From ae22633ad934bba5fb32143f1726cdfbd255b899 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Wed, 25 Feb 2015 21:00:08 +0000 Subject: [PATCH] component: pr3 --- .../contribs/lambdadelta/basic_1/pr3/defs.ma | 2 +- .../contribs/lambdadelta/basic_1/pr3/fwd.ma | 2004 ++++------------- .../contribs/lambdadelta/basic_1/pr3/iso.ma | 300 ++- .../contribs/lambdadelta/basic_1/pr3/pr1.ma | 21 +- .../contribs/lambdadelta/basic_1/pr3/pr3.ma | 95 +- .../contribs/lambdadelta/basic_1/pr3/props.ma | 1437 +++++++++++- .../lambdadelta/basic_1/pr3/subst1.ma | 133 +- .../contribs/lambdadelta/basic_1/pr3/wcpr0.ma | 101 +- 8 files changed, 2118 insertions(+), 1975 deletions(-) diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma index 9619ee1d7..7430c894d 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/defs.ma @@ -14,7 +14,7 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr2/defs.ma". +include "basic_1/pr2/defs.ma". inductive pr3 (c: C): T \to (T \to Prop) \def | pr3_refl: \forall (t: T).(pr3 c t t) diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma index 726c420f0..ca979dd82 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/fwd.ma @@ -14,29 +14,39 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr3/props.ma". +include "basic_1/pr3/defs.ma". -include "Basic-1/pr2/fwd.ma". +include "basic_1/pr2/fwd.ma". + +let rec pr3_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P t +t))) (f0: (\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: +T).((pr3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (t: T) (t0: T) (p: pr3 +c t t0) on p: P t t0 \def match p with [(pr3_refl t1) \Rightarrow (f t1) | +(pr3_sing t2 t1 p0 t3 p1) \Rightarrow (let TMP_1 \def ((pr3_ind c P f f0) t2 +t3 p1) in (f0 t2 t1 p0 t3 p1 TMP_1))]. theorem pr3_gen_sort: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to (eq T x (TSort n))))) \def \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TSort -n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: -T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(pr3_ind c (\lambda -(t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: -T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) (\lambda (t2: -T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda -(_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TSort n)) \to (eq T t3 -t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda -(t: T).(pr2 c t t2)) H1 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t: -T).(eq T t3 t)) (let H6 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TSort n)) -\to (eq T t3 t))) H3 (TSort n) (pr2_gen_sort c t2 n H5)) in (H6 (refl_equal T -(TSort n)))) t1 H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 253 -END *) +n) x)).(let TMP_1 \def (TSort n) in (let TMP_2 \def (\lambda (t: T).(pr3 c t +x)) in (let TMP_3 \def (\lambda (t: T).(eq T x t)) in (let TMP_17 \def +(\lambda (y: T).(\lambda (H0: (pr3 c y x)).(let TMP_4 \def (\lambda (t: +T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0 t)))) in (let TMP_5 \def +(\lambda (t: T).(\lambda (_: (eq T t (TSort n))).(refl_equal T t))) in (let +TMP_16 \def (\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 +t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 +(TSort n)) \to (eq T t3 t2)))).(\lambda (H4: (eq T t1 (TSort n))).(let TMP_6 +\def (\lambda (t: T).(pr2 c t t2)) in (let TMP_7 \def (TSort n) in (let H5 +\def (eq_ind T t1 TMP_6 H1 TMP_7 H4) in (let TMP_8 \def (TSort n) in (let +TMP_9 \def (\lambda (t: T).(eq T t3 t)) in (let TMP_10 \def (\lambda (t: +T).((eq T t (TSort n)) \to (eq T t3 t))) in (let TMP_11 \def (TSort n) in +(let TMP_12 \def (pr2_gen_sort c t2 n H5) in (let H6 \def (eq_ind T t2 TMP_10 +H3 TMP_11 TMP_12) in (let TMP_13 \def (TSort n) in (let TMP_14 \def +(refl_equal T TMP_13) in (let TMP_15 \def (H6 TMP_14) in (eq_ind_r T TMP_8 +TMP_9 TMP_15 t1 H4)))))))))))))))))))) in (pr3_ind c TMP_4 TMP_5 TMP_16 y x +H0)))))) in (insert_eq T TMP_1 TMP_2 TMP_3 TMP_17 H)))))))). theorem pr3_gen_abst: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c @@ -46,73 +56,112 @@ c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))))))) \def \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y: -T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda (t: T).((eq T y (THead -(Bind Abst) u1 t)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) t t2)))))))) (unintro T u1 (\lambda (t: T).(\forall (x0: -T).((eq T y (THead (Bind Abst) t x0)) \to (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x0 t2))))))))) (pr3_ind c +(H: (pr3 c (THead (Bind Abst) u1 t1) x)).(let TMP_1 \def (Bind Abst) in (let +TMP_2 \def (THead TMP_1 u1 t1) in (let TMP_3 \def (\lambda (t: T).(pr3 c t +x)) in (let TMP_11 \def (\lambda (_: T).(let TMP_6 \def (\lambda (u2: +T).(\lambda (t2: T).(let TMP_4 \def (Bind Abst) in (let TMP_5 \def (THead +TMP_4 u2 t2) in (eq T x TMP_5))))) in (let TMP_7 \def (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) in (let TMP_10 \def (\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(let TMP_8 \def (Bind b) +in (let TMP_9 \def (CHead c TMP_8 u) in (pr3 TMP_9 t1 t2))))))) in (ex3_2 T T +TMP_6 TMP_7 TMP_10))))) in (let TMP_112 \def (\lambda (y: T).(\lambda (H0: +(pr3 c y x)).(let TMP_19 \def (\lambda (t: T).((eq T y (THead (Bind Abst) u1 +t)) \to (let TMP_14 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_12 \def +(Bind Abst) in (let TMP_13 \def (THead TMP_12 u2 t2) in (eq T x TMP_13))))) +in (let TMP_15 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) in (let +TMP_18 \def (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: +T).(let TMP_16 \def (Bind b) in (let TMP_17 \def (CHead c TMP_16 u) in (pr3 +TMP_17 t t2))))))) in (ex3_2 T T TMP_14 TMP_15 TMP_18)))))) in (let TMP_27 +\def (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Abst) t x0)) \to +(let TMP_22 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_20 \def (Bind +Abst) in (let TMP_21 \def (THead TMP_20 u2 t2) in (eq T x TMP_21))))) in (let +TMP_23 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) in (let TMP_26 +\def (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(let +TMP_24 \def (Bind b) in (let TMP_25 \def (CHead c TMP_24 u) in (pr3 TMP_25 x0 +t2))))))) in (ex3_2 T T TMP_22 TMP_23 TMP_26))))))) in (let TMP_35 \def (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t -(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2))))))))))) (\lambda (t: T).(\lambda -(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0 -x1))).(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind -Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t2))))) x0 x1 H1 (pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl -(CHead c (Bind b) u) x1)))))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda -(H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda -(H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Abst) x0 x1)) -\to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Bind Abst) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Bind Abst) x0 x1) H4) in (let H6 \def (pr2_gen_abst c x0 x1 -t2 H5) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead -c (Bind b) u) x1 t5))))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) x1 t5)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda -(H7: (eq T t2 (THead (Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 -x2)).(\lambda (H9: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 x3))))).(let H10 \def (eq_ind T t2 (\lambda (t: T).(\forall (x4: -T).(\forall (x5: T).((eq T t (THead (Bind Abst) x4 x5)) \to (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 -t5)))))))))) H3 (THead (Bind Abst) x2 x3) H7) in (let H11 \def (H10 x2 x3 -(refl_equal T (THead (Bind Abst) x2 x3))) in (ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T t4 (THead (Bind Abst) -x4 x5))).(\lambda (H13: (pr3 c x2 x4)).(\lambda (H14: ((\forall (b: -B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(ex3_2_intro T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abst) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) -x4 x5 H12 (pr3_sing c x2 x0 H8 x4 H13) (\lambda (b: B).(\lambda (u: -T).(pr3_sing (CHead c (Bind b) u) x3 x1 (H9 b u) x5 (H14 b u)))))))))) -H11)))))))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 1261 -END *) +(THead (Bind Abst) x0 x1)) \to (let TMP_30 \def (\lambda (u2: T).(\lambda +(t2: T).(let TMP_28 \def (Bind Abst) in (let TMP_29 \def (THead TMP_28 u2 t2) +in (eq T t0 TMP_29))))) in (let TMP_31 \def (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) in (let TMP_34 \def (\lambda (_: T).