T. L1 ⪤[R,ⓑ{p,I}V.T] L2 → L1 ⪤[R,V] L2 ∧ L1.ⓧ ⪤[R,T] L2.ⓧ.
#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind_void … Hf) -Hf
/6 width=6 by sle_sex_trans, sex_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
qed-.
(* Advanced forward lemmas **************************************************)
-lemma rex_fwd_bind_dx_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R, ⓑ{p,I}V.T] L2 →
- L1.ⓧ ⪤[R, T] L2.ⓧ.
+lemma rex_fwd_bind_dx_void (R):
+ ∀p,I,L1,L2,V,T. L1 ⪤[R,ⓑ{p,I}V.T] L2 → L1.ⓧ ⪤[R,T] L2.ⓧ.
#R #p #I #L1 #L2 #V #T #H elim (rex_inv_bind_void … H) -H //
qed-.
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
definition R_fsge_compatible: predicate (relation3 …) ≝ λRN.
- ∀L,T1,T2. RN L T1 T2 → ⦃L, T2⦄ ⊆ ⦃L, T1⦄.
+ ∀L,T1,T2. RN L T1 T2 → ⦃L,T2⦄ ⊆ ⦃L,T1⦄.
definition rex_fsge_compatible: predicate (relation3 …) ≝ λRN.
- ∀L1,L2,T. L1 ⪤[RN, T] L2 → ⦃L2, T⦄ ⊆ ⦃L1, T⦄.
+ ∀L1,L2,T. L1 ⪤[RN,T] L2 → ⦃L2,T⦄ ⊆ ⦃L1,T⦄.
definition rex_fsle_compatible: predicate (relation3 …) ≝ λRN.
- ∀L1,L2,T. L1 ⪤[RN, T] L2 → ⦃L1, T⦄ ⊆ ⦃L2, T⦄.
+ ∀L1,L2,T. L1 ⪤[RN,T] L2 → ⦃L1,T⦄ ⊆ ⦃L2,T⦄.
(* Basic inversions with free variables inclusion for restricted closures ***)
-lemma frees_sex_conf: ∀R. rex_fsge_compatible R →
- ∀L1,T,f1. L1 ⊢ 𝐅*⦃T⦄ ≘ f1 →
- ∀L2. L1 ⪤[cext2 R, cfull, f1] L2 →
- ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 & f2 ⊆ f1.
+lemma frees_sex_conf (R):
+ rex_fsge_compatible R →
+ ∀L1,T,f1. L1 ⊢ 𝐅+⦃T⦄ ≘ f1 →
+ ∀L2. L1 ⪤[cext2 R,cfull,f1] L2 →
+ ∃∃f2. L2 ⊢ 𝐅+⦃T⦄ ≘ f2 & f2 ⊆ f1.
#R #HR #L1 #T #f1 #Hf1 #L2 #H1L
lapply (HR L1 L2 T ?) /2 width=3 by ex2_intro/ #H2L
@(fsle_frees_trans_eq … H2L … Hf1) /3 width=4 by sex_fwd_length, sym_eq/
(* Properties with free variables inclusion for restricted closures *********)
-(* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
-lemma fsge_rex_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
- ∀L2. L1 ⪤[R, T2] L2 → L1 ⪤[R, T1] L2.
+(* Note: we just need lveq_inv_refl: ∀L, n1, n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
+lemma fsge_rex_trans (R):
+ ∀L1,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L1,T2⦄ →
+ ∀L2. L1 ⪤[R,T2] L2 → L1 ⪤[R,T1] L2.
#R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12
elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct
/4 width=5 by rex_inv_frees, sle_sex_trans, ex2_intro/
qed-.
-lemma rex_sym: ∀R. rex_fsge_compatible R →
- (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
- ∀T. symmetric … (rex R T).
+lemma rex_sym (R):
+ rex_fsge_compatible R →
+ (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
+ ∀T. symmetric … (rex R T).
#R #H1R #H2R #T #L1 #L2
* #f1 #Hf1 #HL12
elim (frees_sex_conf … Hf1 … HL12) -Hf1 //
/5 width=5 by sle_sex_trans, sex_sym, cext2_sym, ex2_intro/
qed-.
-lemma rex_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- rex_fsge_compatible R1 →
- ∀L1,L2,V. L1 ⪤[R1, V] L2 → ∀I,T.
- ∃∃L. L1 ⪤[R1, ②{I}V.T] L & L ⪤[R2, V] L2.
+lemma rex_pair_sn_split (R1) (R2):
+ (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
+ rex_fsge_compatible R1 →
+ ∀L1,L2,V. L1 ⪤[R1,V] L2 → ∀I,T.
+ ∃∃L. L1 ⪤[R1,②{I}V.T] L & L ⪤[R2,V] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #V * #f #Hf #HL12 * [ #p ] #I #T
[ elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg
elim (frees_inv_bind … Hg) #y1 #y2 #H #_ #Hy
/4 width=7 by sle_sex_trans, ex2_intro/
qed-.
-lemma rex_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- rex_fsge_compatible R1 →
- ∀L1,L2,T. L1 ⪤[R1, T] L2 → ∀I,V.
- ∃∃L. L1 ⪤[R1, ⓕ{I}V.T] L & L ⪤[R2, T] L2.
+lemma rex_flat_dx_split (R1) (R2):
+ (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
+ rex_fsge_compatible R1 →
+ ∀L1,L2,T. L1 ⪤[R1,T] L2 → ∀I,V.
+ ∃∃L. L1 ⪤[R1,ⓕ{I}V.T] L & L ⪤[R2,T] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #I #V
elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg
elim (frees_inv_flat … Hg) #y1 #y2 #_ #H #Hy
/4 width=7 by sle_sex_trans, ex2_intro/
qed-.
-lemma rex_bind_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- rex_fsge_compatible R1 →
- ∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤[R1, T] L2 → ∀p.
- ∃∃L,V. L1 ⪤[R1, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤[R2, T] L2 & R1 L1 V1 V.
+lemma rex_bind_dx_split (R1) (R2):
+ (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
+ rex_fsge_compatible R1 →
+ ∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤[R1,T] L2 → ∀p.
+ ∃∃L,V. L1 ⪤[R1,ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤[R2,T] L2 & R1 L1 V1 V.
#R1 #R2 #HR1 #HR2 #HR #I #L1 #L2 #V1 #T * #f #Hf #HL12 #p
elim (frees_total L1 (ⓑ{p,I}V1.T)) #g #Hg
elim (frees_inv_bind … Hg) #y1 #y2 #_ #H #Hy
/4 width=7 by sle_sex_trans, ex3_2_intro, ex2_intro/
qed-.
-lemma rex_bind_dx_split_void: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- rex_fsge_compatible R1 →
- ∀L1,L2,T. L1.ⓧ ⪤[R1, T] L2 → ∀p,I,V.
- ∃∃L. L1 ⪤[R1, ⓑ{p,I}V.T] L & L.ⓧ ⪤[R2, T] L2.
+lemma rex_bind_dx_split_void (R1) (R2):
+ (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
+ rex_fsge_compatible R1 →
+ ∀L1,L2,T. L1.ⓧ ⪤[R1,T] L2 → ∀p,I,V.
+ ∃∃L. L1 ⪤[R1,ⓑ{p,I}V.T] L & L.ⓧ ⪤[R2,T] L2.
#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #p #I #V
elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg
elim (frees_inv_bind_void … Hg) #y1 #y2 #_ #H #Hy
(* Main properties with free variables inclusion for restricted closures ****)
-theorem rex_conf: ∀R1,R2.
- rex_fsge_compatible R1 →
- rex_fsge_compatible R2 →
- R_confluent2_rex R1 R2 R1 R2 →
- ∀T. confluent2 … (rex R1 T) (rex R2 T).
+theorem rex_conf (R1) (R2):
+ rex_fsge_compatible R1 → rex_fsge_compatible R2 →
+ R_confluent2_rex R1 R2 R1 R2 →
+ ∀T. confluent2 … (rex R1 T) (rex R2 T).
#R1 #R2 #HR1 #HR2 #HR12 #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02
lapply (frees_mono … Hf1 … Hf) -Hf1 #Hf12
lapply (sex_eq_repl_back … HL01 … Hf12) -f1 #HL01
]
qed-.
-theorem rex_trans_fsle: ∀R1,R2,R3.
- rex_fsle_compatible R1 → f_transitive_next R1 R2 R3 →
- ∀L1,L,T. L1 ⪤[R1, T] L →
- ∀L2. L ⪤[R2, T] L2 → L1 ⪤[R3, T] L2.
+theorem rex_trans_fsle (R1) (R2) (R3):
+ rex_fsle_compatible R1 → f_transitive_next R1 R2 R3 →
+ ∀L1,L,T. L1 ⪤[R1,T] L → ∀L2. L ⪤[R2,T] L2 → L1 ⪤[R3,T] L2.
#R1 #R2 #R3 #H1R #H2R #L1 #L #T #H
lapply (H1R … H) -H1R #H0
cases H -H #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
(* Forward lemmas with length for local environments ************************)
(* Basic_2A1: uses: llpx_sn_fwd_length *)
-lemma rex_fwd_length (R): ∀L1,L2,T. L1 ⪤[R, T] L2 → |L1| = |L2|.
+lemma rex_fwd_length (R): ∀L1,L2,T. L1 ⪤[R,T] L2 → |L1| = |L2|.
#R #L1 #L2 #T * /2 width=4 by sex_fwd_length/
qed-.
(* Properties with length for local environments ****************************)
(* Basic_2A1: uses: llpx_sn_sort *)
-lemma rex_sort_length (R): ∀L1,L2. |L1| = |L2| → ∀s. L1 ⪤[R, ⋆s] L2.
+lemma rex_sort_length (R): ∀L1,L2. |L1| = |L2| → ∀s. L1 ⪤[R,⋆s] L2.
#R #L1 elim L1 -L1
[ #Y #H #s >(length_inv_zero_sn … H) -H //
| #K1 #I1 #IH #Y #H #s
qed.
(* Basic_2A1: uses: llpx_sn_gref *)
-lemma rex_gref_length (R): ∀L1,L2. |L1| = |L2| → ∀l. L1 ⪤[R, §l] L2.
+lemma rex_gref_length (R): ∀L1,L2. |L1| = |L2| → ∀l. L1 ⪤[R,§l] L2.
#R #L1 elim L1 -L1
[ #Y #H #s >(length_inv_zero_sn … H) -H //
| #K1 #I1 #IH #Y #H #s
]
qed.
-lemma rex_unit_length (R): ∀L1,L2. |L1| = |L2| → ∀I. L1.ⓤ{I} ⪤[R, #0] L2.ⓤ{I}.
+lemma rex_unit_length (R): ∀L1,L2. |L1| = |L2| → ∀I. L1.ⓤ{I} ⪤[R,#0] L2.ⓤ{I}.
/3 width=3 by rex_unit, sex_length_isid/ qed.
(* Basic_2A1: uses: llpx_sn_lift_le llpx_sn_lift_ge *)
-lemma rex_lifts_bi (R): d_liftable2_sn … lifts R →
- ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ⪤[R, T] K2 →
- ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- ∀U. ⬆*[f] T ≘ U → L1 ⪤[R, U] L2.
+lemma rex_lifts_bi (R):
+ d_liftable2_sn … lifts R →
+ ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ⪤[R,T] K2 →
+ ∀b,f. ⬇*[b,f] L1 ≘ K1 → ⬇*[b,f] L2 ≘ K2 →
+ ∀U. ⬆*[f] T ≘ U → L1 ⪤[R,U] L2.
#R #HR #L1 #L2 #HL12 #K1 #K2 #T * #f1 #Hf1 #HK12 #b #f #HLK1 #HLK2 #U #HTU
elim (frees_total L1 U) #f2 #Hf2
lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
(* Inversion lemmas with length for local environment ***********************)
-lemma rex_inv_zero_length (R): ∀Y1,Y2. Y1 ⪤[R, #0] Y2 →
- ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 & R L1 V1 V2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
- | ∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
+lemma rex_inv_zero_length (R):
+ ∀Y1,Y2. Y1 ⪤[R,#0] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
+ | ∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
#R #Y1 #Y2 #H elim (rex_inv_zero … H) -H *
/4 width=9 by sex_fwd_length, ex4_5_intro, ex3_3_intro, or3_intro2, or3_intro1, or3_intro0, conj/
qed-.
(* Properties with generic extension of a context-sensitive relation ********)
-lemma rex_lex: ∀R,L1,L2. L1 ⪤[R] L2 → ∀T. L1 ⪤[R, T] L2.
+lemma rex_lex (R):
+ ∀L1,L2. L1 ⪤[R] L2 → ∀T. L1 ⪤[R,T] L2.
#R #L1 #L2 * #f #Hf #HL12 #T
elim (frees_total L1 T) #g #Hg
/4 width=5 by sex_sdj, sdj_isid_sn, ex2_intro/
(* Inversion lemmas with generic extension of a context sensitive relation **)
-lemma rex_inv_lex_req: ∀R. c_reflexive … R →
- rex_fsge_compatible R →
- ∀L1,L2,T. L1 ⪤[R, T] L2 →
- ∃∃L. L1 ⪤[R] L & L ≡[T] L2.
+lemma rex_inv_lex_req (R):
+ c_reflexive … R → rex_fsge_compatible R →
+ ∀L1,L2,T. L1 ⪤[R,T] L2 →
+ ∃∃L. L1 ⪤[R] L & L ≡[T] L2.
#R #H1R #H2R #L1 #L2 #T * #f1 #Hf1 #HL
elim (sex_sdj_split … ceq_ext … HL 𝐈𝐝 ?) -HL
[ #L0 #HL10 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ] -H1R
(* Advanced inversion lemmas ************************************************)
-lemma rex_inv_frees: ∀R,L1,L2,T. L1 ⪤[R, T] L2 →
- ∀f. L1 ⊢ 𝐅*⦃T⦄ ≘ f → L1 ⪤[cext2 R, cfull, f] L2.
+lemma rex_inv_frees (R):
+ ∀L1,L2,T. L1 ⪤[R,T] L2 →
+ ∀f. L1 ⊢ 𝐅+⦃T⦄ ≘ f → L1 ⪤[cext2 R,cfull,f] L2.
#R #L1 #L2 #T * /3 width=6 by frees_mono, sex_eq_repl_back/
qed-.
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: llpx_sn_dec *)
-lemma rex_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀L1,L2,T. Decidable (L1 ⪤[R, T] L2).
+lemma rex_dec (R):
+ (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀L1,L2,T. Decidable (L1 ⪤[R,T] L2).
#R #HR #L1 #L2 #T
elim (frees_total L1 T) #f #Hf
elim (sex_dec (cext2 R) cfull … L1 L2 f)
(* Main properties **********************************************************)
(* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *)
-theorem rex_bind: ∀R,p,I,L1,L2,V1,V2,T.
- L1 ⪤[R, V1] L2 → L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2 →
- L1 ⪤[R, ⓑ{p,I}V1.T] L2.
+theorem rex_bind (R) (p) (I):
+ ∀L1,L2,V1,V2,T. L1 ⪤[R,V1] L2 → L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2 →
+ L1 ⪤[R,ⓑ{p,I}V1.T] L2.
#R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
/3 width=7 by frees_fwd_isfin, frees_bind, sex_join, isfin_tl, ex2_intro/
qed.
(* Basic_2A1: llpx_sn_flat *)
-theorem rex_flat: ∀R,I,L1,L2,V,T.
- L1 ⪤[R, V] L2 → L1 ⪤[R, T] L2 →
- L1 ⪤[R, ⓕ{I}V.T] L2.
+theorem rex_flat (R) (I):
+ ∀L1,L2,V,T. L1 ⪤[R,V] L2 → L1 ⪤[R,T] L2 → L1 ⪤[R,ⓕ{I}V.T] L2.
#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
/3 width=7 by frees_fwd_isfin, frees_flat, sex_join, ex2_intro/
qed.
-theorem rex_bind_void: ∀R,p,I,L1,L2,V,T.
- L1 ⪤[R, V] L2 → L1.ⓧ ⪤[R, T] L2.ⓧ →
- L1 ⪤[R, ⓑ{p,I}V.T] L2.
+theorem rex_bind_void (R) (p) (I):
+ ∀L1,L2,V,T. L1 ⪤[R,V] L2 → L1.ⓧ ⪤[R,T] L2.ⓧ → L1 ⪤[R,ⓑ{p,I}V.T] L2.
#R #p #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
/3 width=7 by frees_fwd_isfin, frees_bind_void, sex_join, isfin_tl, ex2_intro/
(* Negated inversion lemmas *************************************************)
(* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *)
-lemma rnex_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) →
- (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⪤[R, T] L2.ⓑ{I}V → ⊥).
+lemma rnex_inv_bind (R):
+ (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀p,I,L1,L2,V,T. (L1 ⪤[R,ⓑ{p,I}V.T] L2 → ⊥) →
+ ∨∨ (L1 ⪤[R,V] L2 → ⊥) | (L1.ⓑ{I}V ⪤[R,T] L2.ⓑ{I}V → ⊥).
#R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
/4 width=2 by rex_bind, or_intror, or_introl/
qed-.
(* Basic_2A1: uses: nllpx_sn_inv_flat *)
-lemma rnex_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀I,L1,L2,V,T. (L1 ⪤[R, ⓕ{I}V.T] L2 → ⊥) →
- (L1 ⪤[R, V] L2 → ⊥) ∨ (L1 ⪤[R, T] L2 → ⊥).
+lemma rnex_inv_flat (R):
+ (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀I,L1,L2,V,T. (L1 ⪤[R,ⓕ{I}V.T] L2 → ⊥) →
+ ∨∨ (L1 ⪤[R,V] L2 → ⊥) | (L1 ⪤[R,T] L2 → ⊥).
#R #HR #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
/4 width=1 by rex_flat, or_intror, or_introl/
qed-.
-lemma rnex_inv_bind_void: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) →
- (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓧ ⪤[R, T] L2.ⓧ → ⊥).
+lemma rnex_inv_bind_void (R):
+ (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀p,I,L1,L2,V,T. (L1 ⪤[R,ⓑ{p,I}V.T] L2 → ⊥) →
+ ∨∨ (L1 ⪤[R,V] L2 → ⊥) | (L1.ⓧ ⪤[R,T] L2.ⓧ → ⊥).
#R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
/4 width=2 by rex_bind_void, or_intror, or_introl/
qed-.
]
]
qed-.
+
+lemma is_apear_dec (B) (X): Decidable (∃A. ②B.A = X).
+#B * [| #X #A ]
+[| elim (eq_aarity_dec X B) #HX ]
+[| /3 width=2 by ex_intro, or_introl/ ]
+@or_intror * #A #H destruct
+/2 width=1 by/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/lib/arith.ma".
+include "static_2/notation/functions/one_0.ma".
+include "static_2/notation/functions/two_0.ma".
+include "static_2/notation/functions/omega_0.ma".
+
+(* APPLICABILITY CONDITION **************************************************)
+
+(* applicability condition specification *)
+record ac: Type[0] ≝ {
+(* applicability domain *)
+ ad: predicate nat
+}.
+
+(* applicability condition postulates *)
+record ac_props (a): Prop ≝ {
+ ac_dec: ∀m. Decidable (∃∃n. ad a n & m ≤ n)
+}.
+
+(* Notable specifications ***************************************************)
+
+definition apply_top: predicate nat ≝ λn. ⊤.
+
+definition ac_top: ac ≝ mk_ac apply_top.
+
+interpretation "any number (applicability domain)"
+ 'Omega = (ac_top).
+
+lemma ac_top_props: ac_props ac_top ≝ mk_ac_props ….
+/3 width=3 by or_introl, ex2_intro/
+qed.
+
+definition ac_eq (k): ac ≝ mk_ac (eq … k).
+
+interpretation "one (applicability domain)"
+ 'Two = (ac_eq (S O)).
+
+interpretation "zero (applicability domain)"
+ 'One = (ac_eq O).
+
+lemma ac_eq_props (k): ac_props (ac_eq k) ≝ mk_ac_props ….
+#m elim (le_dec m k) #Hm
+[ /3 width=3 by or_introl, ex2_intro/
+| @or_intror * #n #Hn #Hmn destruct /2 width=1 by/
+]
+qed.
+
+definition ac_le (k): ac ≝ mk_ac (λn. n ≤ k).
+
+lemma ac_le_props (k): ac_props (ac_le k) ≝ mk_ac_props ….
+#m elim (le_dec m k) #Hm
+[ /3 width=3 by or_introl, ex2_intro/
+| @or_intror * #n #Hn #Hmn
+ /3 width=3 by transitive_le/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/ac.ma".
+
+(* APPLICABILITY CONDITION PREORDER *****************************************)
+
+definition acle: relation ac ≝
+ λa1,a2. ∀m. ad a1 m → ∃∃n. ad a2 n & m ≤ n.
+
+interpretation "preorder (applicability domain)"
+ 'subseteq a1 a2 = (acle a1 a2).
+
+(* Basic properties *********************************************************)
+
+lemma acle_refl: reflexive … acle.
+/2 width=3 by ex2_intro/ qed.
+
+lemma acle_omega (a): a ⊆ 𝛚.
+/2 width=1 by acle_refl/
+qed.
+
+lemma acle_one (a): ∀n. ad a n → 𝟏 ⊆ a.
+#a #n #Ha #m #Hm destruct
+/2 width=3 by ex2_intro/
+qed.
+
+lemma acle_eq_monotonic_le (k1) (k2):
+ k1 ≤ k2 → (ac_eq k1) ⊆ (ac_eq k2).
+#k1 #k2 #Hk #m #Hm destruct
+/2 width=3 by ex2_intro/
+qed.
+
+lemma acle_le_monotonic_le (k1) (k2):
+ k1 ≤ k2 → (ac_le k1) ⊆ (ac_le k2).
+#k1 #k2 #Hk #m #Hm
+/3 width=3 by acle_refl, transitive_le/
+qed.
+
+lemma acle_eq_le (k): (ac_eq k) ⊆ (ac_le k).
+#k #m #Hm destruct
+/2 width=1 by acle_refl, le_n/
+qed.
+
+lemma acle_le_eq (k): (ac_le k) ⊆ (ac_eq k).
+#k #m #Hm /2 width=3 by ex2_intro/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/acle.ma".
+
+(* APPLICABILITY CONDITION PREORDER *****************************************)
+
+(* Main properties **********************************************************)
+
+theorem acle_trans: Transitive … acle.
+#a1 #a #Ha1 #a2 #Ha2 #m1 #Hm1
+elim (Ha1 … Hm1) -Ha1 -Hm1 #m #Ha #Hm1
+elim (Ha2 … Ha) -Ha2 -Ha #m2 #Ha2 #Hm2
+/3 width=5 by transitive_le, ex2_intro/
+qed-.
(* Basic properties *********************************************************)
(* Basic_1: was: flt_shift *)
-lemma rfw_shift: ∀p,I,K,V,T. ♯{K.ⓑ{I}V, T} < ♯{K, ⓑ{p,I}V.T}.
+lemma rfw_shift: ∀p,I,K,V,T. ♯{K.ⓑ{I}V,T} < ♯{K,ⓑ{p,I}V.T}.
normalize /2 width=1 by monotonic_le_plus_r/
qed.
-lemma rfw_clear: ∀p,I1,I2,K,V,T. ♯{K.ⓤ{I1}, T} < ♯{K, ⓑ{p,I2}V.T}.
+lemma rfw_clear: ∀p,I1,I2,K,V,T. ♯{K.ⓤ{I1},T} < ♯{K,ⓑ{p,I2}V.T}.
normalize /4 width=1 by monotonic_le_plus_r, le_S_S/
qed.
-lemma rfw_tpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L, ②{I}V.T}.
+lemma rfw_tpair_sn: ∀I,L,V,T. ♯{L,V} < ♯{L,②{I}V.T}.
normalize in ⊢ (?→?→?→?→?%%); //
qed.
-lemma rfw_tpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L, ②{I}V.T}.
+lemma rfw_tpair_dx: ∀I,L,V,T. ♯{L,T} < ♯{L,②{I}V.T}.
normalize in ⊢ (?→?→?→?→?%%); //
qed.
-lemma rfw_lpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L.ⓑ{I}V, T}.
+lemma rfw_lpair_sn: ∀I,L,V,T. ♯{L,V} < ♯{L.ⓑ{I}V,T}.
normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
qed.
-lemma rfw_lpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L.ⓑ{I}V, T}.
+lemma rfw_lpair_dx: ∀I,L,V,T. ♯{L,T} < ♯{L.ⓑ{I}V,T}.
normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
qed.
(* Basic properties *********************************************************)
(* Basic_1: was: flt_shift *)
-lemma fw_shift: ∀p,I,G,K,V,T. ♯{G, K.ⓑ{I}V, T} < ♯{G, K, ⓑ{p,I}V.T}.
+lemma fw_shift: ∀p,I,G,K,V,T. ♯{G,K.ⓑ{I}V,T} < ♯{G,K,ⓑ{p,I}V.T}.
normalize /2 width=1 by monotonic_le_plus_r/
qed.
-lemma fw_clear: ∀p,I1,I2,G,K,V,T. ♯{G, K.ⓤ{I1}, T} < ♯{G, K, ⓑ{p,I2}V.T}.
+lemma fw_clear: ∀p,I1,I2,G,K,V,T. ♯{G,K.ⓤ{I1},T} < ♯{G,K,ⓑ{p,I2}V.T}.
normalize /4 width=1 by monotonic_le_plus_r, le_S_S/
qed.
-lemma fw_tpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L, ②{I}V.T}.
+lemma fw_tpair_sn: ∀I,G,L,V,T. ♯{G,L,V} < ♯{G,L,②{I}V.T}.
normalize in ⊢ (?→?→?→?→?→?%%); //
qed.
-lemma fw_tpair_dx: ∀I,G,L,V,T. ♯{G, L, T} < ♯{G, L, ②{I}V.T}.
+lemma fw_tpair_dx: ∀I,G,L,V,T. ♯{G,L,T} < ♯{G,L,②{I}V.T}.
normalize in ⊢ (?→?→?→?→?→?%%); //
qed.
-lemma fw_lpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L.ⓑ{I}V, T}.
+lemma fw_lpair_sn: ∀I,G,L,V,T. ♯{G,L,V} < ♯{G,L.ⓑ{I}V,T}.
normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/syntax/item_sh.ma".
-
-(* SORT DEGREE **************************************************************)
-
-(* sort degree specification *)
-record sd (h:sh): Type[0] ≝ {
- deg : relation nat; (* degree of the sort *)
- deg_total: ∀s. ∃d. deg s d; (* functional relation axioms *)
- deg_mono : ∀s,d1,d2. deg s d1 → deg s d2 → d1 = d2;
- deg_next : ∀s,d. deg s d → deg (next h s) (↓d) (* compatibility condition *)
-}.
-
-(* Notable specifications ***************************************************)
-
-definition deg_O: relation nat ≝ λs,d. d = 0.
-
-definition sd_O: ∀h. sd h ≝ λh. mk_sd h deg_O ….
