From f5dfc6c24a393a4717a7b40689df768d271d9ac0 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Tue, 18 Mar 2008 19:12:57 +0000 Subject: [PATCH] Procedural : tentative update to the new letin cic construction LAMBDA-TYPES: level 1 is stand alone instead of coq dependent level 2 is untested so will not compile for now :) legacy : unpublished --- .../acic_procedural/acic2Procedural.ml | 17 +- .../acic_procedural/proceduralConversion.ml | 12 +- .../acic_procedural/proceduralOptimizer.ml | 67 +- .../contribs/LAMBDA-TYPES/Base-1/ext/arith.ma | 64 +- .../contribs/LAMBDA-TYPES/Base-1/preamble.ma | 97 +-- .../LAMBDA-TYPES/LambdaDelta-1/C/props.ma | 6 +- .../LAMBDA-TYPES/LambdaDelta-1/arity/fwd.ma | 12 +- .../LAMBDA-TYPES/LambdaDelta-1/arity/props.ma | 4 +- .../LambdaDelta-1/arity/subst0.ma | 242 +++---- .../LAMBDA-TYPES/LambdaDelta-1/csubc/drop1.ma | 206 ++---- .../LAMBDA-TYPES/LambdaDelta-1/drop/props.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/flt/props.ma | 16 +- .../LAMBDA-TYPES/LambdaDelta-1/getl/drop.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma | 38 +- .../LAMBDA-TYPES/LambdaDelta-1/lift/props.ma | 66 +- .../LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma | 4 +- .../LAMBDA-TYPES/LambdaDelta-1/nf2/arity.ma | 28 +- .../LAMBDA-TYPES/LambdaDelta-1/nf2/dec.ma | 4 +- .../LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/pc3/dec.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/pr0/fwd.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/pr0/pr0.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma | 30 +- .../LAMBDA-TYPES/LambdaDelta-1/pr2/subst1.ma | 6 +- .../LAMBDA-TYPES/LambdaDelta-1/preamble.ma | 27 - .../LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma | 136 ++-- .../LAMBDA-TYPES/LambdaDelta-1/sn3/fwd.ma | 12 +- .../LAMBDA-TYPES/LambdaDelta-1/sn3/props.ma | 48 +- .../LAMBDA-TYPES/LambdaDelta-1/spare.ma | 12 +- .../LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma | 18 +- .../LambdaDelta-1/subst0/props.ma | 6 +- .../LambdaDelta-1/subst0/subst0.ma | 115 ++-- .../LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma | 184 +++--- .../LAMBDA-TYPES/LambdaDelta-1/subst1/fwd.ma | 2 +- .../LambdaDelta-1/subst1/props.ma | 2 +- .../LambdaDelta-1/subst1/subst1.ma | 4 +- .../LAMBDA-TYPES/LambdaDelta-1/tau0/props.ma | 8 +- .../LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma | 49 +- .../LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma | 10 +- .../LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma | 2 +- .../LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma | 4 +- .../LambdaDelta-1/ty3/pr3_props.ma | 8 +- .../LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma | 193 +++--- .../LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma | 32 +- .../LAMBDA-TYPES/Legacy-1/coq/defs.ma | 99 +++ .../LAMBDA-TYPES/Legacy-1/coq/props.ma | 598 ++++++++++++++++++ .../LAMBDA-TYPES/Legacy-1/definitions.ma | 18 + .../LAMBDA-TYPES/Legacy-1/preamble.ma | 15 + .../contribs/LAMBDA-TYPES/Legacy-1/spare.ma | 18 + .../contribs/LAMBDA-TYPES/Legacy-1/theory.ma | 18 + .../matita/contribs/LAMBDA-TYPES/Makefile | 8 +- .../matita/contribs/LAMBDA-TYPES/root | 1 - helm/software/matita/legacy/Makefile | 2 +- 58 files changed, 1569 insertions(+), 1021 deletions(-) create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma create mode 100644 helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma diff --git a/helm/software/components/acic_procedural/acic2Procedural.ml b/helm/software/components/acic_procedural/acic2Procedural.ml index 18b4f064f..483378ce6 100644 --- a/helm/software/components/acic_procedural/acic2Procedural.ml +++ b/helm/software/components/acic_procedural/acic2Procedural.ml @@ -282,7 +282,7 @@ let rec proc_lambda st name v t = 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 @@ -300,8 +300,7 @@ and proc_letin st what name v t = 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 @@ -385,12 +384,12 @@ and proc_proof st t = {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 diff --git a/helm/software/components/acic_procedural/proceduralConversion.ml b/helm/software/components/acic_procedural/proceduralConversion.ml index b3a247b02..324141af4 100644 --- a/helm/software/components/acic_procedural/proceduralConversion.ml +++ b/helm/software/components/acic_procedural/proceduralConversion.ml @@ -210,9 +210,9 @@ let get_clears c p xtypes = 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 @@ -220,9 +220,9 @@ let get_clears c p xtypes = 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 diff --git a/helm/software/components/acic_procedural/proceduralOptimizer.ml b/helm/software/components/acic_procedural/proceduralOptimizer.ml index 88ad74a66..776d52645 100644 --- a/helm/software/components/acic_procedural/proceduralOptimizer.ml +++ b/helm/software/components/acic_procedural/proceduralOptimizer.ml @@ -30,6 +30,8 @@ module S = CicSubstitution module DTI = DoubleTypeInference module HEL = HExtlib module PEH = ProofEngineHelpers +module TC = CicTypeChecker +module Un = CicUniv module H = ProceduralHelpers module Cl = ProceduralClassify @@ -38,10 +40,16 @@ 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 @@ -61,27 +69,26 @@ let rec add_abst k = 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 @@ -102,8 +109,8 @@ and opt1_appl g es c t vs = 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 @@ -116,7 +123,7 @@ and opt1_appl g es c t vs = | 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 @@ -126,11 +133,11 @@ and opt1_appl g es c t vs = 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)) @@ -143,8 +150,8 @@ and opt1_appl g es c 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 () @@ -191,8 +198,7 @@ and opt1_cast g es c t w = 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 @@ -224,12 +230,10 @@ let eta_expand g tys 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 @@ -261,11 +265,10 @@ and opt2_other g 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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/ext/arith.