(\lambda (t2: +T).(\forall (b: B).(\forall (u: T).(let TMP_32 \def (Bind b) in (let TMP_33 +\def (CHead c TMP_32 u) in (pr3 TMP_33 x1 t2))))))) in (ex3_2 T T TMP_30 +TMP_31 TMP_34))))))))) in (let TMP_47 \def (\lambda (t: T).(\lambda (x0: +T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abst) x0 x1))).(let +TMP_38 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_36 \def (Bind Abst) in +(let TMP_37 \def (THead TMP_36 u2 t2) in (eq T t TMP_37))))) in (let TMP_39 +\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_42 \def +(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(let TMP_40 +\def (Bind b) in (let TMP_41 \def (CHead c TMP_40 u) in (pr3 TMP_41 x1 +t2))))))) in (let TMP_43 \def (pr3_refl c x0) in (let TMP_46 \def (\lambda +(b: B).(\lambda (u: T).(let TMP_44 \def (Bind b) in (let TMP_45 \def (CHead c +TMP_44 u) in (pr3_refl TMP_45 x1))))) in (ex3_2_intro T T TMP_38 TMP_39 +TMP_42 x0 x1 H1 TMP_43 TMP_46)))))))))) in (let TMP_109 \def (\lambda (t2: +T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda +(_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 +(THead (Bind Abst) x0 x1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T t4 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) x1 t5))))))))))).(\lambda (x0: T).(\lambda +(x1: T).(\lambda (H4: (eq T t3 (THead (Bind Abst) x0 x1))).(let TMP_48 \def +(\lambda (t: T).(pr2 c t t2)) in (let TMP_49 \def (Bind Abst) in (let TMP_50 +\def (THead TMP_49 x0 x1) in (let H5 \def (eq_ind T t3 TMP_48 H1 TMP_50 H4) +in (let H6 \def (pr2_gen_abst c x0 x1 t2 H5) in (let TMP_53 \def (\lambda +(u2: T).(\lambda (t5: T).(let TMP_51 \def (Bind Abst) in (let TMP_52 \def +(THead TMP_51 u2 t5) in (eq T t2 TMP_52))))) in (let TMP_54 \def (\lambda +(u2: T).(\lambda (_: T).(pr2 c x0 u2))) in (let TMP_57 \def (\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(let TMP_55 \def (Bind b) +in (let TMP_56 \def (CHead c TMP_55 u) in (pr2 TMP_56 x1 t5))))))) in (let +TMP_60 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_58 \def (Bind Abst) in +(let TMP_59 \def (THead TMP_58 u2 t5) in (eq T t4 TMP_59))))) in (let TMP_61 +\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_64 \def +(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(let TMP_62 +\def (Bind b) in (let TMP_63 \def (CHead c TMP_62 u) in (pr3 TMP_63 x1 +t5))))))) in (let TMP_65 \def (ex3_2 T T TMP_60 TMP_61 TMP_64) in (let +TMP_108 \def (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t2 (THead +(Bind Abst) x2 x3))).(\lambda (H8: (pr2 c x0 x2)).(\lambda (H9: ((\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let TMP_73 \def +(\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind Abst) +x4 x5)) \to (let TMP_68 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_66 +\def (Bind Abst) in (let TMP_67 \def (THead TMP_66 u2 t5) in (eq T t4 +TMP_67))))) in (let TMP_69 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 +u2))) in (let TMP_72 \def (\lambda (_: T).(\lambda (t5: T).(\forall (b: +B).(\forall (u: T).(let TMP_70 \def (Bind b) in (let TMP_71 \def (CHead c +TMP_70 u) in (pr3 TMP_71 x5 t5))))))) in (ex3_2 T T TMP_68 TMP_69 +TMP_72)))))))) in (let TMP_74 \def (Bind Abst) in (let TMP_75 \def (THead +TMP_74 x2 x3) in (let H10 \def (eq_ind T t2 TMP_73 H3 TMP_75 H7) in (let +TMP_76 \def (Bind Abst) in (let TMP_77 \def (THead TMP_76 x2 x3) in (let +TMP_78 \def (refl_equal T TMP_77) in (let H11 \def (H10 x2 x3 TMP_78) in (let +TMP_81 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_79 \def (Bind Abst) in +(let TMP_80 \def (THead TMP_79 u2 t5) in (eq T t4 TMP_80))))) in (let TMP_82 +\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) in (let TMP_85 \def +(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(let TMP_83 +\def (Bind b) in (let TMP_84 \def (CHead c TMP_83 u) in (pr3 TMP_84 x3 +t5))))))) in (let TMP_88 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_86 +\def (Bind Abst) in (let TMP_87 \def (THead TMP_86 u2 t5) in (eq T t4 +TMP_87))))) in (let TMP_89 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) in (let TMP_92 \def (\lambda (_: T).(\lambda (t5: T).(\forall (b: +B).(\forall (u: T).(let TMP_90 \def (Bind b) in (let TMP_91 \def (CHead c +TMP_90 u) in (pr3 TMP_91 x1 t5))))))) in (let TMP_93 \def (ex3_2 T T TMP_88 +TMP_89 TMP_92) in (let TMP_107 \def (\lambda (x4: T).(\lambda (x5: +T).(\lambda (H12: (eq T t4 (THead (Bind Abst) x4 x5))).(\lambda (H13: (pr3 c +x2 x4)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind +b) u) x3 x5))))).(let TMP_96 \def (\lambda (u2: T).(\lambda (t5: T).(let +TMP_94 \def (Bind Abst) in (let TMP_95 \def (THead TMP_94 u2 t5) in (eq T t4 +TMP_95))))) in (let TMP_97 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) in (let TMP_100 \def (\lambda (_: T).(\lambda (t5: T).(\forall (b: +B).(\forall (u: T).(let TMP_98 \def (Bind b) in (let TMP_99 \def (CHead c +TMP_98 u) in (pr3 TMP_99 x1 t5))))))) in (let TMP_101 \def (pr3_sing c x2 x0 +H8 x4 H13) in (let TMP_106 \def (\lambda (b: B).(\lambda (u: T).(let TMP_102 +\def (Bind b) in (let TMP_103 \def (CHead c TMP_102 u) in (let TMP_104 \def +(H9 b u) in (let TMP_105 \def (H14 b u) in (pr3_sing TMP_103 x3 x1 TMP_104 x5 +TMP_105))))))) in (ex3_2_intro T T TMP_96 TMP_97 TMP_100 x4 x5 H12 TMP_101 +TMP_106))))))))))) in (ex3_2_ind T T TMP_81 TMP_82 TMP_85 TMP_93 TMP_107 +H11)))))))))))))))))))))) in (ex3_2_ind T T TMP_53 TMP_54 TMP_57 TMP_65 +TMP_108 H6))))))))))))))))))))))) in (let TMP_110 \def (pr3_ind c TMP_35 +TMP_47 TMP_109 y x H0) in (let TMP_111 \def (unintro T u1 TMP_27 TMP_110) in +(unintro T t1 TMP_19 TMP_111)))))))))) in (insert_eq T TMP_2 TMP_3 TMP_11 +TMP_112 H)))))))))). theorem pr3_gen_cast: \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c @@ -122,102 +171,178 @@ T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (pr3 c t1 x)))))) \def \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 -t2)))) (pr3 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T -t1 (\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c t t2)))) (pr3 c t x)))) (unintro T u1 (\lambda (t: T).(\forall -(x0: T).((eq T y (THead (Flat Cast) t x0)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 -t2)))) (pr3 c x0 x))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall -(x0: T).(\forall (x1: T).((eq T t (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))))))) (\lambda (t: T).(\lambda (x0: -T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) x0 -x1))).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t0: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (pr3 c x1 t0))) (or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c x1 t2)))) (pr3 c x1 (THead (Flat Cast) x0 x1)) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) -x0 x1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c -x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T -(THead (Flat Cast) x0 x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) -(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: -T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: -T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 -t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: -(eq T t3 (THead (Flat Cast) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: -T).(pr2 c t t2)) H1 (THead (Flat Cast) x0 x1) H4) in (let H6 \def -(pr2_gen_cast c x0 x1 t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (pr2 c -x1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H7: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 -c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 \def (eq_ind T t2 (\lambda -(t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) -\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x5 t5)))) (pr3 c x5 t4)))))) H3 (THead (Flat Cast) -x2 x3) H8) in (let H12 \def (H11 x2 x3 (refl_equal T (THead (Flat Cast) x2 -x3))) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5)))) (pr3 c x3 t4) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) +(H: (pr3 c (THead (Flat Cast) u1 t1) x)).(let TMP_1 \def (Flat Cast) in (let +TMP_2 \def (THead TMP_1 u1 t1) in (let TMP_3 \def (\lambda (t: T).(pr3 c t +x)) in (let TMP_11 \def (\lambda (_: T).(let TMP_6 \def (\lambda (u2: +T).(\lambda (t2: T).(let TMP_4 \def (Flat Cast) in (let TMP_5 \def (THead +TMP_4 u2 t2) in (eq T x TMP_5))))) in (let TMP_7 \def (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) in (let TMP_8 \def (\lambda (_: +T).(\lambda (t2: T).