let intro = get_intro name in
proc_proof (add st entry intro) t
-and proc_letin st what name v t =
+and proc_letin st what name v w t =
let intro = get_intro name in
let proceed, dtext = test_depth st in
let script = if proceed then
in
st, C.Decl (H.cic ity), rqv
| None ->
- (*CSC: here we need the type of v *)
- st, C.Def (H.cic v, assert false), [T.LetIn (intro, v, dtext)]
+ st, C.Def (H.cic v, H.cic w), [T.LetIn (intro, v, dtext)]
in
let entry = Some (name, hyp) in
let qt = proc_proof (next (add st entry intro)) t in
{st with context = context; clears = clears; clears_note = note; }
in
match t with
- | C.ALambda (_, name, w, t) -> proc_lambda st name w t
- | C.ALetIn (_, name, v, ty, t) as what -> assert false (*proc_letin (f st) what name v t*)
- | C.ARel _ as what -> proc_rel (f st) what
- | C.AMutConstruct _ as what -> proc_mutconstruct (f st) what
- | C.AAppl (_, hd :: tl) as what -> proc_appl (f st) what hd tl
- | what -> proc_other (f st) what
+ | C.ALambda (_, name, w, t) -> proc_lambda st name w t
+ | C.ALetIn (_, name, v, w, t) as what -> proc_letin (f st) what name v w t
+ | C.ARel _ as what -> proc_rel (f st) what
+ | C.AMutConstruct _ as what -> proc_mutconstruct (f st) what
+ | C.AAppl (_, hd :: tl) as what -> proc_appl (f st) what hd tl
+ | what -> proc_other (f st) what
and proc_bkd_proofs st synth names classes ts =
try
else
hd, names, v
in
- let p = C.LetIn (n, v, assert false, p) in
- let it = C.LetIn (n, v, assert false, it) in
- let et = C.LetIn (n, v, assert false, et) in
+ let p = C.LetIn (n, v, x, p) in
+ let it = C.LetIn (n, v, x, it) in
+ let et = C.LetIn (n, v, x, et) in
aux (hd :: c) names p it et tl
| Some (C.Anonymous as n, C.Decl v) as hd :: tl ->
let p = C.Lambda (n, meta, p) in
let et = C.Lambda (n, meta, et) in
aux (hd :: c) names p it et tl
| Some (C.Anonymous as n, C.Def (v, _)) as hd :: tl ->
- let p = C.LetIn (n, meta, assert false, p) in
- let it = C.LetIn (n, meta, assert false, it) in
- let et = C.LetIn (n, meta, assert false, et) in
+ let p = C.LetIn (n, meta, meta, p) in
+ let it = C.LetIn (n, meta, meta, it) in
+ let et = C.LetIn (n, meta, meta, et) in
aux (hd :: c) names p it et tl
| None :: tl -> assert false
in
module DTI = DoubleTypeInference
module HEL = HExtlib
module PEH = ProofEngineHelpers
+module TC = CicTypeChecker
+module Un = CicUniv
module H = ProceduralHelpers
module Cl = ProceduralClassify
let defined_premise = "DEFINED"
-let define v =
+let get_type msg c bo =
+try
+ let ty, _ = TC.type_of_aux' [] c bo Un.empty_ugraph in
+ ty
+with e -> failwith (msg ^ ": " ^ Printexc.to_string e)
+
+let define c v =
let name = C.Name defined_premise in
- (*CSC: here we need the type of v *)
- C.LetIn (name, v, assert false, C.Rel 1)
+ let ty = get_type "define" c v in
+ C.LetIn (name, v, ty, C.Rel 1)
let clear_absts m =
let rec aux k n = function
| t when k > 0 -> assert false
| t -> C.Lambda (C.Anonymous, C.Implicit None, S.lift 1 t)
-let rec opt1_letin g es c name v t =
+let rec opt1_letin g es c name v w t =
let name = H.mk_fresh_name c name in
- (*CSC: here we need the type of v *)
- let entry = Some (name, C.Def (v, assert false)) in
+ let entry = Some (name, C.Def (v, w)) in
let g t =
if DTI.does_not_occur 1 t then begin
let x = S.lift (-1) t in
HLog.warn "Optimizer: remove 1"; opt1_proof g true c x
end else
let g = function
- | C.LetIn (nname, vv, tyty, tt) when H.is_proof c v ->
- (*CSC: here we need the type of v *)
- let x = C.LetIn (nname, vv, tyty,
- C.LetIn (name, tt, assert false, S.lift_from 2 1 t)) in
+ | C.LetIn (nname, vv, ww, tt) when H.is_proof c v ->
+ let eentry = Some (nname, C.Def (vv, ww)) in
+ let ttw = get_type "opt1_letin 1" (eentry :: c) tt in
+ let x = C.LetIn (nname, vv, ww,
+ C.LetIn (name, tt, ttw, S.lift_from 2 1 t)) in
HLog.warn "Optimizer: swap 1"; opt1_proof g true c x
| v when H.is_proof c v && H.is_atomic v ->
let x = S.subst v t in
HLog.warn "Optimizer: remove 5"; opt1_proof g true c x
- | v ->
- (*CSC: here we need the type of v *)
- g (C.LetIn (name, v, assert false, t))
+ | v ->
+ g (C.LetIn (name, v, w, t))
in
if es then opt1_term g es c v else g v
in
HLog.warn "Optimizer: swap 2"; opt1_proof g true c x
| C.Lambda (name, ww, tt) ->
let v, vs = List.hd vs, List.tl vs in
- (*CSC: here we need the type of v *)
- let x = C.Appl (C.LetIn (name, v, assert false, tt) :: vs) in
+ let w = get_type "opt1_appl 1" c v in
+ let x = C.Appl (C.LetIn (name, v, w, tt) :: vs) in
HLog.warn "Optimizer: remove 2"; opt1_proof g true c x
| C.Appl vvs ->
let x = C.Appl (vvs @ vs) in
| v :: vs, (cc, bb) :: cs ->
if H.is_not_atomic v && I.S.mem 0 cc && bb then begin
HLog.warn "Optimizer: anticipate 1";
- aux true (define v :: rvs) (vs, cs)
+ aux true (define c v :: rvs) (vs, cs)
end else
aux d (v :: rvs) (vs, cs)
| _, [] -> assert false
let csno, vsno = List.length classes, List.length vs in
if csno < vsno then
let vvs, vs = HEL.split_nth csno vs in
- let x = C.Appl (define (C.Appl (t :: vvs)) :: vs) in
+ let x = C.Appl (define c (C.Appl (t :: vvs)) :: vs) in
HLog.warn "Optimizer: anticipate 2"; opt1_proof g true c x
else match conclusion, List.rev vs with
| Some _, rv :: rvs when csno = vsno && H.is_not_atomic rv ->
- let x = C.Appl (t :: List.rev rvs @ [define rv]) in
+ let x = C.Appl (t :: List.rev rvs @ [define c rv]) in
HLog.warn "Optimizer: anticipate 3"; opt1_proof g true c x
| _ (* Some _, _ *) ->
g (C.Appl (t :: vs))
let prev = List.map (S.lift 1) prev in
let vs = List.map (S.lift 1) vs in
let y = C.Appl (t :: List.rev prev @ tt :: vs) in
- (*CSC: here we need the type of vv *)
- let x = C.LetIn (name, vv, assert false, y) in
+ let ww = get_type "opt1_appl 2" c vv in
+ let x = C.LetIn (name, vv, ww, y) in
HLog.warn "Optimizer: swap 3"; opt1_proof g true c x
| v :: vs -> aux h (v :: prev) vs
| [] -> h ()
and opt1_other g es c t = g t
and opt1_proof g es c = function
- (*CSC: what to do now that we have also ty? *)
- | C.LetIn (name, v, ty, t) -> assert false (*opt1_letin g es c name v t*)
+ | C.LetIn (name, v, ty, t) -> opt1_letin g es c name v ty t
| C.Lambda (name, w, t) -> opt1_lambda g es c name w t
| C.Appl (t :: v :: vs) -> opt1_appl g es c t (v :: vs)
| C.Appl [t] -> opt1_proof g es c t
in
g (absts t)
-let rec opt2_letin g c name v t =
- (*CSC: here we need the type of v *)
- let entry = Some (name, C.Def (v, assert false)) in
+let rec opt2_letin g c name v w t =
+ let entry = Some (name, C.Def (v, w)) in
let g t =
- (*CSC: here we need the type of v *)
- let g v = g (C.LetIn (name, v, assert false, t)) in
+ let g v = g (C.LetIn (name, v, w, t)) in
opt2_term g c v
in
opt2_proof g (entry :: c) t
end else g t
and opt2_proof g c = function
- (*CSC: what to do now that we have also ty? *)
- | C.LetIn (name, v, ty, t) -> assert false (*opt2_letin g c name v t*)
- | C.Lambda (name, w, t) -> opt2_lambda g c name w t
- | C.Appl (t :: vs) -> opt2_appl g c t vs
- | t -> opt2_other g c t
+ | C.LetIn (name, v, w, t) -> opt2_letin g c name v w t
+ | C.Lambda (name, w, t) -> opt2_lambda g c name w t
+ | C.Appl (t :: vs) -> opt2_appl g c t vs
+ | t -> opt2_other g c t
and opt2_term g c t =
if H.is_proof c t then opt2_proof g c t else g t
(plus p n)) \to (eq nat m p))))
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat
-(plus m n) (plus p n))).(plus_reg_l n m p (eq_ind_r nat (plus m n) (\lambda
+(plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda
(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0:
-nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_comm n
-p)) (plus m n) H) (plus n m) (plus_comm n m)))))).
+nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_sym n
+p)) (plus m n) H) (plus n m) (plus_sym n m)))))).
theorem minus_Sx_Sy:
\forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
\forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
\def
\lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0:
-nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_comm m n))).
+nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))).
theorem plus_permute_2_in_3:
\forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x
(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat
(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind
nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y)))
-(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_reverse
-x z y)) (plus y z) (plus_comm y z)) (plus (plus x y) z) (plus_assoc_reverse x
-y z)))).
+(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z
+y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))).
theorem plus_permute_2_in_3_assoc:
\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n
\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus
(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r
nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0))
-(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc n k h))
-(plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
+(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k
+h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
theorem plus_O:
\forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat
\def
\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat
(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H)
-(plus (minus m n) n) (plus_comm (minus m n) n)))).
+(plus (minus m n) n) (plus_sym (minus m n) n)))).
theorem le_minus_minus:
\forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z)
\to (le (minus y x) (minus z x))))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z:
-nat).(\lambda (H0: (le y z)).(plus_le_reg_l x (minus y x) (minus z x)
+nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x)
(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat
z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z
(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
\def
\lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n:
nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x))
-(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_comm x (S y)))).
+(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))).
theorem simpl_lt_plus_r:
\forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m
p)) \to (lt n m))))
\def
\lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus
-n p) (plus m p))).(plus_lt_reg_l n m p (let H0 \def (eq_ind nat (plus n p)
-(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_comm n p)) in (let
+n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p)
+(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let
H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0
-(plus p m) (plus_comm m p)) in H1)))))).
+(plus p m) (plus_sym m p)) in H1)))))).
theorem minus_x_Sy:
\forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat
(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x
-y H) (plus (minus y (S x)) x) (plus_comm (minus y (S x)) x)))).
+y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))).
theorem minus_x_SO:
\forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S
O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O))
-(plus_comm x (S O)))))).
+(plus_sym x (S O)))))).
theorem lt_le_e:
\forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P))
\def
\lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq
nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in
-(lt_irrefl y H1))))).
+(lt_n_n y H1))))).
theorem arith0:
\forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n)
(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat
(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus
(plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1)
-(plus_le_compat (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_comm
-h2 d2)) (plus h2 (plus d2 h1)) (plus_assoc h2 d2 h1))) (plus d2 h1)
+(le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2
+d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1)
(minus_plus h2 (plus d2 h1))))))).
theorem O_minus:
nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus
y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n:
nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S
-z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (_: (le (S z0) (S
+z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (H: (le (S z0) (S
x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0))
-(minus O (S z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda
-(n: nat).(le z0 n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H2: (eq
-nat O (S x1))).(\lambda (_: (le z0 x1)).(let H4 \def (eq_ind nat O (\lambda
-(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
-True | (S _) \Rightarrow False])) I (S x1) H2) in (False_ind (eq nat (S x0)
-O) H4))))) (le_gen_S z0 O H0))))) (\lambda (y0: nat).(\lambda (_: (((le (S
-z0) (S x0)) \to ((le (S z0) y0) \to ((eq nat (minus (S x0) (S z0)) (minus y0
-(S z0))) \to (eq nat (S x0) y0)))))).(\lambda (H: (le (S z0) (S
-x0))).(\lambda (H0: (le (S z0) (S y0))).(\lambda (H1: (eq nat (minus (S x0)
-(S z0)) (minus (S y0) (S z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0
-x0 H) (le_S_n z0 y0 H0) H1))))))) y)))) x)))) z).
+(minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (ex2_ind nat (\lambda
+(n: nat).(eq nat O (S n))) (\lambda (n: nat).(le z0 n)) (eq nat (S x0) O)
+(\lambda (x1: nat).(\lambda (H2: (eq nat O (S x1))).(\lambda (_: (le z0
+x1)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S x1) H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0))))))
+(\lambda (y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to
+((eq nat (minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0)
+y0)))))).(\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S
+y0))).(\lambda (H1: (eq nat (minus (S x0) (S z0)) (minus (S y0) (S
+z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0)
+H1))))))) y)))) x)))) z).
theorem plus_plus:
\forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1:
(* *)
(**************************************************************************)
-include "coq.ma".
-
-alias symbol "eq" = "Coq's leibnitz's equality".
-alias symbol "leq" = "Coq's natural 'less or equal to'".
-alias symbol "neq" = "Coq's not equal to (leibnitz)".
-alias symbol "plus" = "Coq's natural plus".
-
-alias id "bool" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1)".
-alias id "conj" = "cic:/Coq/Init/Logic/and.ind#xpointer(1/1/1)".
-alias id "eq_add_S" = "cic:/Coq/Init/Peano/eq_add_S.con".
-alias id "eq" = "cic:/Coq/Init/Logic/eq.ind#xpointer(1/1)".
-alias id "eq_ind" = "cic:/Coq/Init/Logic/eq_ind.con".
-alias id "eq_ind_r" = "cic:/Coq/Init/Logic/eq_ind_r.con".
-alias id "ex2" = "cic:/Coq/Init/Logic/ex2.ind#xpointer(1/1)".
-alias id "ex2_ind" = "cic:/Coq/Init/Logic/ex2_ind.con".
-alias id "ex_intro2" = "cic:/Coq/Init/Logic/ex2.ind#xpointer(1/1/1)".
-alias id "false" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1/2)".
-alias id "False" = "cic:/Coq/Init/Logic/False.ind#xpointer(1/1)".
-alias id "False_ind" = "cic:/Coq/Init/Logic/False_ind.con".
-alias id "I" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1/1)".
-alias id "land" = "cic:/Coq/Init/Logic/and.ind#xpointer(1/1)".
-alias id "le" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1)".
-alias id "le_ind" = "cic:/Coq/Init/Peano/le_ind.con".
-alias id "le_lt_n_Sm" = "cic:/Coq/Arith/Lt/le_lt_n_Sm.con".
-alias id "le_lt_or_eq" = "cic:/Coq/Arith/Lt/le_lt_or_eq.con".
-alias id "le_n" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1/1)".
-alias id "le_n_O_eq" = "cic:/Coq/Arith/Le/le_n_O_eq.con".
-alias id "le_not_lt" = "cic:/Coq/Arith/Lt/le_not_lt.con".
-alias id "le_n_S" = "cic:/Coq/Arith/Le/le_n_S.con".
-alias id "le_O_n" = "cic:/Coq/Arith/Le/le_O_n.con".
-alias id "le_or_lt" = "cic:/Coq/Arith/Lt/le_or_lt.con".
-alias id "le_plus_l" = "cic:/Coq/Arith/Plus/le_plus_l.con".
-alias id "le_plus_minus" = "cic:/Coq/Arith/Minus/le_plus_minus.con".
-alias id "le_plus_minus_r" = "cic:/Coq/Arith/Minus/le_plus_minus_r.con".
-alias id "le_plus_r" = "cic:/Coq/Arith/Plus/le_plus_r.con".
-alias id "le_pred_n" = "cic:/Coq/Arith/Le/le_pred_n.con".
-alias id "le_S" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1/2)".
-alias id "le_S_n" = "cic:/Coq/Arith/Le/le_S_n.con".
-alias id "le_Sn_n" = "cic:/Coq/Arith/Le/le_Sn_n.con".
-alias id "le_trans" = "cic:/Coq/Arith/Le/le_trans.con".
-alias id "lt" = "cic:/Coq/Init/Peano/lt.con".
-alias id "lt_irrefl" = "cic:/Coq/Arith/Lt/lt_irrefl.con".
-alias id "lt_le_S" = "cic:/Coq/Arith/Lt/lt_le_S.con".
-alias id "lt_n_S" = "cic:/Coq/Arith/Lt/lt_n_S.con".
-alias id "minus" = "cic:/Coq/Init/Peano/minus.con".
-alias id "minus_n_O" = "cic:/Coq/Arith/Minus/minus_n_O.con".
-alias id "minus_plus" = "cic:/Coq/Arith/Minus/minus_plus.con".
-alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
-alias id "nat_ind" = "cic:/Coq/Init/Datatypes/nat_ind.con".
-alias id "not" = "cic:/Coq/Init/Logic/not.con".
-alias id "not_eq_S" = "cic:/Coq/Init/Peano/not_eq_S.con".
-alias id "O" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/1)".
-alias id "or" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1)".
-alias id "or_ind" = "cic:/Coq/Init/Logic/or_ind.con".
-alias id "or_introl" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1/1)".
-alias id "or_intror" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1/2)".
-alias id "O_S" = "cic:/Coq/Init/Peano/O_S.con".
-alias id "plus_assoc" = "cic:/Coq/Arith/Plus/plus_assoc.con".
-alias id "plus_assoc_reverse" = "cic:/Coq/Arith/Plus/plus_assoc_reverse.con".
-alias id "plus" = "cic:/Coq/Init/Peano/plus.con".
-alias id "plus_comm" = "cic:/Coq/Arith/Plus/plus_comm.con".
-alias id "plus_le_compat" = "cic:/Coq/Arith/Plus/plus_le_compat.con".
-alias id "plus_le_reg_l" = "cic:/Coq/Arith/Plus/plus_le_reg_l.con".
-alias id "plus_lt_reg_l" = "cic:/Coq/Arith/Plus/plus_lt_reg_l.con".
-alias id "plus_n_Sm" = "cic:/Coq/Init/Peano/plus_n_Sm.con".
-alias id "plus_reg_l" = "cic:/Coq/Arith/Plus/plus_reg_l.con".
-alias id "pred" = "cic:/Coq/Init/Peano/pred.con".
-alias id "refl_equal" = "cic:/Coq/Init/Logic/eq.ind#xpointer(1/1/1)".
-alias id "S" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/2)".
-alias id "true" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1/1)".
-alias id "True" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1)".
-
-theorem f_equal: \forall A,B:Type. \forall f:A \to B.
- \forall x,y:A. x = y \to f x = f y.
- intros. elim H. reflexivity.
-qed.
-
-theorem sym_eq: \forall A:Type. \forall x,y:A. x = y \to y = x.
- intros. rewrite > H. reflexivity.
-qed.
-
-theorem sym_not_eq: \forall A:Type. \forall x,y:A. x \neq y \to y \neq x.
- unfold not. intros. apply H. symmetry. assumption.
-qed.
-
-theorem trans_eq : \forall A:Type. \forall x,y,z:A. x=y \to y=z \to x=z.
- intros. transitivity y; assumption.
-qed.
-
-theorem plus_reg_l: \forall n,m,p. n + m = n + p \to m = p.
- intros. apply plus_reg_l; autobatch.
-qed.
-
-theorem plus_le_reg_l: \forall p,n,m. p + n <= p + m \to n <= m.
- intros. apply plus_le_reg_l; autobatch.
-qed.
+include "Legacy-1/theory.ma".
T).(clt (CHead c k t) (CHead d k t))))))
\def
\lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight
-d))).(\lambda (_: K).(\lambda (t: T).(plus_lt_compat_r (cweight c) (cweight
-d) (tweight t) H))))).
+d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d)
+(tweight t) H))))).
theorem clt_head:
\forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u))))
\def
\lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight
c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u))))
-(plus_le_lt_compat (cweight c) (cweight c) O (tweight u) (le_n (cweight c))
+(le_lt_plus_plus (cweight c) (cweight c) O (tweight u) (le_n (cweight c))
(tweight_lt u)) (cweight c) (plus_n_O (cweight c))))).
theorem clt_wf__q_ind:
include "LambdaDelta-1/leq/asucc.ma".
-include "LambdaDelta-1/leq/fwd.ma".