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/ext/arith.ma index 0d0f2a5df..8be48633f 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/ext/arith.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/ext/arith.ma @@ -61,10 +61,10 @@ theorem simpl_plus_r: (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))) @@ -75,7 +75,7 @@ theorem minus_plus_r: \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 @@ -85,9 +85,8 @@ y) z) (plus (plus x z) y)))) (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 @@ -96,8 +95,8 @@ h) k) (plus n (plus k h))))) \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 @@ -211,14 +210,14 @@ n)))) \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))))))). @@ -306,17 +305,17 @@ theorem lt_x_plus_x_Sy: \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 @@ -356,7 +355,7 @@ theorem lt_plus_minus_r: \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))))) @@ -387,7 +386,7 @@ theorem lt_le_minus: \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)) @@ -442,7 +441,7 @@ theorem lt_neq: \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) @@ -455,8 +454,8 @@ h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2 (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: @@ -504,19 +503,20 @@ nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True 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: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/preamble.ma index d215fd146..16ff2dc44 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/preamble.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Base-1/preamble.ma @@ -12,99 +12,4 @@ (* *) (**************************************************************************) -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". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma index 907c5c373..c93aa0f08 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/C/props.ma @@ -23,15 +23,15 @@ theorem clt_cong: 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: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/fwd.ma index cba7ad939..292c4b65b 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/fwd.ma @@ -18,8 +18,6 @@ include "LambdaDelta-1/arity/defs.ma". include "LambdaDelta-1/leq/asucc.ma". -include "LambdaDelta-1/leq/fwd.ma". - include "LambdaDelta-1/getl/drop.ma". theorem arity_gen_sort: @@ -908,7 +906,7 @@ H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat 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 @@ -965,7 +963,7 @@ a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (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)) @@ -982,7 +980,7 @@ a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u (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) @@ -997,7 +995,7 @@ x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def (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)) @@ -1014,7 +1012,7 @@ t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4 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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/props.ma index 2d00abadf..2e45247f2 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/props.ma @@ -294,8 +294,8 @@ T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_: (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: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma index d62f665f4..930e4c1ba 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/arity/subst0.ma @@ -49,8 +49,8 @@ A).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead d (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 @@ -70,7 +70,7 @@ d0 (Bind Void) u0)) \to (\forall (w: T).(\forall (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 +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 @@ -274,16 +274,16 @@ c (CHead d1 (Bind Abbr) u))).(\lambda (c2: C).(\lambda (t2: T).(\lambda (H1: 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: @@ -298,9 +298,9 @@ i) c2 t2)).(let H_x \def (fsubst0_gen_base c c2 (TLRef i) t2 u0 i0 H4) in (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 @@ -320,7 +320,7 @@ u H14) in (eq_ind T u (\lambda (t: T).(arity g c (lift (S i) O t) a0)) (let 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) @@ -458,9 +458,9 @@ H13 Abbr H20) in (arity_abbr g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abbr) 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: @@ -495,9 +495,9 @@ C).(\lambda (t2: T).(\lambda (H4: (fsubst0 i0 u0 c (TLRef i) c2 t2)).(let H_x (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 @@ -513,7 +513,7 @@ False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) (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 @@ -650,9 +650,9 @@ H13 Abst H20) in (arity_abst g c2 x2 x4 i H24 a0 (H2 d1 u0 (r (Bind Abst) 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: @@ -685,7 +685,7 @@ t) t2 u0 i H6) in (let H7 \def H_x in (or3_ind (land (eq C c c2) (subst0 i u0 (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 @@ -728,7 +728,7 @@ b c u) (CHead d1 (Bind Abbr) u0) i H5) (CHead c (Bind b) x0) x1 (fsubst0_both (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 @@ -737,99 +737,99 @@ i H5 c2 u (fsubst0_fst i u0 c u c2 H10)) t a2 (H4 d1 u0 (S i) (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 @@ -843,7 +843,7 @@ d1 u0 i H4 c x0 (fsubst0_snd i u0 c u x0 H12)) x1 a2 (H3 d1 u0 (S i) 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 @@ -853,7 +853,7 @@ c u (clear_bind Abst c u) (CHead d1 (Bind Abbr) u0) i H4) (CHead c2 (Bind 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))) @@ -915,7 +915,7 @@ Appl) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) (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 @@ -951,13 +951,13 @@ g c t2 a2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (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 @@ -1010,7 +1010,7 @@ Cast) u t) t2 u0 i H5) in (let H6 \def H_x in (or3_ind (land (eq C c c2) (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 @@ -1046,13 +1046,13 @@ g c t2 a0) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (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 @@ -1099,16 +1099,16 @@ g c2 t2 a1)))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda 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))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop1.ma index 1e4e8fb48..6b754a9ab 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop1.