(pr3 c t1 t2))) in (let TMP_9 \def (ex3_2 T T TMP_6 TMP_7 +TMP_8) in (let TMP_10 \def (pr3 c t1 x) in (or TMP_9 TMP_10))))))) in (let +TMP_204 \def (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(let TMP_19 \def +(\lambda (t: T).((eq T y (THead (Flat Cast) u1 t)) \to (let TMP_14 \def +(\lambda (u2: T).(\lambda (t2: T).(let TMP_12 \def (Flat Cast) in (let TMP_13 +\def (THead TMP_12 u2 t2) in (eq T x TMP_13))))) in (let TMP_15 \def (\lambda +(u2: T).(\lambda (_: T).(pr3 c u1 u2))) in (let TMP_16 \def (\lambda (_: +T).(\lambda (t2: T).(pr3 c t t2))) in (let TMP_17 \def (ex3_2 T T TMP_14 +TMP_15 TMP_16) in (let TMP_18 \def (pr3 c t x) in (or TMP_17 TMP_18)))))))) +in (let TMP_27 \def (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Flat +Cast) t x0)) \to (let TMP_22 \def (\lambda (u2: T).(\lambda (t2: T).(let +TMP_20 \def (Flat Cast) in (let TMP_21 \def (THead TMP_20 u2 t2) in (eq T x +TMP_21))))) in (let TMP_23 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c t +u2))) in (let TMP_24 \def (\lambda (_: T).(\lambda (t2: T).(pr3 c x0 t2))) in +(let TMP_25 \def (ex3_2 T T TMP_22 TMP_23 TMP_24) in (let TMP_26 \def (pr3 c +x0 x) in (or TMP_25 TMP_26))))))))) in (let TMP_35 \def (\lambda (t: +T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Flat +Cast) x0 x1)) \to (let TMP_30 \def (\lambda (u2: T).(\lambda (t2: T).(let +TMP_28 \def (Flat Cast) in (let TMP_29 \def (THead TMP_28 u2 t2) in (eq T t0 +TMP_29))))) in (let TMP_31 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) in (let TMP_32 \def (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) in +(let TMP_33 \def (ex3_2 T T TMP_30 TMP_31 TMP_32) in (let TMP_34 \def (pr3 c +x1 t0) in (or TMP_33 TMP_34))))))))))) in (let TMP_71 \def (\lambda (t: +T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Flat Cast) +x0 x1))).(let TMP_36 \def (Flat Cast) in (let TMP_37 \def (THead TMP_36 x0 +x1) in (let TMP_45 \def (\lambda (t0: T).(let TMP_40 \def (\lambda (u2: +T).(\lambda (t2: T).(let TMP_38 \def (Flat Cast) in (let TMP_39 \def (THead +TMP_38 u2 t2) in (eq T t0 TMP_39))))) in (let TMP_41 \def (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_42 \def (\lambda (_: +T).(\lambda (t2: T).(pr3 c x1 t2))) in (let TMP_43 \def (ex3_2 T T TMP_40 +TMP_41 TMP_42) in (let TMP_44 \def (pr3 c x1 t0) in (or TMP_43 TMP_44))))))) +in (let TMP_50 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_46 \def (Flat +Cast) in (let TMP_47 \def (THead TMP_46 x0 x1) in (let TMP_48 \def (Flat +Cast) in (let TMP_49 \def (THead TMP_48 u2 t2) in (eq T TMP_47 TMP_49))))))) +in (let TMP_51 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let +TMP_52 \def (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) in (let TMP_53 +\def (ex3_2 T T TMP_50 TMP_51 TMP_52) in (let TMP_54 \def (Flat Cast) in (let +TMP_55 \def (THead TMP_54 x0 x1) in (let TMP_56 \def (pr3 c x1 TMP_55) in +(let TMP_61 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_57 \def (Flat +Cast) in (let TMP_58 \def (THead TMP_57 x0 x1) in (let TMP_59 \def (Flat +Cast) in (let TMP_60 \def (THead TMP_59 u2 t2) in (eq T TMP_58 TMP_60))))))) +in (let TMP_62 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let +TMP_63 \def (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2))) in (let TMP_64 +\def (Flat Cast) in (let TMP_65 \def (THead TMP_64 x0 x1) in (let TMP_66 \def +(refl_equal T TMP_65) in (let TMP_67 \def (pr3_refl c x0) in (let TMP_68 \def +(pr3_refl c x1) in (let TMP_69 \def (ex3_2_intro T T TMP_61 TMP_62 TMP_63 x0 +x1 TMP_66 TMP_67 TMP_68) in (let TMP_70 \def (or_introl TMP_53 TMP_56 TMP_69) +in (eq_ind_r T TMP_37 TMP_45 TMP_70 t H1))))))))))))))))))))))))) in (let +TMP_201 \def (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 +t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall +(x0: T).(\forall (x1: T).((eq T t2 (THead (Flat Cast) x0 x1)) \to (or (ex3_2 +T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4)) (\lambda (H13: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 -t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 -t5)))) (pr3 c x1 t4)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H14: (eq T -t4 (THead (Flat Cast) x4 x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: -(pr3 c x3 x5)).(eq_ind_r T (THead (Flat Cast) x4 x5) (\lambda (t: T).(or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t))) (or_introl (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Cast) x4 x5) (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 (THead (Flat -Cast) x4 x5)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead -(Flat Cast) x4 x5) (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 -(refl_equal T (THead (Flat Cast) x4 x5)) (pr3_sing c x2 x0 H9 x4 H15) -(pr3_sing c x3 x1 H10 x5 H16))) t4 H14)))))) H13)) (\lambda (H13: (pr3 c x3 -t4)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c -x3 x1 H10 t4 H13))) H12)))))))) H7)) (\lambda (H7: (pr2 c x1 t2)).(or_intror -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Cast) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4) (pr3_sing c t2 x1 H7 t4 -H2))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 2001 -END *) +(t5: T).(pr3 c x1 t5)))) (pr3 c x1 t4))))))).(\lambda (x0: T).(\lambda (x1: +T).(\lambda (H4: (eq T t3 (THead (Flat Cast) x0 x1))).(let TMP_72 \def +(\lambda (t: T).(pr2 c t t2)) in (let TMP_73 \def (Flat Cast) in (let TMP_74 +\def (THead TMP_73 x0 x1) in (let H5 \def (eq_ind T t3 TMP_72 H1 TMP_74 H4) +in (let H6 \def (pr2_gen_cast c x0 x1 t2 H5) in (let TMP_77 \def (\lambda +(u2: T).(\lambda (t5: T).(let TMP_75 \def (Flat Cast) in (let TMP_76 \def +(THead TMP_75 u2 t5) in (eq T t2 TMP_76))))) in (let TMP_78 \def (\lambda +(u2: T).(\lambda (_: T).(pr2 c x0 u2))) in (let TMP_79 \def (\lambda (_: +T).(\lambda (t5: T).(pr2 c x1 t5))) in (let TMP_80 \def (ex3_2 T T TMP_77 +TMP_78 TMP_79) in (let TMP_81 \def (pr2 c x1 t2) in (let TMP_84 \def (\lambda +(u2: T).(\lambda (t5: T).(let TMP_82 \def (Flat Cast) in (let TMP_83 \def +(THead TMP_82 u2 t5) in (eq T t4 TMP_83))))) in (let TMP_85 \def (\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_86 \def (\lambda (_: +T).(\lambda (t5: T).(pr3 c x1 t5))) in (let TMP_87 \def (ex3_2 T T TMP_84 +TMP_85 TMP_86) in (let TMP_88 \def (pr3 c x1 t4) in (let TMP_89 \def (or +TMP_87 TMP_88) in (let TMP_191 \def (\lambda (H7: (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t2 (THead (Flat Cast) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 +t5))))).(let TMP_92 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_90 \def +(Flat Cast) in (let TMP_91 \def (THead TMP_90 u2 t5) in (eq T t2 TMP_91))))) +in (let TMP_93 \def (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) in (let +TMP_94 \def (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5))) in (let TMP_97 +\def (\lambda (u2: T).(\lambda (t5: T).(let TMP_95 \def (Flat Cast) in (let +TMP_96 \def (THead TMP_95 u2 t5) in (eq T t4 TMP_96))))) in (let TMP_98 \def +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_99 \def (\lambda +(_: T).(\lambda (t5: T).(pr3 c x1 t5))) in (let TMP_100 \def (ex3_2 T T +TMP_97 TMP_98 TMP_99) in (let TMP_101 \def (pr3 c x1 t4) in (let TMP_102 \def +(or TMP_100 TMP_101) in (let TMP_190 \def (\lambda (x2: T).(\lambda (x3: +T).(\lambda (H8: (eq T t2 (THead (Flat Cast) x2 x3))).(\lambda (H9: (pr2 c x0 +x2)).(\lambda (H10: (pr2 c x1 x3)).(let TMP_110 \def (\lambda (t: T).(\forall +(x4: T).(\forall (x5: T).((eq T t (THead (Flat Cast) x4 x5)) \to (let TMP_105 +\def (\lambda (u2: T).(\lambda (t5: T).(let TMP_103 \def (Flat Cast) in (let +TMP_104 \def (THead TMP_103 u2 t5) in (eq T t4 TMP_104))))) in (let TMP_106 +\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) in (let TMP_107 \def +(\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5))) in (let TMP_108 \def (ex3_2 +T T TMP_105 TMP_106 TMP_107) in (let TMP_109 \def (pr3 c x5 t4) in (or +TMP_108 TMP_109)))))))))) in (let TMP_111 \def (Flat Cast) in (let TMP_112 +\def (THead TMP_111 x2 x3) in (let H11 \def (eq_ind T t2 TMP_110 H3 TMP_112 +H8) in (let TMP_113 \def (Flat Cast) in (let TMP_114 \def (THead TMP_113 x2 +x3) in (let TMP_115 \def (refl_equal T TMP_114) in (let H12 \def (H11 x2 x3 +TMP_115) in (let TMP_118 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_116 +\def (Flat Cast) in (let TMP_117 \def (THead TMP_116 u2 t5) in (eq T t4 +TMP_117))))) in (let TMP_119 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 +u2))) in (let TMP_120 \def (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) +in (let TMP_121 \def (ex3_2 T T TMP_118 TMP_119 TMP_120) in (let TMP_122 \def +(pr3 c x3 t4) in (let TMP_125 \def (\lambda (u2: T).