-/2 width=2 by le_n_O_to_eq, le_n, ex_intro/ defined.
-
-(* Basic_2A1: includes: deg_SO_pos *)
-inductive deg_SO (h:sh) (s:nat) (s0:nat): predicate nat ≝
-| deg_SO_succ : ∀n. (next h)^n s0 = s → deg_SO h s s0 (↑n)
-| deg_SO_zero: ((∃n. (next h)^n s0 = s) → ⊥) → deg_SO h s s0 0
-.
-
-fact deg_SO_inv_succ_aux: ∀h,s,s0,n0. deg_SO h s s0 n0 → ∀n. n0 = ↑n →
- (next h)^n s0 = s.
-#h #s #s0 #n0 * -n0
-[ #n #Hn #x #H destruct //
-| #_ #x #H destruct
-]
-qed-.
-
-(* Basic_2A1: was: deg_SO_inv_pos *)
-lemma deg_SO_inv_succ: ∀h,s,s0,n. deg_SO h s s0 (↑n) → (next h)^n s0 = s.
-/2 width=3 by deg_SO_inv_succ_aux/ qed-.
-
-lemma deg_SO_refl: ∀h,s. deg_SO h s s 1.
-#h #s @(deg_SO_succ … 0 ?) //
-qed.
-
-lemma deg_SO_gt: ∀h,s1,s2. s1 < s2 → deg_SO h s1 s2 0.
-#h #s1 #s2 #HK12 @deg_SO_zero * #n elim n -n normalize
-[ #H destruct
- elim (lt_refl_false … HK12)
-| #n #_ #H
- lapply (next_lt h ((next h)^n s2)) >H -H #H
- lapply (transitive_lt … H HK12) -s1 #H1
- lapply (nexts_le h s2 n) #H2
- lapply (le_to_lt_to_lt … H2 H1) -h -n #H
- elim (lt_refl_false … H)
-]
-qed.
-
-definition sd_SO: ∀h. nat → sd h ≝ λh,s. mk_sd h (deg_SO h s) ….
-[ #s0
- lapply (nexts_dec h s0 s) *
- [ * /3 width=2 by deg_SO_succ, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
-| #K0 #d1 #d2 * [ #n1 ] #H1 * [1,3: #n2 ] #H2 //
- [ < H2 in H1; -H2 #H
- lapply (nexts_inj … H) -H #H destruct //
- | elim H1 /2 width=2 by ex_intro/
- | elim H2 /2 width=2 by ex_intro/
- ]
-| #s0 #n *
- [ #d #H destruct elim d -d normalize
- /2 width=1 by deg_SO_gt, deg_SO_succ, next_lt/
- | #H1 @deg_SO_zero * #d #H2 destruct
- @H1 -H1 @(ex_intro … (↑d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
- ]
-]
-defined.
-
-rec definition sd_d (h:sh) (s:nat) (d:nat) on d : sd h ≝
- match d with
- [ O ⇒ sd_O h
- | S d ⇒ match d with
- [ O ⇒ sd_SO h s
- | _ ⇒ sd_d h (next h s) d
- ]
- ].
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma deg_inv_pred: ∀h,o,s,d. deg h o (next h s) (↑d) → deg h o s (↑↑d).
-#h #o #s #d #H1
-elim (deg_total h o s) #n #H0
-lapply (deg_next … H0) #H2
-lapply (deg_mono … H1 H2) -H1 -H2 #H >H >S_pred /2 width=2 by ltn_to_ltO/
-qed-.
-
-lemma deg_inv_prec: ∀h,o,s,n,d. deg h o ((next h)^n s) (↑d) → deg h o s (↑(d+n)).
-#h #o #s #n elim n -n normalize /3 width=1 by deg_inv_pred/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma deg_iter: ∀h,o,s,d,n. deg h o s d → deg h o ((next h)^n s) (d-n).
-#h #o #s #d #n elim n -n normalize /3 width=1 by deg_next/
-qed.
-
-lemma deg_next_SO: ∀h,o,s,d. deg h o s (↑d) → deg h o (next h s) d.
-/2 width=1 by deg_next/ qed-.
-
-lemma sd_d_SS: ∀h,s,d. sd_d h s (↑↑d) = sd_d h (next h s) (↑d).
-// qed.
-
-lemma sd_d_correct: ∀h,d,s. deg h (sd_d h s d) s d.
-#h #d elim d -d // #d elim d -d /3 width=1 by deg_inv_pred/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/lib/arith.ma".
-
-(* SORT HIERARCHY ***********************************************************)
-
-(* sort hierarchy specification *)
-record sh: Type[0] ≝ {
- next : nat → nat; (* next sort in the hierarchy *)
- next_lt: ∀s. s < next s (* strict monotonicity condition *)
-}.
-
-definition sh_N: sh ≝ mk_sh S ….
-// defined.
-
-(* Basic properties *********************************************************)
-
-lemma nexts_le: ∀h,s,n. s ≤ (next h)^n s.
-#h #s #n elim n -n // normalize #n #IH
-lapply (next_lt h ((next h)^n s)) #H
-lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
-qed.
-
-lemma nexts_lt: ∀h,s,n. s < (next h)^(↑n) s.
-#h #s #n normalize
-lapply (nexts_le h s n) #H
-@(le_to_lt_to_lt … H) //
-qed.
-
-axiom nexts_dec: ∀h,s1,s2. Decidable (∃n. (next h)^n s1 = s2).
-
-axiom nexts_inj: ∀h,s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2.
(* Basic properties *********************************************************)
-lemma lveq_refl: ∀L. L ≋ⓧ*[0, 0] L.
+lemma lveq_refl: ∀L. L ≋ⓧ*[0,0] L.
#L elim L -L /2 width=1 by lveq_atom, lveq_bind/
qed.
(* Basic inversion lemmas ***************************************************)
-fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
0 = n1 → 0 = n2 →
∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
- | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
+ | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
[1: /3 width=1 by or_introl, conj/
|2: /3 width=7 by ex3_4_intro, or_intror/
]
qed-.
-lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 →
+lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0,0] L2 →
∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
- | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
+ | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
/2 width=5 by lveq_inv_zero_aux/ qed-.
-fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
∀m1. ↑m1 = n1 →
- ∃∃K1. K1 ≋ⓧ*[m1, 0] L2 & K1.ⓧ = L1 & 0 = n2.
+ ∃∃K1. K1 ≋ⓧ*[m1,0] L2 & K1.ⓧ = L1 & 0 = n2.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
[1: #m #H destruct
|2: #I1 #I2 #K1 #K2 #_ #m #H destruct
]
qed-.
-lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1, n2] K2 →
- ∃∃K1. K1 ≋ⓧ*[n1, 0] K2 & K1.ⓧ = L1 & 0 = n2.
+lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1,n2] K2 →
+ ∃∃K1. K1 ≋ⓧ*[n1,0] K2 & K1.ⓧ = L1 & 0 = n2.
/2 width=3 by lveq_inv_succ_sn_aux/ qed-.
-lemma lveq_inv_succ_dx: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1, ↑n2] L2 →
- ∃∃K2. K1 ≋ⓧ*[0, n2] K2 & K2.ⓧ = L2 & 0 = n1.
+lemma lveq_inv_succ_dx: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1,↑n2] L2 →
+ ∃∃K2. K1 ≋ⓧ*[0,n2] K2 & K2.ⓧ = L2 & 0 = n1.
#K1 #L2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H
elim (lveq_inv_succ_sn … H) -H /3 width=3 by lveq_sym, ex3_intro/
qed-.
-fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
∀m1,m2. ↑m1 = n1 → ↑m2 = n2 → ⊥.
#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
[1: #m1 #m2 #H1 #H2 destruct
]
qed-.
-lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1, ↑n2] L2 → ⊥.
+lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1,↑n2] L2 → ⊥.
/2 width=9 by lveq_inv_succ_aux/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lveq_inv_bind: ∀I1,I2,K1,K2. K1.ⓘ{I1} ≋ⓧ*[0, 0] K2.ⓘ{I2} → K1 ≋ⓧ*[0, 0] K2.
+lemma lveq_inv_bind: ∀I1,I2,K1,K2. K1.ⓘ{I1} ≋ⓧ*[0,0] K2.ⓘ{I2} → K1 ≋ⓧ*[0,0] K2.
#I1 #I2 #K1 #K2 #H
elim (lveq_inv_zero … H) -H * [| #Z1 #Z2 #Y1 #Y2 #HY ] #H1 #H2 destruct //
qed-.
-lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1, n2] ⋆ → ∧∧ 0 = n1 & 0 = n2.
+lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1,n2] ⋆ → ∧∧ 0 = n1 & 0 = n2.
* [2: #n1 ] * [2,4: #n2 ] #H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
]
qed-.
-lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1, n2] ⋆ →
- ∃∃m1. K1 ≋ⓧ*[m1, 0] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 0 = n2.
+lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1,n2] ⋆ →
+ ∃∃m1. K1 ≋ⓧ*[m1,0] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 0 = n2.
#I1 #K1 * [2: #n1 ] * [2,4: #n2 ] #H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
]
qed-.
-lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1, n2] K2.ⓘ{I2} →
- ∃∃m2. ⋆ ≋ⓧ*[0, m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2.
+lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1,n2] K2.ⓘ{I2} →
+ ∃∃m2. ⋆ ≋ⓧ*[0,m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2.
#I2 #K2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H
elim (lveq_inv_bind_atom … H) -H
/3 width=3 by lveq_sym, ex4_intro/
qed-.
-lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 →
- ∧∧ K1 ≋ⓧ*[0, 0] K2 & 0 = n1 & 0 = n2.
+lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1,n2] K2.ⓑ{I2}V2 →
+ ∧∧ K1 ≋ⓧ*[0,0] K2 & 0 = n1 & 0 = n2.
#I1 #I2 #K1 #K2 #V1 #V2 * [2: #n1 ] * [2,4: #n2 ] #H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
]
qed-.
-lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1, n2] L2 →
- ∧∧ L1 ≋ ⓧ*[n1, 0] L2 & 0 = n2.
+lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1,n2] L2 →
+ ∧∧ L1 ≋ ⓧ*[n1,0] L2 & 0 = n2.
#L1 #L2 #n1 #n2 #H
elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=1 by conj/
qed-.
-lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, ↑n2] L2.ⓧ →
- ∧∧ L1 ≋ ⓧ*[0, n2] L2 & 0 = n1.
+lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,↑n2] L2.ⓧ →
+ ∧∧ L1 ≋ ⓧ*[0,n2] L2 & 0 = n1.
#L1 #L2 #n1 #n2 #H
lapply (lveq_sym … H) -H #H
elim (lveq_inv_void_succ_sn … H) -H
(* Advanced forward lemmas **************************************************)
-lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
∨∨ 0 = n1 | 0 = n2.
#L1 #L2 * [2: #n1 ] * [2,4: #n2 ] #H
[ elim (lveq_inv_succ … H) ]
/2 width=1 by or_introl, or_intror/
qed-.
-lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2 → 0 = n1.
+lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1,n2] L2 → 0 = n1.
#I1 #K1 #L2 #V1 * [2: #n1 ] // * [2: #n2 ] #H
[ elim (lveq_inv_succ … H)
| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
]
qed-.
-lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 → 0 = n2.
+lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1,n2] K2.ⓑ{I2}V2 → 0 = n2.
/3 width=6 by lveq_fwd_pair_sn, lveq_sym/ qed-.
(* Properties with length for local environments ****************************)
-lemma lveq_length_eq: ∀L1,L2. |L1| = |L2| → L1 ≋ⓧ*[0, 0] L2.
+lemma lveq_length_eq: ∀L1,L2. |L1| = |L2| → L1 ≋ⓧ*[0,0] L2.
#L1 elim L1 -L1
[ #Y2 #H >(length_inv_zero_sn … H) -Y2 /2 width=3 by lveq_atom, ex_intro/
| #K1 #I1 #IH #Y2 #H
(* Forward lemmas with length for local environments ************************)
-lemma lveq_fwd_length_le_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → n1 ≤ |L1|.
+lemma lveq_fwd_length_le_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → n1 ≤ |L1|.
#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
/2 width=1 by le_S_S/
qed-.
-lemma lveq_fwd_length_le_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → n2 ≤ |L2|.
+lemma lveq_fwd_length_le_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → n2 ≤ |L2|.
#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
/2 width=1 by le_S_S/
qed-.
-lemma lveq_fwd_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+lemma lveq_fwd_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
∧∧ |L1|-|L2| = n1 & |L2|-|L1| = n2.
#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 /2 width=1 by conj/
#K1 #K2 #n #_ * #H1 #H2 >length_bind /3 width=1 by minus_Sn_m, conj/
qed-.
-lemma lveq_length_fwd_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → |L1| ≤ |L2| → 0 = n1.
+lemma lveq_length_fwd_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → |L1| ≤ |L2| → 0 = n1.
#L1 #L2 #n1 #n2 #H #HL
elim (lveq_fwd_length … H) -H
>(eq_minus_O … HL) //
qed-.
-lemma lveq_length_fwd_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → |L2| ≤ |L1| → 0 = n2.
+lemma lveq_length_fwd_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → |L2| ≤ |L1| → 0 = n2.
#L1 #L2 #n1 #n2 #H #HL
elim (lveq_fwd_length … H) -H
>(eq_minus_O … HL) //
qed-.
-lemma lveq_inj_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+lemma lveq_inj_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
|L1| = |L2| → ∧∧ 0 = n1 & 0 = n2.
#L1 #L2 #n1 #n2 #H #HL
elim (lveq_fwd_length … H) -H
>HL -HL /2 width=1 by conj/
qed-.
-lemma lveq_fwd_length_plus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+lemma lveq_fwd_length_plus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
|L1| + n2 = |L2| + n1.
#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
/2 width=2 by injective_plus_r/
qed-.
-lemma lveq_fwd_length_eq: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 → |L1| = |L2|.
+lemma lveq_fwd_length_eq: ∀L1,L2. L1 ≋ⓧ*[0,0] L2 → |L1| = |L2|.
/3 width=2 by lveq_fwd_length_plus, injective_plus_l/ qed-.
-lemma lveq_fwd_length_minus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+lemma lveq_fwd_length_minus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
|L1| - n1 = |L2| - n2.
/3 width=3 by lveq_fwd_length_plus, lveq_fwd_length_le_dx, lveq_fwd_length_le_sn, plus_to_minus_2/ qed-.
lemma lveq_fwd_abst_bind_length_le: ∀I1,I2,L1,L2,V1,n1,n2.
- L1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2.ⓘ{I2} → |L1| ≤ |L2|.
+ L1.ⓑ{I1}V1 ≋ⓧ*[n1,n2] L2.ⓘ{I2} → |L1| ≤ |L2|.
#I1 #I2 #L1 #L2 #V1 #n1 #n2 #HL
lapply (lveq_fwd_pair_sn … HL) #H destruct
elim (lveq_fwd_length … HL) -HL >length_bind >length_bind //
qed-.
lemma lveq_fwd_bind_abst_length_le: ∀I1,I2,L1,L2,V2,n1,n2.
- L1.ⓘ{I1} ≋ⓧ*[n1, n2] L2.ⓑ{I2}V2 → |L2| ≤ |L1|.
+ L1.ⓘ{I1} ≋ⓧ*[n1,n2] L2.ⓑ{I2}V2 → |L2| ≤ |L1|.
/3 width=6 by lveq_fwd_abst_bind_length_le, lveq_sym/ qed-.
(* Inversion lemmas with length for local environments **********************)
-lemma lveq_inv_void_dx_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2.ⓧ → |L1| ≤ |L2| →
- ∃∃m2. L1 ≋ ⓧ*[n1, m2] L2 & 0 = n1 & ↑m2 = n2.
+lemma lveq_inv_void_dx_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2.ⓧ → |L1| ≤ |L2| →
+ ∃∃m2. L1 ≋ ⓧ*[n1,m2] L2 & 0 = n1 & ↑m2 = n2.
#L1 #L2 #n1 #n2 #H #HL12
lapply (lveq_fwd_length_plus … H) normalize >plus_n_Sm #H0
lapply (plus2_inv_le_sn … H0 HL12) -H0 -HL12 #H0
elim (lveq_inv_void_succ_dx … H) -H /2 width=3 by ex3_intro/
qed-.
-lemma lveq_inv_void_sn_length: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[n1, n2] L2 → |L2| ≤ |L1| →
- ∃∃m1. L1 ≋ ⓧ*[m1, n2] L2 & ↑m1 = n1 & 0 = n2.
+lemma lveq_inv_void_sn_length: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[n1,n2] L2 → |L2| ≤ |L1| →
+ ∃∃m1. L1 ≋ ⓧ*[m1,n2] L2 & ↑m1 = n1 & 0 = n2.
#L1 #L2 #n1 #n2 #H #HL
lapply (lveq_sym … H) -H #H
elim (lveq_inv_void_dx_length … H HL) -H -HL
(* Main inversion lemmas ****************************************************)
-theorem lveq_inv_bind: ∀K1,K2. K1 ≋ⓧ*[0, 0] K2 →
- ∀I1,I2,m1,m2. K1.ⓘ{I1} ≋ⓧ*[m1, m2] K2.ⓘ{I2} →
+theorem lveq_inv_bind: ∀K1,K2. K1 ≋ⓧ*[0,0] K2 →
+ ∀I1,I2,m1,m2. K1.ⓘ{I1} ≋ⓧ*[m1,m2] K2.ⓘ{I2} →
∧∧ 0 = m1 & 0 = m2.
#K1 #K2 #HK #I1 #I2 #m1 #m2 #H
lapply (lveq_fwd_length_eq … HK) -HK #HK
elim (lveq_inj_length … H) -H normalize /3 width=1 by conj, eq_f/
qed-.
-theorem lveq_inj: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- ∀m1,m2. L1 ≋ⓧ*[m1, m2] L2 →
+theorem lveq_inj: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 →
+ ∀m1,m2. L1 ≋ⓧ*[m1,m2] L2 →
∧∧ n1 = m1 & n2 = m2.
#L1 #L2 #n1 #n2 #Hn #m1 #m2 #Hm
elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct
qed-.
theorem lveq_inj_void_sn_ge: ∀K1,K2. |K2| ≤ |K1| →
- ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
- ∀m1,m2. K1.ⓧ ≋ⓧ*[m1, m2] K2 →
+ ∀n1,n2. K1 ≋ⓧ*[n1,n2] K2 →
+ ∀m1,m2. K1.ⓧ ≋ⓧ*[m1,m2] K2 →
∧∧ ↑n1 = m1 & 0 = m2 & 0 = n2.
#L1 #L2 #HL #n1 #n2 #Hn #m1 #m2 #Hm
elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct
qed-.
theorem lveq_inj_void_dx_le: ∀K1,K2. |K1| ≤ |K2| →
- ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
- ∀m1,m2. K1 ≋ⓧ*[m1, m2] K2.ⓧ →
+ ∀n1,n2. K1 ≋ⓧ*[n1,n2] K2 →
+ ∀m1,m2. K1 ≋ⓧ*[m1,m2] K2.ⓧ →
∧∧ ↑n2 = m2 & 0 = m1 & 0 = n1.
/3 width=5 by lveq_inj_void_sn_ge, lveq_sym/ qed-. (* auto: 2x lveq_sym *)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/sh.ma".
+
+(* SORT DEGREE **************************************************************)
+
+(* sort degree specification *)
+record sd: Type[0] ≝ {
+(* degree of the sort *)
+ deg: relation nat
+}.
+
+(* sort degree postulates *)
+record sd_props (h) (o): Prop ≝ {
+(* functional relation axioms *)
+ deg_total: ∀s. ∃d. deg o s d;
+ deg_mono : ∀s,d1,d2. deg o s d1 → deg o s d2 → d1 = d2;
+(* compatibility condition *)
+ deg_next : ∀s,d. deg o s d → deg o (⫯[h]s) (↓d)
+}.
+
+(* Notable specifications ***************************************************)
+
+definition deg_O: relation nat ≝ λs,d. d = 0.
+
+definition sd_O: sd ≝ mk_sd deg_O.
+
+lemma sd_O_props (h): sd_props h sd_O ≝ mk_sd_props ….
+/2 width=2 by le_n_O_to_eq, le_n, ex_intro/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma deg_inv_pred (h) (o): sd_props h o →
+ ∀s,d. deg o (⫯[h]s) (↑d) → deg o s (↑↑d).
+#h #o #Ho #s #d #H1
+elim (deg_total … Ho s) #d0 #H0
+lapply (deg_next … Ho … H0) #H2
+lapply (deg_mono … Ho … H1 H2) -H1 -H2 #H >H >S_pred
+/2 width=2 by ltn_to_ltO/
+qed-.
+
+lemma deg_inv_prec (h) (o): sd_props h o →
+ ∀s,n,d. deg o ((next h)^n s) (↑d) → deg o s (↑(d+n)).
+#h #o #H0 #s #n elim n -n normalize /3 width=3 by deg_inv_pred/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma deg_iter (h) (o): sd_props h o →
+ ∀s,d,n. deg o s d → deg o ((next h)^n s) (d-n).
+#h #o #Ho #s #d #n elim n -n normalize /3 width=1 by deg_next/
+qed.
+
+lemma deg_next_SO (h) (o): sd_props h o →
+ ∀s,d. deg o s (↑d) → deg o (next h s) d.
+/2 width=1 by deg_next/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/pull/pull_2.ma".
+include "static_2/syntax/sh_props.ma".
+include "static_2/syntax/sd.ma".
+
+(* SORT DEGREE **************************************************************)
+
+(* Basic_2A1: includes: deg_SO_pos *)
+inductive deg_SO (h) (s) (s0): predicate nat ≝
+| deg_SO_succ : ∀n. (next h)^n s0 = s → deg_SO h s s0 (↑n)
+| deg_SO_zero: (∀n. (next h)^n s0 = s → ⊥) → deg_SO h s s0 0
+.
+
+fact deg_SO_inv_succ_aux (h) (s) (s0):
+ ∀n0. deg_SO h s s0 n0 → ∀n. n0 = ↑n → (next h)^n s0 = s.
+#h #s #s0 #n0 * -n0
+[ #n #Hn #x #H destruct //
+| #_ #x #H destruct
+]
+qed-.
+
+(* Basic_2A1: was: deg_SO_inv_pos *)
+lemma deg_SO_inv_succ (h) (s) (s0):
+ ∀n. deg_SO h s s0 (↑n) → (next h)^n s0 = s.
+/2 width=3 by deg_SO_inv_succ_aux/ qed-.
+
+lemma deg_SO_refl (h) (s): deg_SO h s s 1.
+#h #s @(deg_SO_succ … 0 ?) //
+qed.
+
+definition sd_SO (h) (s): sd ≝ mk_sd (deg_SO h s).
+
+lemma sd_SO_props (h) (s): sh_decidable h → sh_acyclic h →
+ sd_props h (sd_SO h s).
+#h #s #Hhd #Hha
+@mk_sd_props
+[ #s0
+ elim (nexts_dec … Hhd s0 s) -Hhd
+ [ * /3 width=2 by deg_SO_succ, ex_intro/
+ | /5 width=2 by deg_SO_zero, ex_intro/
+ ]
+| #s0 #d1 #d2 * [ #n1 ] #H1 * [1,3: #n2 ] #H2
+ [ < H2 in H1; -H2 #H
+ lapply (nexts_inj … Hha … H) -H #H destruct //
+ | elim H1 /2 width=2 by ex_intro/
+ | elim H2 /2 width=2 by ex_intro/
+ | //
+ ]
+| #s0 #d *
+ [ #n #H destruct cases n -n normalize
+ [ @deg_SO_zero #n >iter_n_Sm #H
+ lapply (nexts_inj … Hha … (↑n) 0 H) -H #H destruct
+ | #n @deg_SO_succ >iter_n_Sm //
+ ]
+ | #H0 @deg_SO_zero #n >iter_n_Sm #H destruct
+ /2 width=2 by/
+ ]
+]
+qed.
+
+rec definition sd_d (h:?) (s:?) (d:?) on d: sd ≝
+ match d with
+ [ O ⇒ sd_O
+ | S d ⇒ match d with
+ [ O ⇒ sd_SO h s
+ | _ ⇒ sd_d h (next h s) d
+ ]
+ ].
+
+lemma sd_d_props (h) (s) (d): sh_decidable h → sh_acyclic h →
+ sd_props h (sd_d h s d).
+#h @pull_2 * [ // ]
+#d elim d -d /2 width=1 by sd_SO_props/
+qed.
+
+(* Properties with sd_d *****************************************************)
+
+lemma sd_d_SS (h):
+ ∀s,d. sd_d h s (↑↑d) = sd_d h (⫯[h]s) (↑d).
+// qed.
+
+lemma sd_d_correct (h): sh_decidable h → sh_acyclic h →
+ ∀s,d. deg (sd_d h s d) s d.
+#h #Hhd #Hha @pull_2 #d elim d -d [ // ]
+#d elim d -d /3 width=3 by deg_inv_pred, sd_d_props/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/sh_lt.ma".
+include "static_2/syntax/sd_d.ma".
+
+(* SORT DEGREE **************************************************************)
+
+(* Properties with sh_lt ****************************************************)
+
+lemma deg_SO_gt (h): sh_lt h →
+ ∀s1,s2. s1 < s2 → deg_SO h s1 s2 0.
+#h #Hh #s1 #s2 #Hs12 @deg_SO_zero * normalize
+[ #H destruct
+ elim (lt_refl_false … Hs12)
+| #n #H
+ lapply (next_lt … Hh ((next h)^n s2)) >H -H #H
+ lapply (transitive_lt … H Hs12) -s1 #H1
+ /3 width=5 by lt_le_false, nexts_le/ (* full auto too slow *)
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/lib/arith.ma".
+include "static_2/notation/functions/upspoon_2.ma".
+
+(* SORT HIERARCHY ***********************************************************)
+
+(* sort hierarchy specification *)
+record sh: Type[0] ≝ {
+ next: nat → nat (* next sort in the hierarchy *)
+}.
+
+interpretation "next sort (sort hierarchy)"
+ 'UpSpoon h s = (next h s).
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/sh_props.ma".
+
+(* SORT HIERARCHY ***********************************************************)
+
+(* strict monotonicity condition *)
+record sh_lt (h): Prop ≝
+{
+ next_lt: ∀s. s < ⫯[h]s
+}.
+
+(* Basic properties *********************************************************)
+
+lemma nexts_le (h): sh_lt h → ∀s,n. s ≤ (next h)^n s.
+#h #Hh #s #n elim n -n [ // ] normalize #n #IH
+lapply (next_lt … Hh ((next h)^n s)) #H
+lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
+qed.
+
+lemma nexts_lt (h): sh_lt h → ∀s,n. s < (next h)^(↑n) s.
+#h #Hh #s #n normalize
+lapply (nexts_le … Hh s n) #H
+@(le_to_lt_to_lt … H) /2 width=1 by next_lt/
+qed.