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csubc/drop1.ma @@ -25,85 +25,30 @@ C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (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: @@ -114,83 +59,28 @@ C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (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)). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma index 0ca510275..a0744633e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/drop/props.ma @@ -347,7 +347,7 @@ a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c 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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma index 65636c34f..9c94c2111 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex1/props.ma @@ -460,7 +460,7 @@ x10 t x12)) H33 (TSort O) H35) in (let H38 \def (eq_ind T x11 (\lambda (t: 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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma index c526d5b60..20356f946 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/flt/props.ma @@ -23,8 +23,8 @@ theorem flt_thead_sx: (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: @@ -32,8 +32,8 @@ 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: @@ -46,7 +46,7 @@ k u) t c (THead k u t))))) (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))))))). @@ -69,7 +69,7 @@ K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1) (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: @@ -81,8 +81,8 @@ c1 t1 (CHead c2 k2 t2) (TLRef j))))))))) (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/drop.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/drop.ma index 7f0ade54d..507560f88 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/drop.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/getl/drop.ma @@ -353,7 +353,7 @@ nat).(\lambda (H: (getl i c1 (CHead e (Bind b) u))).(\lambda (c2: C).(\lambda (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: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma index 3dbbd6147..db188a250 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma @@ -144,7 +144,7 @@ i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def 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 @@ -171,16 +171,16 @@ theorem lift_gen_lref_lt: 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 @@ -192,12 +192,12 @@ n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T (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 @@ -208,14 +208,14 @@ n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d 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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma index 168d3e09c..20649f9bf 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/props.ma @@ -105,7 +105,7 @@ theorem lift_lref_gt: (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))))))). @@ -234,30 +234,30 @@ n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1)) 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 @@ -265,8 +265,8 @@ plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2 h2 (sym_eq nat (minus (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 @@ -411,7 +411,7 @@ n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift 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 @@ -474,17 +474,17 @@ nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n))))) (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma index 7e7f6a80a..8f399a0fd 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift1/props.ma @@ -114,14 +114,14 @@ t)) (lift (S (plus (trans hds0 i) h)) O (lift1 (ptrans hds0 i) t)))) (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). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/arity.ma index 0b17ee378..9fe69d766 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/arity.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/arity.ma @@ -176,7 +176,7 @@ nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (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 @@ -213,21 +213,21 @@ 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))))))))).(\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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/dec.ma index 37cc2b116..eaaf4346a 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/dec.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/dec.ma @@ -95,7 +95,7 @@ x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 (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 @@ -128,7 +128,7 @@ x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) 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: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma index 0bafedff2..17e87d94d 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/props.ma @@ -93,7 +93,7 @@ TList).((nfs2 c vs) \to (nf2 c (THeads (Flat Appl) vs (TLRef i))))))) (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/dec.ma index 35ebcfd34..74d45cd17 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/dec.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/dec.ma @@ -135,7 +135,7 @@ u0) x4 x6))))).(let H27 \def (eq_ind T x (\lambda (t: T).