-
include "LambdaDelta-1/getl/drop.ma".
theorem arity_gen_sort:
Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t a1)))) H2
(THead (Flat Cast) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1:
T).(arity g c0 t1 a1)) H1 (THead (Flat Cast) u t) H5) in (let H8 \def (H6
-(refl_equal T (THead (Flat Cast) u t))) in (and_ind (arity g c0 u (asucc g
+(refl_equal T (THead (Flat Cast) u t))) in (land_ind (arity g c0 u (asucc g
a1)) (arity g c0 t a1) (land (arity g c0 u (asucc g a2)) (arity g c0 t a2))
(\lambda (H9: (arity g c0 u (asucc g a1))).(\lambda (H10: (arity g c0 t
a1)).(conj (arity g c0 u (asucc g a2)) (arity g c0 t a2) (arity_repl g c0 u
(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
-i)))).(and_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
+i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
(arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h
(minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h
(minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i)
-(eq T x0 (TLRef (minus i h))))).(and_ind (le (plus x h) i) (eq T x0 (TLRef
+(eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x h) i) (eq T x0 (TLRef
(minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda
(H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda
(t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h)
(lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
(arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
-i)))).(and_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
+i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
(lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
(t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1
(refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt
Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7:
-(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(and_ind (le (plus x
+(land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x
h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le
(plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T
(TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0
(t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2:
A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4:
(arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u
-t0 a2 H4) in (and_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g a
-a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0
+t0 a2 H4) in (land_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g
+a a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0
a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda
(a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3:
A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2:
(v: T).((subst0 i0 w u v) \to (\forall (P: Prop).P)))))))))).(\lambda (d0:
C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0
(Bind Void) u0))).(\lambda (w: T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w
-(TLRef i) v)).(\lambda (P: Prop).(and_ind (eq nat i i0) (eq T v (lift (S i) O
-w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O
+(TLRef i) v)).(\lambda (P: Prop).(land_ind (eq nat i i0) (eq T v (lift (S i)
+O w)) P (\lambda (H5: (eq nat i i0)).(\lambda (_: (eq T v (lift (S i) O
w))).(let H7 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c0 (CHead d0
(Bind Void) u0))) H3 i H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u)
(\lambda (c1: C).(getl i c0 c1)) H0 (CHead d0 (Bind Void) u0) (getl_mono c0
\to (\forall (P: Prop).P)))))))))).(\lambda (d0: C).(\lambda (u0: T).(\lambda
(i0: nat).(\lambda (H3: (getl i0 c0 (CHead d0 (Bind Void) u0))).(\lambda (w:
T).(\lambda (v: T).(\lambda (H4: (subst0 i0 w (TLRef i) v)).(\lambda (P:
-Prop).(and_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq
+Prop).(land_ind (eq nat i i0) (eq T v (lift (S i) O w)) P (\lambda (H5: (eq
nat i i0)).(\lambda (_: (eq T v (lift (S i) O w))).(let H7 \def (eq_ind_r nat
i0 (\lambda (n: nat).(getl n c0 (CHead d0 (Bind Void) u0))) H3 i H5) in (let
H8 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0
n) t2 u i H1) in (let H2 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u
(TSort n) t2)) (land (eq T (TSort n) t2) (csubst0 i u c c2)) (land (subst0 i
u (TSort n) t2) (csubst0 i u c c2)) (arity g c2 t2 (ASort O n)) (\lambda (H3:
-(land (eq C c c2) (subst0 i u (TSort n) t2))).(and_ind (eq C c c2) (subst0 i
+(land (eq C c c2) (subst0 i u (TSort n) t2))).(land_ind (eq C c c2) (subst0 i
u (TSort n) t2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (eq C c
c2)).(\lambda (H5: (subst0 i u (TSort n) t2)).(eq_ind C c (\lambda (c0:
C).(arity g c0 t2 (ASort O n))) (subst0_gen_sort u t2 i n H5 (arity g c t2
(ASort O n))) c2 H4))) H3)) (\lambda (H3: (land (eq T (TSort n) t2) (csubst0
-i u c c2))).(and_ind (eq T (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2
+i u c c2))).(land_ind (eq T (TSort n) t2) (csubst0 i u c c2) (arity g c2 t2
(ASort O n)) (\lambda (H4: (eq T (TSort n) t2)).(\lambda (_: (csubst0 i u c
c2)).(eq_ind T (TSort n) (\lambda (t: T).(arity g c2 t (ASort O n)))
(arity_sort g c2 n) t2 H4))) H3)) (\lambda (H3: (land (subst0 i u (TSort n)
-t2) (csubst0 i u c c2))).(and_ind (subst0 i u (TSort n) t2) (csubst0 i u c
+t2) (csubst0 i u c c2))).(land_ind (subst0 i u (TSort n) t2) (csubst0 i u c
c2) (arity g c2 t2 (ASort O n)) (\lambda (H4: (subst0 i u (TSort n)
t2)).(\lambda (_: (csubst0 i u c c2)).(subst0_gen_sort u t2 i n H4 (arity g
c2 t2 (ASort O n))))) H3)) H2)))))))))))) (\lambda (c: C).(\lambda (d:
(let H5 \def H_x in (or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2))
(land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i)
t2) (csubst0 i0 u0 c c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2)
-(subst0 i0 u0 (TLRef i) t2))).(and_ind (eq C c c2) (subst0 i0 u0 (TLRef i)
+(subst0 i0 u0 (TLRef i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i)
t2) (arity g c2 t2 a0) (\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0
-(TLRef i) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) (and_ind (eq
+(TLRef i) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq
nat i i0) (eq T t2 (lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat
i i0)).(\lambda (H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O
u0) (\lambda (t: T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda
H17 \def (eq_ind_r C d1 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u)))
H16 d H15) in (arity_lift g d u a0 H1 c (S i) O (getl_drop Abbr c d u i
H17))) u0 H14)))) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2 H7)))
-H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))).(and_ind
+H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind
(eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (eq
T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(eq_ind T (TLRef i)
(\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0 (arity g c2 (TLRef i) a0)
d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9:
(le i0 i)).(arity_abbr g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead
d (Bind Abbr) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0
-(TLRef i) t2) (csubst0 i0 u0 c c2))).(and_ind (subst0 i0 u0 (TLRef i) t2)
+(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2)
(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i)
-t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(and_ind (eq nat i i0) (eq T t2
+t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2
(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda
(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t:
T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n:
(or3_ind (land (eq C c c2) (subst0 i0 u0 (TLRef i) t2)) (land (eq T (TLRef i)
t2) (csubst0 i0 u0 c c2)) (land (subst0 i0 u0 (TLRef i) t2) (csubst0 i0 u0 c
c2)) (arity g c2 t2 a0) (\lambda (H6: (land (eq C c c2) (subst0 i0 u0 (TLRef
-i) t2))).(and_ind (eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0)
+i) t2))).(land_ind (eq C c c2) (subst0 i0 u0 (TLRef i) t2) (arity g c2 t2 a0)
(\lambda (H7: (eq C c c2)).(\lambda (H8: (subst0 i0 u0 (TLRef i) t2)).(eq_ind
-C c (\lambda (c0: C).(arity g c0 t2 a0)) (and_ind (eq nat i i0) (eq T t2
+C c (\lambda (c0: C).(arity g c0 t2 a0)) (land_ind (eq nat i i0) (eq T t2
(lift (S i) O u0)) (arity g c t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda
(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t:
T).(arity g c t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n
(Bind Abst) u) i H0 (CHead d1 (Bind Abbr) u0) H11)) in (False_ind (arity g c
(lift (S i) O u0) a0) H13)))) t2 H10))) (subst0_gen_lref u0 t2 i0 i H8)) c2
H7))) H6)) (\lambda (H6: (land (eq T (TLRef i) t2) (csubst0 i0 u0 c
-c2))).(and_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0)
+c2))).(land_ind (eq T (TLRef i) t2) (csubst0 i0 u0 c c2) (arity g c2 t2 a0)
(\lambda (H7: (eq T (TLRef i) t2)).(\lambda (H8: (csubst0 i0 u0 c
c2)).(eq_ind T (TLRef i) (\lambda (t: T).(arity g c2 t a0)) (lt_le_e i i0
(arity g c2 (TLRef i) a0) (\lambda (H9: (lt i i0)).(let H10 \def
d u x4 H22 x2 H23))))))))) H18)) H17)))))))))))) H11)) H10))) (\lambda (H9:
(le i0 i)).(arity_abst g c2 d u i (csubst0_getl_ge i0 i H9 c c2 u0 H8 (CHead
d (Bind Abst) u) H0) a0 H1))) t2 H7))) H6)) (\lambda (H6: (land (subst0 i0 u0
-(TLRef i) t2) (csubst0 i0 u0 c c2))).(and_ind (subst0 i0 u0 (TLRef i) t2)
+(TLRef i) t2) (csubst0 i0 u0 c c2))).(land_ind (subst0 i0 u0 (TLRef i) t2)
(csubst0 i0 u0 c c2) (arity g c2 t2 a0) (\lambda (H7: (subst0 i0 u0 (TLRef i)
-t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(and_ind (eq nat i i0) (eq T t2
+t2)).(\lambda (H8: (csubst0 i0 u0 c c2)).(land_ind (eq nat i i0) (eq T t2
(lift (S i) O u0)) (arity g c2 t2 a0) (\lambda (H9: (eq nat i i0)).(\lambda
(H10: (eq T t2 (lift (S i) O u0))).(eq_ind_r T (lift (S i) O u0) (\lambda (t:
T).(arity g c2 t a0)) (let H11 \def (eq_ind_r nat i0 (\lambda (n:
(THead (Bind b) u t) t2)) (land (eq T (THead (Bind b) u t) t2) (csubst0 i u0
c c2)) (land (subst0 i u0 (THead (Bind b) u t) t2) (csubst0 i u0 c c2))
(arity g c2 t2 a2) (\lambda (H8: (land (eq C c c2) (subst0 i u0 (THead (Bind
-b) u t) t2))).(and_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2)
+b) u t) t2))).(land_ind (eq C c c2) (subst0 i u0 (THead (Bind b) u t) t2)
(arity g c2 t2 a2) (\lambda (H9: (eq C c c2)).(\lambda (H10: (subst0 i u0
(THead (Bind b) u t) t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2))
(or3_ind (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda
(S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c (Bind b) x0)
(csubst0_snd_bind b i u0 u x0 H13 c)))) t2 H12)))))) H11)) (subst0_gen_head
(Bind b) u0 u t t2 i H10)) c2 H9))) H8)) (\lambda (H8: (land (eq T (THead
-(Bind b) u t) t2) (csubst0 i u0 c c2))).(and_ind (eq T (THead (Bind b) u t)
+(Bind b) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T (THead (Bind b) u t)
t2) (csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (eq T (THead (Bind
b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(eq_ind T (THead (Bind b) u
t) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0
(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) t (fsubst0_fst (S i) u0 (CHead c
(Bind b) u) t (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10 u))))
t2 H9))) H8)) (\lambda (H8: (land (subst0 i u0 (THead (Bind b) u t) t2)
-(csubst0 i u0 c c2))).(and_ind (subst0 i u0 (THead (Bind b) u t) t2) (csubst0
-i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead (Bind b) u t)
-t2)).(\lambda (H10: (csubst0 i u0 c c2)).(or3_ind (ex2 T (\lambda (u2: T).(eq
-T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))) (ex2 T
-(\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s
-(Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
-(THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))))
-(arity g c2 t2 a2) (\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t2 (THead
-(Bind b) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda
-(u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))
-(arity g c2 t2 a2) (\lambda (x: T).(\lambda (H12: (eq T t2 (THead (Bind b) x
-t))).(\lambda (H13: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind b) x t)
-(\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 x a1 (H2 d1 u0 i
-H5 c2 x (fsubst0_both i u0 c u x H13 c2 H10)) t a2 (H4 d1 u0 (S i)
-(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
-(Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t (fsubst0_fst (S i) u0 (CHead c
-(Bind b) u) t (CHead c2 (Bind b) x) (csubst0_both_bind b i u0 u x H13 c c2
-H10)))) t2 H12)))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t2
-(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda
-(t3: T).(subst0 (s (Bind b) i) u0 t t3)) (arity g c2 t2 a2) (\lambda (x:
-T).(\lambda (H12: (eq T t2 (THead (Bind b) u x))).(\lambda (H13: (subst0 (s
-(Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind b) u x) (\lambda (t0: T).(arity
-g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2 d1 u0 i H5 c2 u (fsubst0_fst i u0
-c u c2 H10)) x a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u
-(clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) x
-(fsubst0_both (S i) u0 (CHead c (Bind b) u) t x H13 (CHead c2 (Bind b) u)
-(csubst0_fst_bind b i c c2 u0 H10 u)))) t2 H12)))) H11)) (\lambda (H11:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda
-(u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind b) i) u0 t t3))) (arity g c2 t2 a2) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind b) x0 x1))).(\lambda
-(H13: (subst0 i u0 u x0)).(\lambda (H14: (subst0 (s (Bind b) i) u0 t
-x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: T).(arity g c2 t0 a2))
-(arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0 (fsubst0_both i u0 c u x0
-H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c
-u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x0)
-x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1 H14 (CHead c2 (Bind b)
-x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2 H12)))))) H11))
-(subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8)) H7)))))))))))))))))))))
-(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u
-(asucc g a1))).(\lambda (H1: ((\forall (d1: C).(\forall (u0: T).(\forall (i:
-nat).((getl i c (CHead d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2:
-T).((fsubst0 i u0 c u c2 t2) \to (arity g c2 t2 (asucc g
-a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
-(Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1: C).(\forall (u0:
-T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead d1 (Bind Abbr)
-u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 (CHead c (Bind
-Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda (d1: C).(\lambda
-(u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1 (Bind Abbr)
-u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i u0 c (THead
-(Bind Abst) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (THead (Bind
-Abst) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2)
-(subst0 i u0 (THead (Bind Abst) u t) t2)) (land (eq T (THead (Bind Abst) u t)
-t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Bind Abst) u t) t2)
-(csubst0 i u0 c c2)) (arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land (eq C
-c c2) (subst0 i u0 (THead (Bind Abst) u t) t2))).(and_ind (eq C c c2) (subst0
-i u0 (THead (Bind Abst) u t) t2) (arity g c2 t2 (AHead a1 a2)) (\lambda (H8:
-(eq C c c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t) t2)).(eq_ind
-C c (\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind (ex2 T (\lambda
-(u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u
-u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda
-(t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Bind b) u t) t2)
+(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H9: (subst0 i u0 (THead
+(Bind b) u t) t2)).(\lambda (H10: (csubst0 i u0 c c2)).(or3_ind (ex2 T
+(\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0 i
+u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda
+(t3: T).(subst0 (s (Bind b) i) u0 t t3))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2:
T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind Abst) i) u0 t t3)))) (arity g c t2 (AHead a1 a2))
-(\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t)))
-(\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2
-(THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)) (arity g c t2
-(AHead a1 a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x
-t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind Abst) x t)
-(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x a1 (H1 d1 u0
-i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i) (getl_clear_bind
-Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u) (CHead d1 (Bind Abbr)
-u0) i H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i) u0 (CHead c (Bind
-Abst) u) t (CHead c (Bind Abst) x) (csubst0_snd_bind Abst i u0 u x H12 c))))
-t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind
-Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)))).(ex2_ind
-T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3))) (\lambda (t3:
-T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1 a2)) (\lambda
-(x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u x))).(\lambda (H12:
-(subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead (Bind Abst) u x)
-(\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c u a1 H0 x a2
-(H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind
-Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) u) x
-(fsubst0_snd (S i) u0 (CHead c (Bind Abst) u) t x H12))) t2 H11)))) H10))
-(\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t
+T).(subst0 (s (Bind b) i) u0 t t3)))) (arity g c2 t2 a2) (\lambda (H11: (ex2
+T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t))) (\lambda (u2: T).(subst0
+i u0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T t2 (THead (Bind b) u2 t)))
+(\lambda (u2: T).(subst0 i u0 u u2)) (arity g c2 t2 a2) (\lambda (x:
+T).(\lambda (H12: (eq T t2 (THead (Bind b) x t))).(\lambda (H13: (subst0 i u0
+u x)).(eq_ind_r T (THead (Bind b) x t) (\lambda (t0: T).(arity g c2 t0 a2))
+(arity_bind g b H0 c2 x a1 (H2 d1 u0 i H5 c2 x (fsubst0_both i u0 c u x H13
+c2 H10)) t a2 (H4 d1 u0 (S i) (getl_clear_bind b (CHead c (Bind b) u) c u
+(clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c2 (Bind b) x) t
+(fsubst0_fst (S i) u0 (CHead c (Bind b) u) t (CHead c2 (Bind b) x)
+(csubst0_both_bind b i u0 u x H13 c c2 H10)))) t2 H12)))) H11)) (\lambda
+(H11: (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind b) u t3))) (\lambda (t3:
+T).(subst0 (s (Bind b) i) u0 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2
+(THead (Bind b) u t3))) (\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3))
+(arity g c2 t2 a2) (\lambda (x: T).(\lambda (H12: (eq T t2 (THead (Bind b) u
+x))).(\lambda (H13: (subst0 (s (Bind b) i) u0 t x)).(eq_ind_r T (THead (Bind
+b) u x) (\lambda (t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 u a1 (H2
+d1 u0 i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) x a2 (H4 d1 u0 (S i)
+(getl_clear_bind b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1
+(Bind Abbr) u0) i H5) (CHead c2 (Bind b) u) x (fsubst0_both (S i) u0 (CHead c
+(Bind b) u) t x H13 (CHead c2 (Bind b) u) (csubst0_fst_bind b i c c2 u0 H10
+u)))) t2 H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b)
+i) u0 t t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind b) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind b) i) u0 t t3)))
+(arity g c2 t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2
+(THead (Bind b) x0 x1))).(\lambda (H13: (subst0 i u0 u x0)).(\lambda (H14:
+(subst0 (s (Bind b) i) u0 t x1)).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda
+(t0: T).(arity g c2 t0 a2)) (arity_bind g b H0 c2 x0 a1 (H2 d1 u0 i H5 c2 x0
+(fsubst0_both i u0 c u x0 H13 c2 H10)) x1 a2 (H4 d1 u0 (S i) (getl_clear_bind
+b (CHead c (Bind b) u) c u (clear_bind b c u) (CHead d1 (Bind Abbr) u0) i H5)
+(CHead c2 (Bind b) x0) x1 (fsubst0_both (S i) u0 (CHead c (Bind b) u) t x1
+H14 (CHead c2 (Bind b) x0) (csubst0_both_bind b i u0 u x0 H13 c c2 H10)))) t2
+H12)))))) H11)) (subst0_gen_head (Bind b) u0 u t t2 i H9)))) H8))
+H7))))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1:
+A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1:
+C).(\forall (u0: T).(\forall (i: nat).((getl i c (CHead d1 (Bind Abbr) u0))
+\to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0 c u c2 t2) \to (arity g
+c2 t2 (asucc g a1))))))))))).(\lambda (t: T).(\lambda (a2: A).(\lambda (_:
+(arity g (CHead c (Bind Abst) u) t a2)).(\lambda (H3: ((\forall (d1:
+C).(\forall (u0: T).(\forall (i: nat).((getl i (CHead c (Bind Abst) u) (CHead
+d1 (Bind Abbr) u0)) \to (\forall (c2: C).(\forall (t2: T).((fsubst0 i u0
+(CHead c (Bind Abst) u) t c2 t2) \to (arity g c2 t2 a2)))))))))).(\lambda
+(d1: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H4: (getl i c (CHead d1
+(Bind Abbr) u0))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H5: (fsubst0 i
+u0 c (THead (Bind Abst) u t) c2 t2)).(let H_x \def (fsubst0_gen_base c c2
+(THead (Bind Abst) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq
+C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2)) (land (eq T (THead (Bind
+Abst) u t) t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Bind Abst) u
+t) t2) (csubst0 i u0 c c2)) (arity g c2 t2 (AHead a1 a2)) (\lambda (H7: (land
+(eq C c c2) (subst0 i u0 (THead (Bind Abst) u t) t2))).(land_ind (eq C c c2)
+(subst0 i u0 (THead (Bind Abst) u t) t2) (arity g c2 t2 (AHead a1 a2))
+(\lambda (H8: (eq C c c2)).(\lambda (H9: (subst0 i u0 (THead (Bind Abst) u t)
+t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 (AHead a1 a2))) (or3_ind
+(ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2:
+T).(subst0 i u0 u u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u
+t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)))) (arity g c t2
+(AHead a1 a2)) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t2 (THead (Bind
+Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))).(ex2_ind T (\lambda (u2:
+T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2))
+(arity g c t2 (AHead a1 a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead
+(Bind Abst) x t))).(\lambda (H12: (subst0 i u0 u x)).(eq_ind_r T (THead (Bind
+Abst) x t) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g c x
+a1 (H1 d1 u0 i H4 c x (fsubst0_snd i u0 c u x H12)) t a2 (H3 d1 u0 (S i)
+(getl_clear_bind Abst (CHead c (Bind Abst) u) c u (clear_bind Abst c u)
+(CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind Abst) x) t (fsubst0_fst (S i)
+u0 (CHead c (Bind Abst) u) t (CHead c (Bind Abst) x) (csubst0_snd_bind Abst i
+u0 u x H12 c)))) t2 H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3: T).(eq
+T t2 (THead (Bind Abst) u t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u0
+t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead (Bind Abst) u t3)))
+(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3)) (arity g c t2 (AHead a1
+a2)) (\lambda (x: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) u
+x))).(\lambda (H12: (subst0 (s (Bind Abst) i) u0 t x)).(eq_ind_r T (THead
+(Bind Abst) u x) (\lambda (t0: T).(arity g c t0 (AHead a1 a2))) (arity_head g
+c u a1 H0 x a2 (H3 d1 u0 (S i) (getl_clear_bind Abst (CHead c (Bind Abst) u)
+c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c (Bind
+Abst) u) x (fsubst0_snd (S i) u0 (CHead c (Bind Abst) u) t x H12))) t2
+H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i
+u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t
t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u0 u u2)))
(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u0 t t3))) (arity
i) u0 (CHead c (Bind Abst) u) t x1 H13 (CHead c (Bind Abst) x0)
(csubst0_snd_bind Abst i u0 u x0 H12 c)))) t2 H11)))))) H10))
(subst0_gen_head (Bind Abst) u0 u t t2 i H9)) c2 H8))) H7)) (\lambda (H7:
-(land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))).(and_ind (eq T
+(land (eq T (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2))).(land_ind (eq T
(THead (Bind Abst) u t) t2) (csubst0 i u0 c c2) (arity g c2 t2 (AHead a1 a2))
(\lambda (H8: (eq T (THead (Bind Abst) u t) t2)).(\lambda (H9: (csubst0 i u0
c c2)).(eq_ind T (THead (Bind Abst) u t) (\lambda (t0: T).(arity g c2 t0
Abst) u) t (fsubst0_fst (S i) u0 (CHead c (Bind Abst) u) t (CHead c2 (Bind
Abst) u) (csubst0_fst_bind Abst i c c2 u0 H9 u)))) t2 H8))) H7)) (\lambda
(H7: (land (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c
-c2))).(and_ind (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2)
+c2))).(land_ind (subst0 i u0 (THead (Bind Abst) u t) t2) (csubst0 i u0 c c2)
(arity g c2 t2 (AHead a1 a2)) (\lambda (H8: (subst0 i u0 (THead (Bind Abst) u
t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T (\lambda (u2:
T).(eq T t2 (THead (Bind Abst) u2 t))) (\lambda (u2: T).(subst0 i u0 u u2)))
(subst0 i u0 (THead (Flat Appl) u t) t2)) (land (eq T (THead (Flat Appl) u t)
t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Appl) u t) t2)
(csubst0 i u0 c c2)) (arity g c2 t2 a2) (\lambda (H7: (land (eq C c c2)
-(subst0 i u0 (THead (Flat Appl) u t) t2))).(and_ind (eq C c c2) (subst0 i u0
+(subst0 i u0 (THead (Flat Appl) u t) t2))).(land_ind (eq C c c2) (subst0 i u0
(THead (Flat Appl) u t) t2) (arity g c2 t2 a2) (\lambda (H8: (eq C c
c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Appl) u t) t2)).(eq_ind C c
(\lambda (c0: C).(arity g c0 t2 a2)) (or3_ind (ex2 T (\lambda (u2: T).(eq T
(fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c
t x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Appl) u0 u t t2 i H9))
c2 H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Appl) u t) t2) (csubst0
-i u0 c c2))).(and_ind (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2)
+i u0 c c2))).(land_ind (eq T (THead (Flat Appl) u t) t2) (csubst0 i u0 c c2)
(arity g c2 t2 a2) (\lambda (H8: (eq T (THead (Flat Appl) u t) t2)).(\lambda
(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Appl) u t) (\lambda (t0:
T).(arity g c2 t0 a2)) (arity_appl g c2 u a1 (H1 d1 u0 i H4 c2 u (fsubst0_fst
i u0 c u c2 H9)) t a2 (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Appl) u t) t2)
-(csubst0 i u0 c c2))).(and_ind (subst0 i u0 (THead (Flat Appl) u t) t2)
+(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Appl) u t) t2)
(csubst0 i u0 c c2) (arity g c2 t2 a2) (\lambda (H8: (subst0 i u0 (THead
(Flat Appl) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T
(\lambda (u2: T).(eq T t2 (THead (Flat Appl) u2 t))) (\lambda (u2: T).(subst0
(subst0 i u0 (THead (Flat Cast) u t) t2)) (land (eq T (THead (Flat Cast) u t)
t2) (csubst0 i u0 c c2)) (land (subst0 i u0 (THead (Flat Cast) u t) t2)
(csubst0 i u0 c c2)) (arity g c2 t2 a0) (\lambda (H7: (land (eq C c c2)
-(subst0 i u0 (THead (Flat Cast) u t) t2))).(and_ind (eq C c c2) (subst0 i u0
+(subst0 i u0 (THead (Flat Cast) u t) t2))).(land_ind (eq C c c2) (subst0 i u0
(THead (Flat Cast) u t) t2) (arity g c2 t2 a0) (\lambda (H8: (eq C c
c2)).(\lambda (H9: (subst0 i u0 (THead (Flat Cast) u t) t2)).(eq_ind C c
(\lambda (c0: C).(arity g c0 t2 a0)) (or3_ind (ex2 T (\lambda (u2: T).(eq T
(fsubst0_snd i u0 c u x0 H12)) x1 (H3 d1 u0 i H4 c x1 (fsubst0_snd i u0 c t
x1 H13))) t2 H11)))))) H10)) (subst0_gen_head (Flat Cast) u0 u t t2 i H9)) c2
H8))) H7)) (\lambda (H7: (land (eq T (THead (Flat Cast) u t) t2) (csubst0 i
-u0 c c2))).(and_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2)
+u0 c c2))).(land_ind (eq T (THead (Flat Cast) u t) t2) (csubst0 i u0 c c2)
(arity g c2 t2 a0) (\lambda (H8: (eq T (THead (Flat Cast) u t) t2)).(\lambda
(H9: (csubst0 i u0 c c2)).(eq_ind T (THead (Flat Cast) u t) (\lambda (t0:
T).(arity g c2 t0 a0)) (arity_cast g c2 u a0 (H1 d1 u0 i H4 c2 u (fsubst0_fst
i u0 c u c2 H9)) t (H3 d1 u0 i H4 c2 t (fsubst0_fst i u0 c t c2 H9))) t2
H8))) H7)) (\lambda (H7: (land (subst0 i u0 (THead (Flat Cast) u t) t2)
-(csubst0 i u0 c c2))).(and_ind (subst0 i u0 (THead (Flat Cast) u t) t2)
+(csubst0 i u0 c c2))).(land_ind (subst0 i u0 (THead (Flat Cast) u t) t2)
(csubst0 i u0 c c2) (arity g c2 t2 a0) (\lambda (H8: (subst0 i u0 (THead
(Flat Cast) u t) t2)).(\lambda (H9: (csubst0 i u0 c c2)).(or3_ind (ex2 T
(\lambda (u2: T).(eq T t2 (THead (Flat Cast) u2 t))) (\lambda (u2: T).(subst0
c t c2 t2)).(let H_x \def (fsubst0_gen_base c c2 t t2 u i H4) in (let H5 \def
H_x in (or3_ind (land (eq C c c2) (subst0 i u t t2)) (land (eq T t t2)
(csubst0 i u c c2)) (land (subst0 i u t t2) (csubst0 i u c c2)) (arity g c2
-t2 a2) (\lambda (H6: (land (eq C c c2) (subst0 i u t t2))).(and_ind (eq C c
+t2 a2) (\lambda (H6: (land (eq C c c2) (subst0 i u t t2))).(land_ind (eq C c
c2) (subst0 i u t t2) (arity g c2 t2 a2) (\lambda (H7: (eq C c c2)).(\lambda
(H8: (subst0 i u t t2)).(eq_ind C c (\lambda (c0: C).(arity g c0 t2 a2))
(arity_repl g c t2 a1 (H1 d1 u i H3 c t2 (fsubst0_snd i u c t t2 H8)) a2 H2)
-c2 H7))) H6)) (\lambda (H6: (land (eq T t t2) (csubst0 i u c c2))).(and_ind
+c2 H7))) H6)) (\lambda (H6: (land (eq T t t2) (csubst0 i u c c2))).(land_ind
(eq T t t2) (csubst0 i u c c2) (arity g c2 t2 a2) (\lambda (H7: (eq T t
t2)).(\lambda (H8: (csubst0 i u c c2)).(eq_ind T t (\lambda (t0: T).(arity g
c2 t0 a2)) (arity_repl g c2 t a1 (H1 d1 u i H3 c2 t (fsubst0_fst i u c t c2
H8)) a2 H2) t2 H7))) H6)) (\lambda (H6: (land (subst0 i u t t2) (csubst0 i u
-c c2))).(and_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2)
+c c2))).(land_ind (subst0 i u t t2) (csubst0 i u c c2) (arity g c2 t2 a2)
(\lambda (H7: (subst0 i u t t2)).(\lambda (H8: (csubst0 i u c
c2)).(arity_repl g c2 t2 a1 (H1 d1 u i H3 c2 t2 (fsubst0_both i u c t t2 H7
c2 H8)) a2 H2))) H6)) H5))))))))))))))))) c1 t1 a H))))).