(\lambda (t5: T).(let +TMP_123 \def (Flat Cast) in (let TMP_124 \def (THead TMP_123 u2 t5) in (eq T +t4 TMP_124))))) in (let TMP_126 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c +x0 u2))) in (let TMP_127 \def (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 +t5))) in (let TMP_128 \def (ex3_2 T T TMP_125 TMP_126 TMP_127) in (let +TMP_129 \def (pr3 c x1 t4) in (let TMP_130 \def (or TMP_128 TMP_129) in (let +TMP_180 \def (\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq +T t4 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 +u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))))).(let TMP_133 \def +(\lambda (u2: T).(\lambda (t5: T).(let TMP_131 \def (Flat Cast) in (let +TMP_132 \def (THead TMP_131 u2 t5) in (eq T t4 TMP_132))))) in (let TMP_134 +\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) in (let TMP_135 \def +(\lambda (_: T).(\lambda (t5: T).(pr3 c x3 t5))) in (let TMP_138 \def +(\lambda (u2: T).(\lambda (t5: T).(let TMP_136 \def (Flat Cast) in (let +TMP_137 \def (THead TMP_136 u2 t5) in (eq T t4 TMP_137))))) in (let TMP_139 +\def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_140 \def +(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) in (let TMP_141 \def (ex3_2 +T T TMP_138 TMP_139 TMP_140) in (let TMP_142 \def (pr3 c x1 t4) in (let +TMP_143 \def (or TMP_141 TMP_142) in (let TMP_179 \def (\lambda (x4: +T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Flat Cast) x4 +x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: (pr3 c x3 x5)).(let +TMP_144 \def (Flat Cast) in (let TMP_145 \def (THead TMP_144 x4 x5) in (let +TMP_153 \def (\lambda (t: T).(let TMP_148 \def (\lambda (u2: T).(\lambda (t5: +T).(let TMP_146 \def (Flat Cast) in (let TMP_147 \def (THead TMP_146 u2 t5) +in (eq T t TMP_147))))) in (let TMP_149 \def (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) in (let TMP_150 \def (\lambda (_: T).(\lambda (t5: T).(pr3 +c x1 t5))) in (let TMP_151 \def (ex3_2 T T TMP_148 TMP_149 TMP_150) in (let +TMP_152 \def (pr3 c x1 t) in (or TMP_151 TMP_152))))))) in (let TMP_158 \def +(\lambda (u2: T).(\lambda (t5: T).(let TMP_154 \def (Flat Cast) in (let +TMP_155 \def (THead TMP_154 x4 x5) in (let TMP_156 \def (Flat Cast) in (let +TMP_157 \def (THead TMP_156 u2 t5) in (eq T TMP_155 TMP_157))))))) in (let +TMP_159 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) in (let TMP_160 +\def (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) in (let TMP_161 \def +(ex3_2 T T TMP_158 TMP_159 TMP_160) in (let TMP_162 \def (Flat Cast) in (let +TMP_163 \def (THead TMP_162 x4 x5) in (let TMP_164 \def (pr3 c x1 TMP_163) in +(let TMP_169 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_165 \def (Flat +Cast) in (let TMP_166 \def (THead TMP_165 x4 x5) in (let TMP_167 \def (Flat +Cast) in (let TMP_168 \def (THead TMP_167 u2 t5) in (eq T TMP_166 +TMP_168))))))) in (let TMP_170 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c +x0 u2))) in (let TMP_171 \def (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 +t5))) in (let TMP_172 \def (Flat Cast) in (let TMP_173 \def (THead TMP_172 x4 +x5) in (let TMP_174 \def (refl_equal T TMP_173) in (let TMP_175 \def +(pr3_sing c x2 x0 H9 x4 H15) in (let TMP_176 \def (pr3_sing c x3 x1 H10 x5 +H16) in (let TMP_177 \def (ex3_2_intro T T TMP_169 TMP_170 TMP_171 x4 x5 +TMP_174 TMP_175 TMP_176) in (let TMP_178 \def (or_introl TMP_161 TMP_164 +TMP_177) in (eq_ind_r T TMP_145 TMP_153 TMP_178 t4 +H14)))))))))))))))))))))))))) in (ex3_2_ind T T TMP_133 TMP_134 TMP_135 +TMP_143 TMP_179 H13)))))))))))) in (let TMP_189 \def (\lambda (H13: (pr3 c x3 +t4)).(let TMP_183 \def (\lambda (u2: T).(\lambda (t5: T).(let TMP_181 \def +(Flat Cast) in (let TMP_182 \def (THead TMP_181 u2 t5) in (eq T t4 +TMP_182))))) in (let TMP_184 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) in (let TMP_185 \def (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) +in (let TMP_186 \def (ex3_2 T T TMP_183 TMP_184 TMP_185) in (let TMP_187 \def +(pr3 c x1 t4) in (let TMP_188 \def (pr3_sing c x3 x1 H10 t4 H13) in +(or_intror TMP_186 TMP_187 TMP_188)))))))) in (or_ind TMP_121 TMP_122 TMP_130 +TMP_180 TMP_189 H12))))))))))))))))))))))))))) in (ex3_2_ind T T TMP_92 +TMP_93 TMP_94 TMP_102 TMP_190 H7)))))))))))) in (let TMP_200 \def (\lambda +(H7: (pr2 c x1 t2)).(let TMP_194 \def (\lambda (u2: T).(\lambda (t5: T).(let +TMP_192 \def (Flat Cast) in (let TMP_193 \def (THead TMP_192 u2 t5) in (eq T +t4 TMP_193))))) in (let TMP_195 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c +x0 u2))) in (let TMP_196 \def (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 +t5))) in (let TMP_197 \def (ex3_2 T T TMP_194 TMP_195 TMP_196) in (let +TMP_198 \def (pr3 c x1 t4) in (let TMP_199 \def (pr3_sing c t2 x1 H7 t4 H2) +in (or_intror TMP_197 TMP_198 TMP_199)))))))) in (or_ind TMP_80 TMP_81 TMP_89 +TMP_191 TMP_200 H6)))))))))))))))))))))))))))) in (let TMP_202 \def (pr3_ind +c TMP_35 TMP_71 TMP_201 y x H0) in (let TMP_203 \def (unintro T u1 TMP_27 +TMP_202) in (unintro T t1 TMP_19 TMP_203)))))))))) in (insert_eq T TMP_2 +TMP_3 TMP_11 TMP_204 H)))))))))). theorem pr3_gen_lift: \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall @@ -226,38 +351,51 @@ theorem pr3_gen_lift: t2)))))))))) \def \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda -(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(insert_eq T (lift h d t1) -(\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e) -\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e -t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda -(t: T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (ex2 T -(\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t t2))))))) -(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T t (lift h -d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T -t0 (lift h d t2))) (\lambda (t2: T).(pr3 e x0 t2))))))))) (\lambda (t: -T).(\lambda (x0: T).(\lambda (H1: (eq T t (lift h d x0))).(\lambda (e: -C).(\lambda (_: (drop h d c e)).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h -d t2))) (\lambda (t2: T).(pr3 e x0 t2)) x0 H1 (pr3_refl e x0))))))) (\lambda -(t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: +(d: nat).(\lambda (H: (pr3 c (lift h d t1) x)).(let TMP_1 \def (lift h d t1) +in (let TMP_2 \def (\lambda (t: T).(pr3 c t x)) in (let TMP_6 \def (\lambda +(_: T).(\forall (e: C).((drop h d c e) \to (let TMP_4 \def (\lambda (t2: +T).(let TMP_3 \def (lift h d t2) in (eq T x TMP_3))) in (let TMP_5 \def +(\lambda (t2: T).(pr3 e t1 t2)) in (ex2 T TMP_4 TMP_5)))))) in (let TMP_45 +\def (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(let TMP_10 \def (\lambda (t: +T).((eq T y (lift h d t)) \to (\forall (e: C).((drop h d c e) \to (let TMP_8 +\def (\lambda (t2: T).(let TMP_7 \def (lift h d t2) in (eq T x TMP_7))) in +(let TMP_9 \def (\lambda (t2: T).(pr3 e t t2)) in (ex2 T TMP_8 TMP_9))))))) +in (let TMP_14 \def (\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).((eq T +t (lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (let TMP_12 \def +(\lambda (t2: T).(let TMP_11 \def (lift h d t2) in (eq T t0 TMP_11))) in (let +TMP_13 \def (\lambda (t2: T).(pr3 e x0 t2)) in (ex2 T TMP_12 TMP_13))))))))) +in (let TMP_19 \def (\lambda (t: T).(\lambda (x0: T).(\lambda (H1: (eq T t +(lift h d x0))).(\lambda (e: C).(\lambda (_: (drop h d c e)).(let TMP_16 \def +(\lambda (t2: T).(let TMP_15 \def (lift h d t2) in (eq T t TMP_15))) in (let +TMP_17 \def (\lambda (t2: T).(pr3 e x0 t2)) in (let TMP_18 \def (pr3_refl e +x0) in (ex_intro2 T TMP_16 TMP_17 x0 H1 TMP_18))))))))) in (let TMP_43 \def +(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).((eq T t2 (lift h d x0)) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))))))))).(\lambda (x0: T).(\lambda (H4: (eq T t3 (lift h d x0))).(\lambda (e: C).(\lambda (H5: -(drop h d c e)).(let H6 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 -(lift h d x0) H4) in (let H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in -(ex2_ind T (\lambda (t5: T).(eq T t2 (lift h d t5))) (\lambda (t5: T).(pr2 e -x0 t5)) (ex2 T (\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: -T).(pr3 e x0 t5))) (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d -x1))).(\lambda (H9: (pr2 e x0 x1)).(ex2_ind T (\lambda (t5: T).(eq T t4 (lift -h d t5))) (\lambda (t5: T).(pr3 e x1 t5)) (ex2 T (\lambda (t5: T).(eq T t4 -(lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5))) (\lambda (x2: T).(\lambda -(H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(ex_intro2 T -(\lambda (t5: T).