(\forall (t0: 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) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma index 83db5a284..7dc245d1c 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pc3/fwd.ma @@ -82,7 +82,7 @@ Abst) u) t1 t2))))) \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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma index cf86f1393..1ad44c87a 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/dec.ma @@ -203,7 +203,7 @@ H10 \def (eq_ind T t0 (\lambda (t3: T).(subst0 O t t3 (lift (S O) O x))) H3 (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/fwd.ma index e6ca5293d..6a1bee0c7 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/fwd.ma @@ -1950,7 +1950,7 @@ x2))).(\lambda (H7: (eq T (lift (S O) O t2) (lift h (S x1) x3))).(eq_ind_r 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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/pr0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/pr0.ma index 57e9f7ef1..2e70987d7 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/pr0.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/pr0.ma @@ -256,7 +256,7 @@ t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda 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)))))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma index 09a9a3d89..ab6e6ef40 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/clen.ma @@ -100,7 +100,7 @@ v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma index c332a5ffb..fe6666a1d 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/fwd.ma @@ -78,21 +78,21 @@ in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (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))))))) (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/subst1.ma index 936e655f1..35c106e6e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/subst1.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr2/subst1.ma @@ -243,7 +243,7 @@ T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t 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) @@ -262,9 +262,9 @@ x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/preamble.ma index b94953a5b..069fa0d27 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/preamble.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/preamble.ma @@ -13,30 +13,3 @@ (**************************************************************************) 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". diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma index 16393b04b..549441f26 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/arity.ma @@ -232,7 +232,7 @@ is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (AHead a1 a2) c2 (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma index e30101bac..d63d5e3cd 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma @@ -41,14 +41,14 @@ t) \to (arity g c t a))))) \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 @@ -70,8 +70,8 @@ g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall 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) @@ -97,7 +97,7 @@ t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (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)))))))))).(\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 @@ -133,7 +133,7 @@ nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t)))))))))) (\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 @@ -147,9 +147,9 @@ C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall (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 @@ -199,7 +199,7 @@ i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs: 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: @@ -220,8 +220,8 @@ i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (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 @@ -279,12 +279,12 @@ T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) (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 @@ -300,13 +300,13 @@ H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with (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 @@ -328,7 +328,7 @@ is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land (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 @@ -339,7 +339,7 @@ t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0 (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)) @@ -386,7 +386,7 @@ c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c ((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: @@ -412,7 +412,7 @@ PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (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 @@ -421,7 +421,7 @@ d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (and_ind 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 @@ -429,9 +429,9 @@ nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to 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 @@ -448,7 +448,7 @@ I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1) (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 @@ -461,8 +461,8 @@ vs (TLRef i)) (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 (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) @@ -471,7 +471,7 @@ C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_: ((\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 @@ -504,7 +504,7 @@ t) \to (sn3 c t))))) \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 @@ -522,7 +522,7 @@ i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))) \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 @@ -549,7 +549,7 @@ nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: 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 @@ -571,37 +571,37 @@ t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (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 @@ -625,7 +625,7 @@ Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda 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 @@ -652,9 +652,9 @@ TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land 