(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2
e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2
c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2
-e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H1 \def (match H in
-drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
-(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to
-(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2
-c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil
-PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2
-(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1
-e1)) (\lambda (c1: C).(csubc g c2 c1))))) (\lambda (H4: (eq C c2 e2)).(eq_ind
-C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda
-(c1: C).(csubc g c0 c1)))) (let H5 \def (eq_ind_r C e2 (\lambda (c0:
-C).(csubc g c0 e1)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C
-(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c0 c1))))
-(ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g
-c2 c1)) e1 (drop1_nil e1) H5) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c
-c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds0 H2) \Rightarrow (\lambda
-(H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda
-(H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e:
-PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
-\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
-(False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1
-hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 PNil c4 e1)) (\lambda (c4:
-C).(csubc g c2 c4))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList
-PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda
-(n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2:
-C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda
-(c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))).(\lambda
-(c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda
-(e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 \def (match H0 in drop1 return
-(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0
-c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to
-(ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc
-g c2 c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList
-PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c
-e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList
-return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _)
-\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq
-C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda
-(c1: C).(csubc g c2 c1))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds0
-H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0
-p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def
-(f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda
-(_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow
-p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat
-(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with
-[PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0)
-(PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e:
-PList).(match e in PList return (\lambda (_: PList).nat) with [PNil
-\Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0
-p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0
-p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1
-hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1))
-(\lambda (c4: C).(csubc g c2 c4)))))))))) (\lambda (H10: (eq nat d
-n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c2)
-\to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds0 c0 c3) \to (ex2 C
-(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2
-c4))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0:
-PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0
-c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda
-(c4: C).(csubc g c2 c4)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2
-(\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to
-(ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc
-g c2 c4))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c:
-C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c4: C).(drop1
-(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2 c4)))))) (\lambda (H14:
-(drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15
-e1 H1) in (let H16 \def H_x in (ex2_ind C (\lambda (c4: C).(drop1 p c4 e1))
-(\lambda (c4: C).(csubc g c0 c4)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0
-p) c4 e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x: C).(\lambda (H17:
-(drop1 p x e1)).(\lambda (H18: (csubc g c0 x)).(let H_x0 \def
-(drop_csubc_trans g c2 c0 n0 n H14 x H18) in (let H19 \def H_x0 in (ex2_ind C
-(\lambda (c4: C).(drop n n0 c4 x)) (\lambda (c4: C).(csubc g c2 c4)) (ex2 C
-(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c2
-c4))) (\lambda (x0: C).(\lambda (H20: (drop n n0 x0 x)).(\lambda (H21: (csubc
-g c2 x0)).(ex_intro2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1))
-(\lambda (c4: C).(csubc g c2 c4)) x0 (drop1_cons x0 x n n0 H20 e1 p H17)
-H21)))) H19)))))) H16))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2
-H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 H10))) h (sym_eq
-nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n
-n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H_y \def
+(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c:
+C).(csubc g c e1)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1
+e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 (drop1_nil e1) H1)))))))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2:
+C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1)
+\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2
+c1)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n
+n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H_x \def
+(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda
+(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda
+(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))
+(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x
+e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C
+(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g x c1)) (ex2 C
+(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2
+c1))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g
+x x0)).(let H_x1 \def (drop_csubc_trans g c2 x n0 n H3 x0 H7) in (let H8 \def
+H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1:
+C).(csubc g c2 c1)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1))
+(\lambda (c1: C).(csubc g c2 c1))) (\lambda (x1: C).(\lambda (H9: (drop n n0
+x1 x0)).(\lambda (H10: (csubc g c2 x1)).(ex_intro2 C (\lambda (c1: C).(drop1
+(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) x1 (drop1_cons x1 x0
+n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)).
theorem csubc_drop1_conf_rev:
\forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2:
(c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1
e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1
c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2
-e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H1 \def (match H in
-drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda
-(_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to
-(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1
-c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil
-PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2
-(\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1
-e1)) (\lambda (c1: C).(csubc g c1 c2))))) (\lambda (H4: (eq C c2 e2)).(eq_ind
-C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda
-(c1: C).(csubc g c1 c0)))) (let H5 \def (eq_ind_r C e2 (\lambda (c0:
-C).(csubc g e1 c0)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C
-(\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c0))))
-(ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g
-c1 c2)) e1 (drop1_nil e1) H5) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c
-c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds0 H2) \Rightarrow (\lambda
-(H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda
-(H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e:
-PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
-\Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
-(False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1
-hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 PNil c4 e1)) (\lambda (c4:
-C).(csubc g c4 c2))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList
-PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda
-(n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2:
-C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda
-(c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))).(\lambda
-(c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda
-(e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 \def (match H0 in drop1 return
-(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0
-c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to
-(ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc
-g c1 c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList
-PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c
-e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList
-return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _)
-\Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq
-C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda
-(c1: C).(csubc g c1 c2))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds0
-H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0
-p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def
-(f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda
-(_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow
-p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat
-(\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with
-[PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0)
-(PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e:
-PList).(match e in PList return (\lambda (_: PList).nat) with [PNil
-\Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0
-p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0
-p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1
-hds0 c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1))
-(\lambda (c4: C).(csubc g c4 c2)))))))))) (\lambda (H10: (eq nat d
-n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq C c1 c2)
-\to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds0 c0 c3) \to (ex2 C
-(\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4
-c2))))))))) (\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0:
-PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0
-c0 c3) \to (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda
-(c4: C).(csubc g c4 c2)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2
-(\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to
-(ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc
-g c4 c2))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c:
-C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c4: C).(drop1
-(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2)))))) (\lambda (H14:
-(drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15
-e1 H1) in (let H16 \def H_x in (ex2_ind C (\lambda (c4: C).(drop1 p c4 e1))
-(\lambda (c4: C).(csubc g c4 c0)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0
-p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda (x: C).(\lambda (H17:
-(drop1 p x e1)).(\lambda (H18: (csubc g x c0)).(let H_x0 \def
-(csubc_drop_conf_rev g c2 c0 n0 n H14 x H18) in (let H19 \def H_x0 in
-(ex2_ind C (\lambda (c4: C).(drop n n0 c4 x)) (\lambda (c4: C).(csubc g c4
-c2)) (ex2 C (\lambda (c4: C).(drop1 (PCons n n0 p) c4 e1)) (\lambda (c4:
-C).(csubc g c4 c2))) (\lambda (x0: C).(\lambda (H20: (drop n n0 x0
-x)).(\lambda (H21: (csubc g x0 c2)).(ex_intro2 C (\lambda (c4: C).(drop1
-(PCons n n0 p) c4 e1)) (\lambda (c4: C).(csubc g c4 c2)) x0 (drop1_cons x0 x
-n n0 H20 e1 p H17) H21)))) H19)))))) H16))))) c3 (sym_eq C c3 e2 H13))) c1
-(sym_eq C c1 c2 H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0
-H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal
-PList (PCons n n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
+e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H_y \def
+(drop1_gen_pnil c2 e2 H) in (let H1 \def (eq_ind_r C e2 (\lambda (c:
+C).(csubc g e1 c)) H0 c2 H_y) in (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1
+e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 (drop1_nil e1) H1)))))))) (\lambda
+(n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2:
+C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2)
+\to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1
+c2)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n
+n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H_x \def
+(drop1_gen_pcons c2 e2 p n n0 H0) in (let H2 \def H_x in (ex2_ind C (\lambda
+(c3: C).(drop n n0 c2 c3)) (\lambda (c3: C).(drop1 p c3 e2)) (ex2 C (\lambda
+(c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))
+(\lambda (x: C).(\lambda (H3: (drop n n0 c2 x)).(\lambda (H4: (drop1 p x
+e2)).(let H_x0 \def (H x e2 H4 e1 H1) in (let H5 \def H_x0 in (ex2_ind C
+(\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 x)) (ex2 C
+(\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1
+c2))) (\lambda (x0: C).(\lambda (H6: (drop1 p x0 e1)).(\lambda (H7: (csubc g
+x0 x)).(let H_x1 \def (csubc_drop_conf_rev g c2 x n0 n H3 x0 H7) in (let H8
+\def H_x1 in (ex2_ind C (\lambda (c1: C).(drop n n0 c1 x0)) (\lambda (c1:
+C).(csubc g c1 c2)) (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1))
+(\lambda (c1: C).(csubc g c1 c2))) (\lambda (x1: C).(\lambda (H9: (drop n n0
+x1 x0)).(\lambda (H10: (csubc g x1 c2)).(ex_intro2 C (\lambda (c1: C).(drop1
+(PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) x1 (drop1_cons x1 x0
+n n0 H9 e1 p H6) H10)))) H8)))))) H5)))))) H2)))))))))))) hds)).
a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda
(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H_y \def
-(le_n_O_eq (plus d h) H1) in (and_ind (eq nat d O) (eq nat h O) (drop (minus
+(le_n_O_eq (plus d h) H1) in (land_ind (eq nat d O) (eq nat h O) (drop (minus
O h) O e a) (\lambda (H3: (eq nat d O)).(\lambda (H4: (eq nat h O)).(let H5
\def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H3) in (let H6 \def
(eq_ind nat h (\lambda (n: nat).(drop n O a e)) H5 O H4) in (eq_ind_r nat O
T).(pc3 (CHead (CHead (CSort O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O))
(lift (S O) O t) x4)) H31 (TSort O) H35) in (let H39 \def (eq_ind C x10
(\lambda (c: C).(ty3 g c (TSort O) x12)) H37 (CHead (CSort O) (Bind Abst)
-(TSort O)) H36) in (and_ind (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst)
+(TSort O)) H36) in (land_ind (pc3 (CHead (CHead (CHead (CSort O) (Bind Abst)
(TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (TLRef (S (S O)))
x0) (\forall (b: B).(\forall (u0: T).(pc3 (CHead (CHead (CHead (CHead (CSort
O) (Bind Abst) (TSort O)) (Bind Abst) (TSort O)) (Bind Abst) (TLRef O)) (Bind
(THead k u t)))))
\def
\lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t:
-T).(plus_le_lt_compat (cweight c) (cweight c) (tweight u) (S (plus (tweight
-u) (tweight t))) (le_n (cweight c)) (le_n_S (tweight u) (plus (tweight u)
+T).(le_lt_plus_plus (cweight c) (cweight c) (tweight u) (S (plus (tweight u)
+(tweight t))) (le_n (cweight c)) (le_n_S (tweight u) (plus (tweight u)
(tweight t)) (le_plus_l (tweight u) (tweight t))))))).
theorem flt_thead_dx:
(THead k u t)))))
\def
\lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t:
-T).(plus_le_lt_compat (cweight c) (cweight c) (tweight t) (S (plus (tweight
-u) (tweight t))) (le_n (cweight c)) (le_n_S (tweight t) (plus (tweight u)
+T).(le_lt_plus_plus (cweight c) (cweight c) (tweight t) (S (plus (tweight u)
+(tweight t))) (le_n (cweight c)) (le_n_S (tweight t) (plus (tweight u)
(tweight t)) (le_plus_r (tweight u) (tweight t))))))).
theorem flt_shift:
(plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus
(cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight
c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight
-t))) (plus_assoc (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S
+t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S
(plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u)
(tweight t))))))).
(eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n:
nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2)
(tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2)
-(tweight t2)) (S O)) (plus_comm (plus (cweight c2) (tweight t2)) (S
+(tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S
O))))))))))).
theorem flt_arith2:
(H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda
(_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1)
(tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight
-t2)) (S O)) H (plus_le_compat (cweight c2) (plus (cweight c2) (tweight t2))
-(S O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))).
+t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S
+O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))).
theorem flt_wf__q_ind:
\forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C
(H0: (drop (S O) i c1 c2)).(let H3 \def (eq_ind nat (minus (S i) (S O))
(\lambda (n: nat).(drop n O c2 e)) (drop_conf_ge (S i) e c1 (getl_drop b c1 e
u i H) c2 (S O) i H0 (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(le n (S
-i))) (le_n (S i)) (plus i (S O)) (plus_comm i (S O)))) i (minus_Sx_SO i)) in
+i))) (le_n (S i)) (plus i (S O)) (plus_sym i (S O)))) i (minus_Sx_SO i)) in
H3)))))))).
theorem getl_drop_conf_rev:
h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0)))))
(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land
(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h)
-(plus n h)) (eq T (TLRef n) (TLRef n)) (plus_le_compat d n h h H0 (le_n h))
+(plus n h)) (eq T (TLRef n) (TLRef n)) (le_plus_plus d n h h H0 (le_n h))
(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i
H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d:
nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to
d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x
\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d)
(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq
-T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(and_ind (lt
-n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n d)).(\lambda
-(H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0
-(TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2: (land (le
-(plus d h) n) (eq T t (TLRef (minus n h))))).(and_ind (le (plus d h) n) (eq T
-t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus d h)
-n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n
-h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq T (TLRef
-(minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S n) d h
-H))) t H4))) H2)) H1)))))))).
+T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind
+(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n
+d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2:
+(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d
+h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus
+d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef
+(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq
+T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S
+n) d h H))) t H4))) H2)) H1)))))))).
theorem lift_gen_lref_false:
\forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n
(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d
h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n)))
(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land
-(lt n d) (eq T t (TLRef n)))).(and_ind (lt n d) (eq T t (TLRef n)) P (\lambda
-(H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H H4))) H3))
-(\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(and_ind
-(le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda (H4: (le (plus d h)
-n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false (plus d h) n P H4
-H0))) H3)) H2)))))))))).
+(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P
+(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H
+H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n
+h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda
+(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false
+(plus d h) n P H4 H0))) H3)) H2)))))))))).
theorem lift_gen_lref_ge:
\forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall
t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in
(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus
d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n))
-(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(and_ind
+(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind
(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda
(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r
T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq
T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d
(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2))
(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
-h) h))))).(and_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
+h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n
h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda
(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n
h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus
(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus
(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n
(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S
-(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_comm (S O) (pred n)))
+(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_sym (S O) (pred n)))
(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d
(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n))
(S_pred n d H))))))).
t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n
h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t))
(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x
-(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (plus_lt_compat_r n d2 h1 H3) x
-H2))) (\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2:
-T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2
-t2)))) (\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2
-h1) (plus n h1) (plus_le_compat d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus
-(plus d2 h2) h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (plus_lt_compat_r n
-(plus d2 h2) h1 H4) (plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x
-H2 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T
-(TLRef n) (lift h2 d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5
-\def (eq_ind nat (plus n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2
-(plus d2 h1) x))) H2 (plus (minus (plus n h1) h2) h2) (le_plus_minus_sym h2
-(plus n h1) (le_plus_trans h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2
-h2) H4)))) in (eq_ind_r T (TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2
-T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n)
-(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n
-h1) h2)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))
-(TLRef (minus n h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0:
-nat).(eq T (TLRef n0) (lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef
-(plus (minus n h2) h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1))
-t)) (refl_equal T (TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n
-h2))) (lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H
-(le_minus d2 n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans
-h2 (plus d2 h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2)
-h2) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2)))))
+(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (lt_reg_r n d2 h1 H3) x H2)))
+(\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2: T).(eq
+T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
+(\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2 h1) (plus
+n h1) (le_plus_plus d2 n h1 h1 H3 (le_n h1)) (eq_ind_r nat (plus (plus d2 h2)
+h1) (\lambda (n0: nat).(lt (plus n h1) n0)) (lt_reg_r n (plus d2 h2) h1 H4)
+(plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda
+(t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2
+d2 t2)))))) (\lambda (H4: (le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus
+n h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus
+(minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans
+h2 n h1 (le_trans h2 (plus d2 h2) n (le_plus_r d2 h2) H4)))) in (eq_ind_r T
+(TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T
+t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))
+(ex_intro2 T (\lambda (t2: T).(eq T (TLRef (minus (plus n h1) h2)) (lift h1
+d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef (minus n
+h2)) (eq_ind_r nat (plus (minus n h2) h1) (\lambda (n0: nat).(eq T (TLRef n0)
+(lift h1 d1 (TLRef (minus n h2))))) (eq_ind_r T (TLRef (plus (minus n h2)
+h1)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h1)) t)) (refl_equal T
+(TLRef (plus (minus n h2) h1))) (lift h1 d1 (TLRef (minus n h2)))
+(lift_lref_ge (minus n h2) h1 d1 (le_trans d1 d2 (minus n h2) H (le_minus d2
+n h2 H4)))) (minus (plus n h1) h2) (le_minus_plus h2 n (le_trans h2 (plus d2
+h2) n (le_plus_r d2 h2) H4) h1)) (eq_ind_r nat (plus (minus n h2) h2)
+(\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef (minus n0 h2)))))
(eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2) h2)) (\lambda (t:
T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T TLRef (plus (minus
n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2) (f_equal2 nat nat nat
(plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r (minus n h2) h2))
(refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus (minus n h2) h2) h2)))
(lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2 (le_minus d2 (plus
-(minus n h2) h2) h2 (plus_le_compat d2 (minus n h2) h2 h2 (le_minus d2 n h2
-H4) (le_n h2))))) n (le_plus_minus_sym h2 n (le_trans h2 (plus d2 h2) n
+(minus n h2) h2) h2 (le_plus_plus d2 (minus n h2) h2 h2 (le_minus d2 n h2 H4)
+(le_n h2))))) n (le_plus_minus_sym h2 n (le_trans h2 (plus d2 h2) n
(le_plus_r d2 h2) H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus (plus n
h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k:
K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall
nat T TLRef (plus (plus n h) k) (plus n (plus k h))
(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n))
(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge
-(plus n h) k e (le_trans e (plus d h) (plus n h) H (plus_le_compat d n h h H1
+(plus n h) k e (le_trans e (plus d h) (plus n h) H (le_plus_plus d n h h H1
(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda
(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0:
nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T
(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d
(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k)))
-(lift_lref_lt (plus n k) h (plus d k) (plus_lt_compat_r n d k H1)))) (\lambda
-(H1: (le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T
-t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda
+(lift_lref_lt (plus n k) h (plus d k) (lt_reg_r n d k H1)))) (\lambda (H1:
+(le d n)).(eq_ind_r T (TLRef (plus (plus n k) h)) (\lambda (t0: T).(eq T t0
+(lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda
(t0: T).(eq T (TLRef (plus (plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef
(plus (plus n h) k)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0))
(f_equal nat T TLRef (plus (plus n k) h) (plus (plus n h) k) (sym_eq nat
(plus (plus n h) k) (plus (plus n k) h) (plus_permute_2_in_3 n h k))) (lift k
e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_plus_trans e n h H0)))
(lift h d (TLRef n)) (lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus
-n k))) (lift_lref_ge (plus n k) h (plus d k) (plus_le_compat d n k k H1 (le_n
-k)))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n)) (lift_lref_ge n k e
+n k))) (lift_lref_ge (plus n k) h (plus d k) (le_plus_plus d n k k H1 (le_n
+k)))))) (plus k d) (plus_sym k d)) (lift k e (TLRef n)) (lift_lref_ge n k e
H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h:
nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq
T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift h d
(eq_ind_r nat (plus (trans hds0 i) h) (\lambda (n: nat).(eq T (lift (S n) O
(lift1 (ptrans hds0 i) t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans
hds0 i) t)))) (refl_equal T (lift (S (plus (trans hds0 i) h)) O (lift1
-(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_comm h (trans hds0 i)))
+(ptrans hds0 i) t))) (plus h (trans hds0 i)) (plus_sym h (trans hds0 i)))
(plus h (S (trans hds0 i))) (plus_n_Sm h (trans hds0 i))) (lift h d (lift (S
(trans hds0 i)) O (lift1 (ptrans hds0 i) t))) (lift_free (lift1 (ptrans hds0
i) t) (S (trans hds0 i)) h O d (eq_ind nat (S (plus O (trans hds0 i)))
(\lambda (n: nat).(le d n)) (eq_ind_r nat (plus (trans hds0 i) O) (\lambda
(n: nat).(le d (S n))) (le_S d (plus (trans hds0 i) O) (le_plus_trans d
(trans hds0 i) O (bge_le d (trans hds0 i) H0))) (plus O (trans hds0 i))
-(plus_comm O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans
+(plus_sym O (trans hds0 i))) (plus O (S (trans hds0 i))) (plus_n_Sm O (trans
hds0 i))) (le_O_n d)))) x_x))) (lift1 hds0 (lift (S i) O t)) (H i t))))))))
hds).
(TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind
Abst) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind
Abst) u) (TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u
-t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in (and_ind (nf2 c0 u) (nf2
+t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2
(CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0:
T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Appl) u t0))).(let H5 \def
-(nf2_gen_flat Appl c0 u t0 H4) in (and_ind (nf2 c0 u) (nf2 c0 t0) (or3 (ex3_2
-T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead
-(Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda
-(w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda
-(n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat
-(\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads
-(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
-ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda
-(H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def (H3 H7) in (let H8
-\def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0
+(nf2_gen_flat Appl c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 c0 t0) (or3
+(ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0)
(THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
(\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
-(\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws:
-TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
-(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
+(\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList
+nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0)
+(THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
+nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
+i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def
+(H3 H7) in (let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda
+(u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
+T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
+w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat
+(\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
+i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (or3 (ex3_2 T T (\lambda (w:
T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
(clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in
(subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0)
(le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt
-(clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_comm
+(clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_sym
(clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t
(clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5))))
(\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1
H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O)
(clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0))
(\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen
-c0) (S O)) (plus_comm (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x)
+c0) (S O)) (plus_sym (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x)
t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4)))
H3))) H2))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2)
\to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
(Flat Appl) t (TLRef i))))) (\lambda (_: True).H) (\lambda (t: T).(\lambda
(t0: TList).(\lambda (H0: (((nfs2 c t0) \to (nf2 c (THeads (Flat Appl) t0
(TLRef i)))))).(\lambda (H1: (land (nf2 c t) (nfs2 c t0))).(let H2 \def H1 in
-(and_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat
+(land_ind (nf2 c t) (nfs2 c t0) (nf2 c (THead (Flat Appl) t (THeads (Flat
Appl) t0 (TLRef i)))) (\lambda (H3: (nf2 c t)).(\lambda (H4: (nfs2 c
t0)).(let H_y \def (H0 H4) in (\lambda (t2: T).(\lambda (H5: (pr2 c (THead
(Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) t2)).(let H6 \def
T).((eq T t (THead (Bind Abst) x0 t0)) \to (\forall (P: Prop).P)))) H10
(THead (Bind Abst) x5 x6) H24) in (let H28 \def (eq_ind T x (\lambda (t:
T).(nf2 c t)) H5 (THead (Bind Abst) x5 x6) H24) in (let H29 \def
-(nf2_gen_abst c x5 x6 H28) in (and_ind (nf2 c x5) (nf2 (CHead c (Bind Abst)
+(nf2_gen_abst c x5 x6 H28) in (land_ind (nf2 c x5) (nf2 (CHead c (Bind Abst)
x5) x6) False (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c (Bind
Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 x5 H30 u2 H7) in
(H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6)
\def
\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
(H: (pc3 c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2))).(let H_x \def
-(pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (and_ind (pc3 c u u)
+(pc3_gen_abst c u u t1 t2 H) in (let H0 \def H_x in (land_ind (pc3 c u u)
(\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))) (pc3
(CHead c (Bind Abst) u) t1 t2) (\lambda (_: (pc3 c u u)).(\lambda (H2:
((\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) t1 t2))))).(H2
(lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq
T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x)
(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n))
-(le_n (plus (S O) O)) (plus O (S O)) (plus_comm O (S O))) H10 (eq T (THead
+(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H10 (eq T (THead
(Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2
H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P:
Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T
(THead (Bind b) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift
h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S
O) x1) (\lambda (n: nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1))
-(plus x1 (S O)) (plus_comm x1 (S O))) in (let H9 \def (eq_ind nat (S x1)
+(plus x1 (S O)) (plus_sym x1 (S O))) in (let H9 \def (eq_ind nat (S x1)
(\lambda (n: nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O))
H8) in (ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4:
T).(eq T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1
O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t:
T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w
(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n))
-(le_n (plus (S O) O)) (plus O (S O)) (plus_comm O (S O))) H3 (ex2 T (\lambda
+(le_n (plus (S O) O)) (plus O (S O)) (plus_sym O (S O))) H3 (ex2 T (\lambda
(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))))))))
(pr0_gen_lift t4 t3 (S O) O H0)))))))).
(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S
j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead
(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C
-(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(land_ind (eq nat i O) (eq C
(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3)
(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead
d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda
(d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2
(\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in
-(and_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n)) (ex2_2
-C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0))))
-(\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0)))))) (\lambda (H7:
-(eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O u))).(eq_ind_r T (lift (S
-n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
-C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let H9 \def (eq_ind_r
-nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr) u))) H1 n H7) in
-(or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
-C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))) (ex2_2_intro
-C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0))))
-(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))
-d u H9 (refl_equal T (lift (S n) O u))))) t H8))) (subst0_gen_lref u t i n
-H6))) t1 H4))))))))))))) c y x H0))) H)))).
+(land_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n))
+(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr)
+u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))
+(\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O
+u))).(eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n))
+(ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr)
+u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let
+H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr)
+u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T
+(\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0)))))
+(ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind
+Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S
+n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8)))
+(subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))).
theorem pr2_gen_abst:
\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind
T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0)
(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i)
-(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i
+(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i
(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i
x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10))))))
(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0)
(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0
(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d
(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n:
-nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S O)))) x1 x0 H10 x3
+nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S O)))) x1 x0 H10 x3
H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S
-O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S
+O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_sym d0 (S
O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift
(S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12))))))))))
(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0
(**************************************************************************)
include "Base-1/theory.ma".
-
-alias id "and_ind" = "cic:/Coq/Init/Logic/and_ind.con".
-alias id "bool_ind" = "cic:/Coq/Init/Datatypes/bool_ind.con".
-alias id "ex" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1)".
-alias id "ex_ind" = "cic:/Coq/Init/Logic/ex_ind.con".
-alias id "ex_intro" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1/1)".
-alias id "f_equal2" = "cic:/Coq/Init/Logic/f_equal2.con".
-alias id "f_equal3" = "cic:/Coq/Init/Logic/f_equal3.con".
-alias id "le_antisym" = "cic:/Coq/Arith/Le/le_antisym.con".
-alias id "le_lt_trans" = "cic:/Coq/Arith/Lt/le_lt_trans.con".
-alias id "le_plus_trans" = "cic:/Coq/Arith/Plus/le_plus_trans.con".
-alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con".
-alias id "lt_le_weak" = "cic:/Coq/Arith/Lt/lt_le_weak.con".
-alias id "lt_n_Sn" = "cic:/Coq/Arith/Lt/lt_n_Sn.con".
-alias id "lt_S_n" = "cic:/Coq/Arith/Lt/lt_S_n.con".
-alias id "lt_trans" = "cic:/Coq/Arith/Lt/lt_trans.con".
-alias id "lt_wf_ind" = "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con".
-alias id "minus_n_n" = "cic:/Coq/Arith/Minus/minus_n_n.con".
-alias id "minus_Sn_m" = "cic:/Coq/Arith/Minus/minus_Sn_m.con".
-alias id "plus_le_lt_compat" = "cic:/Coq/Arith/Plus/plus_le_lt_compat.con".
-alias id "plus_lt_compat" = "cic:/Coq/Arith/Plus/plus_lt_compat.con".
-alias id "plus_lt_compat_r" = "cic:/Coq/Arith/Plus/plus_lt_compat_r.con".
-alias id "plus_lt_le_compat" = "cic:/Coq/Arith/Plus/plus_lt_le_compat.con".
-alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
-alias id "plus_Snm_nSm" = "cic:/Coq/Arith/Plus/plus_Snm_nSm.con".
-alias id "pred_Sn" = "cic:/Coq/Init/Peano/pred_Sn.con".
-alias id "S_pred" = "cic:/Coq/Arith/Lt/S_pred.con".
(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4:
(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y
\def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2 H5) in (let H6
-\def H_y0 in (and_ind (arity g c2 (lift1 is t0) (AHead a1 a2)) (\forall (d:
+\def H_y0 in (land_ind (arity g c2 (lift1 is t0) (AHead a1 a2)) (\forall (d:
C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d
c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (lift1 is t0)))))))))
(sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0))) (\lambda (_: (arity g c2
\lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind
(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n:
nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
-t))).(let H0 \def H in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (arity g
-c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: (sn3
-c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to (arity
-g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity g c t
-a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d:
+t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (arity
+g c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_:
+(sn3 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to
+(arity g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity
+g c t a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d:
C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in
-(and_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g
+(land_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g
a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat
Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity
g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g
c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda
(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c
t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0
-in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda (H3:
-(arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def
+in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda
+(H3: (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def
(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort1 g n
n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2:
nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k)
(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall
(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is
t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4
-\def H2 in (and_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w:
+\def H2 in (land_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w:
T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity
g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a
(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda
(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h:
nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in
-(and_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t)
+(land_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t)
(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n
n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0))
(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e
(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d
e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c:
C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3
-\def H1 in (and_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall (w:
-T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g a1
-d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t)
+\def H1 in (land_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall
+(w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g
+a1 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t)
(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall
(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is
(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda
TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c:
C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v))
(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda
-(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (and_ind (arity g
+(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (land_ind (arity g
c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat
Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef
i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2:
(lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0
d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat
Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda
-(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (and_ind (arity g
-c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0:
+(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (land_ind (arity
+g c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0:
C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c)
\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift
(S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead
(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1:
(sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2:
(land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat
-Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs
+Appl) vs t)))).(let H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs
u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c
(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads
(Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads
(Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat
-Appl) vs u))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat Appl) vs
+Appl) vs u))).(let H6 \def H2 in (land_ind (arity g c (THeads (Flat Appl) vs
t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads
(Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat
Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat
(Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1
n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads
(Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let
-H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0))
+H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0))
(sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs
(THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs
(THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u)
(ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def
-H2 in (and_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c
-(THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead
+H2 in (land_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3
+c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead
(Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead
(Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort
(S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g
(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall
(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3
-\def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g
+\def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g
a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is:
PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1
is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs
(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d
w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead
(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2
-in (and_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d:
+in (land_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d:
C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c)
\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs
t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t))
((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n
n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c:
C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c
-t))).(let H0 \def H in (and_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c
+t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c
t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2))
H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H:
(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0:
(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t:
T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall
(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1
-d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (and_ind
+d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (land_ind
(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0))))
(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads
(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to
t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0:
C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i))
\to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef
-i))))))))))).(let H5 \def H0 in (and_ind (\forall (c0: C).(\forall (t0:
+i))))))))))).(let H5 \def H0 in (land_ind (\forall (c0: C).(\forall (t0:
T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i:
nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to
((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs
T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs:
TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs
(TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0
-(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (and_ind (arity
-g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to
-(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
+(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (land_ind
+(arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w)
+\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
(lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0
a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to
(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
(THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0
(Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil
(drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst)
-x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (and_ind (sn3
+x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (land_ind (sn3
(CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S
x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef
O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O
\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w
(lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d:
C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is:
-PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (and_ind (\forall (c0:
-C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0:
+PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (land_ind (\forall
+(c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0:
TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl)
vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0
c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl)
((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0
(THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3
c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9
-\def H0 in (and_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to
+\def H0 in (land_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to
(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0:
C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef
i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef
\def
\lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H:
(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def
-H_x in (and_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3
+H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3
c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(let t0 \def (THeads (Flat
Appl) vs (TLRef i)) in (\forall (c0: C).((arity g c0 t0 a) \to ((nf2 c0
(TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a c0 t0)))))))) (sn3 c t) (\lambda
\lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda
(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i))
a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def
-(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (and_ind (\forall (c0:
+(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0:
C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0:
TList).(\forall (i0: nat).(let t \def (THeads (Flat Appl) vs0 (TLRef i0)) in
(\forall (c0: C).((arity g c0 t a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0
T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl)
(lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat
Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0
-in (and_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O
+in (land_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O
vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S
O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t))
(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda
(lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a
d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g
a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs)
-t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (and_ind (arity
-g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a
-a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
+t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (land_ind
+(arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)
+(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall
+(is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat
+Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land
+(arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0))
+(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
+PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
+(Flat Appl) vs (THead (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c
+(Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda
+(H6: ((\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is:
PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl)
-w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land (arity g
-c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d:
+w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity
+g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d:
C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c)
\to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead
-(Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c (Bind b) v) (THeads
-(Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda (H6: ((\forall (d:
-C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d
-(CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
-(Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity g c (THeads (Flat
-Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: C).(\forall (w:
-T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d
-(THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Bind b) v
-t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H3) t vs
-(AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 g a d
-w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def (H1
-(TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs)
-(lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat
-Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is) t))
-(\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1
-is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList (lifts1 (Ss
-is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d (Bind b)
-(lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat Appl) t0
-(lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl) (lifts (S
-O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is v))
-(THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is v))
-(lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S O) O
-(drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is)
+(Bind b) v t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1
+H3) t vs (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3
+g a d w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def
+(H1 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is
+vs) (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead
+(Flat Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is)
+t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl)
+(lifts1 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList
+(lifts1 (Ss is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d
+(Bind b) (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat
+Appl) t0 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl)
+(lifts (S O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is
+v)) (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is
+v)) (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S
+O) O (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is)
(drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts
(S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O
vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is
T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs
(THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead
(Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda
-(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (and_ind (arity g c (THeads
+(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (land_ind (arity g c (THeads
(Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat
Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs
(THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads
PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads
(Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c
v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1
-in (and_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead
-a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is:
-PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
+in (land_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))
+(AHead a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall
+(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is
(THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c
(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead
a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is:
T).(pr3 c x t0)) H7 x H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0:
T).((eq T x t0) \to (\forall (P0: Prop).P0))) H6 x H9) in (H11 (refl_equal T
x) P)))))) (pr3_head_12 c x t2 H7 (Bind b) x0 x0 (pr3_refl (CHead c (Bind b)
-t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (and_ind (sn3 c t2)
+t2) x0)) t2 x0 (refl_equal T (THead (Bind b) t2 x0))) in (land_ind (sn3 c t2)
(sn3 (CHead c (Bind b) t2) x0) (sn3 c t2) (\lambda (H9: (sn3 c t2)).(\lambda
(_: (sn3 (CHead c (Bind b) t2) x0)).H9)) H8)))))) (sn3_sing (CHead c (Bind b)
x) x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P:
T).(pr3 (CHead c (Bind b) x) x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T
t2 (\lambda (t0: T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in
(H11 (refl_equal T x0) P)))))) (pr3_head_12 c x x (pr3_refl c x) (Bind b) x0
-t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (and_ind (sn3 c x) (sn3
+t2 H7) x t2 (refl_equal T (THead (Bind b) x t2))) in (land_ind (sn3 c x) (sn3
(CHead c (Bind b) x) t2) (sn3 (CHead c (Bind b) x) t2) (\lambda (_: (sn3 c
x)).(\lambda (H10: (sn3 (CHead c (Bind b) x) t2)).H10)) H8))))))))))))))) y
H0))))) H))))).
\def (eq_ind_r T t2 (\lambda (t0: T).((eq T x t0) \to (\forall (P0:
Prop).P0))) H6 x H9) in (H11 (refl_equal T x) P)))))) (pr3_head_12 c x t2 H7
(Flat f) x0 x0 (pr3_refl (CHead c (Flat f) t2) x0)) t2 x0 (refl_equal T
-(THead (Flat f) t2 x0))) in (and_ind (sn3 c t2) (sn3 c x0) (sn3 c t2)
+(THead (Flat f) t2 x0))) in (land_ind (sn3 c t2) (sn3 c x0) (sn3 c t2)
(\lambda (H9: (sn3 c t2)).(\lambda (_: (sn3 c x0)).H9)) H8)))))) (sn3_sing c
x0 (\lambda (t2: T).(\lambda (H6: (((eq T x0 t2) \to (\forall (P:
Prop).P)))).(\lambda (H7: (pr3 c x0 t2)).(let H8 \def (H4 (THead (Flat f) x
T).(pr3 c x0 t0)) H7 x0 H9) in (let H11 \def (eq_ind_r T t2 (\lambda (t0:
T).((eq T x0 t0) \to (\forall (P0: Prop).P0))) H6 x0 H9) in (H11 (refl_equal
T x0) P)))))) (pr3_thin_dx c x0 t2 H7 x f) x t2 (refl_equal T (THead (Flat f)
-x t2))) in (and_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c
+x t2))) in (land_ind (sn3 c x) (sn3 c t2) (sn3 c t2) (\lambda (_: (sn3 c
x)).(\lambda (H10: (sn3 c t2)).H10)) H8))))))))))))))) y H0))))) H))))).
theorem sn3_gen_head:
T).(\forall (t: T).((sn3 c (THead k0 u t)) \to (sn3 c u)))))) (\lambda (b:
B).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
(Bind b) u t))).(let H_x \def (sn3_gen_bind b c u t H) in (let H0 \def H_x in
-(and_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3
+(land_ind (sn3 c u) (sn3 (CHead c (Bind b) u) t) (sn3 c u) (\lambda (H1: (sn3
c u)).(\lambda (_: (sn3 (CHead c (Bind b) u) t)).H1)) H0)))))))) (\lambda (f:
F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (sn3 c (THead
(Flat f) u t))).(let H_x \def (sn3_gen_flat f c u t H) in (let H0 \def H_x in
-(and_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_:
+(land_ind (sn3 c u) (sn3 c t) (sn3 c u) (\lambda (H1: (sn3 c u)).(\lambda (_:
(sn3 c t)).H1)) H0)))))))) k).
theorem sn3_gen_cflat:
\def
\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
(sn3 c (THead (Bind b) v t))).(let H_x \def (sn3_gen_bind b c v t H) in (let
-H0 \def H_x in (and_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c
+H0 \def H_x in (land_ind (sn3 c v) (sn3 (CHead c (Bind b) v) t) (sn3 (CHead c
(Bind b) v) t) (\lambda (_: (sn3 c v)).(\lambda (H2: (sn3 (CHead c (Bind b)
v) t)).H2)) H0))))))).
(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0:
C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to
(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def
-(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3
+(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3
(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11:
(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b
(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4)
t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0)
c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda
(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8)
-in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda (_:
-(sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3
+in (let H9 \def H_x0 in (land_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda
+(_: (sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3
H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
\lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
(sn3 c (THead (Flat Appl) u (THead (Bind Abbr) v t)))).(\lambda (w:
T).(\lambda (H0: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THead (Bind
-Abbr) v t) H) in (let H1 \def H_x in (and_ind (sn3 c u) (sn3 c (THead (Bind
+Abbr) v t) H) in (let H1 \def H_x in (land_ind (sn3 c u) (sn3 c (THead (Bind
Abbr) v t)) (sn3 c (THead (Flat Appl) u (THead (Flat Appl) v (THead (Bind
Abst) w t)))) (\lambda (H2: (sn3 c u)).(\lambda (H3: (sn3 c (THead (Bind
Abbr) v t))).(sn3_appl_appl v (THead (Bind Abst) w t) c (sn3_beta c v t H3 w
(sns3 c t1)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t1 (TLRef
i))))))) (\lambda (_: (((sns3 c TNil) \to (sn3 c (THeads (Flat Appl) TNil
(TLRef i)))))).(\lambda (H1: (land (sn3 c t) (sns3 c TNil))).(let H2 \def H1
-in (and_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl)
+in (land_ind (sn3 c t) True (sn3 c (THead (Flat Appl) t (THeads (Flat Appl)
TNil (TLRef i)))) (\lambda (H3: (sn3 c t)).(\lambda (_: True).(sn3_appl_lref
c i H t H3))) H2)))) (\lambda (t1: T).(\lambda (t2: TList).(\lambda (_:
(((((sns3 c t2) \to (sn3 c (THeads (Flat Appl) t2 (TLRef i))))) \to ((land
(sn3 c t) (sns3 c t2)) \to (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) t2
(TLRef i)))))))).(\lambda (H1: (((sns3 c (TCons t1 t2)) \to (sn3 c (THeads
(Flat Appl) (TCons t1 t2) (TLRef i)))))).(\lambda (H2: (land (sn3 c t) (sns3
-c (TCons t1 t2)))).(let H3 \def H2 in (and_ind (sn3 c t) (land (sn3 c t1)
+c (TCons t1 t2)))).(let H3 \def H2 in (land_ind (sn3 c t) (land (sn3 c t1)
(sns3 c t2)) (sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2)
(TLRef i)))) (\lambda (H4: (sn3 c t)).(\lambda (H5: (land (sn3 c t1) (sns3 c
-t2))).(and_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads
+t2))).(land_ind (sn3 c t1) (sns3 c t2) (sn3 c (THead (Flat Appl) t (THeads
(Flat Appl) (TCons t1 t2) (TLRef i)))) (\lambda (H6: (sn3 c t1)).(\lambda
(H7: (sns3 c t2)).(sn3_appl_appls t1 (TLRef i) t2 c (H1 (conj (sn3 c t1)
(sns3 c t2) H6 H7)) t H4 (\lambda (u2: T).(\lambda (H8: (pr3 c (THeads (Flat
(Flat Appl) (TCons t1 t2) u)))).(\lambda (t3: T).(\lambda (H2: (sn3 c (THead
(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) t3)))).(let H_x \def
(sn3_gen_flat Appl c t (THeads (Flat Appl) (TCons t1 t2) t3) H2) in (let H3
-\def H_x in (and_ind (sn3 c t) (sn3 c (THeads (Flat Appl) (TCons t1 t2) t3))
+\def H_x in (land_ind (sn3 c t) (sn3 c (THeads (Flat Appl) (TCons t1 t2) t3))
(sn3 c (THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) (THead (Flat
Cast) u t3)))) (\lambda (_: (sn3 c t)).(\lambda (H5: (sn3 c (THeads (Flat
Appl) (TCons t1 t2) t3))).(let H6 \def H5 in (let H_x0 \def (sn3_gen_flat
Appl c t (THeads (Flat Appl) (TCons t1 t2) u) H1) in (let H7 \def H_x0 in
-(and_ind (sn3 c t) (sn3 c (THeads (Flat Appl) (TCons t1 t2) u)) (sn3 c (THead
-(Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3))))
-(\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c (THeads (Flat Appl) (TCons t1
-t2) u))).(let H10 \def H9 in (sn3_appl_appls t1 (THead (Flat Cast) u t3) t2 c
-(H0 u H10 t3 H6) t H8 (\lambda (u2: T).(\lambda (H11: (pr3 c (THeads (Flat
-Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2)).(\lambda (H12: (((iso
-(THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2) \to (\forall
-(P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads (Flat Appl)
-(TCons t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c (THeads (Flat
-Appl) (TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1 t2) u2 H11
-H12) t Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
+(land_ind (sn3 c t) (sn3 c (THeads (Flat Appl) (TCons t1 t2) u)) (sn3 c
+(THead (Flat Appl) t (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u
+t3)))) (\lambda (H8: (sn3 c t)).(\lambda (H9: (sn3 c (THeads (Flat Appl)
+(TCons t1 t2) u))).(let H10 \def H9 in (sn3_appl_appls t1 (THead (Flat Cast)
+u t3) t2 c (H0 u H10 t3 H6) t H8 (\lambda (u2: T).(\lambda (H11: (pr3 c
+(THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2)).(\lambda
+(H12: (((iso (THeads (Flat Appl) (TCons t1 t2) (THead (Flat Cast) u t3)) u2)
+\to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t (THeads
+(Flat Appl) (TCons t1 t2) t3)) H2 (THead (Flat Appl) t u2) (pr3_thin_dx c
+(THeads (Flat Appl) (TCons t1 t2) t3) u2 (pr3_iso_appls_cast c u t3 (TCons t1
+t2) u2 H11 H12) t Appl))))))))) H7)))))) H3))))))))))) t0))) vs)).