(eq T t4 (lift h d t5))) (\lambda (t5: T).(pr3 e x0 t5)) x2 -H10 (pr3_sing e x1 x0 H9 x2 H11))))) (H3 x1 H8 e H5))))) H7))))))))))))) y x -H0)))) H)))))). -(* COMMENTS -Initial nodes: 689 -END *) +(drop h d c e)).(let TMP_20 \def (\lambda (t: T).(pr2 c t t2)) in (let TMP_21 +\def (lift h d x0) in (let H6 \def (eq_ind T t3 TMP_20 H1 TMP_21 H4) in (let +H7 \def (pr2_gen_lift c x0 t2 h d H6 e H5) in (let TMP_23 \def (\lambda (t5: +T).(let TMP_22 \def (lift h d t5) in (eq T t2 TMP_22))) in (let TMP_24 \def +(\lambda (t5: T).(pr2 e x0 t5)) in (let TMP_26 \def (\lambda (t5: T).(let +TMP_25 \def (lift h d t5) in (eq T t4 TMP_25))) in (let TMP_27 \def (\lambda +(t5: T).(pr3 e x0 t5)) in (let TMP_28 \def (ex2 T TMP_26 TMP_27) in (let +TMP_42 \def (\lambda (x1: T).(\lambda (H8: (eq T t2 (lift h d x1))).(\lambda +(H9: (pr2 e x0 x1)).(let TMP_30 \def (\lambda (t5: T).(let TMP_29 \def (lift +h d t5) in (eq T t4 TMP_29))) in (let TMP_31 \def (\lambda (t5: T).(pr3 e x1 +t5)) in (let TMP_33 \def (\lambda (t5: T).(let TMP_32 \def (lift h d t5) in +(eq T t4 TMP_32))) in (let TMP_34 \def (\lambda (t5: T).(pr3 e x0 t5)) in +(let TMP_35 \def (ex2 T TMP_33 TMP_34) in (let TMP_40 \def (\lambda (x2: +T).(\lambda (H10: (eq T t4 (lift h d x2))).(\lambda (H11: (pr3 e x1 x2)).(let +TMP_37 \def (\lambda (t5: T).(let TMP_36 \def (lift h d t5) in (eq T t4 +TMP_36))) in (let TMP_38 \def (\lambda (t5: T).(pr3 e x0 t5)) in (let TMP_39 +\def (pr3_sing e x1 x0 H9 x2 H11) in (ex_intro2 T TMP_37 TMP_38 x2 H10 +TMP_39))))))) in (let TMP_41 \def (H3 x1 H8 e H5) in (ex2_ind T TMP_30 TMP_31 +TMP_35 TMP_40 TMP_41))))))))))) in (ex2_ind T TMP_23 TMP_24 TMP_28 TMP_42 +H7))))))))))))))))))))) in (let TMP_44 \def (pr3_ind c TMP_14 TMP_19 TMP_43 y +x H0) in (unintro T t1 TMP_10 TMP_44)))))))) in (insert_eq T TMP_1 TMP_2 +TMP_6 TMP_45 H)))))))))). theorem pr3_gen_lref: \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TLRef n) x) \to @@ -267,1338 +405,132 @@ T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T x (lift (S n) O v)))))))))) \def \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr3 c (TLRef -n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr3 c t x)) (\lambda (t: -T).(or (eq T x t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T x (lift (S n) O v)))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y -x)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or -(eq T t0 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t0 (lift (S n) O v)))))))))) (\lambda (t: T).(\lambda (_: (eq T -t (TLRef n))).(or_introl (eq T t t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: -C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (v: T).(eq T t (lift (S n) O v)))))) (refl_equal T t)))) -(\lambda (t2: T).(\lambda (t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: -T).(\lambda (H2: (pr3 c t2 t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or -(eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v)))))))))).(\lambda (H4: (eq T t1 (TLRef -n))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(pr2 c t t2)) H1 (TLRef n) H4) -in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t3 t) (ex3_3 C T T +n) x)).(let TMP_1 \def (TLRef n) in (let TMP_2 \def (\lambda (t: T).(pr3 c t +x)) in (let TMP_12 \def (\lambda (t: T).(let TMP_3 \def (eq T x t) in (let +TMP_6 \def (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(let TMP_4 \def +(Bind Abbr) in (let TMP_5 \def (CHead d TMP_4 u) in (getl n c TMP_5)))))) in +(let TMP_7 \def (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u +v)))) in (let TMP_10 \def (\lambda (_: C).(\lambda (_: T).(\lambda (v: +T).(let TMP_8 \def (S n) in (let TMP_9 \def (lift TMP_8 O v) in (eq T x +TMP_9)))))) in (let TMP_11 \def (ex3_3 C T T TMP_6 TMP_7 TMP_10) in (or TMP_3 +TMP_11))))))) in (let TMP_155 \def (\lambda (y: T).(\lambda (H0: (pr3 c y +x)).(let TMP_22 \def (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to +(let TMP_13 \def (eq T t0 t) in (let TMP_16 \def (\lambda (d: C).(\lambda (u: +T).(\lambda (_: T).(let TMP_14 \def (Bind Abbr) in (let TMP_15 \def (CHead d +TMP_14 u) in (getl n c TMP_15)))))) in (let TMP_17 \def (\lambda (d: +C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_20 \def +(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_18 \def (S n) in +(let TMP_19 \def (lift TMP_18 O v) in (eq T t0 TMP_19)))))) in (let TMP_21 +\def (ex3_3 C T T TMP_16 TMP_17 TMP_20) in (or TMP_13 TMP_21))))))))) in (let +TMP_33 \def (\lambda (t: T).(\lambda (_: (eq T t (TLRef n))).(let TMP_23 \def +(eq T t t) in (let TMP_26 \def (\lambda (d: C).(\lambda (u: T).(\lambda (_: +T).(let TMP_24 \def (Bind Abbr) in (let TMP_25 \def (CHead d TMP_24 u) in +(getl n c TMP_25)))))) in (let TMP_27 \def (\lambda (d: C).(\lambda (u: +T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_30 \def (\lambda (_: +C).(\lambda (_: T).(\lambda (v: T).(let TMP_28 \def (S n) in (let TMP_29 \def +(lift TMP_28 O v) in (eq T t TMP_29)))))) in (let TMP_31 \def (ex3_3 C T T +TMP_26 TMP_27 TMP_30) in (let TMP_32 \def (refl_equal T t) in (or_introl +TMP_23 TMP_31 TMP_32))))))))) in (let TMP_154 \def (\lambda (t2: T).(\lambda +(t1: T).(\lambda (H1: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H2: (pr3 c t2 +t3)).(\lambda (H3: (((eq T t2 (TLRef n)) \to (or (eq T t3 t2) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v)))))))) (let H6 \def (pr2_gen_lref c t2 n H5) in (or_ind (eq T t2 (TLRef -n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) -u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O u))))) (or (eq T -t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (H7: (eq T t2 (TLRef -n))).(let H8 \def (eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or -(eq T t3 t) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))))) H3 (TLRef n) H7) in (let H9 \def -(eq_ind T t2 (\lambda (t: T).(pr3 c t t3)) H2 (TLRef n) H7) in (H8 -(refl_equal T (TLRef n)))))) (\lambda (H7: (ex2_2 C T (\lambda (d: -C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))))).(ex2_2_ind C T (\lambda (d: -C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: -C).(\lambda (u: T).(eq T t2 (lift (S n) O u)))) (or (eq T t3 (TLRef n)) -(ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead -d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u -v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v))))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H8: (getl n c (CHead x0 -(Bind Abbr) x1))).(\lambda (H9: (eq T t2 (lift (S n) O x1))).(let H10 \def -(eq_ind T t2 (\lambda (t: T).((eq T t (TLRef n)) \to (or (eq T t3 t) (ex3_3 C -T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl n c (CHead d (Bind -Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 (lift (S n) O -v))))))))) H3 (lift (S n) O x1) H9) in (let H11 \def (eq_ind T t2 (\lambda -(t: T).(pr3 c t t3)) H2 (lift (S n) O x1) H9) in (let H12 \def (pr3_gen_lift -c x1 t3 (S n) O H11 x0 (getl_drop Abbr c x0 x1 n H8)) in (ex2_ind T (\lambda -(t4: T).(eq T t3 (lift (S n) O t4))) (\lambda (t4: T).(pr3 x0 x1 t4)) (or (eq -T t3 (TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: -T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: -T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda -(v: T).(eq T t3 (lift (S n) O v))))))) (\lambda (x2: T).(\lambda (H13: (eq T -t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(or_intror (eq T t3 -(TLRef n)) (ex3_3 C T T (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(getl -n c (CHead d (Bind Abbr) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (v: -T).(pr3 d u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(eq T t3 -(lift (S n) O v)))))) (ex3_3_intro C T T (\lambda (d: C).(\lambda (u: -T).(\lambda (_: T).(getl n c (CHead d (Bind Abbr) u))))) (\lambda (d: -C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) (\lambda (_: C).(\lambda -(_: T).(\lambda (v: T).