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: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/fwd.ma index c55628d8a..26a719b58 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/fwd.ma @@ -53,7 +53,7 @@ x0) (THead (Bind b) t2 x0) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 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: @@ -66,7 +66,7 @@ x0) (THead (Bind b) x t2) H8) in (let H10 \def (eq_ind_r T t2 (\lambda (t0: 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))))). @@ -105,7 +105,7 @@ H10 \def (eq_ind_r T t2 (\lambda (t0: 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 (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 @@ -117,7 +117,7 @@ in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 | (TLRef _) 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: @@ -128,11 +128,11 @@ 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: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/props.ma index 254589a27..9341f979b 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sn3/props.ma @@ -157,7 +157,7 @@ theorem sn3_shift: \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))))))). @@ -223,7 +223,7 @@ c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s (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) @@ -231,8 +231,8 @@ H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1 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 @@ -2165,7 +2165,7 @@ t)))))))))) \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 @@ -2204,17 +2204,17 @@ c t1) \to (sn3 c (THeads (Flat Appl) t1 (TLRef i))))) \to ((land (sn3 c t) (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 @@ -2258,22 +2258,22 @@ t3))))))))).(\lambda (u: T).(\lambda (H1: (sn3 c (THead (Flat Appl) t (THeads (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: @@ -2308,7 +2308,7 @@ t1))))))))).(\lambda (H2: ((\forall (t1: T).((sn3 (CHead c (Bind b) 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 @@ -2361,7 +2361,7 @@ Appl) (TCons t0 t1) (THead (Bind Abbr) v t))) \to (\forall (w: T).((sn3 c w) (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) @@ -2466,7 +2466,7 @@ i)))))))).(\lambda (H1: (((sn3 c (THeads (Flat Appl) (TCons t t0) (lift (S i) 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 @@ -2487,7 +2487,7 @@ i c d) \to (\forall (ts: TList).((sns3 d ts) \to (sns3 c (lifts h i ts)))))))) 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)))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma index 463e2d7a3..040ab8d46 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/spare.ma @@ -71,14 +71,14 @@ theorem nfs2_tapp: \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))). @@ -404,7 +404,7 @@ u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat 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) @@ -435,9 +435,9 @@ H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift 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))))))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma index 43d9e4df3..4b387483e 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma @@ -320,7 +320,7 @@ t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda 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 @@ -338,7 +338,7 @@ O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (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 @@ -499,13 +499,13 @@ T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i: 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 @@ -573,15 +573,15 @@ nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d (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 @@ -597,9 +597,9 @@ d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u) 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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma index a9ded88cf..bccbf9169 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/props.ma @@ -25,7 +25,7 @@ nat).((subst0 d u t0 t0) \to (\forall (P: Prop).P)))) (\lambda (n: 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: @@ -160,7 +160,7 @@ T).(\lambda (i0: nat).(\lambda (d: nat).(\lambda (H0: (le d i0)).(eq_ind_r T (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: @@ -213,7 +213,7 @@ t1) (lift (S O) d t2)))))))) (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/subst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/subst0.ma index b234990ee..43747913f 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/subst0.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/subst0.ma @@ -293,7 +293,7 @@ T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq 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: @@ -394,10 +394,14 @@ T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) (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: @@ -413,20 +417,24 @@ x1) t))) (\lambda (x: T).(\lambda (H8: (subst0 (s k i2) u2 t2 x)).(\lambda (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 (_: @@ -460,31 +468,40 @@ T).(subst0 i2 u3 (THead k u2 t3) t5)) (\lambda (t5: T).(subst0 i v t t5)))) 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 @@ -498,9 +515,9 @@ T).(\lambda (t2: T).(\lambda (t3: T).(\forall (t4: T).((subst0 n t t2 t4) \to (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: @@ -1360,13 +1377,13 @@ i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S O) i 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)))))))). diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma index 3cad5ab9c..0fc817dcd 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/tlt.ma @@ -52,7 +52,7 @@ i) O v)) (g i)) \to (le (weight_map f (THead (Bind b0) u2 t0)) (weight_map 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 (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 @@ -61,20 +61,20 @@ H3)) 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))))))))) (\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 @@ -102,8 +102,8 @@ t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 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 @@ -118,8 +118,8 @@ t2) (weight_map g t1)))))))).