+
+axiom sh_lt_dec (h): sh_lt h → sh_decidable h.
+
+axiom sh_lt_acyclic (h): sh_lt h → sh_acyclic h.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/sh.ma".
+
+(* SORT HIERARCHY ***********************************************************)
+
+(* acyclicity condition *)
+record sh_acyclic (h): Prop ≝
+{
+ nexts_inj: ∀s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2
+}.
+
+(* decidability condition *)
+record sh_decidable (h): Prop ≝
+{
+ nexts_dec: ∀s1,s2. Decidable (∃n. (next h)^n s1 = s2)
+}.
(* *)
(**************************************************************************)
-include "static_2/notation/relations/stareq_4.ma".
-include "static_2/syntax/item_sd.ma".
+include "static_2/notation/relations/stareq_2.ma".
include "static_2/syntax/term.ma".
-(* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************)
+(* SORT-IRRELEVANT EQUIVALENCE ON TERMS *************************************)
-inductive tdeq (h) (o): relation term ≝
-| tdeq_sort: ∀s1,s2,d. deg h o s1 d → deg h o s2 d → tdeq h o (⋆s1) (⋆s2)
-| tdeq_lref: ∀i. tdeq h o (#i) (#i)
-| tdeq_gref: ∀l. tdeq h o (§l) (§l)
-| tdeq_pair: ∀I,V1,V2,T1,T2. tdeq h o V1 V2 → tdeq h o T1 T2 → tdeq h o (②{I}V1.T1) (②{I}V2.T2)
+inductive tdeq: relation term ≝
+| tdeq_sort: ∀s1,s2. tdeq (⋆s1) (⋆s2)
+| tdeq_lref: ∀i. tdeq (#i) (#i)
+| tdeq_gref: ∀l. tdeq (§l) (§l)
+| tdeq_pair: ∀I,V1,V2,T1,T2. tdeq V1 V2 → tdeq T1 T2 → tdeq (②{I}V1.T1) (②{I}V2.T2)
.
interpretation
- "context-free degree-based equivalence (term)"
- 'StarEq h o T1 T2 = (tdeq h o T1 T2).
+ "context-free sort-irrelevant equivalence (term)"
+ 'StarEq T1 T2 = (tdeq T1 T2).
(* Basic properties *********************************************************)
-lemma tdeq_refl: ∀h,o. reflexive … (tdeq h o).
-#h #o #T elim T -T /2 width=1 by tdeq_pair/
+lemma tdeq_refl: reflexive … tdeq.
+#T elim T -T /2 width=1 by tdeq_pair/
* /2 width=1 by tdeq_lref, tdeq_gref/
-#s elim (deg_total h o s) /2 width=3 by tdeq_sort/
qed.
-lemma tdeq_sym: ∀h,o. symmetric … (tdeq h o).
-#h #o #T1 #T2 #H elim H -T1 -T2
+lemma tdeq_sym: symmetric … tdeq.
+#T1 #T2 #H elim H -T1 -T2
/2 width=3 by tdeq_sort, tdeq_lref, tdeq_gref, tdeq_pair/
qed-.
(* Basic inversion lemmas ***************************************************)
-fact tdeq_inv_sort1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀s1. X = ⋆s1 →
- ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
-#h #o #X #Y * -X -Y
-[ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/
+fact tdeq_inv_sort1_aux: ∀X,Y. X ≛ Y → ∀s1. X = ⋆s1 →
+ ∃s2. Y = ⋆s2.
+#X #Y * -X -Y
+[ #s1 #s2 #s #H destruct /2 width=2 by ex_intro/
| #i #s #H destruct
| #l #s #H destruct
| #I #V1 #V2 #T1 #T2 #_ #_ #s #H destruct
]
qed-.
-lemma tdeq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ≛[h, o] Y →
- ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
-/2 width=3 by tdeq_inv_sort1_aux/ qed-.
+lemma tdeq_inv_sort1: ∀Y,s1. ⋆s1 ≛ Y →
+ ∃s2. Y = ⋆s2.
+/2 width=4 by tdeq_inv_sort1_aux/ qed-.
-fact tdeq_inv_lref1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀i. X = #i → Y = #i.
-#h #o #X #Y * -X -Y //
-[ #s1 #s2 #d #_ #_ #j #H destruct
+fact tdeq_inv_lref1_aux: ∀X,Y. X ≛ Y → ∀i. X = #i → Y = #i.
+#X #Y * -X -Y //
+[ #s1 #s2 #j #H destruct
| #I #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
]
qed-.
-lemma tdeq_inv_lref1: ∀h,o,Y,i. #i ≛[h, o] Y → Y = #i.
+lemma tdeq_inv_lref1: ∀Y,i. #i ≛ Y → Y = #i.
/2 width=5 by tdeq_inv_lref1_aux/ qed-.
-fact tdeq_inv_gref1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀l. X = §l → Y = §l.
-#h #o #X #Y * -X -Y //
-[ #s1 #s2 #d #_ #_ #k #H destruct
+fact tdeq_inv_gref1_aux: ∀X,Y. X ≛ Y → ∀l. X = §l → Y = §l.
+#X #Y * -X -Y //
+[ #s1 #s2 #k #H destruct
| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
]
qed-.
-lemma tdeq_inv_gref1: ∀h,o,Y,l. §l ≛[h, o] Y → Y = §l.
+lemma tdeq_inv_gref1: ∀Y,l. §l ≛ Y → Y = §l.
/2 width=5 by tdeq_inv_gref1_aux/ qed-.
-fact tdeq_inv_pair1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀I,V1,T1. X = ②{I}V1.T1 →
- ∃∃V2,T2. V1 ≛[h, o] V2 & T1 ≛[h, o] T2 & Y = ②{I}V2.T2.
-#h #o #X #Y * -X -Y
-[ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct
+fact tdeq_inv_pair1_aux: ∀X,Y. X ≛ Y → ∀I,V1,T1. X = ②{I}V1.T1 →
+ ∃∃V2,T2. V1 ≛ V2 & T1 ≛ T2 & Y = ②{I}V2.T2.
+#X #Y * -X -Y
+[ #s1 #s2 #J #W1 #U1 #H destruct
| #i #J #W1 #U1 #H destruct
| #l #J #W1 #U1 #H destruct
| #I #V1 #V2 #T1 #T2 #HV #HT #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma tdeq_inv_pair1: ∀h,o,I,V1,T1,Y. ②{I}V1.T1 ≛[h, o] Y →
- ∃∃V2,T2. V1 ≛[h, o] V2 & T1 ≛[h, o] T2 & Y = ②{I}V2.T2.
+lemma tdeq_inv_pair1: ∀I,V1,T1,Y. ②{I}V1.T1 ≛ Y →
+ ∃∃V2,T2. V1 ≛ V2 & T1 ≛ T2 & Y = ②{I}V2.T2.
/2 width=3 by tdeq_inv_pair1_aux/ qed-.
-lemma tdeq_inv_pair2: ∀h,o,I,X1,V2,T2. X1 ≛[h, o] ②{I}V2.T2 →
- ∃∃V1,T1. V1 ≛[h, o] V2 & T1 ≛[h, o] T2 & X1 = ②{I}V1.T1.
-#h #o #I #X1 #V2 #T2 #H
-elim (tdeq_inv_pair1 h o I V2 T2 X1)
+lemma tdeq_inv_sort2: ∀X1,s2. X1 ≛ ⋆s2 →
+ ∃s1. X1 = ⋆s1.
+#X1 #s2 #H
+elim (tdeq_inv_sort1 X1 s2)
+/2 width=2 by tdeq_sym, ex_intro/
+qed-.
+
+lemma tdeq_inv_pair2: ∀I,X1,V2,T2. X1 ≛ ②{I}V2.T2 →
+ ∃∃V1,T1. V1 ≛ V2 & T1 ≛ T2 & X1 = ②{I}V1.T1.
+#I #X1 #V2 #T2 #H
+elim (tdeq_inv_pair1 I V2 T2 X1)
[ #V1 #T1 #HV #HT #H destruct ]
/3 width=5 by tdeq_sym, ex3_2_intro/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma tdeq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ≛[h, o] Y → ∀d. deg h o s1 d →
- ∃∃s2. deg h o s2 d & Y = ⋆s2.
-#h #o #Y #s1 #H #d #Hs1 elim (tdeq_inv_sort1 … H) -H
-#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/
-qed-.
-
-lemma tdeq_inv_sort_deg: ∀h,o,s1,s2. ⋆s1 ≛[h, o] ⋆s2 →
- ∀d1,d2. deg h o s1 d1 → deg h o s2 d2 →
- d1 = d2.
-#h #o #s1 #y #H #d1 #d2 #Hs1 #Hy
-elim (tdeq_inv_sort1_deg … H … Hs1) -s1 #s2 #Hs2 #H destruct
-<(deg_mono h o … Hy … Hs2) -s2 -d1 //
-qed-.
-
-lemma tdeq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ≛[h, o] ②{I2}V2.T2 →
- ∧∧ I1 = I2 & V1 ≛[h, o] V2 & T1 ≛[h, o] T2.
-#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H elim (tdeq_inv_pair1 … H) -H
+lemma tdeq_inv_pair: ∀I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ≛ ②{I2}V2.T2 →
+ ∧∧ I1 = I2 & V1 ≛ V2 & T1 ≛ T2.
+#I1 #I2 #V1 #V2 #T1 #T2 #H elim (tdeq_inv_pair1 … H) -H
#V0 #T0 #HV #HT #H destruct /2 width=1 by and3_intro/
qed-.
-lemma tdeq_inv_pair_xy_x: ∀h,o,I,V,T. ②{I}V.T ≛[h, o] V → ⊥.
-#h #o #I #V elim V -V
+lemma tdeq_inv_pair_xy_x: ∀I,V,T. ②{I}V.T ≛ V → ⊥.
+#I #V elim V -V
[ #J #T #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
| #J #X #Y #IHX #_ #T #H elim (tdeq_inv_pair … H) -H #H #HY #_ destruct /2 width=2 by/
]
qed-.
-lemma tdeq_inv_pair_xy_y: ∀h,o,I,T,V. ②{I}V.T ≛[h, o] T → ⊥.
-#h #o #I #T elim T -T
+lemma tdeq_inv_pair_xy_y: ∀I,T,V. ②{I}V.T ≛ T → ⊥.
+#I #T elim T -T
[ #J #V #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
| #J #X #Y #_ #IHY #V #H elim (tdeq_inv_pair … H) -H #H #_ #HY destruct /2 width=2 by/
]
(* Basic forward lemmas *****************************************************)
-lemma tdeq_fwd_atom1: ∀h,o,I,Y. ⓪{I} ≛[h, o] Y → ∃J. Y = ⓪{J}.
-#h #o * #x #Y #H [ elim (tdeq_inv_sort1 … H) -H ]
+lemma tdeq_fwd_atom1: ∀I,Y. ⓪{I} ≛ Y → ∃J. Y = ⓪{J}.
+* #x #Y #H [ elim (tdeq_inv_sort1 … H) -H ]
/3 width=4 by tdeq_inv_gref1, tdeq_inv_lref1, ex_intro/
qed-.
(* Advanced properties ******************************************************)
-lemma tdeq_dec: ∀h,o,T1,T2. Decidable (T1 ≛[h, o] T2).
-#h #o #T1 elim T1 -T1 [ * #s1 | #I1 #V1 #T1 #IHV #IHT ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
-[ elim (deg_total h o s1) #d1 #H1
- elim (deg_total h o s2) #d2 #H2
- elim (eq_nat_dec d1 d2) #Hd12 destruct /3 width=3 by tdeq_sort, or_introl/
- @or_intror #H
- lapply (tdeq_inv_sort_deg … H … H1 H2) -H -H1 -H2 /2 width=1 by/
+lemma tdeq_dec: ∀T1,T2. Decidable (T1 ≛ T2).
+#T1 elim T1 -T1 [ * #s1 | #I1 #V1 #T1 #IHV #IHT ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
+[ /3 width=1 by tdeq_sort, or_introl/
|2,3,13:
@or_intror #H
- elim (tdeq_inv_sort1 … H) -H #x1 #x2 #_ #_ #H destruct
+ elim (tdeq_inv_sort1 … H) -H #x #H destruct
|4,6,14:
@or_intror #H
lapply (tdeq_inv_lref1 … H) -H #H destruct
(* Negated inversion lemmas *************************************************)
-lemma tdneq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2.
- (②{I1}V1.T1 ≛[h, o] ②{I2}V2.T2 → ⊥) →
+lemma tdneq_inv_pair: ∀I1,I2,V1,V2,T1,T2.
+ (②{I1}V1.T1 ≛ ②{I2}V2.T2 → ⊥) →
∨∨ I1 = I2 → ⊥
- | (V1 ≛[h, o] V2 → ⊥)
- | (T1 ≛[h, o] T2 → ⊥).
-#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H12
+ | (V1 ≛ V2 → ⊥)
+ | (T1 ≛ T2 → ⊥).
+#I1 #I2 #V1 #V2 #T1 #T2 #H12
elim (eq_item2_dec I1 I2) /3 width=1 by or3_intro0/ #H destruct
-elim (tdeq_dec h o V1 V2) /3 width=1 by or3_intro1/
-elim (tdeq_dec h o T1 T2) /4 width=1 by tdeq_pair, or3_intro2/
+elim (tdeq_dec V1 V2) /3 width=1 by or3_intro1/
+elim (tdeq_dec T1 T2) /4 width=1 by tdeq_pair, or3_intro2/
qed-.
(* *)
(**************************************************************************)
-include "static_2/notation/relations/stareq_5.ma".
+include "static_2/notation/relations/stareq_3.ma".
include "static_2/syntax/cext2.ma".
include "static_2/syntax/tdeq.ma".
-(* EXTENDED DEGREE-BASED EQUIVALENCE ****************************************)
+(* EXTENDED SORT-IRRELEVANT EQUIVALENCE *************************************)
-definition tdeq_ext: ∀h. sd h → relation bind ≝
- λh,o. ext2 (tdeq h o).
+definition tdeq_ext: relation bind ≝
+ ext2 tdeq.
-definition cdeq: ∀h. sd h → relation3 lenv term term ≝
- λh,o,L. tdeq h o.
+definition cdeq: relation3 lenv term term ≝
+ λL. tdeq.
-definition cdeq_ext: ∀h. sd h → relation3 lenv bind bind ≝
- λh,o. cext2 (cdeq h o).
+definition cdeq_ext: relation3 lenv bind bind ≝
+ cext2 cdeq.
interpretation
- "context-free degree-based equivalence (binder)"
- 'StarEq h o I1 I2 = (tdeq_ext h o I1 I2).
+ "context-free sort-irrelevant equivalence (binder)"
+ 'StarEq I1 I2 = (tdeq_ext I1 I2).
interpretation
- "context-dependent degree-based equivalence (term)"
- 'StarEq h o L T1 T2 = (cdeq h o L T1 T2).
+ "context-dependent sort-irrelevant equivalence (term)"
+ 'StarEq L T1 T2 = (cdeq L T1 T2).
interpretation
- "context-dependent degree-based equivalence (binder)"
- 'StarEq h o L I1 I2 = (cdeq_ext h o L I1 I2).
+ "context-dependent sort-irrelevant equivalence (binder)"
+ 'StarEq L I1 I2 = (cdeq_ext L I1 I2).
include "static_2/syntax/tdeq.ma".
-(* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************)
+(* SORT-IRRELEVANT EQUIVALENCE ON TERMS *************************************)
(* Main properties **********************************************************)
-theorem tdeq_trans: ∀h,o. Transitive … (tdeq h o).
-#h #o #T1 #T #H elim H -T1 -T
-[ #s1 #s #d #Hs1 #Hs #X #H
- elim (tdeq_inv_sort1_deg … H … Hs) -s /2 width=3 by tdeq_sort/
+theorem tdeq_trans: Transitive … tdeq.
+#T1 #T #H elim H -T1 -T
+[ #s1 #s #X #H
+ elim (tdeq_inv_sort1 … H) -s /2 width=1 by tdeq_sort/
| #i1 #i #H <(tdeq_inv_lref1 … H) -H //
| #l1 #l #H <(tdeq_inv_gref1 … H) -H //
| #I #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H
]
qed-.
-theorem tdeq_canc_sn: ∀h,o. left_cancellable … (tdeq h o).
+theorem tdeq_canc_sn: left_cancellable … tdeq.
/3 width=3 by tdeq_trans, tdeq_sym/ qed-.
-theorem tdeq_canc_dx: ∀h,o. right_cancellable … (tdeq h o).
+theorem tdeq_canc_dx: right_cancellable … tdeq.
/3 width=3 by tdeq_trans, tdeq_sym/ qed-.
-theorem tdeq_repl: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
- ∀U1. T1 ≛[h, o] U1 → ∀U2. T2 ≛[h, o] U2 → U1 ≛[h, o] U2.
+theorem tdeq_repl: ∀T1,T2. T1 ≛ T2 →
+ ∀U1. T1 ≛ U1 → ∀U2. T2 ≛ U2 → U1 ≛ U2.
/3 width=3 by tdeq_canc_sn, tdeq_trans/ qed-.
(* Negated main properies ***************************************************)
-theorem tdeq_tdneq_trans: ∀h,o,T1,T. T1 ≛[h, o] T → ∀T2. (T ≛[h, o] T2 → ⊥) →
- T1 ≛[h, o] T2 → ⊥.
+theorem tdeq_tdneq_trans: ∀T1,T. T1 ≛ T → ∀T2. (T ≛ T2 → ⊥) → T1 ≛ T2 → ⊥.
/3 width=3 by tdeq_canc_sn/ qed-.
-theorem tdneq_tdeq_canc_dx: ∀h,o,T1,T. (T1 ≛[h, o] T → ⊥) → ∀T2. T2 ≛[h, o] T →
- T1 ≛[h, o] T2 → ⊥.
+theorem tdneq_tdeq_canc_dx: ∀T1,T. (T1 ≛ T → ⊥) → ∀T2. T2 ≛ T → T1 ≛ T2 → ⊥.
/3 width=3 by tdeq_trans/ qed-.
(* Basic properties *********************************************************)
+lemma abst_dec (X): ∨∨ ∃∃p,W,T. X = ⓛ{p}W.T
+ | (∀p,W,T. X = ⓛ{p}W.T → ⊥).
+* [ #I | * [ #p * | #I ] #V #T ]
+[3: /3 width=4 by ex1_3_intro, or_introl/ ]
+@or_intror #q #W #U #H destruct
+qed-.
+
(* Basic_1: was: term_dec *)
lemma eq_term_dec: ∀T1,T2:term. Decidable (T1 = T2).
#T1 elim T1 -T1 #I1 [| #V1 #T1 #IHV1 #IHT1 ] * #I2 [2,4: #V2 #T2 ]
(* *)
(**************************************************************************)
+include "ground_2/xoa/ex_1_4.ma".
include "static_2/notation/relations/simple_1.ma".
include "static_2/syntax/term.ma".
* /2 width=2 by ex_intro/
#p #I #V #T #H elim (simple_inv_bind … H)
qed-.
+
+(* Basic properties *********************************************************)
+
+lemma simple_dec_ex (X): ∨∨ 𝐒⦃X⦄ | ∃∃p,I,T,U. X = ⓑ{p,I}T.U.
+* [ /2 width=1 by simple_atom, or_introl/ ]
+* [| /2 width=1 by simple_flat, or_introl/ ]
+/3 width=5 by ex1_4_intro, or_intror/
+qed-.
#T elim T -T //
qed.
+lemma tw_le_pair_dx (I): ∀V,T. ♯{T} < ♯{②{I}V.T}.
+#I #V #T /2 width=1 by le_S_S/
+qed.
+
(* Basic_1: removed theorems 11:
wadd_le wadd_lt wadd_O weight_le weight_eq weight_add_O
weight_add_S tlt_trans tlt_head_sx tlt_head_dx tlt_wf_ind
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/notation/relations/topiso_4.ma".
-include "static_2/syntax/item_sd.ma".
-include "static_2/syntax/term.ma".
-
-(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
-
-(* Basic_2A1: includes: tsts_atom tsts_pair *)
-inductive theq (h) (o): relation term ≝
-| theq_sort: ∀s1,s2,d. deg h o s1 d → deg h o s2 d → theq h o (⋆s1) (⋆s2)
-| theq_lref: ∀i. theq h o (#i) (#i)
-| theq_gref: ∀l. theq h o (§l) (§l)
-| theq_pair: ∀I,V1,V2,T1,T2. theq h o (②{I}V1.T1) (②{I}V2.T2)
-.
-
-interpretation "head equivalence (term)" 'TopIso h o T1 T2 = (theq h o T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact theq_inv_sort1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀s1. X = ⋆s1 →
- ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
-#h #o #X #Y * -X -Y
-[ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/
-| #i #s #H destruct
-| #l #s #H destruct
-| #I #V1 #V2 #T1 #T2 #s #H destruct
-]
-qed-.
-
-(* Basic_1: was just: iso_gen_sort *)
-lemma theq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ⩳[h, o] Y →
- ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
-/2 width=3 by theq_inv_sort1_aux/ qed-.
-
-fact theq_inv_lref1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀i. X = #i → Y = #i.
-#h #o #X #Y * -X -Y //
-[ #s1 #s2 #d #_ #_ #j #H destruct
-| #I #V1 #V2 #T1 #T2 #j #H destruct
-]
-qed-.
-
-(* Basic_1: was: iso_gen_lref *)
-lemma theq_inv_lref1: ∀h,o,Y,i. #i ⩳[h, o] Y → Y = #i.
-/2 width=5 by theq_inv_lref1_aux/ qed-.
-
-fact theq_inv_gref1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀l. X = §l → Y = §l.
-#h #o #X #Y * -X -Y //
-[ #s1 #s2 #d #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #k #H destruct
-]
-qed-.
-
-lemma theq_inv_gref1: ∀h,o,Y,l. §l ⩳[h, o] Y → Y = §l.
-/2 width=5 by theq_inv_gref1_aux/ qed-.
-
-fact theq_inv_pair1_aux: ∀h,o,T1,T2. T1 ⩳[h, o] T2 →
- ∀J,W1,U1. T1 = ②{J}W1.U1 →
- ∃∃W2,U2. T2 = ②{J}W2.U2.
-#h #o #T1 #T2 * -T1 -T2
-[ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct
-| #i #J #W1 #U1 #H destruct
-| #l #J #W1 #U1 #H destruct
-| #I #V1 #V2 #T1 #T2 #J #W1 #U1 #H destruct /2 width=3 by ex1_2_intro/
-]
-qed-.
-
-(* Basic_1: was: iso_gen_head *)
-(* Basic_2A1: was: tsts_inv_pair1 *)
-lemma theq_inv_pair1: ∀h,o,J,W1,U1,T2. ②{J}W1.U1 ⩳[h, o] T2 →
- ∃∃W2,U2. T2 = ②{J}W2. U2.
-/2 width=7 by theq_inv_pair1_aux/ qed-.
-
-fact theq_inv_pair2_aux: ∀h,o,T1,T2. T1 ⩳[h, o] T2 →
- ∀J,W2,U2. T2 = ②{J}W2.U2 →
- ∃∃W1,U1. T1 = ②{J}W1.U1.
-#h #o #T1 #T2 * -T1 -T2
-[ #s1 #s2 #d #_ #_ #J #W2 #U2 #H destruct
-| #i #J #W2 #U2 #H destruct
-| #l #J #W2 #U2 #H destruct
-| #I #V1 #V2 #T1 #T2 #J #W2 #U2 #H destruct /2 width=3 by ex1_2_intro/
-]
-qed-.
-
-(* Basic_2A1: was: tsts_inv_pair2 *)
-lemma theq_inv_pair2: ∀h,o,J,T1,W2,U2. T1 ⩳[h, o] ②{J}W2.U2 →
- ∃∃W1,U1. T1 = ②{J}W1.U1.
-/2 width=7 by theq_inv_pair2_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma theq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ⩳[h, o] Y → ∀d. deg h o s1 d →
- ∃∃s2. deg h o s2 d & Y = ⋆s2.
-#h #o #Y #s1 #H #d #Hs1 elim (theq_inv_sort1 … H) -H
-#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/
-qed-.
-
-lemma theq_inv_sort_deg: ∀h,o,s1,s2. ⋆s1 ⩳[h, o] ⋆s2 →
- ∀d1,d2. deg h o s1 d1 → deg h o s2 d2 →
- d1 = d2.
-#h #o #s1 #y #H #d1 #d2 #Hs1 #Hy
-elim (theq_inv_sort1_deg … H … Hs1) -s1 #s2 #Hs2 #H destruct
-<(deg_mono h o … Hy … Hs2) -s2 -d1 //
-qed-.
-
-lemma theq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ⩳[h, o] ②{I2}V2.T2 →
- I1 = I2.
-#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H elim (theq_inv_pair1 … H) -H
-#V0 #T0 #H destruct //
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: iso_refl *)
-(* Basic_2A1: was: tsts_refl *)
-lemma theq_refl: ∀h,o. reflexive … (theq h o).
-#h #o * //
-* /2 width=1 by theq_lref, theq_gref/
-#s elim (deg_total h o s) /2 width=3 by theq_sort/
-qed.
-
-(* Basic_2A1: was: tsts_sym *)
-lemma theq_sym: ∀h,o. symmetric … (theq h o).
-#h #o #T1 #T2 * -T1 -T2 /2 width=3 by theq_sort/
-qed-.
-
-(* Basic_2A1: was: tsts_dec *)
-lemma theq_dec: ∀h,o,T1,T2. Decidable (T1 ⩳[h, o] T2).
-#h #o * [ * #s1 | #I1 #V1 #T1 ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
-[ elim (deg_total h o s1) #d1 #H1
- elim (deg_total h o s2) #d2 #H2
- elim (eq_nat_dec d1 d2) #Hd12 destruct /3 width=3 by theq_sort, or_introl/
- @or_intror #H
- lapply (theq_inv_sort_deg … H … H1 H2) -H -H1 -H2 /2 width=1 by/
-|2,3,13:
- @or_intror #H
- elim (theq_inv_sort1 … H) -H #x1 #x2 #_ #_ #H destruct
-|4,6,14:
- @or_intror #H
- lapply (theq_inv_lref1 … H) -H #H destruct
-|5:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (theq_inv_lref1 … H) -H #H destruct /2 width=1 by/
-|7,8,15:
- @or_intror #H
- lapply (theq_inv_gref1 … H) -H #H destruct
-|9:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (theq_inv_gref1 … H) -H #H destruct /2 width=1 by/
-|10,11,12:
- @or_intror #H
- elim (theq_inv_pair1 … H) -H #X1 #X2 #H destruct
-|16:
- elim (eq_item2_dec I1 I2) #HI12 destruct
- [ /3 width=1 by theq_pair, or_introl/ ]
- @or_intror #H
- lapply (theq_inv_pair … H) -H /2 width=1 by/
-]
-qed-.
-
-(* Basic_2A1: removed theorems 2:
- tsts_inv_atom1 tsts_inv_atom2
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/syntax/term_simple.ma".