theorem sn3_appls_bind:
\forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u:
T).(\lambda (H3: (sn3 (CHead c (Bind b) u) (THead (Flat Appl) (lift (S O) O
v) (THeads (Flat Appl) (lifts (S O) O (TCons t t0)) t1)))).(let H_x \def
(sn3_gen_flat Appl (CHead c (Bind b) u) (lift (S O) O v) (THeads (Flat Appl)
-(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (and_ind (sn3
+(lifts (S O) O (TCons t t0)) t1) H3) in (let H4 \def H_x in (land_ind (sn3
(CHead c (Bind b) u) (lift (S O) O v)) (sn3 (CHead c (Bind b) u) (THeads
(Flat Appl) (lifts (S O) O (TCons t t0)) t1)) (sn3 c (THead (Flat Appl) v
(THeads (Flat Appl) (TCons t t0) (THead (Bind b) u t1)))) (\lambda (H5: (sn3
(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t))))).(\lambda (w:
T).(\lambda (H2: (sn3 c w)).(let H_x \def (sn3_gen_flat Appl c u (THeads
(Flat Appl) (TCons t0 t1) (THead (Bind Abbr) v t)) H1) in (let H3 \def H_x in
-(and_ind (sn3 c u) (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind
+(land_ind (sn3 c u) (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind
Abbr) v t))) (sn3 c (THead (Flat Appl) u (THeads (Flat Appl) (TCons t0 t1)
(THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H4: (sn3 c
u)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t0 t1) (THead (Bind Abbr)
O w))) \to (sn3 c (THeads (Flat Appl) (TCons t t0) (TLRef i)))))).(\lambda
(H2: (sn3 c (THead (Flat Appl) v (THeads (Flat Appl) (TCons t t0) (lift (S i)
O w))))).(let H_x \def (sn3_gen_flat Appl c v (THeads (Flat Appl) (TCons t
-t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (and_ind (sn3 c v) (sn3 c
+t0) (lift (S i) O w)) H2) in (let H3 \def H_x in (land_ind (sn3 c v) (sn3 c
(THeads (Flat Appl) (TCons t t0) (lift (S i) O w))) (sn3 c (THead (Flat Appl)
v (THeads (Flat Appl) (TCons t t0) (TLRef i)))) (\lambda (H4: (sn3 c
v)).(\lambda (H5: (sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) O
TList).((sns3 d t) \to (sns3 c (lifts h i t)))) (\lambda (H0: True).H0)
(\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: (((sns3 d t0) \to (sns3 c
(lifts h i t0))))).(\lambda (H1: (land (sn3 d t) (sns3 d t0))).(let H2 \def
-H1 in (and_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c
+H1 in (land_ind (sn3 d t) (sns3 d t0) (land (sn3 c (lift h i t)) (sns3 c
(lifts h i t0))) (\lambda (H3: (sn3 d t)).(\lambda (H4: (sns3 d t0)).(conj
(sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0
H4)))) H2)))))) ts)))))).
\def
\lambda (c: C).(\lambda (t: T).(\lambda (ts: TList).(TList_ind (\lambda (t0:
TList).((nfs2 c (TApp t0 t)) \to (land (nfs2 c t0) (nf2 c t)))) (\lambda (H:
-(land (nf2 c t) True)).(let H0 \def H in (and_ind (nf2 c t) True (land True
+(land (nf2 c t) True)).(let H0 \def H in (land_ind (nf2 c t) True (land True
(nf2 c t)) (\lambda (H1: (nf2 c t)).(\lambda (_: True).(conj True (nf2 c t) I
H1))) H0))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (((nfs2 c
(TApp t1 t)) \to (land (nfs2 c t1) (nf2 c t))))).(\lambda (H0: (land (nf2 c
-t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (and_ind (nf2 c t0) (nfs2 c
+t0) (nfs2 c (TApp t1 t)))).(let H1 \def H0 in (land_ind (nf2 c t0) (nfs2 c
(TApp t1 t)) (land (land (nf2 c t0) (nfs2 c t1)) (nf2 c t)) (\lambda (H2:
(nf2 c t0)).(\lambda (H3: (nfs2 c (TApp t1 t))).(let H_x \def (H H3) in (let
-H4 \def H_x in (and_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c
+H4 \def H_x in (land_ind (nfs2 c t1) (nf2 c t) (land (land (nf2 c t0) (nfs2 c
t1)) (nf2 c t)) (\lambda (H5: (nfs2 c t1)).(\lambda (H6: (nf2 c t)).(conj
(land (nf2 c t0) (nfs2 c t1)) (nf2 c t) (conj (nf2 c t0) (nfs2 c t1) H2 H5)
H6))) H4))))) H1)))))) ts))).
Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat
Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t
x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def
-(nf2_gen_abst c x0 x1 H7) in (and_ind (nf2 c x0) (nf2 (CHead c (Bind Abst)
+(nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst)
x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3
c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1)))
(\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0)
i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0
in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (plus O (S i)) j) (eq
T x (TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x
-(TLRef j)))).(and_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt
+(TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt
j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda
-(H5: (land (le (plus O (S i)) j) (eq T x (TLRef (minus j (S i)))))).(and_ind
+(H5: (land (le (plus O (S i)) j) (eq T x (TLRef (minus j (S i)))))).(land_ind
(le (plus O (S i)) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6:
(le (plus O (S i)) j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6))
H5)) H4))))) H2))))))))).
T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef
n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h
(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H
-(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (and_ind (eq nat n i) (eq
+(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (land_ind (eq nat n i) (eq
T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S
(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2:
(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T
(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d))
n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t:
T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S
-(plus i d)) H0)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n
+(plus i d)) H0)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n
h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d))
t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat
(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d
nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1:
(subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P
(\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda
-(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (and_ind
+(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (land_ind
(eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda
(_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0:
nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n
H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n))
(\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d
-H2)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P
+H2)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P
(\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n
h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d
h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h
(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef
n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n))
(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in
-(and_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq T
-x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2)))
+(land_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq
+T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2)))
(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5
\def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus
d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5)))))
(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind
T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h))
-(lift_lref_ge n h d H1)) in (and_ind (eq nat (plus n h) i) (eq T x (lift (S
+(lift_lref_ge n h d H1)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S
(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda
(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n
h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n
h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O
u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0:
nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift
-(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_comm n
+(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n
h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d
-(plus O (S n)) (plus_le_compat O O d (S n) (le_n O) (le_S d n H1))) (le_O_n
+(plus O (S n)) (le_plus_plus O O d (S n) (le_n O) (le_S d n H1))) (le_O_n
d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i
H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k:
K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall
nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TSort n) (TSort
n))).(\lambda (P: Prop).(subst0_gen_sort u (TSort n) d n H P))))) (\lambda
(n: nat).(\lambda (d: nat).(\lambda (H: (subst0 d u (TLRef n) (TLRef
-n))).(\lambda (P: Prop).(and_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O
+n))).(\lambda (P: Prop).(land_ind (eq nat n d) (eq T (TLRef n) (lift (S n) O
u)) P (\lambda (_: (eq nat n d)).(\lambda (H1: (eq T (TLRef n) (lift (S n) O
u))).(lift_gen_lref_false (S n) O n (le_O_n n) (le_n (plus O (S n))) u H1
P))) (subst0_gen_lref u (TLRef n) d n H)))))) (\lambda (k: K).(\lambda (t0:
(plus i0 h) v (TLRef (plus i0 h)) t)) (eq_ind nat (S (plus h i0)) (\lambda
(n: nat).(subst0 (plus i0 h) v (TLRef (plus i0 h)) (lift n O v))) (eq_ind_r
nat (plus h i0) (\lambda (n: nat).(subst0 n v (TLRef n) (lift (S (plus h i0))
-O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_comm i0 h)) (plus h (S
+O v))) (subst0_lref v (plus h i0)) (plus i0 h) (plus_sym i0 h)) (plus h (S
i0)) (plus_n_Sm h i0)) (lift h d (lift (S i0) O v)) (lift_free v (S i0) h O d
(le_S d i0 H0) (le_O_n d))) (lift h d (TLRef i0)) (lift_lref_ge i0 h d
H0)))))) (\lambda (v: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
(plus i (S O)) (\lambda (n: nat).(subst0 n u (lift (S O) d t1) (lift (S O) d
t2))) (subst0_lift_ge t1 t2 u i (S O) H d H0) (S i) (eq_ind_r nat (plus (S O)
i) (\lambda (n: nat).(eq nat n (S i))) (refl_equal nat (S i)) (plus i (S O))
-(plus_comm i (S O)))))))))).
+(plus_sym i (S O)))))))))).
theorem subst0_lift_ge_s:
\forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst0
nat n i2)) \to (ex2 T (\lambda (t5: T).(subst0 i2 u2 t3 t5)) (\lambda (t5:
T).(subst0 n t t4 t5)))))))))))) (\lambda (v: T).(\lambda (i: nat).(\lambda
(t2: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H0: (subst0 i2 u2
-(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(and_ind (eq nat i i2) (eq
+(TLRef i) t2)).(\lambda (H1: (not (eq nat i i2))).(land_ind (eq nat i i2) (eq
T t2 (lift (S i) O u2)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (lift (S i) O v)
t)) (\lambda (t: T).(subst0 i v t2 t))) (\lambda (H2: (eq nat i i2)).(\lambda
(H3: (eq T t2 (lift (S i) O u2))).(let H4 \def (eq_ind nat i (\lambda (n:
(subst0 (s k i2) u2 t2 x0)).(\lambda (H8: (subst0 (s k i) v x x0)).(ex_intro2
T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i
v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u)
-(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (\lambda (H7: (eq nat
-(s k i) (s k i2))).(H3 (s_inj k i i2 H7))))) t4 H5)))) H4)) (\lambda (H4:
-(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5))))
-(\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_:
+(subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (ex2_ind T (\lambda (t:
+T).(subst0 (s k i2) u2 t2 t)) (\lambda (t: T).(subst0 (s k i) v x t)) ((eq
+nat (s k i) (s k i2)) \to False) (\lambda (x0: T).(\lambda (_: (subst0 (s k
+i2) u2 t2 x0)).(\lambda (_: (subst0 (s k i) v x x0)).(\lambda (H9: (eq nat (s
+k i) (s k i2))).(H3 (s_inj k i i2 H9)))))) (H1 x u2 (s k i2) H6 (\lambda (H7:
+(eq nat (s k i) (s k i2))).(H3 (s_inj k i i2 H7))))))) t4 H5)))) H4))
+(\lambda (H4: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k
+u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_:
T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))))).(ex3_2_ind T T (\lambda
(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3:
T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5:
(H9: (subst0 (s k i) v x1 x)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u2
(THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k
x0 x) (subst0_both u2 u x0 i2 H6 k t2 x H8) (subst0_snd k v x x1 i H9 x0)))))
-(H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k i2))).(H3 (s_inj k i
-i2 H8))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3 t4 i2
-H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda
-(i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1: ((\forall (t2:
-T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) \to ((not (eq
-nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t:
-T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2: T).(\lambda (t3:
-T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3: ((\forall (t4:
-T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4) \to ((not (eq
-nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3 t)) (\lambda (t:
-T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4: T).(\lambda (u3:
-T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k u0 t2)
-t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq
-T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T
+(H1 x1 u2 (s k i2) H7 (ex2_ind T (\lambda (t: T).(subst0 (s k i2) u2 t2 t))
+(\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq nat (s k i) (s k i2)) \to
+False) (\lambda (x: T).(\lambda (_: (subst0 (s k i2) u2 t2 x)).(\lambda (_:
+(subst0 (s k i) v x1 x)).(\lambda (H10: (eq nat (s k i) (s k i2))).(H3 (s_inj
+k i i2 H10)))))) (H1 x1 u2 (s k i2) H7 (\lambda (H8: (eq nat (s k i) (s k
+i2))).(H3 (s_inj k i i2 H8))))))) t4 H5)))))) H4)) (subst0_gen_head k u2 u t3
+t4 i2 H2))))))))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u2:
+T).(\lambda (i: nat).(\lambda (H0: (subst0 i v u0 u2)).(\lambda (H1:
+((\forall (t2: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2)
+\to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t))
+(\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2:
+T).(\lambda (t3: T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3:
+((\forall (t4: T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4)
+\to ((not (eq nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3
+t)) (\lambda (t: T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4:
+T).(\lambda (u3: T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k
+u0 t2) t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4:
+T).(eq T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4))) (ex2 T
(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2)
u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4
t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_:
x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda
(t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3
i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8
-(\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H9))))) t4 H7))))
-H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4
+(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0
+(s k i) v x t)) ((eq nat (s k i) (s k i2)) \to False) (\lambda (x0:
+T).(\lambda (_: (subst0 (s k i2) u3 t3 x0)).(\lambda (_: (subst0 (s k i) v x
+x0)).(\lambda (H11: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H11))))))
+(H3 x u3 (s k i2) H8 (\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i
+i2 H9))))))) t4 H7)))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4:
+T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: T).(\lambda (_:
+T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2)
+u3 t2 t5))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4
(THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4)))
-(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))))).(ex3_2_ind T
-T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4:
-T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5:
-T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k
-u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H7: (eq T t4 (THead k x0 x1))).(\lambda (H8: (subst0 i2 u3 u0
-x0)).(\lambda (H9: (subst0 (s k i2) u3 t2 x1)).(eq_ind_r T (THead k x0 x1)
-(\lambda (t: T).(ex2 T (\lambda (t5: T).(subst0 i2 u3 (THead k u2 t3) t5))
-(\lambda (t5: T).(subst0 i v t t5)))) (ex2_ind T (\lambda (t: T).(subst0 i2
-u3 u2 t)) (\lambda (t: T).(subst0 i v x0 t)) (ex2 T (\lambda (t: T).(subst0
-i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t)))
-(\lambda (x: T).(\lambda (H10: (subst0 i2 u3 u2 x)).(\lambda (H11: (subst0 i
-v x0 x)).(ex2_ind T (\lambda (t: T).(subst0 (s k i2) u3 t3 t)) (\lambda (t:
-T).(subst0 (s k i) v x1 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2
-t3) t)) (\lambda (t: T).(subst0 i v (THead k x0 x1) t))) (\lambda (x2:
-T).(\lambda (H12: (subst0 (s k i2) u3 t3 x2)).(\lambda (H13: (subst0 (s k i)
-v x1 x2)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t))
-(\lambda (t: T).(subst0 i v (THead k x0 x1) t)) (THead k x x2) (subst0_both
-u3 u2 x i2 H10 k t3 x2 H12) (subst0_both v x0 x i H11 k x1 x2 H13))))) (H3 x1
-u3 (s k i2) H9 (\lambda (H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2
-H12)))))))) (H1 x0 u3 i2 H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2
-t4 i2 H4)))))))))))))))))) i1 u1 t0 t1 H))))).
+(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda
+(t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t)))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t4 (THead k x0
+x1))).(\lambda (H8: (subst0 i2 u3 u0 x0)).(\lambda (H9: (subst0 (s k i2) u3
+t2 x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(ex2 T (\lambda (t5:
+T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5))))
+(ex2_ind T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v
+x0 t)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t:
+T).(subst0 i v (THead k x0 x1) t))) (\lambda (x: T).(\lambda (H10: (subst0 i2
+u3 u2 x)).(\lambda (H11: (subst0 i v x0 x)).(ex2_ind T (\lambda (t:
+T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) (ex2 T
+(\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v
+(THead k x0 x1) t))) (\lambda (x2: T).(\lambda (H12: (subst0 (s k i2) u3 t3
+x2)).(\lambda (H13: (subst0 (s k i) v x1 x2)).(ex_intro2 T (\lambda (t:
+T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x0
+x1) t)) (THead k x x2) (subst0_both u3 u2 x i2 H10 k t3 x2 H12) (subst0_both
+v x0 x i H11 k x1 x2 H13))))) (H3 x1 u3 (s k i2) H9 (ex2_ind T (\lambda (t:
+T).(subst0 (s k i2) u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v x1 t)) ((eq
+nat (s k i) (s k i2)) \to False) (\lambda (x2: T).(\lambda (_: (subst0 (s k
+i2) u3 t3 x2)).(\lambda (_: (subst0 (s k i) v x1 x2)).(\lambda (H14: (eq nat
+(s k i) (s k i2))).(H5 (s_inj k i i2 H14)))))) (H3 x1 u3 (s k i2) H9 (\lambda
+(H12: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H12)))))))))) (H1 x0 u3 i2
+H8 H5)) t4 H7)))))) H6)) (subst0_gen_head k u3 u0 t2 t4 i2
+H4)))))))))))))))))) i1 u1 t0 t1 H))))).
theorem subst0_confluence_eq:
\forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst0
(or4 (eq T t3 t4) (ex2 T (\lambda (t5: T).(subst0 n t t3 t5)) (\lambda (t5:
T).(subst0 n t t4 t5))) (subst0 n t t3 t4) (subst0 n t t4 t3)))))))) (\lambda
(v: T).(\lambda (i0: nat).(\lambda (t2: T).(\lambda (H0: (subst0 i0 v (TLRef
-i0) t2)).(and_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T (lift
-(S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t))
-(\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2)
+i0) t2)).(land_ind (eq nat i0 i0) (eq T t2 (lift (S i0) O v)) (or4 (eq T
+(lift (S i0) O v) t2) (ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v)
+t)) (\lambda (t: T).(subst0 i0 v t2 t))) (subst0 i0 v (lift (S i0) O v) t2)
(subst0 i0 v t2 (lift (S i0) O v))) (\lambda (_: (eq nat i0 i0)).(\lambda
(H2: (eq T t2 (lift (S i0) O v))).(or4_intro0 (eq T (lift (S i0) O v) t2)
(ex2 T (\lambda (t: T).(subst0 i0 v (lift (S i0) O v) t)) (\lambda (t:
t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1)
x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S
O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O))
-(plus_comm i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift
+(plus_sym i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift
(S O) i t2) (lift (S O) i t1))).(subst0_gen_lift_false t2 u (lift (S O) i t1)
(S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n))
-(le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O))) H1 (eq T t1 t2)))
+(le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2)))
(\lambda (H1: (subst0 i u (lift (S O) i t1) (lift (S O) i
t2))).(subst0_gen_lift_false t1 u (lift (S O) i t2) (S O) i i (le_n i)
(eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O)
-i)) (plus i (S O)) (plus_comm i (S O))) H1 (eq T t1 t2)))
+i)) (plus i (S O)) (plus_sym i (S O))) H1 (eq T t1 t2)))
(subst0_confluence_eq t0 (lift (S O) i t2) u i H0 (lift (S O) i t1) H)))))))).
m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S
(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
-(plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
g u1))) (\lambda (n: nat).(wadd_le f g H2 (S (weight_map f u2)) (S
nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
(weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
-t0)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
-O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O)
-(wadd g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda
-(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall
-(m: nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O
-v)) (g i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
-(weight_map g u1) (weight_map (wadd g O) t0)) (plus_le_compat (weight_map f
-u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0)
-(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le
-f g H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to
+t0)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
+t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd
+g O) (\lambda (n: nat).(wadd_le f g H2 O O (le_n O) n))))))))) (\lambda (f:
+((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
+nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
+(weight_map g u1) (weight_map (wadd g O) t0)) (le_plus_plus (weight_map f u2)
+(weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0) (H1 f
+g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g
+H2 O O (le_n O) n))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to
nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
i))).(le_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g
-u1) (weight_map g t0)) (plus_le_compat (weight_map f0 u2) (weight_map g u1)
+u1) (weight_map g t0)) (le_plus_plus (weight_map f0 u2) (weight_map g u1)
(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g
H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v:
T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1
nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
-t1)) (plus_le_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd f
-(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
+t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f (S
+(weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le
nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f
m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
i))).(le_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus
-(weight_map g u0) (weight_map (wadd g O) t1)) (plus_le_compat (weight_map f
-u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(weight_map g u0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u0)
+(weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f
g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda
(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v))
\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u0)
(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
-t1)) (plus_le_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd f
-O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd
-g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat
-(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3
-(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i)
-f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2:
-T).(\lambda (t1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v t1
-t2)).(\lambda (H1: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift
-(S i) O v)) (g i)) \to (le (weight_map f0 t2) (weight_map g
-t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat
-\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda
-(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map
-f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1))
-(plus_le_compat (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2)
-(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k))
-(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
-(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall
-(g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt
-(weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f u2) (weight_map
-g u1)))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t1:
-T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to (((\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
-\to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s k0 i))) \to (le
-(weight_map f t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to
-nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m) (g m))))
-\to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead
-k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b: B).(B_ind
-(\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s (Bind b0) i) v
-t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S
-(s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f t2)
-(weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g: ((nat
+t1)) (le_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f O)
+t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd g
+O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat (weight_map
+f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3 (weight_map (wadd f O)
+(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f)))))))))))))))) b))
+(\lambda (_: F).(\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda
+(i: nat).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H1: ((\forall (f0: ((nat
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g
+m)))) \to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (le (weight_map f0
+t2) (weight_map g t1)))))))).(\lambda (u0: T).(\lambda (f0: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
+m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
+i))).(le_n_S (plus (weight_map f0 u0) (weight_map f0 t2)) (plus (weight_map g
+u0) (weight_map g t1)) (le_plus_plus (weight_map f0 u0) (weight_map g u0)
+(weight_map f0 t2) (weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2
+H3))))))))))))))) k)) (\lambda (v: T).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall
+(f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f
+m) (g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le
+(weight_map f u2) (weight_map g u1)))))))).(\lambda (k: K).(K_ind (\lambda
+(k0: K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k0 i) v t1 t2) \to
+(((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m:
+nat).(le (f m) (g m)))) \to ((lt (weight_map f (lift (S (s k0 i)) O v)) (g (s
+k0 i))) \to (le (weight_map f t2) (weight_map g t1))))))) \to (\forall (f:
+((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
+(g m)))) \to ((lt (weight_map f (lift (S i) O v)) (g i)) \to (le (weight_map
+f (THead k0 u2 t2)) (weight_map g (THead k0 u1 t1)))))))))))) (\lambda (b:
+B).(B_ind (\lambda (b0: B).(\forall (t1: T).(\forall (t2: T).((subst0 (s
+(Bind b0) i) v t1 t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat
\to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map f
-(lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2))
+(lift (S (s (Bind b0) i)) O v)) (g (s (Bind b0) i))) \to (le (weight_map f
+t2) (weight_map g t1))))))) \to (\forall (f: ((nat \to nat))).(\forall (g:
+((nat \to nat))).(((\forall (m: nat).(le (f m) (g m)))) \to ((lt (weight_map
+f (lift (S i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t2))
(weight_map g (THead (Bind b0) u1 t1)))))))))))) (\lambda (t1: T).(\lambda
(t2: T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f:
((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f m)
m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f
u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1)))
-t1)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
-(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1
-f g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1)))
+t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1 f
+g H4 H5) (H3 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map g u1)))
(\lambda (m: nat).(wadd_le f g H4 (S (weight_map f u2)) (S (weight_map g u1))
(le_n_S (weight_map f u2) (weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat
(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5
\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
i))).(le_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
-(weight_map g u1) (weight_map (wadd g O) t1)) (plus_le_compat (weight_map f
-u2) (weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
-(H1 f g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O
-O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
+(weight_map g u1) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map f u2)
+(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f
+g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f g H4 O O
+(le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n:
nat).(lt n (g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v))
(lift_weight_add_O O v (S i) f))))))))))))) (\lambda (t1: T).(\lambda (t2:
T).(\lambda (_: (subst0 (S i) v t1 t2)).(\lambda (H3: ((\forall (f: ((nat \to
\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5:
(lt (weight_map f (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f u2)
(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
-t1)) (plus_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
-O) t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
+t1)) (le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
+t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
(\lambda (m: nat).(wadd_le f g H4 O O (le_n O) m)) (eq_ind nat (weight_map f
(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O)
(lift (S (S i)) O v)) (lift_weight_add_O O v (S i) f))))))))))))) b))
t1)))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to
nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda (H5:
(lt (weight_map f0 (lift (S i) O v)) (g i))).(le_n_S (plus (weight_map f0 u2)
-(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1))
-(plus_le_compat (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2)
-(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t
-z H))))).