(\lambda (u0: T).(\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 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)) @@ -132,41 +132,40 @@ t1)))))))).(\lambda (u0: T).(\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 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) @@ -176,9 +175,9 @@ 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 (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 @@ -191,10 +190,10 @@ i))) \to (le (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \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 @@ -204,8 +203,8 @@ t2) (weight_map g t1)))))))).(\lambda (f: ((nat \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) +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)) @@ -216,10 +215,9 @@ nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \to ((lt (weight_map f0 (lift 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 @@ -251,7 +249,7 @@ i) O v)) (g i)) \to (lt (weight_map f (THead (Bind b0) u2 t0)) (weight_map 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 (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 @@ -260,23 +258,23 @@ H3)) 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) (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 @@ -303,8 +301,8 @@ t1))))))))))))))) (\lambda (v: T).(\lambda (t2: T).(\lambda (t1: T).(\lambda 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 @@ -319,8 +317,8 @@ t2) (weight_map g t1)))))))).(\lambda (u0: T).(\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 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)) @@ -333,9 +331,9 @@ t1)))))))).(\lambda (u0: T).(\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 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: @@ -347,7 +345,7 @@ 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))).(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 @@ -377,10 +375,10 @@ 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 (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 @@ -393,8 +391,8 @@ t1)))))))).(\lambda (f: ((nat \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) +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) @@ -406,13 +404,13 @@ i))) \to (lt (weight_map f t2) (weight_map g t1)))))))).(\lambda (f: ((nat \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) @@ -420,7 +418,7 @@ nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).(le (f0 m) (g m)))) \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))))). @@ -432,7 +430,7 @@ theorem subst0_tlt_head: 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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/fwd.ma index 8610452f1..285a870e2 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/fwd.ma @@ -38,7 +38,7 @@ i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift (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) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/props.ma index 5a4059bfc..cb13ac644 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/props.ma @@ -138,7 +138,7 @@ nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/subst1.ma index c683a4940..50f389929 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/subst1.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst1/subst1.ma @@ -176,7 +176,7 @@ t2))))) (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (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 @@ -188,7 +188,7 @@ t3)) (\lambda (y0: T).(\lambda (H5: (subst1 i u t0 y0)).(subst1_ind i u t0 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))) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/props.ma index 36c004667..236215b22 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tau0/props.ma @@ -71,10 +71,10 @@ O w)))) (eq_ind_r T (lift (plus h (S i)) O w) (\lambda (t: T).(tau0 g c0 (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: @@ -114,10 +114,10 @@ O v)))) (eq_ind_r T (lift (plus h (S i)) O v) (\lambda (t: T).(tau0 g c0 (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: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma index bd6794ad0..72b13d733 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/tlt/props.ma @@ -88,7 +88,7 @@ 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 (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)) @@ -100,31 +100,30 @@ t0) (weight_map g t0))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (f: (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma index ee064039c..7a71c7d35 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/fsubst0.ma @@ -78,7 +78,7 @@ t0)))))))))).(\lambda (i: nat).(\lambda (u0: T).(\lambda (c2: C).(\lambda (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: @@ -237,7 +237,7 @@ H14)) H13))))))))))) H8)) H7))) (\lambda (H6: (le i n)).(ty3_abbr g n c3 d u (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: @@ -271,7 +271,7 @@ c2 t2)).(fsubst0_ind i u0 c (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 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 @@ -440,8 +440,8 @@ H7))) (\lambda (H6: (le i n)).