-include "static_2/syntax/theq.ma".
-
-(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
-
-(* Properies with simple (neutral) terms ************************************)
-
-(* Basic_2A1: was: simple_tsts_repl_dx *)
-lemma simple_theq_repl_dx: ∀h,o,T1,T2. T1 ⩳[h, o] T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
-#h #o #T1 #T2 * -T1 -T2 //
-#I #V1 #V2 #T1 #T2 #H
-elim (simple_inv_pair … H) -H #J #H destruct //
-qed-.
-
-(* Basic_2A1: was: simple_tsts_repl_sn *)
-lemma simple_theq_repl_sn: ∀h,o,T1,T2. T1 ⩳[h, o] T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
-/3 width=5 by simple_theq_repl_dx, theq_sym/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/syntax/term_vector.ma".
-include "static_2/syntax/theq_simple.ma".
-
-(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
-
-(* Advanced inversion lemmas with simple (neutral) terms ********************)
-
-(* Basic_1: was only: iso_flats_lref_bind_false iso_flats_flat_bind_false *)
-(* Basic_2A1: was: tsts_inv_bind_applv_simple *)
-lemma theq_inv_applv_bind_simple: ∀h,o,p,I,Vs,V2,T1,T2. ⒶVs.T1 ⩳[h, o] ⓑ{p,I}V2.T2 →
- 𝐒⦃T1⦄ → ⊥.
-#h #o #p #I #Vs #V2 #T1 #T2 #H elim (theq_inv_pair2 … H) -H
-#V0 #T0 elim Vs -Vs normalize
-[ #H destruct #H /2 width=5 by simple_inv_bind/
-| #V #Vs #_ #H destruct
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/syntax/tdeq.ma".
-include "static_2/syntax/theq.ma".
-
-(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
-
-(* Properties with degree-based equivalence for terms ***********************)
-
-lemma tdeq_theq: ∀h,o,T1,T2. T1 ≛[h, o] T2 → T1 ⩳[h, o] T2.
-#h #o #T1 #T2 * -T1 -T2 /2 width=3 by theq_sort, theq_pair/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "static_2/syntax/theq.ma".
-
-(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: iso_trans *)
-(* Basic_2A1: was: tsts_trans *)
-theorem theq_trans: ∀h,o. Transitive … (theq h o).
-#h #o #T1 #T * -T1 -T
-[ #s1 #s #d #Hs1 #Hs #X #H
- elim (theq_inv_sort1_deg … H … Hs) -s /2 width=3 by theq_sort/
-| #i1 #i #H <(theq_inv_lref1 … H) -H //
-| #l1 #l #H <(theq_inv_gref1 … H) -H //
-| #I #V1 #V #T1 #T #X #H
- elim (theq_inv_pair1 … H) -H #V2 #T2 #H destruct //
-]
-qed-.
-
-(* Basic_2A1: was: tsts_canc_sn *)
-theorem theq_canc_sn: ∀h,o. left_cancellable … (theq h o).
-/3 width=3 by theq_trans, theq_sym/ qed-.
-
-(* Basic_2A1: was: tsts_canc_dx *)
-theorem theq_canc_dx: ∀h,o. right_cancellable … (theq h o).
-/3 width=3 by theq_trans, theq_sym/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/notation/relations/topiso_2.ma".
+include "static_2/syntax/term.ma".
+
+(* SORT-IRRELEVANT OUTER EQUIVALENCE FOR TERMS ******************************)
+
+(* Basic_2A1: includes: tsts_atom tsts_pair *)
+inductive toeq: relation term ≝
+| toeq_sort: ∀s1,s2. toeq (⋆s1) (⋆s2)
+| toeq_lref: ∀i. toeq (#i) (#i)
+| toeq_gref: ∀l. toeq (§l) (§l)
+| toeq_pair: ∀I,V1,V2,T1,T2. toeq (②{I}V1.T1) (②{I}V2.T2)
+.
+
+interpretation
+ "sort-irrelevant outer equivalence (term)"
+ 'TopIso T1 T2 = (toeq T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact toeq_inv_sort1_aux: ∀X,Y. X ⩳ Y → ∀s1. X = ⋆s1 →
+ ∃s2. Y = ⋆s2.
+#X #Y * -X -Y
+[ #s1 #s2 #s #H destruct /2 width=2 by ex_intro/
+| #i #s #H destruct
+| #l #s #H destruct
+| #I #V1 #V2 #T1 #T2 #s #H destruct
+]
+qed-.
+
+(* Basic_1: was just: iso_gen_sort *)
+lemma toeq_inv_sort1: ∀Y,s1. ⋆s1 ⩳ Y →
+ ∃s2. Y = ⋆s2.
+/2 width=4 by toeq_inv_sort1_aux/ qed-.
+
+fact toeq_inv_lref1_aux: ∀X,Y. X ⩳ Y → ∀i. X = #i → Y = #i.
+#X #Y * -X -Y //
+[ #s1 #s2 #j #H destruct
+| #I #V1 #V2 #T1 #T2 #j #H destruct
+]
+qed-.
+
+(* Basic_1: was: iso_gen_lref *)
+lemma toeq_inv_lref1: ∀Y,i. #i ⩳ Y → Y = #i.
+/2 width=5 by toeq_inv_lref1_aux/ qed-.
+
+fact toeq_inv_gref1_aux: ∀X,Y. X ⩳ Y → ∀l. X = §l → Y = §l.
+#X #Y * -X -Y //
+[ #s1 #s2 #k #H destruct
+| #I #V1 #V2 #T1 #T2 #k #H destruct
+]
+qed-.
+
+lemma toeq_inv_gref1: ∀Y,l. §l ⩳ Y → Y = §l.
+/2 width=5 by toeq_inv_gref1_aux/ qed-.
+
+fact toeq_inv_pair1_aux: ∀T1,T2. T1 ⩳ T2 →
+ ∀J,W1,U1. T1 = ②{J}W1.U1 →
+ ∃∃W2,U2. T2 = ②{J}W2.U2.
+#T1 #T2 * -T1 -T2
+[ #s1 #s2 #J #W1 #U1 #H destruct
+| #i #J #W1 #U1 #H destruct
+| #l #J #W1 #U1 #H destruct
+| #I #V1 #V2 #T1 #T2 #J #W1 #U1 #H destruct /2 width=3 by ex1_2_intro/
+]
+qed-.
+
+(* Basic_1: was: iso_gen_head *)
+(* Basic_2A1: was: tsts_inv_pair1 *)
+lemma toeq_inv_pair1: ∀J,W1,U1,T2. ②{J}W1.U1 ⩳ T2 →
+ ∃∃W2,U2. T2 = ②{J}W2. U2.
+/2 width=7 by toeq_inv_pair1_aux/ qed-.
+
+fact toeq_inv_pair2_aux: ∀T1,T2. T1 ⩳ T2 →
+ ∀J,W2,U2. T2 = ②{J}W2.U2 →
+ ∃∃W1,U1. T1 = ②{J}W1.U1.
+#T1 #T2 * -T1 -T2
+[ #s1 #s2 #J #W2 #U2 #H destruct
+| #i #J #W2 #U2 #H destruct
+| #l #J #W2 #U2 #H destruct
+| #I #V1 #V2 #T1 #T2 #J #W2 #U2 #H destruct /2 width=3 by ex1_2_intro/
+]
+qed-.
+
+(* Basic_2A1: was: tsts_inv_pair2 *)
+lemma toeq_inv_pair2: ∀J,T1,W2,U2. T1 ⩳ ②{J}W2.U2 →
+ ∃∃W1,U1. T1 = ②{J}W1.U1.
+/2 width=7 by toeq_inv_pair2_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma toeq_inv_pair: ∀I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ⩳ ②{I2}V2.T2 →
+ I1 = I2.
+#I1 #I2 #V1 #V2 #T1 #T2 #H elim (toeq_inv_pair1 … H) -H
+#V0 #T0 #H destruct //
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: iso_refl *)
+(* Basic_2A1: was: tsts_refl *)
+lemma toeq_refl: reflexive … toeq.
+* //
+* /2 width=1 by toeq_lref, toeq_gref/
+qed.
+
+(* Basic_2A1: was: tsts_sym *)
+lemma toeq_sym: symmetric … toeq.
+#T1 #T2 * -T1 -T2 /2 width=3 by toeq_sort/
+qed-.
+
+(* Basic_2A1: was: tsts_dec *)
+lemma toeq_dec: ∀T1,T2. Decidable (T1 ⩳ T2).
+* [ * #s1 | #I1 #V1 #T1 ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
+[ /3 width=1 by toeq_sort, or_introl/
+|2,3,13:
+ @or_intror #H
+ elim (toeq_inv_sort1 … H) -H #x #H destruct
+|4,6,14:
+ @or_intror #H
+ lapply (toeq_inv_lref1 … H) -H #H destruct
+|5:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (toeq_inv_lref1 … H) -H #H destruct /2 width=1 by/
+|7,8,15:
+ @or_intror #H
+ lapply (toeq_inv_gref1 … H) -H #H destruct
+|9:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (toeq_inv_gref1 … H) -H #H destruct /2 width=1 by/
+|10,11,12:
+ @or_intror #H
+ elim (toeq_inv_pair1 … H) -H #X1 #X2 #H destruct
+|16:
+ elim (eq_item2_dec I1 I2) #HI12 destruct
+ [ /3 width=1 by toeq_pair, or_introl/ ]
+ @or_intror #H
+ lapply (toeq_inv_pair … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Basic_2A1: removed theorems 2:
+ tsts_inv_atom1 tsts_inv_atom2
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/term_simple.ma".
+include "static_2/syntax/toeq.ma".
+
+(* SORT-IRRELEVANT OUTER EQUIVALENCE FOR TERMS ******************************)
+
+(* Properies with simple (neutral) terms ************************************)
+
+(* Basic_2A1: was: simple_tsts_repl_dx *)
+lemma simple_toeq_repl_dx: ∀T1,T2. T1 ⩳ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#T1 #T2 * -T1 -T2 //
+#I #V1 #V2 #T1 #T2 #H
+elim (simple_inv_pair … H) -H #J #H destruct //
+qed-.
+
+(* Basic_2A1: was: simple_tsts_repl_sn *)
+lemma simple_toeq_repl_sn: ∀T1,T2. T1 ⩳ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+/3 width=3 by simple_toeq_repl_dx, toeq_sym/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/term_vector.ma".
+include "static_2/syntax/toeq_simple.ma".
+
+(* SORT-IRRELEVANT OUTER EQUIVALENCE FOR TERMS ******************************)
+
+(* Advanced inversion lemmas with simple (neutral) terms ********************)
+
+(* Basic_1: was only: iso_flats_lref_bind_false iso_flats_flat_bind_false *)
+(* Basic_2A1: was: tsts_inv_bind_applv_simple *)
+lemma toeq_inv_applv_bind_simple (p) (I):
+ ∀Vs,V2,T1,T2. ⒶVs.T1 ⩳ ⓑ{p,I}V2.T2 → 𝐒⦃T1⦄ → ⊥.
+#p #I #Vs #V2 #T1 #T2 #H elim (toeq_inv_pair2 … H) -H
+#V0 #T0 elim Vs -Vs normalize
+[ #H destruct #H /2 width=5 by simple_inv_bind/
+| #V #Vs #_ #H destruct
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tdeq.ma".
+include "static_2/syntax/toeq.ma".
+
+(* SORT-IRRELEVANT OUTER EQUIVALENCE FOR TERMS ******************************)
+
+(* Properties with sort-irrelevant equivalence for terms ********************)
+
+lemma tdeq_toeq: ∀T1,T2. T1 ≛ T2 → T1 ⩳ T2.
+#T1 #T2 * -T1 -T2 /2 width=1 by toeq_sort, toeq_pair/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/toeq.ma".
+
+(* SORT-IRRELEVANT OUTER EQUIVALENCE FOR TERMS ******************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: iso_trans *)
+(* Basic_2A1: was: tsts_trans *)
+theorem toeq_trans: Transitive … toeq.
+#T1 #T * -T1 -T
+[ #s1 #s #X #H
+ elim (toeq_inv_sort1 … H) -s /2 width=1 by toeq_sort/
+| #i1 #i #H <(toeq_inv_lref1 … H) -H //
+| #l1 #l #H <(toeq_inv_gref1 … H) -H //
+| #I #V1 #V #T1 #T #X #H
+ elim (toeq_inv_pair1 … H) -H #V2 #T2 #H destruct //
+]
+qed-.
+
+(* Basic_2A1: was: tsts_canc_sn *)
+theorem toeq_canc_sn: left_cancellable … toeq.
+/3 width=3 by toeq_trans, toeq_sym/ qed-.
+
+(* Basic_2A1: was: tsts_canc_dx *)
+theorem toeq_canc_dx: right_cancellable … toeq.
+/3 width=3 by toeq_trans, toeq_sym/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/notation/relations/approxeq_2.ma".
+include "static_2/syntax/term_weight.ma".
+
+(* SORT-IRRELEVANT WHD EQUIVALENCE ON TERMS *********************************)
+
+inductive tweq: relation term ≝
+| tweq_sort: ∀s1,s2. tweq (⋆s1) (⋆s2)
+| tweq_lref: ∀i. tweq (#i) (#i)
+| tweq_gref: ∀l. tweq (§l) (§l)
+| tweq_abbr: ∀p,V1,V2,T1,T2. (p=Ⓣ→tweq T1 T2) → tweq (ⓓ{p}V1.T1) (ⓓ{p}V2.T2)
+| tweq_abst: ∀p,V1,V2,T1,T2. tweq (ⓛ{p}V1.T1) (ⓛ{p}V2.T2)
+| tweq_appl: ∀V1,V2,T1,T2. tweq T1 T2 → tweq (ⓐV1.T1) (ⓐV2.T2)
+| tweq_cast: ∀V1,V2,T1,T2. tweq V1 V2 → tweq T1 T2 → tweq (ⓝV1.T1) (ⓝV2.T2)
+.
+
+interpretation
+ "context-free tail sort-irrelevant equivalence (term)"
+ 'ApproxEq T1 T2 = (tweq T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tweq_abbr_pos: ∀V1,V2,T1,T2. T1 ≅ T2 → +ⓓV1.T1 ≅ +ⓓV2.T2.
+/3 width=1 by tweq_abbr/ qed.
+
+lemma tweq_abbr_neg: ∀V1,V2,T1,T2. -ⓓV1.T1 ≅ -ⓓV2.T2.
+#V1 #V2 #T1 #T2
+@tweq_abbr #H destruct
+qed.
+
+lemma tweq_refl: reflexive … tweq.
+#T elim T -T * [||| #p * | * ]
+/2 width=1 by tweq_sort, tweq_lref, tweq_gref, tweq_abbr, tweq_abst, tweq_appl, tweq_cast/
+qed.
+
+lemma tweq_sym: symmetric … tweq.
+#T1 #T2 #H elim H -T1 -T2
+/3 width=3 by tweq_sort, tweq_lref, tweq_gref, tweq_abbr, tweq_abst, tweq_appl, tweq_cast/
+qed-.
+
+(* Left basic inversion lemmas **********************************************)
+
+fact tweq_inv_sort_sn_aux:
+ ∀X,Y. X ≅ Y → ∀s1. X = ⋆s1 → ∃s2. Y = ⋆s2.
+#X #Y * -X -Y
+[1 : #s1 #s2 #s #H destruct /2 width=2 by ex_intro/
+|2,3: #i #s #H destruct
+|4 : #p #V1 #V2 #T1 #T2 #_ #s #H destruct
+|5 : #p #V1 #V2 #T1 #T2 #s #H destruct
+|6 : #V1 #V2 #T1 #T2 #_ #s #H destruct
+|7 : #V1 #V2 #T1 #T2 #_ #_ #s #H destruct
+]
+qed-.
+
+lemma tweq_inv_sort_sn:
+ ∀Y,s1. ⋆s1 ≅ Y → ∃s2. Y = ⋆s2.
+/2 width=4 by tweq_inv_sort_sn_aux/ qed-.
+
+fact tweq_inv_lref_sn_aux:
+ ∀X,Y. X ≅ Y → ∀i. X = #i → Y = #i.
+#X #Y * -X -Y
+[1 : #s1 #s2 #j #H destruct
+|2,3: //
+|4 : #p #V1 #V2 #T1 #T2 #_ #j #H destruct
+|5 : #p #V1 #V2 #T1 #T2 #j #H destruct
+|6 : #V1 #V2 #T1 #T2 #_ #j #H destruct
+|7 : #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
+]
+qed-.
+
+lemma tweq_inv_lref_sn: ∀Y,i. #i ≅ Y → Y = #i.
+/2 width=5 by tweq_inv_lref_sn_aux/ qed-.
+
+fact tweq_inv_gref_sn_aux:
+ ∀X,Y. X ≅ Y → ∀l. X = §l → Y = §l.
+#X #Y * -X -Y
+[1 : #s1 #s2 #k #H destruct
+|2,3: //
+|4 : #p #V1 #V2 #T1 #T2 #_ #k #H destruct
+|5 : #p #V1 #V2 #T1 #T2 #k #H destruct
+|6 : #V1 #V2 #T1 #T2 #_ #k #H destruct
+|7 : #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
+]
+qed-.
+
+lemma tweq_inv_gref_sn:
+ ∀Y,l. §l ≅ Y → Y = §l.
+/2 width=5 by tweq_inv_gref_sn_aux/ qed-.
+
+fact tweq_inv_abbr_sn_aux:
+ ∀X,Y. X ≅ Y → ∀p,V1,T1. X = ⓓ{p}V1.T1 →
+ ∃∃V2,T2. p = Ⓣ → T1 ≅ T2 & Y = ⓓ{p}V2.T2.
+#X #Y * -X -Y
+[1 : #s1 #s2 #q #W1 #U1 #H destruct
+|2,3: #i #q #W1 #U1 #H destruct
+|4 : #p #V1 #V2 #T1 #T2 #HT #q #W1 #U1 #H destruct /3 width=4 by ex2_2_intro/
+|5 : #p #V1 #V2 #T1 #T2 #q #W1 #U1 #H destruct
+|6 : #V1 #V2 #T1 #T2 #_ #q #W1 #U1 #H destruct
+|7 : #V1 #V2 #T1 #T2 #_ #_ #q #W1 #U1 #H destruct
+]
+qed-.
+
+lemma tweq_inv_abbr_sn:
+ ∀p,V1,T1,Y. ⓓ{p}V1.T1 ≅ Y →
+ ∃∃V2,T2. p = Ⓣ → T1 ≅ T2 & Y = ⓓ{p}V2.T2.
+/2 width=4 by tweq_inv_abbr_sn_aux/ qed-.
+
+fact tweq_inv_abst_sn_aux:
+ ∀X,Y. X ≅ Y → ∀p,V1,T1. X = ⓛ{p}V1.T1 →
+ ∃∃V2,T2. Y = ⓛ{p}V2.T2.
+#X #Y * -X -Y
+[1 : #s1 #s2 #q #W1 #U1 #H destruct
+|2,3: #i #q #W1 #U1 #H destruct
+|4 : #p #V1 #V2 #T1 #T2 #_ #q #W1 #U1 #H destruct
+|5 : #p #V1 #V2 #T1 #T2 #q #W1 #U1 #H destruct /2 width=3 by ex1_2_intro/
+|6 : #V1 #V2 #T1 #T2 #_ #q #W1 #U1 #H destruct
+|7 : #V1 #V2 #T1 #T2 #_ #_ #q #W1 #U1 #H destruct
+]
+qed-.
+
+lemma tweq_inv_abst_sn:
+ ∀p,V1,T1,Y. ⓛ{p}V1.T1 ≅ Y →
+ ∃∃V2,T2. Y = ⓛ{p}V2.T2.
+/2 width=5 by tweq_inv_abst_sn_aux/ qed-.
+
+fact tweq_inv_appl_sn_aux:
+ ∀X,Y. X ≅ Y → ∀V1,T1. X = ⓐV1.T1 →
+ ∃∃V2,T2. T1 ≅ T2 & Y = ⓐV2.T2.
+#X #Y * -X -Y
+[1 : #s1 #s2 #W1 #U1 #H destruct
+|2,3: #i #W1 #U1 #H destruct
+|4 : #p #V1 #V2 #T1 #T2 #HT #W1 #U1 #H destruct
+|5 : #p #V1 #V2 #T1 #T2 #W1 #U1 #H destruct
+|6 : #V1 #V2 #T1 #T2 #HT #W1 #U1 #H destruct /2 width=4 by ex2_2_intro/
+|7 : #V1 #V2 #T1 #T2 #_ #_ #W1 #U1 #H destruct
+]
+qed-.
+
+lemma tweq_inv_appl_sn:
+ ∀V1,T1,Y. ⓐV1.T1 ≅ Y →
+ ∃∃V2,T2. T1 ≅ T2 & Y = ⓐV2.T2.
+/2 width=4 by tweq_inv_appl_sn_aux/ qed-.
+
+fact tweq_inv_cast_sn_aux:
+ ∀X,Y. X ≅ Y → ∀V1,T1. X = ⓝV1.T1 →
+ ∃∃V2,T2. V1 ≅ V2 & T1 ≅ T2 & Y = ⓝV2.T2.
+#X #Y * -X -Y
+[1 : #s1 #s2 #W1 #U1 #H destruct
+|2,3: #i #W1 #U1 #H destruct
+|4 : #p #V1 #V2 #T1 #T2 #_ #W1 #U1 #H destruct
+|5 : #p #V1 #V2 #T1 #T2 #W1 #U1 #H destruct
+|6 : #V1 #V2 #T1 #T2 #_ #W1 #U1 #H destruct
+|7 : #V1 #V2 #T1 #T2 #HV #HT #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma tweq_inv_cast_sn:
+ ∀V1,T1,Y. ⓝV1.T1 ≅ Y →
+ ∃∃V2,T2. V1 ≅ V2 & T1 ≅ T2 & Y = ⓝV2.T2.
+/2 width=3 by tweq_inv_cast_sn_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma tweq_inv_abbr_pos_sn:
+ ∀V1,T1,Y. +ⓓV1.T1 ≅ Y → ∃∃V2,T2. T1 ≅ T2 & Y = +ⓓV2.T2.
+#V1 #V2 #Y #H
+elim (tweq_inv_abbr_sn … H) -H #V2 #T2
+/3 width=4 by ex2_2_intro/
+qed-.
+
+lemma tweq_inv_abbr_neg_sn:
+ ∀V1,T1,Y. -ⓓV1.T1 ≅ Y → ∃∃V2,T2. Y = -ⓓV2.T2.
+#V1 #V2 #Y #H
+elim (tweq_inv_abbr_sn … H) -H #V2 #T2 #_
+/2 width=3 by ex1_2_intro/
+qed-.
+
+lemma tweq_inv_abbr_pos_bi:
+ ∀V1,V2,T1,T2. +ⓓV1.T1 ≅ +ⓓV2.T2 → T1 ≅ T2.
+#V1 #V2 #T1 #T2 #H
+elim (tweq_inv_abbr_pos_sn … H) -H #W2 #U2 #HTU #H destruct //
+qed-.
+
+lemma tweq_inv_appl_bi:
+ ∀V1,V2,T1,T2. ⓐV1.T1 ≅ ⓐV2.T2 → T1 ≅ T2.
+#V1 #V2 #T1 #T2 #H
+elim (tweq_inv_appl_sn … H) -H #W2 #U2 #HTU #H destruct //
+qed-.
+
+lemma tweq_inv_cast_bi:
+ ∀V1,V2,T1,T2. ⓝV1.T1 ≅ ⓝV2.T2 → ∧∧ V1 ≅ V2 & T1 ≅ T2.
+#V1 #V2 #T1 #T2 #H
+elim (tweq_inv_cast_sn … H) -H #W2 #U2 #HVW #HTU #H destruct
+/2 width=1 by conj/
+qed-.
+
+lemma tweq_inv_cast_xy_y: ∀T,V. ⓝV.T ≅ T → ⊥.
+@(f_ind … tw) #n #IH #T #Hn #V #H destruct
+elim (tweq_inv_cast_sn … H) -H #X1 #X2 #_ #HX2 #H destruct -V
+/2 width=4 by/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma tweq_fwd_pair_sn (I):
+ ∀V1,T1,X2. ②{I}V1.T1 ≅ X2 → ∃∃V2,T2. X2 = ②{I}V2.T2.
+* [ #p ] * [ cases p -p ] #V1 #T1 #X2 #H
+[ elim (tweq_inv_abbr_pos_sn … H) -H #V2 #T2 #_ #H
+| elim (tweq_inv_abbr_neg_sn … H) -H #V2 #T2 #H
+| elim (tweq_inv_abst_sn … H) -H #V2 #T2 #H
+| elim (tweq_inv_appl_sn … H) -H #V2 #T2 #_ #H
+| elim (tweq_inv_cast_sn … H) -H #V2 #T2 #_ #_ #H
+] /2 width=3 by ex1_2_intro/
+qed-.
+
+lemma tweq_fwd_pair_bi (I1) (I2):
+ ∀V1,V2,T1,T2. ②{I1}V1.T1 ≅ ②{I2}V2.T2 → I1 = I2.
+#I1 #I2 #V1 #V2 #T1 #T2 #H
+elim (tweq_fwd_pair_sn … H) -H #W2 #U2 #H destruct //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma tweq_dec: ∀T1,T2. Decidable (T1 ≅ T2).
+#T1 elim T1 -T1 [ * #s1 | #I1 #V1 #T1 #IHV #IHT ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
+[ /3 width=1 by tweq_sort, or_introl/
+|2,3,13:
+ @or_intror #H
+ elim (tweq_inv_sort_sn … H) -H #x #H destruct
+|4,6,14:
+ @or_intror #H
+ lapply (tweq_inv_lref_sn … H) -H #H destruct
+|5:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (tweq_inv_lref_sn … H) -H #H destruct /2 width=1 by/
+|7,8,15:
+ @or_intror #H
+ lapply (tweq_inv_gref_sn … H) -H #H destruct
+|9:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (tweq_inv_gref_sn … H) -H #H destruct /2 width=1 by/
+|10,11,12:
+ @or_intror #H
+ elim (tweq_fwd_pair_sn … H) -H #X1 #X2 #H destruct
+|16:
+ elim (eq_item2_dec I1 I2) #HI12 destruct
+ [ cases I2 -I2 [ #p ] * [ cases p -p ]
+ [ elim (IHT T2) -IHT #HT12
+ [ /3 width=1 by tweq_abbr_pos, or_introl/
+ | /4 width=3 by tweq_inv_abbr_pos_bi, or_intror/
+ ]
+ | /3 width=1 by tweq_abbr_neg, or_introl/
+ | /3 width=1 by tweq_abst, or_introl/
+ | elim (IHT T2) -IHT #HT12
+ [ /3 width=1 by tweq_appl, or_introl/
+ | /4 width=3 by tweq_inv_appl_bi, or_intror/
+ ]
+ | elim (IHV V2) -IHV #HV12
+ elim (IHT T2) -IHT #HT12
+ [1: /3 width=1 by tweq_cast, or_introl/
+ |*: @or_intror #H
+ elim (tweq_inv_cast_bi … H) -H #HV12 #HT12
+ /2 width=1 by/
+ ]
+ ]
+ | /4 width=5 by tweq_fwd_pair_bi, or_intror/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/term_simple.ma".