+(weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1)) (le_plus_plus
+(weight_map f0 u2) (weight_map g u1) (weight_map f0 t2) (weight_map g t1) (H1
+f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t z H))))).
theorem subst0_weight_lt:
\forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S
(plus (weight_map f u2) (weight_map (wadd f (S (weight_map f u2))) t0)) (plus
(weight_map g u1) (weight_map (wadd g (S (weight_map g u1))) t0))
-(plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
+(lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f (S
(weight_map f u2))) t0) (weight_map (wadd g (S (weight_map g u1))) t0) (H1 f
g H2 H3) (weight_le t0 (wadd f (S (weight_map f u2))) (wadd g (S (weight_map
g u1))) (\lambda (n: nat).(wadd_lt f g H2 (S (weight_map f u2)) (S
nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3: (lt
(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
(weight_map (wadd f O) t0)) (plus (weight_map g u1) (weight_map (wadd g O)
-t0)) (plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd
-f O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O)
+t0)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+O) t0) (weight_map (wadd g O) t0) (H1 f g H2 H3) (weight_le t0 (wadd f O)
(wadd g O) (\lambda (n: nat).(le_S_n (wadd f O n) (wadd g O n) (le_n_S (wadd
f O n) (wadd g O n) (wadd_le f g H2 O O (le_n O) n))))))))))) (\lambda (f:
((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m:
nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t0)) (plus
-(weight_map g u1) (weight_map (wadd g O) t0)) (plus_lt_le_compat (weight_map
-f u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0)
+(weight_map g u1) (weight_map (wadd g O) t0)) (lt_le_plus_plus (weight_map f
+u2) (weight_map g u1) (weight_map (wadd f O) t0) (weight_map (wadd g O) t0)
(H1 f g H2 H3) (weight_le t0 (wadd f O) (wadd g O) (\lambda (n: nat).(le_S_n
(wadd f O n) (wadd g O n) (le_n_S (wadd f O n) (wadd g O n) (wadd_le f g H2 O
O (le_n O) n))))))))))) b)) (\lambda (_: F).(\lambda (f0: ((nat \to
nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0
m) (g m))))).(\lambda (H3: (lt (weight_map f0 (lift (S i) O v)) (g
i))).(lt_n_S (plus (weight_map f0 u2) (weight_map f0 t0)) (plus (weight_map g
-u1) (weight_map g t0)) (plus_lt_le_compat (weight_map f0 u2) (weight_map g
-u1) (weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g
+u1) (weight_map g t0)) (lt_le_plus_plus (weight_map f0 u2) (weight_map g u1)
+(weight_map f0 t0) (weight_map g t0) (H1 f0 g H2 H3) (weight_le t0 f0 g
H2)))))))) k))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (v:
T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k0 i) v t1
t2) \to (((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
nat).(le (f m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f (S (weight_map f
u0))) t2)) (plus (weight_map g u0) (weight_map (wadd g (S (weight_map g u0)))
-t1)) (plus_le_lt_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd
-f (S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
+t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f
+(S (weight_map f u0))) t2) (weight_map (wadd g (S (weight_map g u0))) t1)
(weight_le u0 f g H2) (H1 (wadd f (S (weight_map f u0))) (wadd g (S
(weight_map g u0))) (\lambda (m: nat).(wadd_le f g H2 (S (weight_map f u0))
(S (weight_map g u0)) (le_n_S (weight_map f u0) (weight_map g u0) (weight_le
nat))).(\lambda (g: ((nat \to nat))).(\lambda (H2: ((\forall (m: nat).(le (f
m) (g m))))).(\lambda (H3: (lt (weight_map f (lift (S i) O v)) (g
i))).(lt_n_S (plus (weight_map f u0) (weight_map (wadd f O) t2)) (plus
-(weight_map g u0) (weight_map (wadd g O) t1)) (plus_le_lt_compat (weight_map
-f u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
+(weight_map g u0) (weight_map (wadd g O) t1)) (le_lt_plus_plus (weight_map f
+u0) (weight_map g u0) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
(weight_le u0 f g H2) (H1 (wadd f O) (wadd g O) (\lambda (m: nat).(wadd_le f
g H2 O O (le_n O) m)) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda
(n: nat).(lt n (g i))) H3 (weight_map (wadd f O) (lift (S (S i)) O v))
\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H3:
(lt (weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u0)
(weight_map (wadd f O) t2)) (plus (weight_map g u0) (weight_map (wadd g O)
-t1)) (plus_le_lt_compat (weight_map f u0) (weight_map g u0) (weight_map (wadd
-f O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O)
-(wadd g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat
+t1)) (le_lt_plus_plus (weight_map f u0) (weight_map g u0) (weight_map (wadd f
+O) t2) (weight_map (wadd g O) t1) (weight_le u0 f g H2) (H1 (wadd f O) (wadd
+g O) (\lambda (m: nat).(wadd_le f g H2 O O (le_n O) m)) (eq_ind nat
(weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H3
(weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v (S i)
f)))))))))))))))) b)) (\lambda (_: F).(\lambda (v: T).(\lambda (t2:
\to nat))).(\lambda (H2: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda
(H3: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map
f0 u0) (weight_map f0 t2)) (plus (weight_map g u0) (weight_map g t1))
-(plus_le_lt_compat (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2)
+(le_lt_plus_plus (weight_map f0 u0) (weight_map g u0) (weight_map f0 t2)
(weight_map g t1) (weight_le u0 f0 g H2) (H1 f0 g H2 H3))))))))))))))) k))
(\lambda (v: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda
(_: (subst0 i v u1 u2)).(\lambda (H1: ((\forall (f: ((nat \to nat))).(\forall
m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f (S (weight_map f
u2))) t2)) (plus (weight_map g u1) (weight_map (wadd g (S (weight_map g u1)))
-t1)) (plus_lt_le_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd
-f (S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1)
-(H1 f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f
-u2))) (wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S
+t1)) (lt_le_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f
+(S (weight_map f u2))) t2) (weight_map (wadd g (S (weight_map g u1))) t1) (H1
+f g H4 H5) (subst0_weight_le v t1 t2 (S i) H2 (wadd f (S (weight_map f u2)))
+(wadd g (S (weight_map g u1))) (\lambda (m: nat).(wadd_lt f g H4 (S
(weight_map f u2)) (S (weight_map g u1)) (lt_n_S (weight_map f u2)
(weight_map g u1) (H1 f g H4 H5)) m)) (eq_ind nat (weight_map f (lift (S i) O
v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f (S (weight_map f
nat))).(\lambda (H4: ((\forall (m: nat).(le (f m) (g m))))).(\lambda (H5: (lt
(weight_map f (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map f u2)
(weight_map (wadd f O) t2)) (plus (weight_map g u1) (weight_map (wadd g O)
-t1)) (plus_lt_compat (weight_map f u2) (weight_map g u1) (weight_map (wadd f
-O) t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
+t1)) (lt_plus_plus (weight_map f u2) (weight_map g u1) (weight_map (wadd f O)
+t2) (weight_map (wadd g O) t1) (H1 f g H4 H5) (H3 (wadd f O) (wadd g O)
(\lambda (m: nat).(le_S_n (wadd f O m) (wadd g O m) (le_n_S (wadd f O m)
(wadd g O m) (wadd_le f g H4 O O (le_n O) m)))) (eq_ind nat (weight_map f
(lift (S i) O v)) (\lambda (n: nat).(lt n (g i))) H5 (weight_map (wadd f O)
\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H4: ((\forall (m: nat).(le
(f m) (g m))))).(\lambda (H5: (lt (weight_map f (lift (S i) O v)) (g
i))).(lt_n_S (plus (weight_map f u2) (weight_map (wadd f O) t2)) (plus
-(weight_map g u1) (weight_map (wadd g O) t1)) (plus_lt_compat (weight_map f
-u2) (weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1)
-(H1 f g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O
-m) (wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n
-O) m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n
-(g i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O
-v (S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2:
+(weight_map g u1) (weight_map (wadd g O) t1)) (lt_plus_plus (weight_map f u2)
+(weight_map g u1) (weight_map (wadd f O) t2) (weight_map (wadd g O) t1) (H1 f
+g H4 H5) (H3 (wadd f O) (wadd g O) (\lambda (m: nat).(le_S_n (wadd f O m)
+(wadd g O m) (le_n_S (wadd f O m) (wadd g O m) (wadd_le f g H4 O O (le_n O)
+m)))) (eq_ind nat (weight_map f (lift (S i) O v)) (\lambda (n: nat).(lt n (g
+i))) H5 (weight_map (wadd f O) (lift (S (S i)) O v)) (lift_weight_add_O O v
+(S i) f))))))))))))) b)) (\lambda (_: F).(\lambda (t1: T).(\lambda (t2:
T).(\lambda (_: (subst0 i v t1 t2)).(\lambda (H3: ((\forall (f0: ((nat \to
nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m))))
\to ((lt (weight_map f0 (lift (S i) O v)) (g i)) \to (lt (weight_map f0 t2)
\to nat))).(\lambda (H4: ((\forall (m: nat).(le (f0 m) (g m))))).(\lambda
(H5: (lt (weight_map f0 (lift (S i) O v)) (g i))).(lt_n_S (plus (weight_map
f0 u2) (weight_map f0 t2)) (plus (weight_map g u1) (weight_map g t1))
-(plus_lt_compat (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2)
+(lt_plus_plus (weight_map f0 u2) (weight_map g u1) (weight_map f0 t2)
(weight_map g t1) (H1 f0 g H4 H5) (H3 f0 g H4 H5)))))))))))) k)))))))) d u t
z H))))).
z)).(lt_n_S (plus (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z)) (plus
(weight_map (\lambda (_: nat).O) u) (weight_map (wadd (\lambda (_: nat).O) (S
-(weight_map (\lambda (_: nat).O) u))) t)) (plus_le_lt_compat (weight_map
+(weight_map (\lambda (_: nat).O) u))) t)) (le_lt_plus_plus (weight_map
(\lambda (_: nat).O) u) (weight_map (\lambda (_: nat).O) u) (weight_map (wadd
(\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) z) (weight_map
(wadd (\lambda (_: nat).O) (S (weight_map (\lambda (_: nat).O) u))) t) (le_n
(eq T t (TLRef n)) (land (eq nat n i) (eq T t (lift (S n) O v))))) (or_introl
(eq T (TLRef n) (TLRef n)) (land (eq nat n i) (eq T (TLRef n) (lift (S n) O
v))) (refl_equal T (TLRef n))) (\lambda (t2: T).(\lambda (H0: (subst0 i v
-(TLRef n) t2)).(and_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2
+(TLRef n) t2)).(land_ind (eq nat n i) (eq T t2 (lift (S n) O v)) (or (eq T t2
(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v)))) (\lambda (H1: (eq
nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O v))).(or_intror (eq T t2
(TLRef n)) (land (eq nat n i) (eq T t2 (lift (S n) O v))) (conj (eq nat n i)
(TLRef h) (TLRef n) (lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n))
(TLRef (plus h (S n))) (lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n))
(sym_eq nat (S (plus h n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h)
-(plus_comm n h)) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n))
+(plus_sym n h)) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n))
(lift_lref_ge n (S h) n (le_n n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt
n (S h) (S n) (le_n (S n)))) i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T
(TLRef (plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i
(t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4
\def (sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u
(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda
-(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O)))
+(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_sym i (S O)))
H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1:
(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3:
T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2
T).(subst0 i u t (lift (S O) i t1))) H4 (lift (S O) i t3) H6) in
(subst0_gen_lift_false t3 u (lift (S O) i t1) (S O) i i (le_n i) (eq_ind_r
nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i
-(S O)) (plus_comm i (S O))) H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6:
+(S O)) (plus_sym i (S O))) H7 (eq T t1 t3)))) (\lambda (t4: T).(\lambda (H6:
(subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let H8 \def
(eq_ind T t4 (\lambda (t: T).(subst0 i u t0 t)) H6 (lift (S O) i t3) H7) in
(sym_eq T t3 t1 (subst0_confluence_lift t0 t3 u i H8 t1 H4)))))) y0 H5)))
(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(tau0 g
c0 (TLRef (plus i h)) (lift n O w))) (tau0_abbr g c0 d v (plus i h)
(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abbr) v) H0 H4) w H1) (plus
-h (S i)) (plus_comm h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i)
+h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O w)) (lift_free w (S i)
h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
-i (S O)) (plus_comm i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
+i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
H4)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda
(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) v))).(\lambda (w:
T).(\lambda (H1: (tau0 g d v w)).(\lambda (H2: ((\forall (c0: C).(\forall (h:
(TLRef (plus i h)) t)) (eq_ind_r nat (plus (S i) h) (\lambda (n: nat).(tau0 g
c0 (TLRef (plus i h)) (lift n O v))) (tau0_abst g c0 d v (plus i h)
(drop_getl_trans_ge i c0 c d0 h H3 (CHead d (Bind Abst) v) H0 H4) w H1) (plus
-h (S i)) (plus_comm h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i)
+h (S i)) (plus_sym h (S i))) (lift h d0 (lift (S i) O v)) (lift_free v (S i)
h O d0 (le_S d0 i H4) (le_O_n d0))) (plus i (S O)) (eq_ind_r nat (plus (S O)
i) (\lambda (n: nat).(eq nat (S i) n)) (refl_equal nat (plus (S O) i)) (plus
-i (S O)) (plus_comm i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
+i (S O)) (plus_sym i (S O)))) (lift h d0 (TLRef i)) (lift_lref_ge i h d0
H4)))))))))))))))) (\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda
(t3: T).(\lambda (t4: T).(\lambda (_: (tau0 g (CHead c (Bind b) v) t3
t4)).(\lambda (H1: ((\forall (c0: C).(\forall (h: nat).(\forall (d:
nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus
(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)) (plus
(weight_map g t0) (weight_map (wadd g (S (weight_map g t0))) t1))
-(plus_le_compat (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S
+(le_plus_plus (weight_map f t0) (weight_map g t0) (weight_map (wadd f (S
(weight_map f t0))) t1) (weight_map (wadd g (S (weight_map g t0))) t1) (H f g
H1) (H0 (wadd f (S (weight_map f t0))) (wadd g (S (weight_map g t0)))
(\lambda (n: nat).(wadd_le f g H1 (S (weight_map f t0)) (S (weight_map g t0))
(g n)))) \to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat
\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le
(f n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1))
-(plus (weight_map g t0) (weight_map (wadd g O) t1)) (plus_le_compat
-(weight_map f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map
-(wadd g O) t1) (H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n:
-nat).(wadd_le f g H1 O O (le_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H:
-((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
-nat).(le (f n) (g n)))) \to (le (weight_map f t0) (weight_map g
-t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: ((nat \to
+(plus (weight_map g t0) (weight_map (wadd g O) t1)) (le_plus_plus (weight_map
+f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map (wadd g O) t1)
+(H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O
+(le_n O) n)))))))))))) (\lambda (t0: T).(\lambda (H: ((\forall (f: ((nat \to
nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f n) (g n))))
-\to (le (weight_map f t1) (weight_map g t1))))))).(\lambda (f: ((nat \to
-nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le (f
-n) (g n))))).(le_n_S (plus (weight_map f t0) (weight_map (wadd f O) t1))
-(plus (weight_map g t0) (weight_map (wadd g O) t1)) (plus_le_compat
-(weight_map f t0) (weight_map g t0) (weight_map (wadd f O) t1) (weight_map
-(wadd g O) t1) (H f g H1) (H0 (wadd f O) (wadd g O) (\lambda (n:
-nat).(wadd_le f g H1 O O (le_n O) n)))))))))))) b)) (\lambda (_: F).(\lambda
-(t0: T).(\lambda (H: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to
-nat))).(((\forall (n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t0)
-(weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f0: ((nat
-\to nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g
-n)))) \to (le (weight_map f0 t1) (weight_map g t1))))))).(\lambda (f0: ((nat
-\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (n: nat).(le
-(f0 n) (g n))))).(le_n_S (plus (weight_map f0 t0) (weight_map f0 t1)) (plus
-(weight_map g t0) (weight_map g t1)) (plus_le_compat (weight_map f0 t0)
-(weight_map g t0) (weight_map f0 t1) (weight_map g t1) (H f0 g H1) (H0 f0 g
-H1))))))))))) k)) t).
+\to (le (weight_map f t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda
+(H0: ((\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(n: nat).(le (f n) (g n)))) \to (le (weight_map f t1) (weight_map g
+t1))))))).(\lambda (f: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H1: ((\forall (n: nat).(le (f n) (g n))))).(le_n_S (plus
+(weight_map f t0) (weight_map (wadd f O) t1)) (plus (weight_map g t0)
+(weight_map (wadd g O) t1)) (le_plus_plus (weight_map f t0) (weight_map g t0)
+(weight_map (wadd f O) t1) (weight_map (wadd g O) t1) (H f g H1) (H0 (wadd f
+O) (wadd g O) (\lambda (n: nat).(wadd_le f g H1 O O (le_n O) n))))))))))))
+b)) (\lambda (_: F).(\lambda (t0: T).(\lambda (H: ((\forall (f0: ((nat \to
+nat))).(\forall (g: ((nat \to nat))).(((\forall (n: nat).(le (f0 n) (g n))))
+\to (le (weight_map f0 t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda
+(H0: ((\forall (f0: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(n: nat).(le (f0 n) (g n)))) \to (le (weight_map f0 t1) (weight_map g
+t1))))))).(\lambda (f0: ((nat \to nat))).(\lambda (g: ((nat \to
+nat))).(\lambda (H1: ((\forall (n: nat).(le (f0 n) (g n))))).(le_n_S (plus
+(weight_map f0 t0) (weight_map f0 t1)) (plus (weight_map g t0) (weight_map g
+t1)) (le_plus_plus (weight_map f0 t0) (weight_map g t0) (weight_map f0 t1)
+(weight_map g t1) (H f0 g H1) (H0 f0 g H1))))))))))) k)) t).