(ty3_abst g n c3 d u (csubst0_getl_ge i n H6 c 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: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma index f76e49289..0c23d2956 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/nf2.ma @@ -257,7 +257,7 @@ x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(ex3_2_ind T T (\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: diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma index b689bf35c..659b8fd84 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3.ma @@ -250,7 +250,7 @@ t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind 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 @@ -396,7 +396,7 @@ g (CHead c2 (Bind Abst) u0) t4 t5))) (ty3 g c2 (THead (Bind Abbr) v2 t4) (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) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma index 14ce7c088..20d795000 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props.ma @@ -118,7 +118,7 @@ nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda \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: @@ -154,7 +154,7 @@ x1 (S n)) x4) t H17) (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4)) 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 @@ -182,7 +182,7 @@ nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda \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: @@ -224,7 +224,7 @@ n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus x1 (S n)) O (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma index eeff4dc80..bf9f641b0 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/props.ma @@ -80,10 +80,10 @@ t)))) (eq_ind_r T (lift (plus h (S n)) O t) (\lambda (t0: T).(ty3 g c0 (TLRef (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 @@ -123,10 +123,10 @@ u)))) (eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t0: T).(ty3 g c0 (TLRef (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 @@ -492,101 +492,102 @@ 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 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) diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma index 6dfdb48e4..dc7c1df85 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma @@ -234,10 +234,10 @@ 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) 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: @@ -371,10 +371,10 @@ 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) 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: @@ -739,12 +739,12 @@ n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S 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: @@ -856,9 +856,9 @@ O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda (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 diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma new file mode 100644 index 000000000..7d4696229 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/defs.ma @@ -0,0 +1,99 @@ +(**************************************************************************) +(* ___ *) +(* ||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))))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma new file mode 100644 index 000000000..e9d29ff6d --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/coq/props.ma @@ -0,0 +1,598 @@ +(**************************************************************************) +(* ___ *) +(* ||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)))). + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma new file mode 100644 index 000000000..63fc85890 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/definitions.ma @@ -0,0 +1,18 @@ +(**************************************************************************) +(* ___ *) +(* ||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". + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma new file mode 100644 index 000000000..96c1bc1fa --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/preamble.ma @@ -0,0 +1,15 @@ +(**************************************************************************) +(* ___ *) +(* ||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 . diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma new file mode 100644 index 000000000..77939a1b6 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/spare.ma @@ -0,0 +1,18 @@ +(**************************************************************************) +(* ___ *) +(* ||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". + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma new file mode 100644 index 000000000..4ee597e09 --- /dev/null +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Legacy-1/theory.ma @@ -0,0 +1,18 @@ +(**************************************************************************) +(* ___ *) +(* ||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". + diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Makefile b/helm/software/matita/contribs/LAMBDA-TYPES/Makefile index 214ce0711..0ccf2514a 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Makefile +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Makefile @@ -4,7 +4,7 @@ MATITAOPTIONS=$(MATITAUSEROPTIONS) -onepass DIR=$(shell basename $$PWD) -MMAS = $(shell find Base-2 -name "*.mma") +MMAS = $(shell find -name "*.mma") # Base-2 MAS = $(MMAS:%.mma=%.ma) XMAS = Base-2/theory.ma LambdaDelta-2/theory.ma @@ -39,11 +39,11 @@ clean.ma: depend: @echo matitadep $(H)../../matitadep $(foreach FILE,$(XMAS),-exclude $(FILE)) - $(H)cat Base-2/depends >> depends +# $(H)cat Base-2/depends >> depends depend.opt: @echo matitadep.opt $(H)../../matitadep.opt $(foreach FILE,$(XMAS),-exclude $(FILE)) - $(H)cat Base-2/depends >> depends +# $(H)cat Base-2/depends >> depends depends: depend.opt @@ -52,4 +52,4 @@ depends: depend.opt $(H)../../matitac.opt $(MATITAOPTIONS) -dump $@ $< 2> /dev/null $(H)echo $@ `../../matitadep.opt -stdout $@` >> depends -include Base-2/.depend +#include Base-2/.depend diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/root b/helm/software/matita/contribs/LAMBDA-TYPES/root index 6819b272f..ca1729d65 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/root +++ b/helm/software/matita/contribs/LAMBDA-TYPES/root @@ -1,2 +1 @@ baseuri=cic:/matita/LAMBDA-TYPES -include_paths= ../../legacy diff --git a/helm/software/matita/legacy/Makefile b/helm/software/matita/legacy/Makefile index a8d99c3dd..9dd21dd3b 100644 --- a/helm/software/matita/legacy/Makefile +++ b/helm/software/matita/legacy/Makefile @@ -1,5 +1,5 @@ DIR=$(shell basename $$PWD) -MATITAOPTIONS=$(MATITAUSEROPTIONS) -onepass +MATITAOPTIONS=-onepass $(DIR) all: ../matitac $(MATITAOPTIONS) -- 2.39.2