+include "static_2/syntax/tweq.ma".
+
+(* SORT-IRRELEVANT WHD EQUIVALENCE ON TERMS *********************************)
+
+(* Properties with simple terms *********************************************)
+
+lemma tweq_simple_trans:
+ ∀T1,T2. T1 ≅ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#T1 #T2 * -T1 -T2
+[4,5: #p #V1 #V2 #T1 #T2 [ #_ ] #H
+ elim (simple_inv_bind … H)
+|* : /1 width=1 by simple_atom, simple_flat/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tdeq.ma".
+include "static_2/syntax/tweq.ma".
+
+(* SORT-IRRELEVANT WHD EQUIVALENCE ON TERMS *********************************)
+
+(* Properties with sort-irrelevant equivalence for terms ********************)
+
+lemma tdeq_tweq: ∀T1,T2. T1 ≛ T2 → T1 ≅ T2.
+#T1 #T2 #H elim H -T1 -T2 [||| * [ #p ] * #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT ]
+[ /1 width=1 by tweq_sort/
+| /1 width=1 by tweq_lref/
+| /1 width=1 by tweq_gref/
+| cases p -p /2 width=1 by tweq_abbr_pos, tweq_abbr_neg/
+| /1 width=1 by tweq_abst/
+| /2 width=1 by tweq_appl/
+| /2 width=1 by tweq_cast/
+]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tweq.ma".
+
+(* SORT-IRRELEVANT WHD EQUIVALENCE ON TERMS *********************************)
+
+(* Main properties **********************************************************)
+
+theorem tweq_trans: Transitive … tweq.
+#T1 #T #H elim H -T1 -T
+[ #s1 #s #X #H
+ elim (tweq_inv_sort_sn … H) -s #s2 destruct
+ /2 width=1 by tweq_sort/
+| #i1 #i #H //
+| #l1 #l #H //
+| #p #V1 #V #T1 #T #_ #IHT #X #H
+ elim (tweq_inv_abbr_sn … H) -H #V2 #T2 #HT #H destruct
+ /4 width=1 by tweq_abbr/
+| #p #V1 #V #T1 #T #X #H
+ elim (tweq_inv_abst_sn … H) -H #V2 #T2 #H destruct
+ /2 width=1 by tweq_abst/
+| #V1 #V #T1 #T #_ #IHT #X #H
+ elim (tweq_inv_appl_sn … H) -H #V2 #T2 #HT #H destruct
+ /3 width=1 by tweq_appl/
+| #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H
+ elim (tweq_inv_cast_sn … H) -H #V2 #T2 #HV #HT #H destruct
+ /3 width=1 by tweq_cast/
+]
+qed-.
+
+theorem tweq_canc_sn: left_cancellable … tweq.
+/3 width=3 by tweq_trans, tweq_sym/ qed-.
+
+theorem tweq_canc_dx: right_cancellable … tweq.
+/3 width=3 by tweq_trans, tweq_sym/ qed-.
+
+theorem tweq_repl:
+ ∀T1,T2. T1 ≅ T2 → ∀U1. T1 ≅ U1 → ∀U2. T2 ≅ U2 → U1 ≅ U2.
+/3 width=3 by tweq_canc_sn, tweq_trans/ qed-.
]
[ { "atomic arity assignment" * } {
[ [ "restricted refinement for lenvs" ] "lsuba" + "( ? ⊢ ? ⫃⁝ ? )" "lsuba_drops" + "lsuba_lsubr" + "lsuba_aaa" + "lsuba_lsuba" * ]
- [ [ "for terms" ] "aaa" + "( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_drops" + "aaa_fqus" + "aaa_rdeq" + "aaa_fdeq" + "aaa_aaa" * ]
+ [ [ "for terms" ] "aaa" + "( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_drops" + "aaa_fqus" + "aaa_rdeq" + "aaa_fdeq" + "aaa_aaa" + "aaa_dec" * ]
}
]
[ { "degree-based equivalence" * } {
- [ [ "for closures on referred entries" ] "fdeq" + "( ⦃?,?,?⦄ ≛[?,?] ⦃?,?,?⦄ )" "fdeq_fqup" + "fdeq_fqus" + "fdeq_req" + "fdeq_fdeq" * ]
- [ [ "for lenvs on referred entries" ] "rdeq" + "( ? ≛[?,?,?] ? )" "rdeq_length" + "rdeq_drops" + "rdeq_fqup" + "rdeq_fqus" + "rdeq_req" + "rdeq_rdeq" * ]
+ [ [ "for closures on referred entries" ] "fdeq" + "( ⦃?,?,?⦄ ≛ ⦃?,?,?⦄ )" "fdeq_fqup" + "fdeq_fqus" + "fdeq_req" + "fdeq_fdeq" * ]
+ [ [ "for lenvs on referred entries" ] "rdeq" + "( ? ≛[?] ? )" "rdeq_length" + "rdeq_drops" + "rdeq_fqup" + "rdeq_fqus" + "rdeq_req" + "rdeq_rdeq" * ]
}
]
[ { "syntactic equivalence" * } {
]
[ { "context-sensitive free variables" * } {
[ [ "inclusion for restricted closures" ] "fsle" + "( ⦃?,?⦄ ⊆ ⦃?,?⦄ )" "fsle_length" + "fsle_drops" + "fsle_fqup" + "fsle_fsle" * ]
- [ [ "restricted refinement for lenvs" ] "lsubf" + "( ⦃?,?⦄ ⫃𝐅* ⦃?,?⦄ )" "lsubf_lsubr" + "lsubf_frees" + "lsubf_lsubf" * ]
- [ [ "for terms" ] "frees" + "( ? ⊢ 𝐅*⦃?⦄ ≘ ? )" "frees_append" + "frees_drops" + "frees_fqup" + "frees_frees" * ]
+ [ [ "restricted refinement for lenvs" ] "lsubf" + "( ⦃?,?⦄ ⫃𝐅+ ⦃?,?⦄ )" "lsubf_lsubr" + "lsubf_frees" + "lsubf_lsubf" * ]
+ [ [ "for terms" ] "frees" + "( ? ⊢ 𝐅+⦃?⦄ ≘ ? )" "frees_append" + "frees_drops" + "frees_fqup" + "frees_frees" * ]
}
]
[ { "local environments" * } {
class "grass"
[ { "s-computation" * } {
[ { "iterated structural successor" * } {
- [ [ "for closures" ] "fqus" + "( â¦\83?,?,?â¦\84 â\8a\90*[?] â¦\83?,?,?â¦\84 )" + "( â¦\83?,?,?â¦\84 â\8a\90* ⦃?,?,?⦄ )" "fqus_weight" + "fqus_drops" + "fqus_fqup" + "fqus_fqus" * ]
- [ [ "proper for closures" ] "fqup" + "( â¦\83?,?,?â¦\84 â\8a\90+[?] â¦\83?,?,?â¦\84 )" + "( â¦\83?,?,?â¦\84 â\8a\90+ ⦃?,?,?⦄ )" "fqup_weight" + "fqup_drops" + "fqup_fqup" * ]
+ [ [ "for closures" ] "fqus" + "( â¦\83?,?,?â¦\84 â¬\82*[?] â¦\83?,?,?â¦\84 )" + "( â¦\83?,?,?â¦\84 â¬\82* ⦃?,?,?⦄ )" "fqus_weight" + "fqus_drops" + "fqus_fqup" + "fqus_fqus" * ]
+ [ [ "proper for closures" ] "fqup" + "( â¦\83?,?,?â¦\84 â¬\82+[?] â¦\83?,?,?â¦\84 )" + "( â¦\83?,?,?â¦\84 â¬\82+ ⦃?,?,?⦄ )" "fqup_weight" + "fqup_drops" + "fqup_fqup" * ]
}
]
}
class "yellow"
[ { "s-transition" * } {
[ { "structural successor" * } {
- [ [ "for closures" ] "fquq" + "( â¦\83?,?,?â¦\84 â\8a\90⸮[?] â¦\83?,?,?â¦\84 )" + "( â¦\83?,?,?â¦\84 â\8a\90⸮ ⦃?,?,?⦄ )" "fquq_length" + "fquq_weight" * ]
- [ [ "proper for closures" ] "fqu" + "( â¦\83?,?,?â¦\84 â\8a\90[?] â¦\83?,?,?â¦\84 )" + "( â¦\83?,?,?â¦\84 â\8a\90 ⦃?,?,?⦄ )" "fqu_length" + "fqu_weight" + "fqu_tdeq" * ]
+ [ [ "for closures" ] "fquq" + "( â¦\83?,?,?â¦\84 â¬\82⸮[?] â¦\83?,?,?â¦\84 )" + "( â¦\83?,?,?â¦\84 â¬\82⸮ ⦃?,?,?⦄ )" "fquq_length" + "fquq_weight" * ]
+ [ [ "proper for closures" ] "fqu" + "( â¦\83?,?,?â¦\84 â¬\82[?] â¦\83?,?,?â¦\84 )" + "( â¦\83?,?,?â¦\84 â¬\82 ⦃?,?,?⦄ )" "fqu_length" + "fqu_weight" + "fqu_tdeq" * ]
}
]
}
]
class "orange"
[ { "relocation" * } {
- [ { "generic slicing" * } {
+ [ { "generic and uniform slicing" * } {
[ [ "for lenvs" ] "drops" + "( ⬇*[?,?] ? ≘ ? )" + "( ⬇*[?] ? ≘ ? )" "drops_ctc" + "drops_ltc" + "drops_weight" + "drops_length" + "drops_cext2" + "drops_sex" + "drops_lex" + "drops_seq" + "drops_drops" + "drops_vector" * ]
}
]
- [ { "generic relocation" * } {
+ [ { "basic relocation" * } {
+ [ [ "for terms" ] "lifts_basic" + "( ⬆[?,?] ? ≘ ? )" * ]
+ }
+ ]
+ [ { "generic and uniform relocation" * } {
[ [ "for binders" ] "lifts_bind" + "( ⬆*[?] ? ≘ ? )" "lifts_weight_bind" + "lifts_lifts_bind" * ]
[ [ "for term vectors" ] "lifts_vector" + "( ⬆*[?] ? ≘ ? )" "lifts_lifts_vector" * ]
- [ [ "for terms" ] "lifts" + "( ⬆*[?] ? ≘ ? )" "lifts_simple" + "lifts_weight" + "lifts_tdeq" + "lifts_lifts" * ]
+ [ [ "for terms" ] "lifts" + "( ⬆*[?] ? ≘ ? )" "lifts_simple" + "lifts_weight" + "lifts_tdeq" + "lifts_tweq" + "lifts_toeq" + "lifts_lifts" * ]
}
]
[ { "syntactic equivalence" * } {
]
class "red"
[ { "syntax" * } {
+ [ { "applicability condition" * } {
+ [ [ "preorder" ] "acle" + "( ? ⊆ ? )" "acle_acle" * ]
+ [ [ "properties" ] "ac" + "( 𝟏 )" + "( 𝟐 )" + "( 𝛚 )" * ]
+ }
+ ]
[ { "equivalence up to exclusion binders" * } {
[ [ "for lenvs" ] "lveq" + "( ? ≋ⓧ*[?,?] ? )" "lveq_length" + "lveq_lveq" * ]
}
[ [ "for lenvs" ] "append" + "( ? + ? )" "append_length" * ]
}
]
- [ { "head equivalence" * } {
- [ [ "for terms" ] "theq" + "( ? ⩳[?,?] ? )" "theq_simple" + "theq_tdeq" + "theq_theq" + "theq_simple_vector" * ]
+ [ { "sort-irrelevant outer equivalence" * } {
+ [ [ "for terms" ] "toeq" + "( ? ⩳ ? )" "toeq_simple" + "toeq_tdeq" + "toeq_toeq" + "toeq_simple_vector" * ]
}
]
- [ { "degree-based equivalence" * } {
- [ [ "" ] "tdeq_ext" + "( ? ≛[?,?] ? )" + "( ? ⊢ ? ≛[?,?] ? )" * ]
- [ [ "" ] "tdeq" + "( ? ≛[?,?] ? )" "tdeq_tdeq" * ]
+ [ { "sort-irrelevant whd equivalence" * } {
+ [ [ "for terms" ] "tweq" + "( ? ≅ ? )" "tweq_simple" + "tweq_tdeq" + "tweq_tueq" * ]
+ }
+ ]
+ [ { "sort-irrelevant equivalence" * } {
+ [ [ "" ] "tdeq_ext" + "( ? ≛ ? )" + "( ? ⊢ ? ≛ ? )" * ]
+ [ [ "" ] "tdeq" + "( ? ≛ ? )" "tdeq_tdeq" * ]
}
]
[ { "closures" * } {
}
]
[ { "items" * } {
- [ [ "" ] "item_sd" * ]
- [ [ "" ] "item_sh" * ]
[ [ "" ] "item" * ]
}
]
+ [ { "sorts" * } {
+ [ [ "degree" ] "sd" "sd_d" + "sd_lt" * ]
+ [ [ "hierarchy" ] "sh" + "( ⫯[?]? )" "sh_props" + "sh_lt" * ]
+ }
+ ]
[ { "atomic arities" * } {
[ [ "" ] "aarity" * ]
}
<!-- ================= Tactics ========================= -->
-<!--
<chapter id="sec_declarative_tactics">
<title>Declarative Tactics</title>
<sect1 id="tac_assume">
<title>assume</title>
<titleabbrev>assume</titleabbrev>
- <para><userinput>assume x : t</userinput></para>
+ <para><userinput>assume x : T that is equivalent to T'</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">assume</emphasis> &id; <emphasis role="bold"> : </emphasis> &sterm;</para>
+ <para><emphasis role="bold">assume</emphasis> &id; <emphasis role="bold"> : </emphasis>
+ &sterm; [ <emphasis role="bold">that is equivalent to</emphasis> &term; ]</para>
</listitem>
</varlistentry>
<varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>It adds to the context of the current sequent to prove a new
- declaration <command>x : T </command>. The new conclusion becomes
- <command>P</command>.</para>
+ <para>It adds to the context of the current sequent to prove a new declaration <command>x : T
+ </command>. The new conclusion becomes <command>P</command>. Alternatively, if a type
+ <command>T'</command> is supplied and <command>T</command> and <command>T'</command> are beta equivalent the new declaration that is added to the context is
+ <command>x:T'</command>.</para>
</listitem>
</varlistentry>
<varlistentry>
</para>
</sect1>
- <sect1 id="tac_byinduction">
- <title>by induction hypothesis we know</title>
- <titleabbrev>by induction hypothesis we know</titleabbrev>
- <para><userinput>by induction hypothesis we know t (id)</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem><para><emphasis role="bold">by induction hypothesis we know</emphasis> &term; <emphasis role="bold"> (</emphasis> &id; <emphasis role="bold">)</emphasis></para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-condition:</term>
- <listitem>
- <para>To be used in a proof by induction to state the inductive
- hypothesis.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para> Introduces the inductive hypothesis. </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>None.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
+ <sect1 id="tac_suppose">
+ <title>suppose</title>
+ <titleabbrev>suppose</titleabbrev>
+ <para><userinput>suppose T (x) that is equivalent to T'</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para><emphasis role="bold">suppose</emphasis> &term; <emphasis role="bold"> (</emphasis> &id;
+ <emphasis role="bold">) </emphasis> [ <emphasis role="bold">that is equivalent to</emphasis> &term; ]</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para>The conclusion of the current proof must be
+ <command>∀x:T.P</command> or
+ <command>T→P</command> where <command>T</command> is
+ a proposition (i.e. <command>T</command> has type
+ <command>Prop</command> or <command>CProp</command>).</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It adds to the context of the current sequent to prove a new declaration <command>x : T
+ </command>. The new conclusion becomes <command>P</command>. Alternatively, if a type
+ <command>T'</command> is supplied and <command>T</command> and <command>T'</command> are beta equivalent the new declaration that is added to the context is
+ <command>x:T'</command>.
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
- <sect1 id="tac_case">
- <title>case</title>
- <titleabbrev>case</titleabbrev>
- <para><userinput>case id (id1:t1) … (idn:tn)</userinput></para>
+<sect1 id="tac_let">
+ <title>letin</title>
+ <titleabbrev>letin</titleabbrev>
+ <para><userinput>let x := T </userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para><emphasis role="bold">let</emphasis> &id; <emphasis role="bold"> = </emphasis> &term;</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para>None</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It adds a new local definition <command>x := T</command> to the context of the sequent to prove.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_bytermweproved">
+ <title>we proved</title>
+ <titleabbrev>we proved</titleabbrev>
+ <para><userinput>justification we proved T (id) that is equivalent to T'</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para><emphasis role="bold">case</emphasis> &id; [<emphasis role="bold">(</emphasis> &id; <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">)</emphasis>] … </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-condition:</term>
- <listitem>
- <para>To be used in a proof by induction or by cases to start
- a new case</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>Starts the new case <command>id</command> declaring
- the local parameters <command>(id1:t1) … (idn:tn)</command></para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>None</para>
+ <para>&justification; <emphasis role="bold">we proved</emphasis> &term;
+ <emphasis role="bold">(</emphasis> &id;
+ <emphasis role="bold">)</emphasis> [ <emphasis role="bold">that is equivalent to</emphasis> &term;] [ <emphasis role="bold">done</emphasis>]</para>
</listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para><command>T</command> must have type <command>Prop</command>.
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It derives <command>T</command>
+ using the justification and labels the conclusion with
+ <command>id</command>. Alternatively, if a proposition
+ <command>T'</command> is supplied and <command>T</command> and <command>T'</command> are beta equivalent the new hypothesis that is added to the context is
+ <command>id:T'</command>.
+
+ If the user does not supply a label and ends the command with <command>done</command> then if T is alpha equivalent to the conclusion of the current sequent then it closes it (if <command>T'</command> is supplied this must be alpha equivalent to the conclusion, but in this case <command>T</command> does not need to).
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequent to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
<sect1 id="tac_bydone">
<title>done</title>
</para>
</sect1>
-
- <sect1 id="tac_exitselim">
+ <sect1 id="tac_existselim">
<title>let such that</title>
<titleabbrev>let such that</titleabbrev>
<para><userinput>justification let x:t such that p (id)</userinput>
</para>
</sect1>
- <sect1 id="tac_obtain">
- <title>obtain</title>
- <titleabbrev>obtain</titleabbrev>
- <para><userinput>obtain H t1 = t2 justification</userinput></para>
+ <sect1 id="tac_andelim">
+ <title>we have</title>
+ <titleabbrev>we have</titleabbrev>
+ <para><userinput>justification we have t1 (id1) and t2 (id2)</userinput>
+ </para>
<para>
<variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para>[<emphasis role="bold">obtain</emphasis> &id; | <emphasis role="bold">conclude</emphasis> &term;] <emphasis role="bold">=</emphasis> &term; [&autoparams; | <emphasis role="bold">using</emphasis> &term; | <emphasis role="bold">using once</emphasis> &term; | <emphasis role="bold">proof</emphasis>] [<emphasis role="bold">done</emphasis>]</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-condition:</term>
- <listitem>
- <para><command>conclude</command> can be used only if the current
- sequent is stating an equality. The left hand side must be omitted
- in an equality chain.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>Starts or continues an equality chain. If the chain starts
- with <command>obtain H</command> a new subproof named
- <command>H</command> is started.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequent to prove:</term>
- <listitem>
- <para>If the chain starts
- with <command>obtain H</command> a nre sequent for
- <command>t2 = ?</command> is opened.
- </para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
-
- <sect1 id="tac_suppose">
- <title>suppose</title>
- <titleabbrev>suppose</titleabbrev>
- <para><userinput>suppose t1 (x) that is equivalent to t2</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">suppose</emphasis> &term; <emphasis role="bold"> (</emphasis> &id;
- <emphasis role="bold">) </emphasis> [ <emphasis role="bold">that is equivalent to</emphasis> &term; ]</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-condition:</term>
- <listitem>
- <para>The conclusion of the current proof must be
- <command>∀x:T.P</command> or
- <command>T→P</command> where <command>T</command> is
- a proposition (i.e. <command>T</command> has type
- <command>Prop</command> or <command>CProp</command>).</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
+ <varlistentry role="tactic_synopsis">
+ <term>Synopsis:</term>
<listitem>
- <para>It adds to the context of the current sequent to prove a new
- declaration <command>x : T </command>. The new conclusion becomes
- <command>P</command>.</para>
+ <para>&justification; <emphasis role="bold">we have</emphasis> &term;
+ <emphasis role="bold">( </emphasis> &id; <emphasis role="bold"> ) and </emphasis> &term;
+ <emphasis role="bold"> ( </emphasis> &id; <emphasis role="bold">)</emphasis></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para></para>
</listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>None.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>It derives <command>t1∧t2</command> using the
+ <command>justification</command> then it introduces in the context
+ <command>t1</command> labelled with <command>id1</command> and
+ <command>t2</command> labelled with <command>id2</command>.
+ </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequent to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
</sect1>
- <sect1 id="tac_thesisbecomes">
- <title>the thesis becomes</title>
- <titleabbrev>the thesis becomes</titleabbrev>
- <para><userinput>the thesis becomes t</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role ="bold">the thesis becomes</emphasis> &term; </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-condition:</term>
- <listitem>
- <para>The provided term <command>t</command> must be convertible with
- current sequent.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>It changes the current goal to the one provided.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequent to prove:</term>
- <listitem>
- <para>None.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
-
<sect1 id="tac_weneedtoprove">
<title>we need to prove</title>
<titleabbrev>we need to prove</titleabbrev>
</para>
</sect1>
-
- <sect1 id="tac_andelim">
- <title>we have</title>
- <titleabbrev>we have</titleabbrev>
- <para><userinput>justification we have t1 (id1) and t2 (id2)</userinput>
- </para>
+ <sect1 id="tac_weproceedbyinduction">
+ <title>we proceed by induction on</title>
+ <titleabbrev>we proceed by induction on</titleabbrev>
+ <para><userinput>we proceed by induction on t to prove th</userinput></para>
<para>
<variablelist>
- <varlistentry role="tactic_synopsis">
+ <varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para>&justification; <emphasis role="bold">we have</emphasis> &term;
- <emphasis role="bold">( </emphasis> &id; <emphasis role="bold"> ) and </emphasis> &term;
- <emphasis role="bold"> ( </emphasis> &id; <emphasis role="bold">)</emphasis></para>
- </listitem>
+ <para><emphasis role="bold">we proceed by induction on</emphasis> &term; <emphasis role="bold"> to prove </emphasis> &term; </para>
+ </listitem>
</varlistentry>
- <varlistentry>
+ <varlistentry>
<term>Pre-condition:</term>
<listitem>
- <para></para>
+ <para><command>t</command> must inhabitant of an inductive type and
+ <command>th</command> must be the conclusion to be proved by induction.
+ </para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
- <listitem>
- <para>It derives <command>t1∧t2</command> using the
- <command>justification</command> then it introduces in the context
- <command>t1</command> labelled with <command>id1</command> and
- <command>t2</command> labelled with <command>id2</command>.
- </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequent to prove:</term>
- <listitem>
- <para>None.</para>
- </listitem>
+ <listitem>
+ <para>It proceed by induction on <command>t</command>.</para>
+ </listitem>
</varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>It opens one new sequent for each constructor of the
+ type of <command>t</command>.</para>
+ </listitem>
+ </varlistentry>
</variablelist>
</para>
</sect1>
</variablelist>
</para>
</sect1>
-
- <sect1 id="tac_weproceedbyinduction">
- <title>we proceed by induction on</title>
- <titleabbrev>we proceed by induction on</titleabbrev>
- <para><userinput>we proceed by induction on t to prove th</userinput></para>
- <para>
- <variablelist>
- <varlistentry role="tactic.synopsis">
- <term>Synopsis:</term>
- <listitem>
- <para><emphasis role="bold">we proceed by induction on</emphasis> &term; <emphasis role="bold"> to prove </emphasis> &term; </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Pre-condition:</term>
- <listitem>
- <para><command>t</command> must inhabitant of an inductive type and
- <command>th</command> must be the conclusion to be proved by induction.
- </para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>Action:</term>
- <listitem>
- <para>It proceed by induction on <command>t</command>.</para>
- </listitem>
- </varlistentry>
- <varlistentry>
- <term>New sequents to prove:</term>
- <listitem>
- <para>It opens one new sequent for each constructor of the
- type of <command>t</command>.</para>
- </listitem>
- </varlistentry>
- </variablelist>
- </para>
- </sect1>
-
- <sect1 id="tac_bytermweproved">
- <title>we proved</title>
- <titleabbrev>we proved</titleabbrev>
- <para><userinput>justification we proved t (id)</userinput></para>
+ <sect1 id="tac_case">
+ <title>case</title>
+ <titleabbrev>case</titleabbrev>
+ <para><userinput>case id (id1:t1) … (idn:tn)</userinput></para>
<para>
<variablelist>
<varlistentry role="tactic.synopsis">
<term>Synopsis:</term>
<listitem>
- <para>&justification; <emphasis role="bold">we proved</emphasis> &term;
- <emphasis role="bold">(</emphasis> &id;
- <emphasis role="bold">)</emphasis></para>
+ <para><emphasis role="bold">case</emphasis> &id; [<emphasis role="bold">(</emphasis> &id; <emphasis role="bold">:</emphasis> &term; <emphasis role="bold">)</emphasis>] … </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para>To be used in a proof by induction or by cases to start
+ a new case</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>Starts the new case <command>id</command> declaring
+ the local parameters <command>(id1:t1) … (idn:tn)</command></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None</para>
</listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_byinduction">
+ <title>by induction hypothesis we know</title>
+ <titleabbrev>by induction hypothesis we know</titleabbrev>
+ <para><userinput>by induction hypothesis we know t (id)</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem><para><emphasis role="bold">by induction hypothesis we know</emphasis> &term; <emphasis role="bold"> (</emphasis> &id; <emphasis role="bold">)</emphasis></para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para>To be used in a proof by induction to state the inductive
+ hypothesis.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para> Introduces the inductive hypothesis. </para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequents to prove:</term>
+ <listitem>
+ <para>None.</para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+ <sect1 id="tac_thesisbecomes">
+ <title>the thesis becomes</title>
+ <titleabbrev>the thesis becomes</titleabbrev>
+ <para><userinput>the thesis becomes t</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para><emphasis role ="bold">the thesis becomes</emphasis> &term; </para>
+ </listitem>
</varlistentry>
<varlistentry>
<term>Pre-condition:</term>
<listitem>
- <para><command>t</command>must have type <command>Prop</command>.
- </para>
+ <para>The provided term <command>t</command> must be convertible with
+ current sequent.</para>
</listitem>
</varlistentry>
<varlistentry>
<term>Action:</term>
<listitem>
- <para>It derives <command>t</command>
- using the justification and labels the conclusion with
- <command>id</command>.