theorem weight_eq:
\forall (t: T).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
(TLRef n) (\lambda (c0: C).(\lambda (t3: T).(\forall (e: C).((getl i c (CHead
e (Bind Abbr) u0)) \to (ty3 g c0 t3 (lift (S n) O t0)))))) (\lambda (t3:
T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (e: C).(\lambda (H5:
-(getl i c (CHead e (Bind Abbr) u0))).(and_ind (eq nat n i) (eq T t3 (lift (S
+(getl i c (CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S
n) O u0)) (ty3 g c t3 (lift (S n) O t0)) (\lambda (H6: (eq nat n i)).(\lambda
(H7: (eq T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4:
T).(ty3 g c t4 (lift (S n) O t0))) (let H8 \def (eq_ind_r nat i (\lambda (n0:
(csubst0_getl_ge i n H6 c c3 u0 H4 (CHead d (Bind Abbr) u) H0) t0 H1)))))))
(\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3:
C).(\lambda (H5: (csubst0 i u0 c c3)).(\lambda (e: C).(\lambda (H6: (getl i c
-(CHead e (Bind Abbr) u0))).(and_ind (eq nat n i) (eq T t3 (lift (S n) O u0))
+(CHead e (Bind Abbr) u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0))
(ty3 g c3 t3 (lift (S n) O t0)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq
T t3 (lift (S n) O u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3
g c3 t4 (lift (S n) O t0))) (let H9 \def (eq_ind_r nat i (\lambda (n0:
T).(\forall (e: C).((getl i c (CHead e (Bind Abbr) u0)) \to (ty3 g c0 t3
(lift (S n) O u)))))) (\lambda (t3: T).(\lambda (H4: (subst0 i u0 (TLRef n)
t3)).(\lambda (e: C).(\lambda (H5: (getl i c (CHead e (Bind Abbr)
-u0))).(and_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S
+u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c t3 (lift (S
n) O u)) (\lambda (H6: (eq nat n i)).(\lambda (H7: (eq T t3 (lift (S n) O
u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c t4 (lift (S n)
O u))) (let H8 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e
c3 u0 H4 (CHead d (Bind Abst) u) H0) t0 H1))))))) (\lambda (t3: T).(\lambda
(H4: (subst0 i u0 (TLRef n) t3)).(\lambda (c3: C).(\lambda (H5: (csubst0 i u0
c c3)).(\lambda (e: C).(\lambda (H6: (getl i c (CHead e (Bind Abbr)
-u0))).(and_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift (S
-n) O u)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O
+u0))).(land_ind (eq nat n i) (eq T t3 (lift (S n) O u0)) (ty3 g c3 t3 (lift
+(S n) O u)) (\lambda (H7: (eq nat n i)).(\lambda (H8: (eq T t3 (lift (S n) O
u0))).(eq_ind_r T (lift (S n) O u0) (\lambda (t4: T).(ty3 g c3 t4 (lift (S n)
O u))) (let H9 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead e
(Bind Abbr) u0))) H6 n H7) in (let H10 \def (eq_ind_r nat i (\lambda (n0:
(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
(x4: T).(\lambda (x5: T).(\lambda (H13: (pc3 c (THead (Bind Abst) x0 x4)
(THead (Bind Abst) w u))).(\lambda (_: (ty3 g c x0 x5)).(\lambda (H15: (ty3 g
-(CHead c (Bind Abst) x0) x1 x4)).(and_ind (pc3 c x0 w) (\forall (b:
+(CHead c (Bind Abst) x0) x1 x4)).(land_ind (pc3 c x0 w) (\forall (b:
B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x4 u))) (ex4_2 T T (\lambda
(v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w
v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
T x0 (\lambda (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O
(subst1_gen_lift_eq t4 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat
(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S
-O)) (plus_comm O (S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3)
+O)) (plus_sym O (S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3)
(THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3
x H22) t4 x1 H28 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead
(Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2
(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x2: T).(\lambda
(x3: T).(\lambda (H22: (pc3 c2 (THead (Bind Abst) u0 x2) (THead (Bind Abst) u
t0))).(\lambda (H23: (ty3 g c2 u0 x3)).(\lambda (H24: (ty3 g (CHead c2 (Bind
-Abst) u0) t4 x2)).(and_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1:
+Abst) u0) t4 x2)).(land_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1:
T).(pc3 (CHead c2 (Bind b) u1) x2 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4)
(THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H25: (pc3 c2 u0
u)).(\lambda (H26: ((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b)
\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1
h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7:
-(land (lt n x1) (eq T x0 (TLRef n)))).(and_ind (lt n x1) (eq T x0 (TLRef n))
+(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n))
(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda
(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0
(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2:
n e x3 x2 H12 x4 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16)))))))))
(getl_drop_conf_lt Abbr c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9)))
H7)) (\lambda (H7: (land (le (plus x1 h) n) (eq T x0 (TLRef (minus n
-h))))).(and_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T
+h))))).(land_ind (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))) (ex2 T
(\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
T).(ty3 g e x0 t2))) (\lambda (H8: (le (plus x1 h) n)).(\lambda (H9: (eq T x0
(TLRef (minus n h)))).(eq_ind_r T (TLRef (minus n h)) (\lambda (t0: T).(ex2 T
\def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1
h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7:
-(land (lt n x1) (eq T x0 (TLRef n)))).(and_ind (lt n x1) (eq T x0 (TLRef n))
+(land (lt n x1) (eq T x0 (TLRef n)))).(land_ind (lt n x1) (eq T x0 (TLRef n))
(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda
(t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0
(TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2:
(le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1
(le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst
c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land
-(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(and_ind (le (plus x1 h)
+(le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(land_ind (le (plus x1 h)
n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le
(plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T
(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
(TLRef (plus n h)) (lift n0 O t))) (ty3_abbr g (plus n h) c0 d u
(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abbr) u) H0 H4) t H1) (plus
-h (S n)) (plus_comm h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n)
+h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O t)) (lift_free t (S n)
h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
-(plus n (S O)) (plus_comm n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
+(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
d0 H4)))))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d:
C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda
(t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c0: C).(\forall
(plus n h)) t0)) (eq_ind_r nat (plus (S n) h) (\lambda (n0: nat).(ty3 g c0
(TLRef (plus n h)) (lift n0 O u))) (ty3_abst g (plus n h) c0 d u
(drop_getl_trans_ge n c0 c d0 h H3 (CHead d (Bind Abst) u) H0 H4) t H1) (plus
-h (S n)) (plus_comm h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n)
+h (S n)) (plus_sym h (S n))) (lift h d0 (lift (S n) O u)) (lift_free u (S n)
h O d0 (le_S d0 n H4) (le_O_n d0))) (plus n (S O)) (eq_ind_r nat (plus (S O)
n) (\lambda (n0: nat).(eq nat (S n) n0)) (refl_equal nat (plus (S O) n))
-(plus n (S O)) (plus_comm n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
+(plus n (S O)) (plus_sym n (S O)))) (lift h d0 (TLRef n)) (lift_lref_ge n h
d0 H4)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t:
T).(\lambda (_: (ty3 g c u t)).(\lambda (H1: ((\forall (c0: C).(\forall (d:
nat).(\forall (h: nat).((drop h d c0 c) \to (ty3 g c0 (lift h d u) (lift h d
T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
-(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(and_ind (eq nat n i)
-(eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda
-(H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def
-(eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n
-H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1: C).(getl
-n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n
-H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b)
-v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in
-((let H10 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
-C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k
-in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v)
-(getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in
-((let H11 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2]))
-(CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
-Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: (eq B Abbr
-b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v (\lambda (t2:
-T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda
-(t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0
-(\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C
-d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def
-(eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abbr
-H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v
-H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n H3))))))))))))))))))
-(\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda
-(H0: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t0: T).(\lambda (H1:
-(ty3 g d u0 t0)).(\lambda (_: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
-nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d0: C).(\forall (v:
-T).((getl i d (CHead d0 (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d0 v
-w))))))))))))).(\lambda (v0: T).(\lambda (t1: T).(\lambda (i: nat).(\lambda
-(H3: (subst0 i v0 (TLRef n) t1)).(\lambda (b: B).(\lambda (d0: C).(\lambda
-(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(and_ind (eq nat n i)
-(eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w))) (\lambda
-(H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7 \def
-(eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n
-H5) in (let H8 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c1: C).(getl
-n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n
-H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d |
-(CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u0) (CHead d0 (Bind b)
-v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in
-((let H10 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
-C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k
-in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abst])])) (CHead d (Bind Abst) u0) (CHead d0 (Bind b) v)
-(getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in
-((let H11 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2]))
+(v: T).(\lambda (H4: (getl i c0 (CHead d0 (Bind b) v))).(land_ind (eq nat n
+i) (eq T t1 (lift (S n) O v0)) (ex T (\lambda (w: T).(ty3 g d0 v w)))
+(\lambda (H5: (eq nat n i)).(\lambda (_: (eq T t1 (lift (S n) O v0))).(let H7
+\def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c0 (CHead d0 (Bind b) v)))
+H4 n H5) in (let H8 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c1:
+C).(getl n c0 c1)) H0 (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (let H9 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0)
+(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0
+(Bind b) v) H7)) in ((let H10 \def (f_equal C B (\lambda (e: C).(match e in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0)
+(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0
+(Bind b) v) H7)) in ((let H11 \def (f_equal C T (\lambda (e: C).(match e in C
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t2)
+\Rightarrow t2])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind b) v) (getl_mono
+c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12:
+(eq B Abbr b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v
+(\lambda (t2: T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T
+u0 (\lambda (t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def
+(eq_ind_r C d0 (\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d
+H13) in (eq_ind C d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w))))
+(let H16 \def (eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0)
+u0))) H15 Abbr H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0
+H13)) v H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n
+H3)))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
+u0))).(\lambda (t0: T).(\lambda (H1: (ty3 g d u0 t0)).(\lambda (_: ((\forall
+(v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to (\forall
+(b: B).(\forall (d0: C).(\forall (v: T).((getl i d (CHead d0 (Bind b) v)) \to
+(ex T (\lambda (w: T).(ty3 g d0 v w))))))))))))).(\lambda (v0: T).(\lambda
+(t1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v0 (TLRef n) t1)).(\lambda
+(b: B).(\lambda (d0: C).(\lambda (v: T).(\lambda (H4: (getl i c0 (CHead d0
+(Bind b) v))).(land_ind (eq nat n i) (eq T t1 (lift (S n) O v0)) (ex T
+(\lambda (w: T).(ty3 g d0 v w))) (\lambda (H5: (eq nat n i)).(\lambda (_: (eq
+T t1 (lift (S n) O v0))).(let H7 \def (eq_ind_r nat i (\lambda (n0:
+nat).(getl n0 c0 (CHead d0 (Bind b) v))) H4 n H5) in (let H8 \def (eq_ind C
+(CHead d (Bind Abst) u0) (\lambda (c1: C).(getl n c0 c1)) H0 (CHead d0 (Bind
+b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind b) v) H7))
+in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
-Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in (\lambda (H12: (eq B Abst
-b)).(\lambda (H13: (eq C d d0)).(let H14 \def (eq_ind_r T v (\lambda (t2:
-T).(getl n c0 (CHead d0 (Bind b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda
-(t2: T).(ex T (\lambda (w: T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0
-(\lambda (c1: C).(getl n c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C
-d (\lambda (c1: C).(ex T (\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def
-(eq_ind_r B b (\lambda (b0: B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst
-H12) in (ex_intro T (\lambda (w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v
-H11))))) H10)) H9)))))) (subst0_gen_lref v0 t1 i n H3))))))))))))))))))
-(\lambda (c0: C).(\lambda (u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0
-t0)).(\lambda (H1: ((\forall (v0: T).(\forall (t1: T).(\forall (i:
-nat).((subst0 i v0 u0 t1) \to (\forall (b: B).(\forall (d: C).(\forall (v:
-T).((getl i c0 (CHead d (Bind b) v)) \to (ex T (\lambda (w: T).(ty3 g d v
-w))))))))))))).(\lambda (b: B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_:
-(ty3 g (CHead c0 (Bind b) u0) t1 t2)).(\lambda (H3: ((\forall (v0:
-T).(\forall (t3: T).(\forall (i: nat).((subst0 i v0 t1 t3) \to (\forall (b0:
-B).(\forall (d: C).(\forall (v: T).((getl i (CHead c0 (Bind b) u0) (CHead d
-(Bind b0) v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda
-(v0: T).(\lambda (t3: T).(\lambda (i: nat).(\lambda (H4: (subst0 i v0 (THead
-(Bind b) u0 t1) t3)).(\lambda (b0: B).(\lambda (d: C).(\lambda (v:
-T).(\lambda (H5: (getl i c0 (CHead d (Bind b0) v))).(or3_ind (ex2 T (\lambda
+Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H10 \def (f_equal C B
+(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
+\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])]))
+(CHead d (Bind Abst) u0) (CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind
+Abst) u0) n H0 (CHead d0 (Bind b) v) H7)) in ((let H11 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u0 | (CHead _ _ t2) \Rightarrow t2])) (CHead d (Bind Abst) u0)
+(CHead d0 (Bind b) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0
+(Bind b) v) H7)) in (\lambda (H12: (eq B Abst b)).(\lambda (H13: (eq C d
+d0)).(let H14 \def (eq_ind_r T v (\lambda (t2: T).(getl n c0 (CHead d0 (Bind
+b) t2))) H8 u0 H11) in (eq_ind T u0 (\lambda (t2: T).(ex T (\lambda (w:
+T).(ty3 g d0 t2 w)))) (let H15 \def (eq_ind_r C d0 (\lambda (c1: C).(getl n
+c0 (CHead c1 (Bind b) u0))) H14 d H13) in (eq_ind C d (\lambda (c1: C).(ex T
+(\lambda (w: T).(ty3 g c1 u0 w)))) (let H16 \def (eq_ind_r B b (\lambda (b0:
+B).(getl n c0 (CHead d (Bind b0) u0))) H15 Abst H12) in (ex_intro T (\lambda
+(w: T).(ty3 g d u0 w)) t0 H1)) d0 H13)) v H11))))) H10)) H9))))))
+(subst0_gen_lref v0 t1 i n H3)))))))))))))))))) (\lambda (c0: C).(\lambda
+(u0: T).(\lambda (t0: T).(\lambda (_: (ty3 g c0 u0 t0)).(\lambda (H1:
+((\forall (v0: T).(\forall (t1: T).(\forall (i: nat).((subst0 i v0 u0 t1) \to
+(\forall (b: B).(\forall (d: C).(\forall (v: T).((getl i c0 (CHead d (Bind b)
+v)) \to (ex T (\lambda (w: T).(ty3 g d v w))))))))))))).(\lambda (b:
+B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
+u0) t1 t2)).(\lambda (H3: ((\forall (v0: T).(\forall (t3: T).(\forall (i:
+nat).((subst0 i v0 t1 t3) \to (\forall (b0: B).(\forall (d: C).(\forall (v:
+T).((getl i (CHead c0 (Bind b) u0) (CHead d (Bind b0) v)) \to (ex T (\lambda
+(w: T).(ty3 g d v w))))))))))))).(\lambda (v0: T).(\lambda (t3: T).(\lambda
+(i: nat).(\lambda (H4: (subst0 i v0 (THead (Bind b) u0 t1) t3)).(\lambda (b0:
+B).(\lambda (d: C).(\lambda (v: T).(\lambda (H5: (getl i c0 (CHead d (Bind
+b0) v))).(or3_ind (ex2 T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1)))
+(\lambda (u2: T).(subst0 i v0 u0 u2))) (ex2 T (\lambda (t4: T).(eq T t3
+(THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_:
+T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))) (ex T (\lambda (w:
+T).(ty3 g d v w))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t3 (THead
+(Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0 u2)))).(ex2_ind T (\lambda
(u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i v0 u0
-u2))) (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4:
-T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s (Bind b)
-i) v0 t1 t4)))) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (H6: (ex2 T
-(\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1))) (\lambda (u2: T).(subst0 i
-v0 u0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t3 (THead (Bind b) u2 t1)))
-(\lambda (u2: T).(subst0 i v0 u0 u2)) (ex T (\lambda (w: T).(ty3 g d v w)))
-(\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b) x t1))).(\lambda (H8:
-(subst0 i v0 u0 x)).(H1 v0 x i H8 b0 d v H5)))) H6)) (\lambda (H6: (ex2 T
+u2)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T
+t3 (THead (Bind b) x t1))).(\lambda (H8: (subst0 i v0 u0 x)).(H1 v0 x i H8 b0
+d v H5)))) H6)) (\lambda (H6: (ex2 T (\lambda (t4: T).(eq T t3 (THead (Bind
+b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)))).(ex2_ind T
(\lambda (t4: T).(eq T t3 (THead (Bind b) u0 t4))) (\lambda (t4: T).(subst0
-(s (Bind b) i) v0 t1 t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead (Bind
-b) u0 t4))) (\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4)) (ex T (\lambda
-(w: T).(ty3 g d v w))) (\lambda (x: T).(\lambda (_: (eq T t3 (THead (Bind b)
-u0 x))).(\lambda (H8: (subst0 (s (Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d
-v (getl_head (Bind b) i c0 (CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda
-(H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_:
-T).(\lambda (t4: T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda
-(u2: T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
+(s (Bind b) i) v0 t1 t4)) (ex T (\lambda (w: T).(ty3 g d v w))) (\lambda (x:
+T).(\lambda (_: (eq T t3 (THead (Bind b) u0 x))).(\lambda (H8: (subst0 (s
+(Bind b) i) v0 t1 x)).(H3 v0 x (S i) H8 b0 d v (getl_head (Bind b) i c0
+(CHead d (Bind b0) v) H5 u0))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
+T).(subst0 (s (Bind b) i) v0 t1 t4))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t4: T).(eq T t3 (THead (Bind b) u2 t4)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i v0 u0 u2))) (\lambda (_: T).(\lambda (t4:
T).(subst0 (s (Bind b) i) v0 t1 t4))) (ex T (\lambda (w: T).(ty3 g d v w)))
(\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (eq T t3 (THead (Bind b) x0
x1))).(\lambda (H8: (subst0 i v0 u0 x0)).(\lambda (_: (subst0 (s (Bind b) i)
O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0
n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a
(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6
-(plus d0 (S O)) (plus_comm d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O
-d0 n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_comm (S O) (minus
-n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n
-(S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
+(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0
+n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S
+O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S
+O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
H6))))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e:
O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0
n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abst) u) H0) a
(S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6
-(plus d0 (S O)) (plus_comm d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O
-d0 n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_comm (S O) (minus
-n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n
-(S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
+(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0
+n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S
+O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S
+O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
H6))))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t:
T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: ((\forall (e: C).(\forall (u0:
T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0:
O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t)))
(ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr)
u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le
-n0 n)) H5 (plus d0 (S O)) (plus_comm d0 (S O)))) t H1) n (minus_x_SO n
-(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O)))
-(plus_comm (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal
-nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n
-(le_O_n d0) H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda
-(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
+n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n
+(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) (plus_sym
+(S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus
+O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0)
+H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst)
u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e:
C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u0))
\to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1:
(eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n
(S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge
n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0)
-(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_comm d0 (S O)))) t H1)
+(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1)
n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n
-(S O))) (plus_comm (S O) (minus n (S O)))) (S (plus O (minus n (S O))))
+(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O))))
(refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n
(le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0:
C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda
--- /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 automatically generated: do not edit *********************)
+
+include "Legacy-1/preamble.ma".
+
+inductive eq (A: Set) (x: A): A \to Prop \def
+| refl_equal: eq A x x.
+
+inductive True: Prop \def
+| I: True.
+
+inductive land (A: Prop) (B: Prop): Prop \def
+| conj: A \to (B \to (land A B)).
+
+inductive or (A: Prop) (B: Prop): Prop \def
+| or_introl: A \to (or A B)
+| or_intror: B \to (or A B).
+
+inductive ex (A: Set) (P: A \to Prop): Prop \def
+| ex_intro: \forall (x: A).((P x) \to (ex A P)).
+
+inductive ex2 (A: Set) (P: A \to Prop) (Q: A \to Prop): Prop \def
+| ex_intro2: \forall (x: A).((P x) \to ((Q x) \to (ex2 A P Q))).
+
+definition not:
+ Prop \to Prop
+\def
+ \lambda (A: Prop).(A \to False).
+
+inductive bool: Set \def
+| true: bool
+| false: bool.
+
+inductive nat: Set \def
+| O: nat
+| S: nat \to nat.
+
+inductive le (n: nat): nat \to Prop \def
+| le_n: le n n
+| le_S: \forall (m: nat).((le n m) \to (le n (S m))).
+
+definition lt:
+ nat \to (nat \to Prop)
+\def
+ \lambda (n: nat).(\lambda (m: nat).(le (S n) m)).
+
+definition IsSucc:
+ nat \to Prop
+\def
+ \lambda (n: nat).(match n with [O \Rightarrow False | (S _) \Rightarrow
+True]).
+
+definition pred:
+ nat \to nat
+\def
+ \lambda (n: nat).(match n with [O \Rightarrow O | (S u) \Rightarrow u]).
+
+definition plus:
+ nat \to (nat \to nat)
+\def
+ let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n
+with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus.
+
+definition minus:
+ nat \to (nat \to nat)
+\def
+ let rec minus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n
+with [O \Rightarrow O | (S k) \Rightarrow (match m with [O \Rightarrow (S k)
+| (S l) \Rightarrow (minus k l)])])) in minus.
+
+inductive Acc (A: Set) (R: A \to (A \to Prop)): A \to Prop \def
+| Acc_intro: \forall (x: A).(((\forall (y: A).((R y x) \to (Acc A R y)))) \to
+(Acc A R x)).
+
+definition well_founded:
+ \forall (A: Set).(((A \to (A \to Prop))) \to Prop)
+\def
+ \lambda (A: Set).(\lambda (R: ((A \to (A \to Prop)))).(\forall (a: A).(Acc A
+R a))).
+
+definition ltof:
+ \forall (A: Set).(((A \to nat)) \to (A \to (A \to Prop)))
+\def
+ \lambda (A: Set).(\lambda (f: ((A \to nat))).(\lambda (a: A).(\lambda (b:
+A).(lt (f a) (f b))))).
+
--- /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 automatically generated: do not edit *********************)
+
+include "Legacy-1/coq/defs.ma".
+
+theorem f_equal:
+ \forall (A: Set).(\forall (B: Set).(\forall (f: ((A \to B))).(\forall (x:
+A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y)))))))
+\def
+ \lambda (A: Set).(\lambda (B: Set).(\lambda (f: ((A \to B))).(\lambda (x:
+A).(\lambda (y: A).(\lambda (H: (eq A x y)).(eq_ind A x (\lambda (a: A).(eq B
+(f x) (f a))) (refl_equal B (f x)) y H)))))).
+
+theorem f_equal2:
+ \forall (A1: Set).(\forall (A2: Set).(\forall (B: Set).(\forall (f: ((A1 \to
+(A2 \to B)))).(\forall (x1: A1).(\forall (y1: A1).(\forall (x2: A2).(\forall
+(y2: A2).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to (eq B (f x1 x2) (f y1
+y2)))))))))))
+\def
+ \lambda (A1: Set).(\lambda (A2: Set).(\lambda (B: Set).(\lambda (f: ((A1 \to
+(A2 \to B)))).(\lambda (x1: A1).(\lambda (y1: A1).(\lambda (x2: A2).(\lambda
+(y2: A2).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: A1).((eq A2
+x2 y2) \to (eq B (f x1 x2) (f a y2)))) (\lambda (H0: (eq A2 x2 y2)).(eq_ind
+A2 x2 (\lambda (a: A2).(eq B (f x1 x2) (f x1 a))) (refl_equal B (f x1 x2)) y2
+H0)) y1 H))))))))).
+
+theorem f_equal3:
+ \forall (A1: Set).(\forall (A2: Set).(\forall (A3: Set).(\forall (B:
+Set).(\forall (f: ((A1 \to (A2 \to (A3 \to B))))).(\forall (x1: A1).(\forall
+(y1: A1).(\forall (x2: A2).(\forall (y2: A2).(\forall (x3: A3).(\forall (y3:
+A3).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (eq B (f x1 x2
+x3) (f y1 y2 y3)))))))))))))))
+\def
+ \lambda (A1: Set).(\lambda (A2: Set).(\lambda (A3: Set).(\lambda (B:
+Set).(\lambda (f: ((A1 \to (A2 \to (A3 \to B))))).(\lambda (x1: A1).(\lambda
+(y1: A1).(\lambda (x2: A2).(\lambda (y2: A2).(\lambda (x3: A3).(\lambda (y3:
+A3).(\lambda (H: (eq A1 x1 y1)).(eq_ind A1 x1 (\lambda (a: A1).((eq A2 x2 y2)
+\to ((eq A3 x3 y3) \to (eq B (f x1 x2 x3) (f a y2 y3))))) (\lambda (H0: (eq
+A2 x2 y2)).(eq_ind A2 x2 (\lambda (a: A2).((eq A3 x3 y3) \to (eq B (f x1 x2
+x3) (f x1 a y3)))) (\lambda (H1: (eq A3 x3 y3)).(eq_ind A3 x3 (\lambda (a:
+A3).(eq B (f x1 x2 x3) (f x1 x2 a))) (refl_equal B (f x1 x2 x3)) y3 H1)) y2
+H0)) y1 H)))))))))))).
+
+theorem sym_eq:
+ \forall (A: Set).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y
+x))))
+\def
+ \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x
+y)).(eq_ind A x (\lambda (a: A).(eq A a x)) (refl_equal A x) y H)))).
+
+theorem eq_ind_r:
+ \forall (A: Set).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to
+(\forall (y: A).((eq A y x) \to (P y))))))
+\def
+ \lambda (A: Set).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(\lambda (H:
+(P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0) in
+eq return (\lambda (a: A).(\lambda (_: (eq ? ? a)).(P a))) with [refl_equal
+\Rightarrow H])))))).
+
+theorem trans_eq:
+ \forall (A: Set).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A x y)
+\to ((eq A y z) \to (eq A x z))))))
+\def
+ \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (z: A).(\lambda
+(H: (eq A x y)).(\lambda (H0: (eq A y z)).(eq_ind A y (\lambda (a: A).(eq A x
+a)) H z H0)))))).
+
+theorem sym_not_eq:
+ \forall (A: Set).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to (not
+(eq A y x)))))
+\def
+ \lambda (A: Set).(\lambda (x: A).(\lambda (y: A).(\lambda (h1: (not (eq A x
+y))).(\lambda (h2: (eq A y x)).(h1 (eq_ind A y (\lambda (a: A).(eq A a y))
+(refl_equal A y) x h2)))))).
+
+theorem nat_double_ind:
+ \forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to
+(((\forall (n: nat).(R (S n) O))) \to (((\forall (n: nat).(\forall (m:
+nat).((R n m) \to (R (S n) (S m)))))) \to (\forall (n: nat).(\forall (m:
+nat).(R n m))))))
+\def
+ \lambda (R: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (n: nat).(R O
+n)))).(\lambda (H0: ((\forall (n: nat).(R (S n) O)))).(\lambda (H1: ((\forall
+(n: nat).(\forall (m: nat).((R n m) \to (R (S n) (S m))))))).(\lambda (n:
+nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) H (\lambda (n0:
+nat).(\lambda (H2: ((\forall (m: nat).(R n0 m)))).(\lambda (m: nat).(nat_ind
+(\lambda (n1: nat).(R (S n0) n1)) (H0 n0) (\lambda (n1: nat).(\lambda (_: (R
+(S n0) n1)).(H1 n0 n1 (H2 n1)))) m)))) n))))).
+
+theorem eq_add_S:
+ \forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S
+m))).(f_equal nat nat pred (S n) (S m) H))).
+
+theorem O_S:
+ \forall (n: nat).(not (eq nat O (S n)))
+\def
+ \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(eq_ind nat (S n) (\lambda
+(n0: nat).(IsSucc n0)) I O (sym_eq nat O (S n) H))).
+
+theorem not_eq_S:
+ \forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S
+n) (S m)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda
+(H0: (eq nat (S n) (S m))).(H (eq_add_S n m H0))))).
+
+theorem pred_Sn:
+ \forall (m: nat).(eq nat m (pred (S m)))
+\def
+ \lambda (m: nat).(refl_equal nat (pred (S m))).
+
+theorem S_pred:
+ \forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt m n)).(le_ind (S m)
+(\lambda (n0: nat).(eq nat n0 (S (pred n0)))) (refl_equal nat (S (pred (S
+m)))) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (_: (eq nat m0
+(S (pred m0)))).(refl_equal nat (S (pred (S m0))))))) n H))).
+
+theorem le_trans:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p)
+\to (le n p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n
+m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(le n n0)) H
+(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (le n m0)).(le_S n
+m0 IHle)))) p H0))))).
+
+theorem le_trans_S:
+ \forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(le_trans n (S
+n) m (le_S n n (le_n n)) H))).
+
+theorem le_n_S:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
+(n0: nat).(le (S n) (S n0))) (le_n (S n)) (\lambda (m0: nat).(\lambda (_: (le
+n m0)).(\lambda (IHle: (le (S n) (S m0))).(le_S (S n) (S m0) IHle)))) m H))).
+
+theorem le_O_n:
+ \forall (n: nat).(le O n)
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le O n0)) (le_n O) (\lambda
+(n0: nat).(\lambda (IHn: (le O n0)).(le_S O n0 IHn))) n).
+
+theorem le_S_n:
+ \forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) (S m))).(le_ind (S
+n) (\lambda (n0: nat).(le (pred (S n)) (pred n0))) (le_n n) (\lambda (m0:
+nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S
+n m0 H0)))) (S m) H))).
+
+theorem le_Sn_O:
+ \forall (n: nat).(not (le (S n) O))
+\def
+ \lambda (n: nat).(\lambda (H: (le (S n) O)).(le_ind (S n) (\lambda (n0:
+nat).(IsSucc n0)) I (\lambda (m: nat).(\lambda (_: (le (S n) m)).(\lambda (_:
+(IsSucc m)).I))) O H)).
+
+theorem le_Sn_n:
+ \forall (n: nat).(not (le (S n) n))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(not (le (S n0) n0))) (le_Sn_O
+O) (\lambda (n0: nat).(\lambda (IHn: (not (le (S n0) n0))).(\lambda (H: (le
+(S (S n0)) (S n0))).(IHn (le_S_n (S n0) n0 H))))) n).
+
+theorem le_antisym:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n
+m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (h: (le n m)).(le_ind n (\lambda
+(n0: nat).((le n0 n) \to (eq nat n n0))) (\lambda (_: (le n n)).(refl_equal
+nat n)) (\lambda (m0: nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n)
+\to (eq nat n m0)))).(\lambda (H1: (le (S m0) n)).(False_ind (eq nat n (S
+m0)) (let H2 \def (le_trans (S m0) n m0 H1 H) in ((let H3 \def (le_Sn_n m0)
+in (\lambda (H4: (le (S m0) m0)).(H3 H4))) H2))))))) m h))).
+
+theorem le_n_O_eq:
+ \forall (n: nat).((le n O) \to (eq nat O n))
+\def
+ \lambda (n: nat).(\lambda (H: (le n O)).(le_antisym O n (le_O_n n) H)).
+
+theorem le_elim_rel:
+ \forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to
+(((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p q) \to (P (S p) (S
+q))))))) \to (\forall (n: nat).(\forall (m: nat).((le n m) \to (P n m))))))
+\def
+ \lambda (P: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (p: nat).(P O
+p)))).(\lambda (H0: ((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p
+q) \to (P (S p) (S q)))))))).(\lambda (n: nat).(nat_ind (\lambda (n0:
+nat).(\forall (m: nat).((le n0 m) \to (P n0 m)))) (\lambda (m: nat).(\lambda
+(_: (le O m)).(H m))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (m:
+nat).((le n0 m) \to (P n0 m))))).(\lambda (m: nat).(\lambda (Le: (le (S n0)
+m)).(le_ind (S n0) (\lambda (n1: nat).(P (S n0) n1)) (H0 n0 n0 (le_n n0) (IHn
+n0 (le_n n0))) (\lambda (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_:
+(P (S n0) m0)).(H0 n0 m0 (le_trans_S n0 m0 H1) (IHn m0 (le_trans_S n0 m0
+H1)))))) m Le))))) n)))).
+
+theorem lt_n_n:
+ \forall (n: nat).(not (lt n n))
+\def
+ le_Sn_n.
+
+theorem lt_n_S:
+ \forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_n_S (S n) m
+H))).
+
+theorem lt_n_Sn:
+ \forall (n: nat).(lt n (S n))
+\def
+ \lambda (n: nat).(le_n (S n)).
+
+theorem lt_S_n:
+ \forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(le_S_n (S
+n) m H))).
+
+theorem lt_n_O:
+ \forall (n: nat).(not (lt n O))
+\def
+ le_Sn_O.
+
+theorem lt_trans:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p)
+\to (lt n p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n
+m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0)) (le_S
+(S n) m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt
+n m0)).(le_S (S n) m0 IHle)))) p H0))))).
+
+theorem lt_O_Sn:
+ \forall (n: nat).(lt O (S n))
+\def
+ \lambda (n: nat).(le_n_S O n (le_O_n n)).
+
+theorem lt_le_S:
+ \forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p)))
+\def
+ \lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)).
+
+theorem le_not_lt:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
+(n0: nat).(not (lt n0 n))) (lt_n_n n) (\lambda (m0: nat).(\lambda (_: (le n
+m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(IHle
+(le_trans_S (S m0) n H1)))))) m H))).
+
+theorem le_lt_n_Sm:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))).
+
+theorem le_lt_trans:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p)
+\to (lt n p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n
+m)).(\lambda (H0: (lt m p)).(le_ind (S m) (\lambda (n0: nat).(lt n n0))
+(le_n_S n m H) (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle:
+(lt n m0)).(le_S (S n) m0 IHle)))) p H0))))).
+
+theorem lt_le_trans:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p)
+\to (lt n p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n
+m)).(\lambda (H0: (le m p)).(le_ind m (\lambda (n0: nat).(lt n n0)) H
+(\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle: (lt n m0)).(le_S
+(S n) m0 IHle)))) p H0))))).
+
+theorem lt_le_weak:
+ \forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m
+H))).
+
+theorem lt_n_Sm_le:
+ \forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m
+H))).
+
+theorem le_lt_or_eq:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
+(n0: nat).(or (lt n n0) (eq nat n n0))) (or_intror (lt n n) (eq nat n n)
+(refl_equal nat n)) (\lambda (m0: nat).(\lambda (H0: (le n m0)).(\lambda (_:
+(or (lt n m0) (eq nat n m0))).(or_introl (lt n (S m0)) (eq nat n (S m0))
+(le_n_S n m0 H0))))) m H))).
+
+theorem le_or_lt:
+ \forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(nat_double_ind (\lambda (n0:
+nat).(\lambda (n1: nat).(or (le n0 n1) (lt n1 n0)))) (\lambda (n0:
+nat).(or_introl (le O n0) (lt n0 O) (le_O_n n0))) (\lambda (n0:
+nat).(or_intror (le (S n0) O) (lt O (S n0)) (lt_le_S O (S n0) (lt_O_Sn n0))))
+(\lambda (n0: nat).(\lambda (m0: nat).(\lambda (H: (or (le n0 m0) (lt m0
+n0))).(or_ind (le n0 m0) (lt m0 n0) (or (le (S n0) (S m0)) (lt (S m0) (S
+n0))) (\lambda (H0: (le n0 m0)).(or_introl (le (S n0) (S m0)) (lt (S m0) (S
+n0)) (le_n_S n0 m0 H0))) (\lambda (H0: (lt m0 n0)).(or_intror (le (S n0) (S
+m0)) (lt (S m0) (S n0)) (le_n_S (S m0) n0 H0))) H)))) n m)).
+
+theorem plus_n_O:
+ \forall (n: nat).(eq nat n (plus n O))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (plus n0 O)))
+(refl_equal nat O) (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0
+O))).(f_equal nat nat S n0 (plus n0 O) H))) n).
+
+theorem plus_n_Sm:
+ \forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m))))
+\def
+ \lambda (m: nat).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat (S
+(plus n0 n)) (plus n0 (S n)))) (refl_equal nat (S n)) (\lambda (n0:
+nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S n)))).(f_equal nat nat
+S (S (plus n0 n)) (plus n0 (S n)) H))) m)).
+
+theorem plus_sym:
+ \forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(eq nat (plus
+n0 m) (plus m n0))) (plus_n_O m) (\lambda (y: nat).(\lambda (H: (eq nat (plus
+y m) (plus m y))).(eq_ind nat (S (plus m y)) (\lambda (n0: nat).(eq nat (S
+(plus y m)) n0)) (f_equal nat nat S (plus y m) (plus m y) H) (plus m (S y))
+(plus_n_Sm m y)))) n)).
+
+theorem plus_Snm_nSm:
+ \forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(eq_ind_r nat (plus m n) (\lambda (n0:
+nat).(eq nat (S n0) (plus n (S m)))) (eq_ind_r nat (plus (S m) n) (\lambda
+(n0: nat).(eq nat (S (plus m n)) n0)) (refl_equal nat (plus (S m) n)) (plus n
+(S m)) (plus_sym n (S m))) (plus n m) (plus_sym n m))).
+
+theorem plus_assoc_l:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m
+p)) (plus (plus n m) p))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0:
+nat).(eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))) (refl_equal nat
+(plus m p)) (\lambda (n0: nat).(\lambda (H: (eq nat (plus n0 (plus m p))
+(plus (plus n0 m) p))).(f_equal nat nat S (plus n0 (plus m p)) (plus (plus n0
+m) p) H))) n))).
+
+theorem plus_assoc_r:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n
+m) p) (plus n (plus m p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(sym_eq nat (plus n
+(plus m p)) (plus (plus n m) p) (plus_assoc_l n m p)))).
+
+theorem simpl_plus_l:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m)
+(plus n p)) \to (eq nat m p))))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(\forall (p:
+nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m p))))) (\lambda (m:
+nat).(\lambda (p: nat).(\lambda (H: (eq nat m p)).H))) (\lambda (n0:
+nat).(\lambda (IHn: ((\forall (m: nat).(\forall (p: nat).((eq nat (plus n0 m)
+(plus n0 p)) \to (eq nat m p)))))).(\lambda (m: nat).(\lambda (p:
+nat).(\lambda (H: (eq nat (S (plus n0 m)) (S (plus n0 p)))).(IHn m p (IHn
+(plus n0 m) (plus n0 p) (f_equal nat nat (plus n0) (plus n0 m) (plus n0 p)
+(eq_add_S (plus n0 m) (plus n0 p) H))))))))) n).
+
+theorem minus_n_O:
+ \forall (n: nat).(eq nat n (minus n O))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat n0 (minus n0 O)))
+(refl_equal nat O) (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0
+O))).(refl_equal nat (S n0)))) n).
+
+theorem minus_n_n:
+ \forall (n: nat).(eq nat O (minus n n))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(eq nat O (minus n0 n0)))
+(refl_equal nat O) (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0
+n0))).IHn)) n).
+
+theorem minus_Sn_m:
+ \forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m))
+(minus (S n) m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le m n)).(le_elim_rel
+(\lambda (n0: nat).(\lambda (n1: nat).(eq nat (S (minus n1 n0)) (minus (S n1)
+n0)))) (\lambda (p: nat).(f_equal nat nat S (minus p O) p (sym_eq nat p
+(minus p O) (minus_n_O p)))) (\lambda (p: nat).(\lambda (q: nat).(\lambda (_:
+(le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match p with [O \Rightarrow
+(S q) | (S l) \Rightarrow (minus q l)]))).H0)))) m n Le))).
+
+theorem plus_minus:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p))
+\to (eq nat p (minus n m)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_double_ind
+(\lambda (n0: nat).(\lambda (n1: nat).((eq nat n1 (plus n0 p)) \to (eq nat p
+(minus n1 n0))))) (\lambda (n0: nat).(\lambda (H: (eq nat n0 p)).(eq_ind nat
+n0 (\lambda (n1: nat).(eq nat p n1)) (sym_eq nat n0 p H) (minus n0 O)
+(minus_n_O n0)))) (\lambda (n0: nat).(\lambda (H: (eq nat O (S (plus n0
+p)))).(False_ind (eq nat p O) (let H0 \def H in ((let H1 \def (O_S (plus n0
+p)) in (\lambda (H2: (eq nat O (S (plus n0 p)))).(H1 H2))) H0))))) (\lambda
+(n0: nat).(\lambda (m0: nat).(\lambda (H: (((eq nat m0 (plus n0 p)) \to (eq
+nat p (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(H
+(eq_add_S m0 (plus n0 p) H0)))))) m n))).
+
+theorem minus_plus:
+ \forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(sym_eq nat m (minus (plus n m) n)
+(plus_minus (plus n m) n m (refl_equal nat (plus n m))))).
+
+theorem le_pred_n:
+ \forall (n: nat).(le (pred n) n)
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(le (pred n0) n0)) (le_n O)
+(\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(le_S (pred (S n0)) n0
+(le_n n0)))) n).
+
+theorem le_plus_l:
+ \forall (n: nat).(\forall (m: nat).(le n (plus n m)))
+\def
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (m: nat).(le n0 (plus
+n0 m)))) (\lambda (m: nat).(le_O_n m)) (\lambda (n0: nat).(\lambda (IHn:
+((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(le_n_S n0 (plus
+n0 m) (IHn m))))) n).
+
+theorem le_plus_r:
+ \forall (n: nat).(\forall (m: nat).(le m (plus n m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(nat_ind (\lambda (n0: nat).(le m (plus
+n0 m))) (le_n m) (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(le_S m
+(plus n0 m) H))) n)).
+
+theorem simpl_le_plus_l:
+ \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p
+m)) \to (le n m))))
+\def
+ \lambda (p: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (m:
+nat).((le (plus n n0) (plus n m)) \to (le n0 m))))) (\lambda (n:
+nat).(\lambda (m: nat).(\lambda (H: (le n m)).H))) (\lambda (p0:
+nat).(\lambda (IHp: ((\forall (n: nat).(\forall (m: nat).((le (plus p0 n)
+(plus p0 m)) \to (le n m)))))).(\lambda (n: nat).(\lambda (m: nat).(\lambda
+(H: (le (S (plus p0 n)) (S (plus p0 m)))).(IHp n m (le_S_n (plus p0 n) (plus
+p0 m) H))))))) p).
+
+theorem le_plus_trans:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n
+(plus m p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n
+m)).(le_trans n m (plus m p) H (le_plus_l m p))))).
+
+theorem le_reg_l:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus
+p n) (plus p m)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0:
+nat).((le n m) \to (le (plus n0 n) (plus n0 m)))) (\lambda (H: (le n m)).H)
+(\lambda (p0: nat).(\lambda (IHp: (((le n m) \to (le (plus p0 n) (plus p0
+m))))).(\lambda (H: (le n m)).(le_n_S (plus p0 n) (plus p0 m) (IHp H)))))
+p))).
+
+theorem le_plus_plus:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le
+n m) \to ((le p q) \to (le (plus n p) (plus m q)))))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q:
+nat).(\lambda (H: (le n m)).(\lambda (H0: (le p q)).(le_ind n (\lambda (n0:
+nat).(le (plus n p) (plus n0 q))) (le_reg_l p q n H0) (\lambda (m0:
+nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus m0 q))).(le_S
+(plus n p) (plus m0 q) H2)))) m H)))))).
+
+theorem le_plus_minus:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m
+n)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le n m)).(le_elim_rel
+(\lambda (n0: nat).(\lambda (n1: nat).(eq nat n1 (plus n0 (minus n1 n0)))))
+(\lambda (p: nat).(minus_n_O p)) (\lambda (p: nat).(\lambda (q: nat).(\lambda
+(_: (le p q)).(\lambda (H0: (eq nat q (plus p (minus q p)))).(f_equal nat nat
+S q (plus p (minus q p)) H0))))) n m Le))).
+
+theorem le_plus_minus_r:
+ \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m
+n)) m)))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(sym_eq nat m
+(plus n (minus m n)) (le_plus_minus n m H)))).
+
+theorem simpl_lt_plus_l:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p
+m)) \to (lt n m))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0:
+nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) (\lambda (H: (lt n m)).H)
+(\lambda (p0: nat).(\lambda (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n
+m)))).(\lambda (H: (lt (S (plus p0 n)) (S (plus p0 m)))).(IHp (le_S_n (S
+(plus p0 n)) (plus p0 m) H))))) p))).
+
+theorem lt_reg_l:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus
+p n) (plus p m)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(nat_ind (\lambda (n0:
+nat).((lt n m) \to (lt (plus n0 n) (plus n0 m)))) (\lambda (H: (lt n m)).H)
+(\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to (lt (plus p0 n) (plus p0
+m))))).(\lambda (H: (lt n m)).(lt_n_S (plus p0 n) (plus p0 m) (IHp H)))))
+p))).
+
+theorem lt_reg_r:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus
+n p) (plus m p)))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n
+m)).(eq_ind_r nat (plus p n) (\lambda (n0: nat).(lt n0 (plus m p))) (eq_ind_r
+nat (plus p m) (\lambda (n0: nat).(lt (plus p n) n0)) (nat_ind (\lambda (n0:
+nat).(lt (plus n0 n) (plus n0 m))) H (\lambda (n0: nat).(\lambda (_: (lt
+(plus n0 n) (plus n0 m))).(lt_reg_l n m (S n0) H))) p) (plus m p) (plus_sym m
+p)) (plus n p) (plus_sym n p))))).
+
+theorem le_lt_plus_plus:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le
+n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q:
+nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(eq_ind_r nat (plus n
+(S p)) (\lambda (n0: nat).(le n0 (plus m q))) (le_plus_plus n m (S p) q H H0)
+(plus (S n) p) (plus_Snm_nSm n p))))))).
+
+theorem lt_le_plus_plus:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt
+n m) \to ((le p q) \to (lt (plus n p) (plus m q)))))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q:
+nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(le_plus_plus (S n) m
+p q H H0)))))).
+
+theorem lt_plus_plus:
+ \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt
+n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
+\def
+ \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q:
+nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(lt_le_plus_plus n m p q
+H (lt_le_weak p q H0))))))).
+
+theorem well_founded_ltof:
+ \forall (A: Set).(\forall (f: ((A \to nat))).(well_founded A (ltof A f)))
+\def
+ \lambda (A: Set).(\lambda (f: ((A \to nat))).(let H \def (\lambda (n:
+nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to (Acc A
+(ltof A f) a)))) (\lambda (a: A).(\lambda (H: (lt (f a) O)).(False_ind (Acc A
+(ltof A f) a) (let H0 \def H in ((let H1 \def (lt_n_O (f a)) in (\lambda (H2:
+(lt (f a) O)).(H1 H2))) H0))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall
+(a: A).((lt (f a) n0) \to (Acc A (ltof A f) a))))).(\lambda (a: A).(\lambda
+(ltSma: (lt (f a) (S n0))).(Acc_intro A (ltof A f) a (\lambda (b: A).(\lambda
+(ltfafb: (lt (f b) (f a))).(IHn b (lt_le_trans (f b) (f a) n0 ltfafb
+(lt_n_Sm_le (f a) n0 ltSma)))))))))) n)) in (\lambda (a: A).(H (S (f a)) a
+(le_n (S (f a))))))).
+
+theorem lt_wf:
+ well_founded nat lt
+\def
+ well_founded_ltof nat (\lambda (m: nat).m).
+
+theorem lt_wf_ind:
+ \forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n:
+nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n)))) \to (P p)))
+\def
+ \lambda (p: nat).(\lambda (P: ((nat \to Prop))).(\lambda (H: ((\forall (n:
+nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n))))).(Acc_ind nat lt
+(\lambda (n: nat).(P n)) (\lambda (x: nat).(\lambda (_: ((\forall (y:
+nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: ((\forall (y: nat).((lt y
+x) \to (P y))))).(H x H1)))) p (lt_wf p)))).
+
--- /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 automatically generated: do not edit *********************)
+
+include "Legacy-1/coq/defs.ma".
+
--- /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 *)
+(* *)
+(**************************************************************************)
+
+inductive False: Prop \def .
--- /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 automatically generated: do not edit *********************)
+
+include "Legacy-1/theory.ma".
+
--- /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 automatically generated: do not edit *********************)
+
+include "Legacy-1/coq/props.ma".
+
DIR=$(shell basename $$PWD)
-MMAS = $(shell find Base-2 -name "*.mma")
+MMAS = $(shell find -name "*.mma") # Base-2
MAS = $(MMAS:%.mma=%.ma)
XMAS = Base-2/theory.ma LambdaDelta-2/theory.ma
depend:
@echo matitadep
$(H)../../matitadep $(foreach FILE,$(XMAS),-exclude $(FILE))
- $(H)cat Base-2/depends >> depends
+# $(H)cat Base-2/depends >> depends
depend.opt:
@echo matitadep.opt
$(H)../../matitadep.opt $(foreach FILE,$(XMAS),-exclude $(FILE))
- $(H)cat Base-2/depends >> depends
+# $(H)cat Base-2/depends >> depends
depends: depend.opt
$(H)../../matitac.opt $(MATITAOPTIONS) -dump $@ $< 2> /dev/null
$(H)echo $@ `../../matitadep.opt -stdout $@` >> depends
-include Base-2/.depend
+#include Base-2/.depend
baseuri=cic:/matita/LAMBDA-TYPES
-include_paths= ../../legacy
DIR=$(shell basename $$PWD)
-MATITAOPTIONS=$(MATITAUSEROPTIONS) -onepass
+MATITAOPTIONS=-onepass
$(DIR) all:
../matitac $(MATITAOPTIONS)
projects / helm.git / commitdiff