- </para>
+ <para>It changes the current goal to the one provided.</para>
</listitem>
</varlistentry>
<varlistentry>
</varlistentry>
</variablelist>
</para>
- </sect1>
+ </sect1>
+ <sect1 id="tac_obtain">
+ <title>conclude/obtain</title>
+ <titleabbrev>conclude/obtain</titleabbrev>
+ <para><userinput>conclude/obtain (H) t1 = t2 justification</userinput></para>
+ <para>
+ <variablelist>
+ <varlistentry role="tactic.synopsis">
+ <term>Synopsis:</term>
+ <listitem>
+ <para>[<emphasis role="bold">obtain</emphasis> &id; | <emphasis role="bold">conclude</emphasis> &term;] <emphasis role="bold">=</emphasis> &term; [&autoparams; | <emphasis role="bold">using</emphasis> &term; | <emphasis role="bold">using once</emphasis> &term; | <emphasis role="bold">proof</emphasis>] [<emphasis role="bold">done</emphasis>]</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Pre-condition:</term>
+ <listitem>
+ <para><command>conclude</command> can be used only if the current
+ sequent is stating an equality. The left hand side must be omitted
+ in an equality chain.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>Action:</term>
+ <listitem>
+ <para>Starts or continues an equality chain. If the chain starts
+ with <command>obtain H</command> a new subproof named
+ <command>H</command> is started.</para>
+ </listitem>
+ </varlistentry>
+ <varlistentry>
+ <term>New sequent to prove:</term>
+ <listitem>
+ <para>If the chain starts
+ with <command>obtain H</command> a nre sequent for
+ <command>t2 = ?</command> is opened.
+ </para>
+ </listitem>
+ </varlistentry>
+ </variablelist>
+ </para>
+ </sect1>
+
+
+
+
+
+
+
</chapter>
--->
notation "hvbox(a break ≬ b)" non associative with precedence 45
for @{ 'overlaps $a $b }. (* \between *)
-notation "hvbox(a break ⊆ b)" non associative with precedence 45
-for @{ 'subseteq $a $b }. (* \subseteq *)
-
notation "hvbox(a break ∩ b)" left associative with precedence 60
for @{ 'intersects $a $b }. (* \cap *)
(* other notations **********************************************************)
-notation "hvbox(a break \approx b)" non associative with precedence 45
- for @{ 'napart $a $b}.
-
-notation "hvbox(a break # b)" non associative with precedence 45
- for @{ 'apart $a $b}.
-
notation < "term 76 a \sup term 90 b" non associative with precedence 75 for @{ 'exp $a $b}.
notation > "a \sup term 90 b" non associative with precedence 75 for @{ 'exp $a $b}.
notation > "a ^ term 90 b" non associative with precedence 75 for @{ 'exp $a $b}.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+(* Core notation *******************************************************)
+
+notation "hvbox(a break # b)" non associative with precedence 45
+ for @{ 'apart $a $b}.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+(* Core notation *******************************************************)
+
+notation "hvbox(a break \approx b)" non associative with precedence 45
+ for @{ 'napart $a $b}.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+(* Core notation *******************************************************)
+
+notation "hvbox(a break ⊆ b)" non associative with precedence 45
+for @{ 'subseteq $a $b }. (* \subseteq *)
include "basics/logic.ma".
include "basics/core_notation/compose_2.ma".
+include "basics/core_notation/subseteq_2.ma".
(********** predicates *********)
include "basics/logic.ma".
include "basics/core_notation/singl_1.ma".
+include "basics/core_notation/subseteq_2.ma".
(**** a subset of A is just an object of type A→Prop ****)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+(* Logic system *)
+
+include "basics/pts.ma".
+include "hints_declaration.ma".
+
+inductive Imply (A,B:Prop) : Prop ≝
+| Imply_intro: (A → B) → Imply A B.
+
+definition Imply_elim ≝ λA,B:Prop.λf:Imply A B. λa:A.
+ match f with [ Imply_intro g ⇒ g a].
+
+inductive And (A,B:Prop) : Prop ≝
+| And_intro: A → B → And A B.
+
+definition And_elim_l ≝ λA,B.λc:And A B.
+ match c with [ And_intro a b ⇒ a ].
+
+definition And_elim_r ≝ λA,B.λc:And A B.
+ match c with [ And_intro a b ⇒ b ].
+
+inductive Or (A,B:Prop) : Prop ≝
+| Or_intro_l: A → Or A B
+| Or_intro_r: B → Or A B.
+
+definition Or_elim ≝ λA,B,C:Prop.λc:Or A B.λfa: A → C.λfb: B → C.
+ match c with
+ [ Or_intro_l a ⇒ fa a
+ | Or_intro_r b ⇒ fb b].
+
+inductive Top : Prop :=
+| Top_intro : Top.
+
+inductive Bot : Prop := .
+
+definition Bot_elim ≝ λP:Prop.λx:Bot.
+ match x in Bot return λx.P with [].
+
+definition Not := λA:Prop.Imply A Bot.
+
+definition Not_intro : ∀A.(A → Bot) → Not A ≝ λA.
+ Imply_intro A Bot.
+
+definition Not_elim : ∀A.Not A → A → Bot ≝ λA.
+ Imply_elim ? Bot.
+
+definition Discharge := λA:Prop.λa:A.
+ a.
+
+axiom Raa : ∀A.(Not A → Bot) → A.
+
+axiom sort : Type[0].
+
+inductive Exists (A:Type[0]) (P:A→Prop) : Prop ≝
+ Exists_intro: ∀w:A. P w → Exists A P.
+
+definition Exists_elim ≝
+ λA:Type[0].λP:A→Prop.λC:Prop.λc:Exists A P.λH:(Πx.P x → C).
+ match c with [ Exists_intro w p ⇒ H w p ].
+
+inductive Forall (A:Type[0]) (P:A→Prop) : Prop ≝
+ Forall_intro: (∀n:A. P n) → Forall A P.
+
+definition Forall_elim ≝
+ λA:Type[0].λP:A→Prop.λn:A.λf:Forall A P.match f with [ Forall_intro g ⇒ g n ].
+
+(* Dummy proposition *)
+axiom unit : Prop.
+
+(* Notations *)
+notation "hbox(a break ⇒ b)" right associative with precedence 20
+for @{ 'Imply $a $b }.
+interpretation "Imply" 'Imply a b = (Imply a b).
+interpretation "constructive or" 'or x y = (Or x y).
+interpretation "constructive and" 'and x y = (And x y).
+notation "⊤" non associative with precedence 90 for @{'Top}.
+interpretation "Top" 'Top = Top.
+notation "⊥" non associative with precedence 90 for @{'Bot}.
+interpretation "Bot" 'Bot = Bot.
+interpretation "Not" 'not a = (Not a).
+notation "✶" non associative with precedence 90 for @{'unit}.
+interpretation "dummy prop" 'unit = unit.
+notation > "\exists list1 ident x sep , . term 19 Px" with precedence 20
+for ${ fold right @{$Px} rec acc @{'myexists (λ${ident x}.$acc)} }.
+notation < "hvbox(\exists ident i break . p)" with precedence 20
+for @{ 'myexists (\lambda ${ident i} : $ty. $p) }.
+interpretation "constructive ex" 'myexists \eta.x = (Exists sort x).
+notation > "\forall ident x.break term 19 Px" with precedence 20
+for @{ 'Forall (λ${ident x}.$Px) }.
+notation < "\forall ident x.break term 19 Px" with precedence 20
+for @{ 'Forall (λ${ident x}:$tx.$Px) }.
+interpretation "Forall" 'Forall \eta.Px = (Forall ? Px).
+
+(* Variables *)
+axiom A : Prop.
+axiom B : Prop.
+axiom C : Prop.
+axiom D : Prop.
+axiom E : Prop.
+axiom F : Prop.
+axiom G : Prop.
+axiom H : Prop.
+axiom I : Prop.
+axiom J : Prop.
+axiom K : Prop.
+axiom L : Prop.
+axiom M : Prop.
+axiom N : Prop.
+axiom O : Prop.
+axiom x: sort.
+axiom y: sort.
+axiom z: sort.
+axiom w: sort.
+
+(* Every formula user provided annotates its proof:
+ `A` becomes `(show A ?)` *)
+definition show : ΠA.A→A ≝ λA:Prop.λa:A.a.
+
+(* When something does not fit, this daemon is used *)
+axiom cast: ΠA,B:Prop.B → A.
+
+(* begin a proof: draws the root *)
+notation > "'prove' p" non associative with precedence 19
+for @{ 'prove $p }.
+interpretation "prove KO" 'prove p = (cast ? ? (show p ?)).
+interpretation "prove OK" 'prove p = (show p ?).
+
+(* Leaves *)
+notation < "\infrule (t\atop ⋮) a ?" with precedence 19
+for @{ 'leaf_ok $a $t }.
+interpretation "leaf OK" 'leaf_ok a t = (show a t).
+notation < "\infrule (t\atop ⋮) mstyle color #ff0000 (a) ?" with precedence 19
+for @{ 'leaf_ko $a $t }.
+interpretation "leaf KO" 'leaf_ko a t = (cast ? ? (show a t)).
+
+(* discharging *)
+notation < "[ a ] \sup mstyle color #ff0000 (H)" with precedence 19
+for @{ 'discharge_ko_1 $a $H }.
+interpretation "discharge_ko_1" 'discharge_ko_1 a H =
+ (show a (cast ? ? (Discharge ? H))).
+notation < "[ mstyle color #ff0000 (a) ] \sup mstyle color #ff0000 (H)" with precedence 19
+for @{ 'discharge_ko_2 $a $H }.
+interpretation "discharge_ko_2" 'discharge_ko_2 a H =
+ (cast ? ? (show a (cast ? ? (Discharge ? H)))).
+
+notation < "[ a ] \sup H" with precedence 19
+for @{ 'discharge_ok_1 $a $H }.
+interpretation "discharge_ok_1" 'discharge_ok_1 a H =
+ (show a (Discharge ? H)).
+notation < "[ mstyle color #ff0000 (a) ] \sup H" with precedence 19
+for @{ 'discharge_ok_2 $a $H }.
+interpretation "discharge_ok_2" 'discharge_ok_2 a H =
+ (cast ? ? (show a (Discharge ? H))).
+
+notation > "'discharge' [H]" with precedence 19
+for @{ 'discharge $H }.
+interpretation "discharge KO" 'discharge H = (cast ? ? (Discharge ? H)).
+interpretation "discharge OK" 'discharge H = (Discharge ? H).
+
+(* ⇒ introduction *)
+notation < "\infrule hbox(\emsp b \emsp) ab (mstyle color #ff0000 (⇒\sub\i \emsp) ident H) " with precedence 19
+for @{ 'Imply_intro_ko_1 $ab (λ${ident H}:$p.$b) }.
+interpretation "Imply_intro_ko_1" 'Imply_intro_ko_1 ab \eta.b =
+ (show ab (cast ? ? (Imply_intro ? ? b))).
+
+notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (mstyle color #ff0000 (⇒\sub\i \emsp) ident H) " with precedence 19
+for @{ 'Imply_intro_ko_2 $ab (λ${ident H}:$p.$b) }.
+interpretation "Imply_intro_ko_2" 'Imply_intro_ko_2 ab \eta.b =
+ (cast ? ? (show ab (cast ? ? (Imply_intro ? ? b)))).
+
+notation < "maction (\infrule hbox(\emsp b \emsp) ab (⇒\sub\i \emsp ident H) ) (\vdots)" with precedence 19
+for @{ 'Imply_intro_ok_1 $ab (λ${ident H}:$p.$b) }.
+interpretation "Imply_intro_ok_1" 'Imply_intro_ok_1 ab \eta.b =
+ (show ab (Imply_intro ? ? b)).
+
+notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (⇒\sub\i \emsp ident H) " with precedence 19
+for @{ 'Imply_intro_ok_2 $ab (λ${ident H}:$p.$b) }.
+interpretation "Imply_intro_ok_2" 'Imply_intro_ok_2 ab \eta.b =
+ (cast ? ? (show ab (Imply_intro ? ? b))).
+
+notation > "⇒#'i' [ident H] term 90 b" with precedence 19
+for @{ 'Imply_intro $b (λ${ident H}.show $b ?) }.
+
+interpretation "Imply_intro KO" 'Imply_intro b pb =
+ (cast ? (Imply unit b) (Imply_intro ? b pb)).
+interpretation "Imply_intro OK" 'Imply_intro b pb =
+ (Imply_intro ? b pb).
+
+(* ⇒ elimination *)
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b mstyle color #ff0000 (⇒\sub\e) " with precedence 19
+for @{ 'Imply_elim_ko_1 $ab $a $b }.
+interpretation "Imply_elim_ko_1" 'Imply_elim_ko_1 ab a b =
+ (show b (cast ? ? (Imply_elim ? ? (cast ? ? ab) (cast ? ? a)))).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) mstyle color #ff0000 (⇒\sub\e) " with precedence 19
+for @{ 'Imply_elim_ko_2 $ab $a $b }.
+interpretation "Imply_elim_ko_2" 'Imply_elim_ko_2 ab a b =
+ (cast ? ? (show b (cast ? ? (Imply_elim ? ? (cast ? ? ab) (cast ? ? a))))).
+
+notation < "maction (\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (⇒\sub\e) ) (\vdots)" with precedence 19
+for @{ 'Imply_elim_ok_1 $ab $a $b }.
+interpretation "Imply_elim_ok_1" 'Imply_elim_ok_1 ab a b =
+ (show b (Imply_elim ? ? ab a)).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) (⇒\sub\e) " with precedence 19
+for @{ 'Imply_elim_ok_2 $ab $a $b }.
+interpretation "Imply_elim_ok_2" 'Imply_elim_ok_2 ab a b =
+ (cast ? ? (show b (Imply_elim ? ? ab a))).
+
+notation > "⇒#'e' term 90 ab term 90 a" with precedence 19
+for @{ 'Imply_elim (show $ab ?) (show $a ?) }.
+interpretation "Imply_elim KO" 'Imply_elim ab a =
+ (cast ? ? (Imply_elim ? ? (cast (Imply unit unit) ? ab) (cast unit ? a))).
+interpretation "Imply_elim OK" 'Imply_elim ab a =
+ (Imply_elim ? ? ab a).
+
+(* ∧ introduction *)
+notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab mstyle color #ff0000 (∧\sub\i)" with precedence 19
+for @{ 'And_intro_ko_1 $a $b $ab }.
+interpretation "And_intro_ko_1" 'And_intro_ko_1 a b ab =
+ (show ab (cast ? ? (And_intro ? ? a b))).
+
+notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∧\sub\i)" with precedence 19
+for @{ 'And_intro_ko_2 $a $b $ab }.
+interpretation "And_intro_ko_2" 'And_intro_ko_2 a b ab =
+ (cast ? ? (show ab (cast ? ? (And_intro ? ? a b)))).
+
+notation < "maction (\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab (∧\sub\i)) (\vdots)" with precedence 19
+for @{ 'And_intro_ok_1 $a $b $ab }.
+interpretation "And_intro_ok_1" 'And_intro_ok_1 a b ab =
+ (show ab (And_intro ? ? a b)).
+
+notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) mstyle color #ff0000 (ab) (∧\sub\i)" with precedence 19
+for @{ 'And_intro_ok_2 $a $b $ab }.
+interpretation "And_intro_ok_2" 'And_intro_ok_2 a b ab =
+ (cast ? ? (show ab (And_intro ? ? a b))).
+
+notation > "∧#'i' term 90 a term 90 b" with precedence 19
+for @{ 'And_intro (show $a ?) (show $b ?) }.
+interpretation "And_intro KO" 'And_intro a b = (cast ? ? (And_intro ? ? a b)).
+interpretation "And_intro OK" 'And_intro a b = (And_intro ? ? a b).
+
+(* ∧ elimination *)
+notation < "\infrule hbox(\emsp ab \emsp) a mstyle color #ff0000 (∧\sub(\e_\l))" with precedence 19
+for @{ 'And_elim_l_ko_1 $ab $a }.
+interpretation "And_elim_l_ko_1" 'And_elim_l_ko_1 ab a =
+ (show a (cast ? ? (And_elim_l ? ? (cast ? ? ab)))).
+
+notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) mstyle color #ff0000 (∧\sub(\e_\l))" with precedence 19
+for @{ 'And_elim_l_ko_2 $ab $a }.
+interpretation "And_elim_l_ko_2" 'And_elim_l_ko_2 ab a =
+ (cast ? ? (show a (cast ? ? (And_elim_l ? ? (cast ? ? ab))))).
+
+notation < "maction (\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\l))) (\vdots)" with precedence 19
+for @{ 'And_elim_l_ok_1 $ab $a }.
+interpretation "And_elim_l_ok_1" 'And_elim_l_ok_1 ab a =
+ (show a (And_elim_l ? ? ab)).
+
+notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) (∧\sub(\e_\l))" with precedence 19
+for @{ 'And_elim_l_ok_2 $ab $a }.
+interpretation "And_elim_l_ok_2" 'And_elim_l_ok_2 ab a =
+ (cast ? ? (show a (And_elim_l ? ? ab))).
+
+notation > "∧#'e_l' term 90 ab" with precedence 19
+for @{ 'And_elim_l (show $ab ?) }.
+interpretation "And_elim_l KO" 'And_elim_l a = (cast ? ? (And_elim_l ? ? (cast (And unit unit) ? a))).
+interpretation "And_elim_l OK" 'And_elim_l a = (And_elim_l ? ? a).
+
+notation < "\infrule hbox(\emsp ab \emsp) a mstyle color #ff0000 (∧\sub(\e_\r))" with precedence 19
+for @{ 'And_elim_r_ko_1 $ab $a }.
+interpretation "And_elim_r_ko_1" 'And_elim_r_ko_1 ab a =
+ (show a (cast ? ? (And_elim_r ? ? (cast ? ? ab)))).
+
+notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) mstyle color #ff0000 (∧\sub(\e_\r))" with precedence 19
+for @{ 'And_elim_r_ko_2 $ab $a }.
+interpretation "And_elim_r_ko_2" 'And_elim_r_ko_2 ab a =
+ (cast ? ? (show a (cast ? ? (And_elim_r ? ? (cast ? ? ab))))).
+
+notation < "maction (\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\r))) (\vdots)" with precedence 19
+for @{ 'And_elim_r_ok_1 $ab $a }.
+interpretation "And_elim_r_ok_1" 'And_elim_r_ok_1 ab a =
+ (show a (And_elim_r ? ? ab)).
+
+notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) (∧\sub(\e_\r))" with precedence 19
+for @{ 'And_elim_r_ok_2 $ab $a }.
+interpretation "And_elim_r_ok_2" 'And_elim_r_ok_2 ab a =
+ (cast ? ? (show a (And_elim_r ? ? ab))).
+
+notation > "∧#'e_r' term 90 ab" with precedence 19
+for @{ 'And_elim_r (show $ab ?) }.
+interpretation "And_elim_r KO" 'And_elim_r a = (cast ? ? (And_elim_r ? ? (cast (And unit unit) ? a))).
+interpretation "And_elim_r OK" 'And_elim_r a = (And_elim_r ? ? a).
+
+(* ∨ introduction *)
+notation < "\infrule hbox(\emsp a \emsp) ab mstyle color #ff0000 (∨\sub(\i_\l))" with precedence 19
+for @{ 'Or_intro_l_ko_1 $a $ab }.
+interpretation "Or_intro_l_ko_1" 'Or_intro_l_ko_1 a ab =
+ (show ab (cast ? ? (Or_intro_l ? ? a))).
+
+notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∨\sub(\i_\l))" with precedence 19
+for @{ 'Or_intro_l_ko_2 $a $ab }.
+interpretation "Or_intro_l_ko_2" 'Or_intro_l_ko_2 a ab =
+ (cast ? ? (show ab (cast ? ? (Or_intro_l ? ? a)))).
+
+notation < "maction (\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\l))) (\vdots)" with precedence 19
+for @{ 'Or_intro_l_ok_1 $a $ab }.
+interpretation "Or_intro_l_ok_1" 'Or_intro_l_ok_1 a ab =
+ (show ab (Or_intro_l ? ? a)).
+
+notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\l))" with precedence 19
+for @{ 'Or_intro_l_ok_2 $a $ab }.
+interpretation "Or_intro_l_ok_2" 'Or_intro_l_ok_2 a ab =
+ (cast ? ? (show ab (Or_intro_l ? ? a))).
+
+notation > "∨#'i_l' term 90 a" with precedence 19
+for @{ 'Or_intro_l (show $a ?) }.
+interpretation "Or_intro_l KO" 'Or_intro_l a = (cast ? (Or ? unit) (Or_intro_l ? ? a)).
+interpretation "Or_intro_l OK" 'Or_intro_l a = (Or_intro_l ? ? a).
+
+notation < "\infrule hbox(\emsp a \emsp) ab mstyle color #ff0000 (∨\sub(\i_\r))" with precedence 19
+for @{ 'Or_intro_r_ko_1 $a $ab }.
+interpretation "Or_intro_r_ko_1" 'Or_intro_r_ko_1 a ab =
+ (show ab (cast ? ? (Or_intro_r ? ? a))).
+
+notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∨\sub(\i_\r))" with precedence 19
+for @{ 'Or_intro_r_ko_2 $a $ab }.
+interpretation "Or_intro_r_ko_2" 'Or_intro_r_ko_2 a ab =
+ (cast ? ? (show ab (cast ? ? (Or_intro_r ? ? a)))).
+
+notation < "maction (\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\r))) (\vdots)" with precedence 19
+for @{ 'Or_intro_r_ok_1 $a $ab }.
+interpretation "Or_intro_r_ok_1" 'Or_intro_r_ok_1 a ab =
+ (show ab (Or_intro_r ? ? a)).
+
+notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\r))" with precedence 19
+for @{ 'Or_intro_r_ok_2 $a $ab }.
+interpretation "Or_intro_r_ok_2" 'Or_intro_r_ok_2 a ab =
+ (cast ? ? (show ab (Or_intro_r ? ? a))).
+
+notation > "∨#'i_r' term 90 a" with precedence 19
+for @{ 'Or_intro_r (show $a ?) }.
+interpretation "Or_intro_r KO" 'Or_intro_r a = (cast ? (Or unit ?) (Or_intro_r ? ? a)).
+interpretation "Or_intro_r OK" 'Or_intro_r a = (Or_intro_r ? ? a).
+
+(* ∨ elimination *)
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) c (mstyle color #ff0000 (∨\sub\e \emsp) ident Ha \emsp ident Hb)" with precedence 19
+for @{ 'Or_elim_ko_1 $ab $c (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) }.
+interpretation "Or_elim_ko_1" 'Or_elim_ko_1 ab c \eta.ac \eta.bc =
+ (show c (cast ? ? (Or_elim ? ? ? (cast ? ? ab) (cast ? ? ac) (cast ? ? bc)))).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) mstyle color #ff0000 (c) (mstyle color #ff0000 (∨\sub\e) \emsp ident Ha \emsp ident Hb)" with precedence 19
+for @{ 'Or_elim_ko_2 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
+interpretation "Or_elim_ko_2" 'Or_elim_ko_2 ab \eta.ac \eta.bc c =
+ (cast ? ? (show c (cast ? ? (Or_elim ? ? ? (cast ? ? ab) (cast ? ? ac) (cast ? ? bc))))).
+
+notation < "maction (\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) c (∨\sub\e \emsp ident Ha \emsp ident Hb)) (\vdots)" with precedence 19
+for @{ 'Or_elim_ok_1 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
+interpretation "Or_elim_ok_1" 'Or_elim_ok_1 ab \eta.ac \eta.bc c =
+ (show c (Or_elim ? ? ? ab ac bc)).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) mstyle color #ff0000 (c) (∨\sub\e \emsp ident Ha \emsp ident Hb)" with precedence 19
+for @{ 'Or_elim_ok_2 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
+interpretation "Or_elim_ok_2" 'Or_elim_ok_2 ab \eta.ac \eta.bc c =
+ (cast ? ? (show c (Or_elim ? ? ? ab ac bc))).
+
+definition unit_to ≝ λx:Prop.unit → x.
+
+notation > "∨#'e' term 90 ab [ident Ha] term 90 cl [ident Hb] term 90 cr" with precedence 19
+for @{ 'Or_elim (show $ab ?) (λ${ident Ha}.show $cl ?) (λ${ident Hb}.show $cr ?) }.
+interpretation "Or_elim KO" 'Or_elim ab ac bc =
+ (cast ? ? (Or_elim ? ? ?
+ (cast (Or unit unit) ? ab)
+ (cast (unit_to unit) (unit_to ?) ac)
+ (cast (unit_to unit) (unit_to ?) bc))).
+interpretation "Or_elim OK" 'Or_elim ab ac bc = (Or_elim ? ? ? ab ac bc).
+
+(* ⊤ introduction *)
+notation < "\infrule \nbsp ⊤ mstyle color #ff0000 (⊤\sub\i)" with precedence 19
+for @{'Top_intro_ko_1}.
+interpretation "Top_intro_ko_1" 'Top_intro_ko_1 =
+ (show ? (cast ? ? Top_intro)).
+
+notation < "\infrule \nbsp mstyle color #ff0000 (⊤) mstyle color #ff0000 (⊤\sub\i)" with precedence 19
+for @{'Top_intro_ko_2}.
+interpretation "Top_intro_ko_2" 'Top_intro_ko_2 =
+ (cast ? ? (show ? (cast ? ? Top_intro))).
+
+notation < "maction (\infrule \nbsp ⊤ (⊤\sub\i)) (\vdots)" with precedence 19
+for @{'Top_intro_ok_1}.
+interpretation "Top_intro_ok_1" 'Top_intro_ok_1 = (show ? Top_intro).
+
+notation < "maction (\infrule \nbsp ⊤ (⊤\sub\i)) (\vdots)" with precedence 19
+for @{'Top_intro_ok_2 }.
+interpretation "Top_intro_ok_2" 'Top_intro_ok_2 = (cast ? ? (show ? Top_intro)).
+
+notation > "⊤#'i'" with precedence 19 for @{ 'Top_intro }.
+interpretation "Top_intro KO" 'Top_intro = (cast ? ? Top_intro).
+interpretation "Top_intro OK" 'Top_intro = Top_intro.
+
+(* ⊥ introduction *)
+notation < "\infrule b a mstyle color #ff0000 (⊥\sub\e)" with precedence 19
+for @{'Bot_elim_ko_1 $a $b}.
+interpretation "Bot_elim_ko_1" 'Bot_elim_ko_1 a b =
+ (show a (Bot_elim ? (cast ? ? b))).
+
+notation < "\infrule b mstyle color #ff0000 (a) mstyle color #ff0000 (⊥\sub\e)" with precedence 19
+for @{'Bot_elim_ko_2 $a $b}.
+interpretation "Bot_elim_ko_2" 'Bot_elim_ko_2 a b =
+ (cast ? ? (show a (Bot_elim ? (cast ? ? b)))).
+
+notation < "maction (\infrule b a (⊥\sub\e)) (\vdots)" with precedence 19
+for @{'Bot_elim_ok_1 $a $b}.
+interpretation "Bot_elim_ok_1" 'Bot_elim_ok_1 a b =
+ (show a (Bot_elim ? b)).
+
+notation < "\infrule b mstyle color #ff0000 (a) (⊥\sub\e)" with precedence 19
+for @{'Bot_elim_ok_2 $a $b}.
+interpretation "Bot_elim_ok_2" 'Bot_elim_ok_2 a b =
+ (cast ? ? (show a (Bot_elim ? b))).
+
+notation > "⊥#'e' term 90 b" with precedence 19
+for @{ 'Bot_elim (show $b ?) }.
+interpretation "Bot_elim KO" 'Bot_elim a = (Bot_elim ? (cast ? ? a)).
+interpretation "Bot_elim OK" 'Bot_elim a = (Bot_elim ? a).
+
+(* ¬ introduction *)
+notation < "\infrule hbox(\emsp b \emsp) ab (mstyle color #ff0000 (\lnot\sub(\emsp\i)) \emsp ident H)" with precedence 19
+for @{ 'Not_intro_ko_1 $ab (λ${ident H}:$p.$b) }.
+interpretation "Not_intro_ko_1" 'Not_intro_ko_1 ab \eta.b =
+ (show ab (cast ? ? (Not_intro ? (cast ? ? b)))).
+
+notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (mstyle color #ff0000 (\lnot\sub(\emsp\i)) \emsp ident H)" with precedence 19
+for @{ 'Not_intro_ko_2 $ab (λ${ident H}:$p.$b) }.
+interpretation "Not_intro_ko_2" 'Not_intro_ko_2 ab \eta.b =
+ (cast ? ? (show ab (cast ? ? (Not_intro ? (cast ? ? b))))).
+
+notation < "maction (\infrule hbox(\emsp b \emsp) ab (\lnot\sub(\emsp\i) \emsp ident H) ) (\vdots)" with precedence 19
+for @{ 'Not_intro_ok_1 $ab (λ${ident H}:$p.$b) }.
+interpretation "Not_intro_ok_1" 'Not_intro_ok_1 ab \eta.b =
+ (show ab (Not_intro ? b)).
+
+notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (\lnot\sub(\emsp\i) \emsp ident H) " with precedence 19
+for @{ 'Not_intro_ok_2 $ab (λ${ident H}:$p.$b) }.
+interpretation "Not_intro_ok_2" 'Not_intro_ok_2 ab \eta.b =
+ (cast ? ? (show ab (Not_intro ? b))).
+
+notation > "¬#'i' [ident H] term 90 b" with precedence 19
+for @{ 'Not_intro (λ${ident H}.show $b ?) }.
+interpretation "Not_intro KO" 'Not_intro a = (cast ? ? (Not_intro ? (cast ? ? a))).
+interpretation "Not_intro OK" 'Not_intro a = (Not_intro ? a).
+
+(* ¬ elimination *)
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b mstyle color #ff0000 (\lnot\sub(\emsp\e)) " with precedence 19
+for @{ 'Not_elim_ko_1 $ab $a $b }.
+interpretation "Not_elim_ko_1" 'Not_elim_ko_1 ab a b =
+ (show b (cast ? ? (Not_elim ? (cast ? ? ab) (cast ? ? a)))).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) mstyle color #ff0000 (\lnot\sub(\emsp\e)) " with precedence 19
+for @{ 'Not_elim_ko_2 $ab $a $b }.
+interpretation "Not_elim_ko_2" 'Not_elim_ko_2 ab a b =
+ (cast ? ? (show b (cast ? ? (Not_elim ? (cast ? ? ab) (cast ? ? a))))).
+
+notation < "maction (\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (\lnot\sub(\emsp\e)) ) (\vdots)" with precedence 19
+for @{ 'Not_elim_ok_1 $ab $a $b }.
+interpretation "Not_elim_ok_1" 'Not_elim_ok_1 ab a b =
+ (show b (Not_elim ? ab a)).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) (\lnot\sub(\emsp\e)) " with precedence 19
+for @{ 'Not_elim_ok_2 $ab $a $b }.
+interpretation "Not_elim_ok_2" 'Not_elim_ok_2 ab a b =
+ (cast ? ? (show b (Not_elim ? ab a))).
+
+notation > "¬#'e' term 90 ab term 90 a" with precedence 19
+for @{ 'Not_elim (show $ab ?) (show $a ?) }.
+interpretation "Not_elim KO" 'Not_elim ab a =
+ (cast ? ? (Not_elim unit (cast ? ? ab) (cast ? ? a))).
+interpretation "Not_elim OK" 'Not_elim ab a =
+ (Not_elim ? ab a).
+
+(* RAA *)
+notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (\RAA) \emsp ident x)" with precedence 19
+for @{ 'RAA_ko_1 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "RAA_ko_1" 'RAA_ko_1 Px Pn =
+ (show Pn (cast ? ? (Raa ? (cast ? ? Px)))).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (mstyle color #ff0000 (\RAA) \emsp ident x)" with precedence 19
+for @{ 'RAA_ko_2 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "RAA_ko_2" 'RAA_ko_2 Px Pn =
+ (cast ? ? (show Pn (cast ? ? (Raa ? (cast ? ? Px))))).
+
+notation < "maction (\infrule hbox(\emsp Px \emsp) Pn (\RAA \emsp ident x)) (\vdots)" with precedence 19
+for @{ 'RAA_ok_1 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "RAA_ok_1" 'RAA_ok_1 Px Pn =
+ (show Pn (Raa ? Px)).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (\RAA \emsp ident x)" with precedence 19
+for @{ 'RAA_ok_2 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "RAA_ok_2" 'RAA_ok_2 Px Pn =
+ (cast ? ? (show Pn (Raa ? Px))).
+
+notation > "'RAA' [ident H] term 90 b" with precedence 19
+for @{ 'Raa (λ${ident H}.show $b ?) }.
+interpretation "RAA KO" 'Raa p = (cast ? unit (Raa ? (cast ? (unit_to ?) p))).
+interpretation "RAA OK" 'Raa p = (Raa ? p).
+
+(* ∃ introduction *)
+notation < "\infrule hbox(\emsp Pn \emsp) Px mstyle color #ff0000 (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro_ko_1 $Pn $Px }.
+interpretation "Exists_intro_ko_1" 'Exists_intro_ko_1 Pn Px =
+ (show Px (cast ? ? (Exists_intro ? ? ? (cast ? ? Pn)))).
+
+notation < "\infrule hbox(\emsp Pn \emsp) mstyle color #ff0000 (Px) mstyle color #ff0000 (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro_ko_2 $Pn $Px }.
+interpretation "Exists_intro_ko_2" 'Exists_intro_ko_2 Pn Px =
+ (cast ? ? (show Px (cast ? ? (Exists_intro ? ? ? (cast ? ? Pn))))).
+
+notation < "maction (\infrule hbox(\emsp Pn \emsp) Px (∃\sub\i)) (\vdots)" with precedence 19
+for @{ 'Exists_intro_ok_1 $Pn $Px }.
+interpretation "Exists_intro_ok_1" 'Exists_intro_ok_1 Pn Px =
+ (show Px (Exists_intro ? ? ? Pn)).
+
+notation < "\infrule hbox(\emsp Pn \emsp) mstyle color #ff0000 (Px) (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro_ok_2 $Pn $Px }.
+interpretation "Exists_intro_ok_2" 'Exists_intro_ok_2 Pn Px =
+ (cast ? ? (show Px (Exists_intro ? ? ? Pn))).
+
+notation >"∃#'i' {term 90 t} term 90 Pt" non associative with precedence 19
+for @{'Exists_intro $t (λw.? w) (show $Pt ?)}.
+interpretation "Exists_intro KO" 'Exists_intro t P Pt =
+ (cast ? ? (Exists_intro sort P t (cast ? ? Pt))).
+interpretation "Exists_intro OK" 'Exists_intro t P Pt =
+ (Exists_intro sort P t Pt).
+
+(* ∃ elimination *)
+notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) c (mstyle color #ff0000 (∃\sub\e) \emsp ident n \emsp ident HPn)" with precedence 19
+for @{ 'Exists_elim_ko_1 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ko_1" 'Exists_elim_ko_1 ExPx Pc c =
+ (show c (cast ? ? (Exists_elim ? ? ? (cast ? ? ExPx) (cast ? ? Pc)))).
+
+notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) mstyle color #ff0000 (c) (mstyle color #ff0000 (∃\sub\e) \emsp ident n \emsp ident HPn)" with precedence 19
+for @{ 'Exists_elim_ko_2 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ko_2" 'Exists_elim_ko_2 ExPx Pc c =
+ (cast ? ? (show c (cast ? ? (Exists_elim ? ? ? (cast ? ? ExPx) (cast ? ? Pc))))).
+
+notation < "maction (\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) c (∃\sub\e \emsp ident n \emsp ident HPn)) (\vdots)" with precedence 19
+for @{ 'Exists_elim_ok_1 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ok_1" 'Exists_elim_ok_1 ExPx Pc c =
+ (show c (Exists_elim ? ? ? ExPx Pc)).
+
+notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) mstyle color #ff0000 (c) (∃\sub\e \emsp ident n \emsp ident HPn)" with precedence 19
+for @{ 'Exists_elim_ok_2 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ok_2" 'Exists_elim_ok_2 ExPx Pc c =
+ (cast ? ? (show c (Exists_elim ? ? ? ExPx Pc))).
+
+definition ex_concl := λx:sort → Prop.Πy:sort.unit → x y.
+definition ex_concl_dummy := Πy:sort.unit → unit.
+definition fake_pred := λx:sort.unit.
+
+notation >"∃#'e' term 90 ExPt {ident t} [ident H] term 90 c" non associative with precedence 19
+for @{'Exists_elim (λx.? x) (show $ExPt ?) (λ${ident t}:sort.λ${ident H}.show $c ?)}.
+interpretation "Exists_elim KO" 'Exists_elim P ExPt c =
+ (cast ? ? (Exists_elim sort P ?
+ (cast (Exists ? P) ? ExPt)
+ (cast ex_concl_dummy (ex_concl ?) c))).
+interpretation "Exists_elim OK" 'Exists_elim P ExPt c =
+ (Exists_elim sort P ? ExPt c).
+
+(* ∀ introduction *)
+
+notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (∀\sub\i) \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ko_1 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ko_1" 'Forall_intro_ko_1 Px Pn =
+ (show Pn (cast ? ? (Forall_intro ? ? (cast ? ? Px)))).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000(Pn) (mstyle color #ff0000 (∀\sub\i) \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ko_2 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ko_2" 'Forall_intro_ko_2 Px Pn =
+ (cast ? ? (show Pn (cast ? ? (Forall_intro ? ? (cast ? ? Px))))).
+
+notation < "maction (\infrule hbox(\emsp Px \emsp) Pn (∀\sub\i \emsp ident x)) (\vdots)" with precedence 19
+for @{ 'Forall_intro_ok_1 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ok_1" 'Forall_intro_ok_1 Px Pn =
+ (show Pn (Forall_intro ? ? Px)).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (∀\sub\i \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ok_2 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ok_2" 'Forall_intro_ok_2 Px Pn =
+ (cast ? ? (show Pn (Forall_intro ? ? Px))).
+
+notation > "∀#'i' {ident y} term 90 Px" non associative with precedence 19
+for @{ 'Forall_intro (λ_.?) (λ${ident y}.show $Px ?) }.
+interpretation "Forall_intro KO" 'Forall_intro P Px =
+ (cast ? ? (Forall_intro sort P (cast ? ? Px))).
+interpretation "Forall_intro OK" 'Forall_intro P Px =
+ (Forall_intro sort P Px).
+
+(* ∀ elimination *)
+notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (∀\sub\e))" with precedence 19
+for @{ 'Forall_elim_ko_1 $Px $Pn }.
+interpretation "Forall_elim_ko_1" 'Forall_elim_ko_1 Px Pn =
+ (show Pn (cast ? ? (Forall_elim ? ? ? (cast ? ? Px)))).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000(Pn) (mstyle color #ff0000 (∀\sub\e))" with precedence 19
+for @{ 'Forall_elim_ko_2 $Px $Pn }.
+interpretation "Forall_elim_ko_2" 'Forall_elim_ko_2 Px Pn =
+ (cast ? ? (show Pn (cast ? ? (Forall_elim ? ? ? (cast ? ? Px))))).
+
+notation < "maction (\infrule hbox(\emsp Px \emsp) Pn (∀\sub\e)) (\vdots)" with precedence 19
+for @{ 'Forall_elim_ok_1 $Px $Pn }.
+interpretation "Forall_elim_ok_1" 'Forall_elim_ok_1 Px Pn =
+ (show Pn (Forall_elim ? ? ? Px)).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (∀\sub\e)" with precedence 19
+for @{ 'Forall_elim_ok_2 $Px $Pn }.
+interpretation "Forall_elim_ok_2" 'Forall_elim_ok_2 Px Pn =
+ (cast ? ? (show Pn (Forall_elim ? ? ? Px))).
+
+notation > "∀#'e' {term 90 t} term 90 Pn" non associative with precedence 19
+for @{ 'Forall_elim (λ_.?) $t (show $Pn ?) }.
+interpretation "Forall_elim KO" 'Forall_elim P t Px =
+ (cast ? unit (Forall_elim sort P t (cast ? ? Px))).
+interpretation "Forall_elim OK" 'Forall_elim P t Px =
+ (Forall_elim sort P t Px).
+
+(* already proved lemma *)
+definition hide_args : ΠA:Type[0].A→A := λA:Type[0].λa:A.a.
+notation < "t" non associative with precedence 90 for @{'hide_args $t}.
+interpretation "hide 0 args" 'hide_args t = (hide_args ? t).
+interpretation "hide 1 args" 'hide_args t = (hide_args ? t ?).
+interpretation "hide 2 args" 'hide_args t = (hide_args ? t ? ?).
+interpretation "hide 3 args" 'hide_args t = (hide_args ? t ? ? ?).
+interpretation "hide 4 args" 'hide_args t = (hide_args ? t ? ? ? ?).
+interpretation "hide 5 args" 'hide_args t = (hide_args ? t ? ? ? ? ?).
+interpretation "hide 6 args" 'hide_args t = (hide_args ? t ? ? ? ? ? ?).
+interpretation "hide 7 args" 'hide_args t = (hide_args ? t ? ? ? ? ? ? ?).
+
+(* more args crashes the pattern matcher *)
+
+(* already proved lemma, 0 assumptions *)
+definition Lemma : ΠA.A→A ≝ λA:Prop.λa:A.a.
+
+notation < "\infrule
+ (\infrule
+ (\emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma_ko_1 $p ($H : $_) }.
+interpretation "lemma_ko_1" 'lemma_ko_1 p H =
+ (show p (cast ? ? (Lemma ? (cast ? ? H)))).
+
+notation < "\infrule
+ (\infrule
+ (\emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma_ko_2 $p ($H : $_) }.
+interpretation "lemma_ko_2" 'lemma_ko_2 p H =
+ (cast ? ? (show p (cast ? ?
+ (Lemma ? (cast ? ? H))))).
+
+
+notation < "\infrule
+ (\infrule
+ (\emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma_ok_1 $p ($H : $_) }.
+interpretation "lemma_ok_1" 'lemma_ok_1 p H =
+ (show p (Lemma ? H)).
+
+notation < "\infrule
+ (\infrule
+ (\emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma_ok_2 $p ($H : $_) }.
+interpretation "lemma_ok_2" 'lemma_ok_2 p H =
+ (cast ? ? (show p (Lemma ? H))).
+
+notation > "'lem' 0 term 90 l" non associative with precedence 19
+for @{ 'Lemma (hide_args ? $l : ?) }.
+interpretation "lemma KO" 'Lemma l =
+ (cast ? ? (Lemma unit (cast unit ? l))).
+interpretation "lemma OK" 'Lemma l = (Lemma ? l).
+
+
+(* already proved lemma, 1 assumption *)
+definition Lemma1 : ΠA,B. (A ⇒ B) → A → B ≝
+ λA,B:Prop.λf:A⇒B.λa:A.
+ Imply_elim A B f a.
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma1_ko_1 $a $p ($H : $_) }.
+interpretation "lemma1_ko_1" 'lemma1_ko_1 a p H =
+ (show p (cast ? ? (Lemma1 ? ? (cast ? ? H) (cast ? ? a)))).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma1_ko_2 $a $p ($H : $_) }.
+interpretation "lemma1_ko_2" 'lemma1_ko_2 a p H =
+ (cast ? ? (show p (cast ? ?
+ (Lemma1 ? ? (cast ? ? H) (cast ? ? a))))).
+
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma1_ok_1 $a $p ($H : $_) }.
+interpretation "lemma1_ok_1" 'lemma1_ok_1 a p H =
+ (show p (Lemma1 ? ? H a)).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma1_ok_2 $a $p ($H : $_) }.
+interpretation "lemma1_ok_2" 'lemma1_ok_2 a p H =
+ (cast ? ? (show p (Lemma1 ? ? H a))).
+
+
+notation > "'lem' 1 term 90 l term 90 p" non associative with precedence 19
+for @{ 'Lemma1 (hide_args ? $l : ?) (show $p ?) }.
+interpretation "lemma 1 KO" 'Lemma1 l p =
+ (cast ? ? (Lemma1 unit unit (cast (Imply unit unit) ? l) (cast unit ? p))).
+interpretation "lemma 1 OK" 'Lemma1 l p = (Lemma1 ? ? l p).
+
+(* already proved lemma, 2 assumptions *)
+definition Lemma2 : ΠA,B,C. (A ⇒ B ⇒ C) → A → B → C ≝
+ λA,B,C:Prop.λf:A⇒B⇒C.λa:A.λb:B.
+ Imply_elim B C (Imply_elim A (B⇒C) f a) b.
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma2_ko_1 $a $b $p ($H : $_) }.
+interpretation "lemma2_ko_1" 'lemma2_ko_1 a b p H =
+ (show p (cast ? ? (Lemma2 ? ? ? (cast ? ? H) (cast ? ? a) (cast ? ? b)))).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma2_ko_2 $a $b $p ($H : $_) }.
+interpretation "lemma2_ko_2" 'lemma2_ko_2 a b p H =
+ (cast ? ? (show p (cast ? ?
+ (Lemma2 ? ? ? (cast ? ? H) (cast ? ? a) (cast ? ? b))))).
+
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma2_ok_1 $a $b $p ($H : $_) }.
+interpretation "lemma2_ok_1" 'lemma2_ok_1 a b p H =
+ (show p (Lemma2 ? ? ? H a b)).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma2_ok_2 $a $b $p ($H : $_) }.
+interpretation "lemma2_ok_2" 'lemma2_ok_2 a b p H =
+ (cast ? ? (show p (Lemma2 ? ? ? H a b))).
+
+notation > "'lem' 2 term 90 l term 90 p term 90 q" non associative with precedence 19
+for @{ 'Lemma2 (hide_args ? $l : ?) (show $p ?) (show $q ?) }.
+interpretation "lemma 2 KO" 'Lemma2 l p q =
+ (cast ? ? (Lemma2 unit unit unit (cast (Imply unit (Imply unit unit)) ? l) (cast unit ? p) (cast unit ? q))).
+interpretation "lemma 2 OK" 'Lemma2 l p q = (Lemma2 ? ? ? l p q).
+
+(* already proved lemma, 3 assumptions *)
+definition Lemma3 : ΠA,B,C,D. (A ⇒ B ⇒ C ⇒ D) → A → B → C → D ≝
+ λA,B,C,D:Prop.λf:A⇒B⇒C⇒D.λa:A.λb:B.λc:C.
+ Imply_elim C D (Imply_elim B (C⇒D) (Imply_elim A (B⇒C⇒D) f a) b) c.
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma3_ko_1 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ko_1" 'lemma3_ko_1 a b c p H =
+ (show p (cast ? ?
+ (Lemma3 ? ? ? ? (cast ? ? H) (cast ? ? a) (cast ? ? b) (cast ? ? c)))).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp)
+ (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma3_ko_2 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ko_2" 'lemma3_ko_2 a b c p H =
+ (cast ? ? (show p (cast ? ?
+ (Lemma3 ? ? ? ? (cast ? ? H) (cast ? ? a) (cast ? ? b) (cast ? ? c))))).
+
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ p \nbsp"
+non associative with precedence 19
+for @{ 'lemma3_ok_1 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ok_1" 'lemma3_ok_1 a b c p H =
+ (show p (Lemma3 ? ? ? ? H a b c)).
+
+notation < "\infrule
+ (\infrule
+ (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp)
+ (╲ mstyle mathsize normal (H) ╱) \nbsp)
+ mstyle color #ff0000 (p) \nbsp"
+non associative with precedence 19
+for @{ 'lemma3_ok_2 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ok_2" 'lemma3_ok_2 a b c p H =
+ (cast ? ? (show p (Lemma3 ? ? ? ? H a b c))).
+
+notation > "'lem' 3 term 90 l term 90 p term 90 q term 90 r" non associative with precedence 19
+for @{ 'Lemma3 (hide_args ? $l : ?) (show $p ?) (show $q ?) (show $r ?) }.
+interpretation "lemma 3 KO" 'Lemma3 l p q r =
+ (cast ? ? (Lemma3 unit unit unit unit (cast (Imply unit (Imply unit (Imply unit unit))) ? l) (cast unit ? p) (cast unit ? q) (cast unit ? r))).
+interpretation "lemma 3 OK" 'Lemma3 l p q r = (Lemma3 ? ? ? ? l p q r).
(**************************************************************************)
include "pts_dummy/rc_hsat.ma".
+include "basics/core_notation/napart_2.ma".
(*
(* THE EVALUATION *************************************************************)
(**************************************************************************)
include "pts_dummy/rc_hsat.ma".
+include "basics/core_notation/napart_2.ma".
(*
(* THE EVALUATION *************************************************************)
include "arithmetics/sigma_pi.ma".
include "arithmetics/bounded_quantifiers.ma".
include "reverse_complexity/big_O.ma".
+include "basics/core_notation/napart_2.ma".
(************************* notation for minimization *****************************)
notation "μ_{ ident i < n } p"
@Hmono @(mono_h_of2 … Hr Hmono … ltin)
]
qed.
-
\ No newline at end of file
+
include "arithmetics/sigma_pi.ma".
include "arithmetics/bounded_quantifiers.ma".
include "reverse_complexity/big_O.ma".
+include "basics/core_notation/napart_2.ma".
(************************* notation for minimization *****************************)
notation "μ_{ ident i < n } p"
@Hmono @(mono_h_of2 … Hr Hmono … ltin)
]
qed.
-
\ No newline at end of file
+
include "arithmetics/sigma_pi.ma".
include "arithmetics/bounded_quantifiers.ma".
include "reverse_complexity/big_O.ma".
+include "basics/core_notation/napart_2.ma".
(************************* notation for minimization *****************************)
notation "μ_{ ident i < n } p"
@Hmono @(mono_h_of2 … Hr Hmono … ltin)
]
qed.
-
\ No newline at end of file
+
<keyword>inversion</keyword>
<keyword>lapply</keyword>
<keyword>destruct</keyword>
+ <keyword>assume</keyword>
+ <keyword>suppose</keyword>
+ <keyword>that</keyword>
+ <keyword>is</keyword>
+ <keyword>equivalent</keyword>
+ <keyword>to</keyword>
+ <keyword>we</keyword>
+ <keyword>need</keyword>
+ <keyword>prove</keyword>
+ <keyword>or</keyword>
+ <keyword>equivalently</keyword>
+ <keyword>by</keyword>
+ <keyword>done</keyword>
+ <keyword>proved</keyword>
+ <keyword>have</keyword>
+ <keyword>such</keyword>
+ <keyword>the</keyword>
+ <keyword>thesis</keyword>
+ <keyword>becomes</keyword>
+ <keyword>conclude</keyword>
+ <keyword>obtain</keyword>
+ <keyword>proceed</keyword>
+ <keyword>induction</keyword>
+ <keyword>case</keyword>
+ <keyword>hypothesis</keyword>
+ <keyword>know</keyword>
<!-- commands -->
<keyword>alias</keyword>
(* Copyright (C) 2005, HELM Team.
- *
+ *
* This file is part of HELM, an Hypertextual, Electronic
* Library of Mathematics, developed at the Computer Science
* Department, University of Bologna, Italy.
- *
+ *
* HELM is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public License
* as published by the Free Software Foundation; either version 2
* of the License, or (at your option) any later version.
- *
+ *
* HELM is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* along with HELM; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place - Suite 330, Boston,
* MA 02111-1307, USA.
- *
+ *
* For details, see the HELM World-Wide-Web page,
* http://helm.cs.unibo.it/
- *)
+*)
(* $Id$ *)
+module G = GrafiteAst
open Printf
class status baseuri =
- object
- inherit GrafiteTypes.status baseuri
- inherit ApplyTransformation.status
- end
+ object
+ inherit GrafiteTypes.status baseuri
+ inherit ApplyTransformation.status
+ end
exception TryingToAdd of string Lazy.t
exception EnrichedWithStatus of exn * status
let slash_n_RE = Pcre.regexp "\\n" ;;
-let pp_ast_statement status stm =
- let stm = GrafiteAstPp.pp_statement status stm
- ~map_unicode_to_tex:(Helm_registry.get_bool "matita.paste_unicode_as_tex")
+let first_line = ref true ;;
+
+let cases_or_induction_context stack =
+ match stack with
+ [] -> false
+ | (_g,_t,_k,_tag,p)::_tl -> try
+ let s = List.assoc "context" p in
+ s = "cases" || s = "induction"
+ with
+ Not_found -> false
+;;
+
+let has_focused_goal stack =
+ match stack with
+ [] -> false
+ | (g,_t,_k,_tag,_p)::_tl -> (List.length g) > 0
+;;
+
+let get_indentation status _statement =
+ let base_ind =
+ match status#stack with
+ [] -> 0
+ | s -> List.length(s) * 2
in
- let stm = Pcre.replace ~rex:slash_n_RE stm in
- let stm =
+ if cases_or_induction_context status#stack then
+ (
+ if has_focused_goal status#stack then
+ base_ind + 2
+ else
+ base_ind
+ )
+ else
+ base_ind
+;;
+
+let pp_ind s n =
+ let rec aux s n =
+ match n with
+ 0 -> s
+ | n -> " " ^ (aux s (n-1))
+ in
+ aux s n
+
+let write_ast_to_file status fname statement =
+ let indentation = get_indentation status statement in
+ let str = match statement with
+ G.Comment _ -> GrafiteAstPp.pp_statement status statement
+ ~map_unicode_to_tex:(Helm_registry.get_bool "matita.paste_unicode_as_tex")
+ | G.Executable (_,code) ->
+ (
+ match code with
+ G.NTactic _ -> GrafiteAstPp.pp_statement status statement
+ ~map_unicode_to_tex:(Helm_registry.get_bool "matita.paste_unicode_as_tex")
+ | G.NCommand (_,cmd) ->
+ (
+ match cmd with
+ | G.NObj (_,obj,_) ->
+ (
+ match obj with
+ NotationPt.Theorem _ -> "\n" ^ GrafiteAstPp.pp_statement status statement
+ ~map_unicode_to_tex:(Helm_registry.get_bool "matita.paste_unicode_as_tex")
+ | _ -> ""
+ )
+ | G.NQed _ -> GrafiteAstPp.pp_statement status statement
+ ~map_unicode_to_tex:(Helm_registry.get_bool "matita.paste_unicode_as_tex")
+ | _ -> ""
+ )
+ | _ -> ""
+ )
+ in
+ if str <> "" then
+ (
+ let s = pp_ind str indentation in
+ let flaglist = if !first_line = false then [Open_wronly; Open_append; Open_creat]
+ else (first_line := false; [Open_wronly; Open_trunc; Open_creat])
+ in
+ let out_channel =
+ Stdlib.open_out_gen flaglist 0o0644 fname in
+ let _ = Stdlib.output_string out_channel ((if str.[0] <> '\n' then s else str) ^ "\n") in
+ let _ = Stdlib.close_out out_channel in
+ str
+ )
+ else
+ str
+;;
+
+let pp_ast_statement status stm ~fname =
+ let stm = write_ast_to_file status (fname ^ ".parsed.ma") stm in
+ if stm <> "" then
+ (
+ let stm = Pcre.replace ~rex:slash_n_RE stm in
+ let stm =
if String.length stm > 50 then String.sub stm 0 50 ^ " ..."
else stm
- in
+ in
HLog.debug ("Executing: ``" ^ stm ^ "''")
+ )
+ else
+ HLog.debug ("Executing: `` Unprintable statement ''")
;;
let clean_exit baseuri exn =
raise (FailureCompiling (baseuri,exn))
;;
-let cut prefix s =
+let cut prefix s =
let lenp = String.length prefix in
let lens = String.length s in
assert (lens > lenp);
String.sub s lenp (lens-lenp)
;;
-let print_string =
- let indent = ref 0 in
- let print_string ~right_justify s =
- let ss =
- match right_justify with
- None -> ""
- | Some (ss,len_ss) ->
- let i = 80 - !indent - len_ss - String.length s in
- if i > 0 then String.make i ' ' ^ ss else ss
- in
- assert (!indent >=0);
- print_string (String.make !indent ' ' ^ s ^ ss) in
- fun enter ?right_justify s ->
- if enter then (print_string ~right_justify s; incr indent) else (decr indent; print_string ~right_justify s)
+let print_string =
+ let indent = ref 0 in
+ let print_string ~right_justify s =
+ let ss =
+ match right_justify with
+ None -> ""
+ | Some (ss,len_ss) ->
+ let i = 80 - !indent - len_ss - String.length s in
+ if i > 0 then String.make i ' ' ^ ss else ss
+ in
+ assert (!indent >=0);
+ print_string (String.make !indent ' ' ^ s ^ ss) in
+ fun enter ?right_justify s ->
+ if enter then (print_string ~right_justify s; incr indent) else (decr indent; print_string ~right_justify s)
;;
-let pp_times ss fname rc big_bang big_bang_u big_bang_s =
+let pp_times ss fname rc big_bang big_bang_u big_bang_s =
if not (Helm_registry.get_bool "matita.verbose") then
let { Unix.tms_utime = u ; Unix.tms_stime = s} = Unix.times () in
let r = Unix.gettimeofday () -. big_bang in
let u = u -. big_bang_u in
let s = s -. big_bang_s in
let extra = try Sys.getenv "BENCH_EXTRA_TEXT" with Not_found -> "" in
- let rc =
+ let rc =
if rc then "\e[0;32mOK\e[0m" else "\e[0;31mFAIL\e[0m" in
- let times =
- let fmt t =
+ let times =
+ let fmt t =
let seconds = int_of_float t in
let cents = int_of_float ((t -. floor t) *. 100.0) in
let minutes = seconds / 60 in
;;
let eval_ast ~include_paths ?do_heavy_checks status (text,prefix_len,ast) =
- let baseuri = status#baseuri in
- let new_aliases,new_status =
- GrafiteDisambiguate.eval_with_new_aliases status
- (fun status ->
- let time0 = Unix.gettimeofday () in
- let status =
- GrafiteEngine.eval_ast ~include_paths ?do_heavy_checks status
- (text,prefix_len,ast) in
- let time1 = Unix.gettimeofday () in
- HLog.debug ("... grafite_engine done in " ^ string_of_float (time1 -. time0) ^ "s");
- status
- ) in
- let _,intermediate_states =
- List.fold_left
- (fun (status,acc) (k,value) ->
- let v = GrafiteAst.description_of_alias value in
- let b =
- try
- let NReference.Ref (uri,_) = NReference.reference_of_string v in
- NUri.baseuri_of_uri uri = baseuri
- with
- NReference.IllFormedReference _ ->
- false (* v is a description, not a URI *)
- in
- if b then
- status,acc
- else
- let status =
- GrafiteDisambiguate.set_proof_aliases status ~implicit_aliases:false
- GrafiteAst.WithPreferences [k,value]
- in
- status, (status ,Some (k,value))::acc
- ) (status,[]) new_aliases (* WARNING: this must be the old status! *)
- in
+ let baseuri = status#baseuri in
+ let new_aliases,new_status =
+ GrafiteDisambiguate.eval_with_new_aliases status
+ (fun status ->
+ let time0 = Unix.gettimeofday () in
+ let status =
+ GrafiteEngine.eval_ast ~include_paths ?do_heavy_checks status
+ (text,prefix_len,ast) in
+ let time1 = Unix.gettimeofday () in
+ HLog.debug ("... grafite_engine done in " ^ string_of_float (time1 -. time0) ^ "s");
+ status
+ ) in
+ let _,intermediate_states =
+ List.fold_left
+ (fun (status,acc) (k,value) ->
+ let v = GrafiteAst.description_of_alias value in
+ let b =
+ try
+ let NReference.Ref (uri,_) = NReference.reference_of_string v in
+ NUri.baseuri_of_uri uri = baseuri
+ with
+ NReference.IllFormedReference _ ->
+ false (* v is a description, not a URI *)
+ in
+ if b then
+ status,acc
+ else
+ let status =
+ GrafiteDisambiguate.set_proof_aliases status ~implicit_aliases:false
+ GrafiteAst.WithPreferences [k,value]
+ in
+ status, (status ,Some (k,value))::acc
+ ) (status,[]) new_aliases (* WARNING: this must be the old status! *)
+ in
(new_status,None)::intermediate_states
;;
let baseuri_of_script ~include_paths fname =
- try Librarian.baseuri_of_script ~include_paths fname
- with
- Librarian.NoRootFor _ ->
+ try Librarian.baseuri_of_script ~include_paths fname
+ with
+ Librarian.NoRootFor _ ->
HLog.error ("The included file '"^fname^"' has no root file,");
HLog.error "please create it.";
raise (Failure ("No root file for "^fname))
- | Librarian.FileNotFound _ ->
+ | Librarian.FileNotFound _ ->
raise (Failure ("File not found: "^fname))
;;
in
let rc = root :: includes in
List.iter (HLog.debug) rc; rc
- with Librarian.NoRootFor _ | Librarian.FileNotFound _ ->
- []
+ with Librarian.NoRootFor _ | Librarian.FileNotFound _ ->
+ []
;;
-let rec get_ast status ~compiling ~asserted ~include_paths strm =
+let rec get_ast status ~compiling ~asserted ~include_paths strm =
match GrafiteParser.parse_statement status strm with
- (GrafiteAst.Executable
+ (GrafiteAst.Executable
(_,GrafiteAst.NCommand (_,GrafiteAst.Include (_,_,mafilename)))) as cmd
- ->
- let already_included = NCicLibrary.get_transitively_included status in
- let asserted,_ =
- assert_ng ~already_included ~compiling ~asserted ~include_paths
- mafilename
- in
- asserted,cmd
- | cmd -> asserted,cmd
+ ->
+ let already_included = NCicLibrary.get_transitively_included status in
+ let asserted,_ =
+ assert_ng ~already_included ~compiling ~asserted ~include_paths
+ mafilename
+ in
+ asserted,cmd
+ | cmd -> asserted,cmd
and eval_from_stream ~compiling ~asserted ~include_paths ?do_heavy_checks status str cb =
- let matita_debug = Helm_registry.get_bool "matita.debug" in
- let rec loop asserted status str =
- let asserted,stop,status,str =
- try
- let cont =
- try Some (get_ast status ~compiling ~asserted ~include_paths str)
- with End_of_file -> None in
- match cont with
- | None -> asserted, true, status, str
- | Some (asserted,ast) ->
- cb status ast;
- let new_statuses =
- eval_ast ~include_paths ?do_heavy_checks status ("",0,ast) in
- let status =
- match new_statuses with
- [s,None] -> s
- | _::(_,Some (_,value))::_ ->
- raise (TryingToAdd (lazy (GrafiteAstPp.pp_alias value)))
- | _ -> assert false in
- (* CSC: complex patch to re-build the lexer since the tokens may
- have changed. Note: this way we loose look-ahead tokens.
- Hence the "include" command must be terminated (no look-ahead) *)
- let str =
- match ast with
- (GrafiteAst.Executable
- (_,GrafiteAst.NCommand
- (_,(GrafiteAst.Include _ | GrafiteAst.Notation _)))) ->
+ let matita_debug = Helm_registry.get_bool "matita.debug" in
+ let rec loop asserted status str =
+ let asserted,stop,status,str =
+ try
+ let cont =
+ try Some (get_ast status ~compiling ~asserted ~include_paths str)
+ with End_of_file -> None in
+ match cont with
+ | None -> asserted, true, status, str
+ | Some (asserted,ast) ->
+ cb status ast;
+ let new_statuses =
+ eval_ast ~include_paths ?do_heavy_checks status ("",0,ast) in
+ let status =
+ match new_statuses with
+ [s,None] -> s
+ | _::(_,Some (_,value))::_ ->
+ raise (TryingToAdd (lazy (GrafiteAstPp.pp_alias value)))
+ | _ -> assert false in
+ (* CSC: complex patch to re-build the lexer since the tokens may
+ have changed. Note: this way we loose look-ahead tokens.
+ Hence the "include" command must be terminated (no look-ahead) *)
+ let str =
+ match ast with
+ (GrafiteAst.Executable
+ (_,GrafiteAst.NCommand
+ (_,(GrafiteAst.Include _ | GrafiteAst.Notation _)))) ->
GrafiteParser.parsable_statement status
- (GrafiteParser.strm_of_parsable str)
- | _ -> str
- in
- asserted, false, status, str
- with exn when not matita_debug ->
- raise (EnrichedWithStatus (exn, status))
+ (GrafiteParser.strm_of_parsable str)
+ | _ -> str
+ in
+ asserted, false, status, str
+ with exn when not matita_debug ->
+ raise (EnrichedWithStatus (exn, status))
+ in
+ if stop then asserted,status else loop asserted status str
in
- if stop then asserted,status else loop asserted status str
- in
loop asserted status str
and compile ~compiling ~asserted ~include_paths fname =
if List.mem fname compiling then raise (CircularDependency fname);
let compiling = fname::compiling in
let matita_debug = Helm_registry.get_bool "matita.debug" in
- let root,baseuri,fname,_tgt =
+ let root,baseuri,fname,_tgt =
Librarian.baseuri_of_script ~include_paths fname in
if Http_getter_storage.is_read_only baseuri then assert false;
(* MATITA 1.0: debbo fare time_travel sulla ng_library? *)
let ocamldirname = Filename.dirname fname in
let ocamlfname = Filename.chop_extension (Filename.basename fname) in
let status,ocamlfname =
- Common.modname_of_filename status false ocamlfname in
+ Common.modname_of_filename status false ocamlfname in
let ocamlfname = ocamldirname ^ "/" ^ ocamlfname ^ ".ml" in
let status = OcamlExtraction.open_file status ~baseuri ocamlfname in
let big_bang = Unix.gettimeofday () in
- let { Unix.tms_utime = big_bang_u ; Unix.tms_stime = big_bang_s} =
- Unix.times ()
+ let { Unix.tms_utime = big_bang_u ; Unix.tms_stime = big_bang_s} =
+ Unix.times ()
in
let time = Unix.time () in
- let cc =
- let rex = Str.regexp ".*opt$" in
- if Str.string_match rex Sys.argv.(0) 0 then "matitac.opt"
- else "matitac" in
+ let cc =
+ let rex = Str.regexp ".*opt$" in
+ if Str.string_match rex Sys.argv.(0) 0 then "matitac.opt"
+ else "matitac" in
let s = Printf.sprintf "%s %s" cc (cut (root^"/") fname) in
try
(* cleanup of previously compiled objects *)
if (not (Http_getter_storage.is_empty ~local:true baseuri))
- then begin
+ then begin
HLog.message ("baseuri " ^ baseuri ^ " is not empty");
HLog.message ("cleaning baseuri " ^ baseuri);
LibraryClean.clean_baseuris [baseuri];
end;
HLog.message ("compiling " ^ Filename.basename fname ^ " in " ^ baseuri);
if not (Helm_registry.get_bool "matita.verbose") then
- (print_string true (s ^ "\n"); flush stdout);
+ (print_string true (s ^ "\n"); flush stdout);
(* we dalay this error check until we print 'matitac file ' *)
assert (Http_getter_storage.is_empty ~local:true baseuri);
(* create dir for XML files *)
if not (Helm_registry.get_opt_default Helm_registry.bool "matita.nodisk"
- ~default:false)
+ ~default:false)
then
- HExtlib.mkdir
- (Filename.dirname
- (Http_getter.filename ~local:true ~writable:true (baseuri ^
- "foo.con")));
+ HExtlib.mkdir
+ (Filename.dirname
+ (Http_getter.filename ~local:true ~writable:true (baseuri ^
+ "foo.con")));
let buf =
- GrafiteParser.parsable_statement status
- (Ulexing.from_utf8_channel (open_in fname))
+ GrafiteParser.parsable_statement status
+ (Ulexing.from_utf8_channel (open_in fname))
in
let print_cb =
- if not (Helm_registry.get_bool "matita.verbose") then (fun _ _ -> ())
- else pp_ast_statement
+ if not (Helm_registry.get_bool "matita.verbose") then fun _ _ -> ()
+ else pp_ast_statement ~fname
in
let asserted, status =
- eval_from_stream ~compiling ~asserted ~include_paths status buf print_cb in
+ eval_from_stream ~compiling ~asserted ~include_paths status buf print_cb in
let status = OcamlExtraction.close_file status in
let elapsed = Unix.time () -. time in
- (if Helm_registry.get_bool "matita.moo" then begin
- GrafiteTypes.Serializer.serialize ~baseuri:(NUri.uri_of_string baseuri)
- status
- end;
+ (if Helm_registry.get_bool "matita.moo" then begin
+ GrafiteTypes.Serializer.serialize ~baseuri:(NUri.uri_of_string baseuri)
+ status
+ end;
let tm = Unix.gmtime elapsed in
let sec = string_of_int tm.Unix.tm_sec ^ "''" in
- let min =
- if tm.Unix.tm_min > 0 then (string_of_int tm.Unix.tm_min^"' ") else ""
+ let min =
+ if tm.Unix.tm_min > 0 then (string_of_int tm.Unix.tm_min^"' ") else ""
in
- let hou =
+ let hou =
if tm.Unix.tm_hour > 0 then (string_of_int tm.Unix.tm_hour^"h ") else ""
in
- HLog.message
+ HLog.message
(sprintf "execution of %s completed in %s." fname (hou^min^sec));
pp_times s fname true big_bang big_bang_u big_bang_s;
(*CSC: bad, one imperative bit is still there!
to be moved into functional status *)
NCicMetaSubst.pushmaxmeta ();
-(* MATITA 1.0: debbo fare time_travel sulla ng_library?
- LexiconSync.time_travel
- ~present:lexicon_status ~past:initial_lexicon_status;
-*)
+ (* MATITA 1.0: debbo fare time_travel sulla ng_library?
+ LexiconSync.time_travel
+ ~present:lexicon_status ~past:initial_lexicon_status;
+ *)
asserted)
- with
+ with
(* all exceptions should be wrapped to allow lexicon-undo (LS.time_travel) *)
| exn when not matita_debug ->
-(* MATITA 1.0: debbo fare time_travel sulla ng_library?
- LexiconSync.time_travel ~present:lexicon ~past:initial_lexicon_status;
- * *)
- (*CSC: bad, one imperative bit is still there!
- to be moved into functional status *)
- NCicMetaSubst.pushmaxmeta ();
- pp_times s fname false big_bang big_bang_u big_bang_s;
- clean_exit baseuri exn
+ (* MATITA 1.0: debbo fare time_travel sulla ng_library?
+ LexiconSync.time_travel ~present:lexicon ~past:initial_lexicon_status;
+ * *)
+ (*CSC: bad, one imperative bit is still there!
+ to be moved into functional status *)
+ NCicMetaSubst.pushmaxmeta ();
+ pp_times s fname false big_bang big_bang_u big_bang_s;
+ clean_exit baseuri exn
and assert_ng ~already_included ~compiling ~asserted ~include_paths mapath =
- let root,baseuri,fullmapath,_ =
- Librarian.baseuri_of_script ~include_paths mapath in
- if List.mem fullmapath asserted then asserted,false
- else
- begin
- let include_paths =
- let includes =
- try
- Str.split (Str.regexp " ")
- (List.assoc "include_paths" (Librarian.load_root_file (root^"/root")))
- with Not_found -> []
- in
- root::includes @
- Helm_registry.get_list Helm_registry.string "matita.includes" in
- let baseuri = NUri.uri_of_string baseuri in
- let ngtime_of baseuri =
- let ngpath = NCicLibrary.ng_path_of_baseuri baseuri in
- try
- Some (Unix.stat ngpath).Unix.st_mtime
- with Unix.Unix_error (Unix.ENOENT, "stat", f) when f = ngpath -> None in
- let matime =
- try (Unix.stat fullmapath).Unix.st_mtime
- with Unix.Unix_error (Unix.ENOENT, "stat", f) when f = fullmapath -> assert false
- in
- let ngtime = ngtime_of baseuri in
- let asserted,to_be_compiled =
- match ngtime with
- Some ngtime ->
- let preamble = GrafiteTypes.Serializer.dependencies_of baseuri in
- let asserted,children_bad =
- List.fold_left
- (fun (asserted,b) mapath ->
- let asserted,b1 =
- try
- assert_ng ~already_included ~compiling ~asserted ~include_paths
- mapath
- with Librarian.NoRootFor _ | Librarian.FileNotFound _ ->
- asserted, true
- in
- asserted, b || b1
- || let _,baseuri,_,_ =
- (*CSC: bug here? include_paths should be empty and
- mapath should be absolute *)
- Librarian.baseuri_of_script ~include_paths mapath in
- let baseuri = NUri.uri_of_string baseuri in
- (match ngtime_of baseuri with
- Some child_ngtime -> child_ngtime > ngtime
- | None -> assert false)
- ) (asserted,false) preamble
+ let root,baseuri,fullmapath,_ =
+ Librarian.baseuri_of_script ~include_paths mapath in
+ if List.mem fullmapath asserted then asserted,false
+ else
+ begin
+ let include_paths =
+ let includes =
+ try
+ Str.split (Str.regexp " ")
+ (List.assoc "include_paths" (Librarian.load_root_file (root^"/root")))
+ with Not_found -> []
in
- asserted, children_bad || matime > ngtime
- | None -> asserted,true
- in
- if not to_be_compiled then fullmapath::asserted,false
- else
- if List.mem baseuri already_included then
- (* maybe recompiling it I would get the same... *)
- raise (AlreadyLoaded (lazy mapath))
- else
- let asserted = compile ~compiling ~asserted ~include_paths fullmapath in
- fullmapath::asserted,true
- end
+ root::includes @
+ Helm_registry.get_list Helm_registry.string "matita.includes" in
+ let baseuri = NUri.uri_of_string baseuri in
+ let ngtime_of baseuri =
+ let ngpath = NCicLibrary.ng_path_of_baseuri baseuri in
+ try
+ Some (Unix.stat ngpath).Unix.st_mtime
+ with Unix.Unix_error (Unix.ENOENT, "stat", f) when f = ngpath -> None in
+ let matime =
+ try (Unix.stat fullmapath).Unix.st_mtime
+ with Unix.Unix_error (Unix.ENOENT, "stat", f) when f = fullmapath -> assert false
+ in
+ let ngtime = ngtime_of baseuri in
+ let asserted,to_be_compiled =
+ match ngtime with
+ Some ngtime ->
+ let preamble = GrafiteTypes.Serializer.dependencies_of baseuri in
+ let asserted,children_bad =
+ List.fold_left
+ (fun (asserted,b) mapath ->
+ let asserted,b1 =
+ try
+ assert_ng ~already_included ~compiling ~asserted ~include_paths
+ mapath
+ with Librarian.NoRootFor _ | Librarian.FileNotFound _ ->
+ asserted, true
+ in
+ asserted, b || b1
+ || let _,baseuri,_,_ =
+ (*CSC: bug here? include_paths should be empty and
+ mapath should be absolute *)
+ Librarian.baseuri_of_script ~include_paths mapath in
+ let baseuri = NUri.uri_of_string baseuri in
+ (match ngtime_of baseuri with
+ Some child_ngtime -> child_ngtime > ngtime
+ | None -> assert false)
+ ) (asserted,false) preamble
+ in
+ asserted, children_bad || matime > ngtime
+ | None -> asserted,true
+ in
+ if not to_be_compiled then fullmapath::asserted,false
+ else
+ if List.mem baseuri already_included then
+ (* maybe recompiling it I would get the same... *)
+ raise (AlreadyLoaded (lazy mapath))
+ else
+ let asserted = compile ~compiling ~asserted ~include_paths fullmapath in
+ fullmapath::asserted,true
+ end
;;
let assert_ng ~include_paths mapath =
- snd (assert_ng ~include_paths ~already_included:[] ~compiling:[] ~asserted:[]
- mapath)
+ snd (assert_ng ~include_paths ~already_included:[] ~compiling:[] ~asserted:[]
+ mapath)
let get_ast status ~include_paths strm =
- snd (get_ast status ~compiling:[] ~asserted:[] ~include_paths strm)
+ snd (get_ast status ~compiling:[] ~asserted:[] ~include_paths strm)
let stack_goals = Stack.open_goals status#stack in
let proof_goals = List.map fst metasenv in
if
- HExtlib.list_uniq (List.sort Pervasives.compare stack_goals)
- <> List.sort Pervasives.compare proof_goals
+ HExtlib.list_uniq (List.sort compare stack_goals)
+ <> List.sort compare proof_goals
then begin
prerr_endline ("STACK GOALS = " ^ String.concat " " (List.map string_of_int stack_goals));
prerr_endline ("PROOF GOALS = " ^ String.concat " " (List.map string_of_int proof_goals));
let content = Http_getter.ls ~local:false dir in
let l =
List.fast_sort
- Pervasives.compare
+ compare
(List.map
(function
| Http_getter_types.Ls_section s -> "dir", s
["+"; "⨭"; "⨮"; "⨁"; "⊕"; "⊞"; ];
["-"; "÷"; "⊢"; "⊩"; "⧟"; "⊟"; ];
["="; "≝"; "≡"; "≘"; "≗"; "≐"; "≑"; "≛"; "≚"; "≙"; "⌆"; "⧦"; "⊜"; "≋"; "⩳"; "≅"; "⩬"; "≂"; "≃"; "≈"; ];
- ["→"; "↦"; "⇝"; "⤞"; "⇾"; "⤍"; "⤏"; "⤳"; ] ;
- ["⇒"; "⤇"; "➾"; "⇨"; "➡"; "⬈"; "➤"; "➸"; "⇉"; "⥰"; ] ;
+ ["â\86\92"; "⥲"; "â\86¦"; "â\87\9d"; "â¤\9e"; "â\87¾"; "â¤\8d"; "â¤\8f"; "⤳"; ] ;
+ ["â\87\92"; "â¤\87"; "â\9e¾"; "â\87¨"; "â¬\80"; "â\9e¡"; "â¬\88"; "â\9e¤"; "â\9e¸"; "â\87\89"; "⥰"; ] ;
["^"; "↑"; "⇡"; ] ;
["⇑"; "⇧"; "⬆"; ] ;
- ["⇓"; "⇩"; "⬇"; "⬊"; "➷"; ] ;
+ ["â\87\93"; "â\87©"; "â¬\82"; "â¬\87"; "â¬\8a"; "â\9e·"; ] ;
["⇕"; "⇳"; "⬍"; "↕"; ];
["↔"; "⇔"; "⬄"; "⬌"; ] ;
["≤"; "≲"; "≼"; "≰"; "≴"; "⋠"; "⊆"; "⫃"; "⊑"; ] ;
["Y"; "ϒ"; "𝕐"; "𝐘"; "𝚼"; "Ⓨ"; ] ;
["z"; "ζ"; "𝕫"; "𝐳"; "𝛇"; "ⓩ"; ] ;
["Z"; "ℨ"; "ℤ"; "𝐙"; "Ⓩ"; ] ;
- ["0"; "𝟘"; "⓪"; ] ;
- ["1"; "𝟙"; "①"; "⓵"; ] ;
- ["2"; "𝟚"; "②"; "⓶"; ] ;
- ["3"; "𝟛"; "③"; "⓷"; ] ;
- ["4"; "𝟜"; "④"; "⓸"; ] ;
- ["5"; "𝟝"; "⑤"; "⓹"; ] ;
- ["6"; "𝟞"; "⑥"; "⓺"; ] ;
- ["7"; "𝟟"; "⑦"; "⓻"; ] ;
- ["8"; "𝟠"; "⑧"; "⓼"; "∞"; ] ;
- ["9"; "𝟡"; "⑨"; "⓽"; ] ;
+ ["0"; "𝟘"; "⓪"; "𝟎"; ] ;
+ ["1"; "𝟙"; "①"; "⓵"; "𝟏"; ] ;
+ ["2"; "𝟚"; "②"; "⓶"; "𝟐"; ] ;
+ ["3"; "𝟛"; "③"; "⓷"; "𝟑"; ] ;
+ ["4"; "𝟜"; "④"; "⓸"; "𝟒"; ] ;
+ ["5"; "𝟝"; "⑤"; "⓹"; "𝟓"; ] ;
+ ["6"; "𝟞"; "⑥"; "⓺"; "𝟔"; ] ;
+ ["7"; "𝟟"; "⑦"; "⓻"; "𝟕"; ] ;
+ ["8"; "𝟠"; "⑧"; "⓼"; "𝟖"; "∞"; ] ;
+ ["9"; "𝟡"; "⑨"; "⓽"; "𝟗"; ] ;
]
;;
projects / helm.git / commitdiff