(eq T t3 (lift (S n) O v))))) x0 x1 x2 H8 H14 H13))))) -H12)))))))) H7)) H6)) t1 H4))))))))) y x H0))) H)))). -(* COMMENTS -Initial nodes: 1515 -END *) - -theorem pr3_gen_void: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 -(lift (S O) O x))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c -(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y -x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 -(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c -(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t -(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 -(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: -T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind -Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 -(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T -(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead -(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) -(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) -u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c -t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall -(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: -T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let -H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 -x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) -(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O -t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind -Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda -(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead -c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq -T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) -O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind -Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind -T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind -Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O -t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 -(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c -(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda -(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 -x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall -(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c -(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 -(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead -c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: -(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) -(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind -Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift -(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: -((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O -t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead -c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) -(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S -O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) -O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 2645 -END *) - -theorem pr3_gen_abbr: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) -u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda -(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) -O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind -Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 -(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: -T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind -Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda -(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 -x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 -x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) -(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: -T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: -T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 -t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 -(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 -(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: -T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda -(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 -(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind -b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead -c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z -t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 -(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: -T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda -(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda -(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 -c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c -(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 -(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: -T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: -T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) -z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c -(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 -(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 -z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) -(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c -(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: -B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind -T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind -Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 -(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 -x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) -in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind -Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c -(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 -t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 -t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c -(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: -T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c -x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T -(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 -x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 -(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 -H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O -t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) -x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) -O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: -T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 -x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c -(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: -T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) -x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall -(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S -O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 -(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S -O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda -(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 -x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) -x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 -x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) -(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 -x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c -(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 -x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O -t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T -(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) -(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: -T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: -T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: -T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c -(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq -T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 -t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: -T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 -x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) -x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall -(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S -O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 -(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S -O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda -(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6 -x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) -x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) -x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S -O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda -(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 -(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 -x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 -H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind -Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 -(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) -(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O -t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 -(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) -x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 -(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing -(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 -(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) -H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda -(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) -x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing -(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) -(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c -(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 5983 -END *) - -theorem pr3_gen_appl: - \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c -(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) -\def - \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda -(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 -t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 -t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 -(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda -(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: -T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall -(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 -T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) -(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t -(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda -(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead -(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind -Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) -(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat -Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 -c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) -(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 -(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: -T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 -x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda -(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 -t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat -Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 -z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 -(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c -t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 -t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq -T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead -(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 -T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 -t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) -t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 -c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 -x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 -\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t -(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: -T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 -(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda -(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 -t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) -(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 -(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2: -T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 -t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat -Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) -t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 -c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: 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(\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))) 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x4 x5)))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) -(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl) -x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c -x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T -(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10 -x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 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T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: -T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind -Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 -t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10 -(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T -T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c -(THead (Bind b) y2 (THead (Flat Appl) (lift (S 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T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: -T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3 -c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9 -(THead (Flat Appl) 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(u2: T).(\lambda (y2: T).(pr3 c -(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) -x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 -(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) -(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind -Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind -Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) -z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 -(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq -T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 -x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall -(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 -x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T -t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c -(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 -\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t -(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: -T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T -T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 -c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: -T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c -(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda -(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind -Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) -x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 -(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: -B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T -t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) -(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 -y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: -T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) -(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) -u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: -T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda -(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) -t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 -c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead -(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) -y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda -(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: -T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T -x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 -(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 -x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) -x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) -(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: -T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 -(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda -(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind -b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda -(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead -(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 -(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) -x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead -(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) -(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) -(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall -(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T -T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda -(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 -c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda -(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind -b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda -(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 -(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let -H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 -(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T -(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda -(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: -T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind -Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: -T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: -T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead -(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: -T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) -z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) -(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda -(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 -z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: -T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat -Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 -u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T -(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: -T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c -(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda -(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) -t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: -T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: -B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda -(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: -T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) -y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) -H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) -x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))). -(* COMMENTS -Initial nodes: 12691 -END *) - -theorem pr3_gen_bind: - \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: -T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind -b) u1) t1 (lift (S O) O x))))))))) -\def - \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall -(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind -b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 -(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B -Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: -T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def -(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) -u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T -T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) -t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 -t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind -Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T -(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) -(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) -t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x -(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 -(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 -H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S -O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 -(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: -(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 -\def (match (H (refl_equal B Abst)) in False return (\lambda (_: False).(or -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 -t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind Abst) u1) t1 t2)))) (pr3 (CHead c -(Bind Abst) u1) t1 (lift (S O) O x)))) with []) in H1))))))) (\lambda (_: -(not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: -T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 -\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall -(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c -(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda -(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) -u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda -(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) -u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: -T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) -u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x -(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead -c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 -t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: -T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 -x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall -(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: -T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 -(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S -O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead -(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) -(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 -H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 -(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 -c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 -t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b). -(* COMMENTS -Initial nodes: 1721 -END *) +v)))))))))).(\lambda (H4: (eq T t1 (TLRef n))).(let TMP_34 \def (\lambda (t: +T).(pr2 c t t2)) in (let TMP_35 \def (TLRef n) in (let H5 \def (eq_ind T t1 +TMP_34 H1 TMP_35 H4) in (let TMP_36 \def (TLRef n) in (let TMP_46 \def +(\lambda (t: T).(let TMP_37 \def (eq T t3 t) in (let TMP_40 \def (\lambda (d: +C).(\lambda (u: T).(\lambda (_: T).(let TMP_38 \def (Bind Abbr) in (let +TMP_39 \def (CHead d TMP_38 u) in (getl n c TMP_39)))))) in (let TMP_41 \def +(\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_44 +\def (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_42 \def (S n) +in (let TMP_43 \def (lift TMP_42 O v) in (eq T t3 TMP_43)))))) in (let TMP_45 +\def (ex3_3 C T T TMP_40 TMP_41 TMP_44) in (or TMP_37 TMP_45))))))) in (let +H6 \def (pr2_gen_lref c t2 n H5) in (let TMP_47 \def (TLRef n) in (let TMP_48 +\def (eq T t2 TMP_47) in (let TMP_51 \def (\lambda (d: C).(\lambda (u: +T).(let TMP_49 \def (Bind Abbr) in (let TMP_50 \def (CHead d TMP_49 u) in +(getl n c TMP_50))))) in (let TMP_54 \def (\lambda (_: C).(\lambda (u: +T).(let TMP_52 \def (S n) in (let TMP_53 \def (lift TMP_52 O u) in (eq T t2 +TMP_53))))) in (let TMP_55 \def (ex2_2 C T TMP_51 TMP_54) in (let TMP_56 \def +(TLRef n) in (let TMP_57 \def (eq T t3 TMP_56) in (let TMP_60 \def (\lambda +(d: C).(\lambda (u: T).(\lambda (_: T).(let TMP_58 \def (Bind Abbr) in (let +TMP_59 \def (CHead d TMP_58 u) in (getl n c TMP_59)))))) in (let TMP_61 \def +(\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_64 +\def (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_62 \def (S n) +in (let TMP_63 \def (lift TMP_62 O v) in (eq T t3 TMP_63)))))) in (let TMP_65 +\def (ex3_3 C T T TMP_60 TMP_61 TMP_64) in (let TMP_66 \def (or TMP_57 +TMP_65) in (let TMP_82 \def (\lambda (H7: (eq T t2 (TLRef n))).(let TMP_76 +\def (\lambda (t: T).((eq T t (TLRef n)) \to (let TMP_67 \def (eq T t3 t) in +(let TMP_70 \def (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(let TMP_68 +\def (Bind Abbr) in (let TMP_69 \def (CHead d TMP_68 u) in (getl n c +TMP_69)))))) in (let TMP_71 \def (\lambda (d: C).(\lambda (u: T).(\lambda (v: +T).(pr3 d u v)))) in (let TMP_74 \def (\lambda (_: C).(\lambda (_: +T).(\lambda (v: T).(let TMP_72 \def (S n) in (let TMP_73 \def (lift TMP_72 O +v) in (eq T t3 TMP_73)))))) in (let TMP_75 \def (ex3_3 C T T TMP_70 TMP_71 +TMP_74) in (or TMP_67 TMP_75)))))))) in (let TMP_77 \def (TLRef n) in (let H8 +\def (eq_ind T t2 TMP_76 H3 TMP_77 H7) in (let TMP_78 \def (\lambda (t: +T).(pr3 c t t3)) in (let TMP_79 \def (TLRef n) in (let H9 \def (eq_ind T t2 +TMP_78 H2 TMP_79 H7) in (let TMP_80 \def (TLRef n) in (let TMP_81 \def +(refl_equal T TMP_80) in (H8 TMP_81)))))))))) in (let TMP_152 \def (\lambda +(H7: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind +Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S n) O +u)))))).(let TMP_85 \def (\lambda (d: C).(\lambda (u: T).(let TMP_83 \def +(Bind Abbr) in (let TMP_84 \def (CHead d TMP_83 u) in (getl n c TMP_84))))) +in (let TMP_88 \def (\lambda (_: C).(\lambda (u: T).(let TMP_86 \def (S n) in +(let TMP_87 \def (lift TMP_86 O u) in (eq T t2 TMP_87))))) in (let TMP_89 +\def (TLRef n) in (let TMP_90 \def (eq T t3 TMP_89) in (let TMP_93 \def +(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(let TMP_91 \def (Bind Abbr) +in (let TMP_92 \def (CHead d TMP_91 u) in (getl n c TMP_92)))))) in (let +TMP_94 \def (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in +(let TMP_97 \def (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_95 +\def (S n) in (let TMP_96 \def (lift TMP_95 O v) in (eq T t3 TMP_96)))))) in +(let TMP_98 \def (ex3_3 C T T TMP_93 TMP_94 TMP_97) in (let TMP_99 \def (or +TMP_90 TMP_98) in (let TMP_151 \def (\lambda (x0: C).(\lambda (x1: +T).(\lambda (H8: (getl n c (CHead x0 (Bind Abbr) x1))).(\lambda (H9: (eq T t2 +(lift (S n) O x1))).(let TMP_109 \def (\lambda (t: T).((eq T t (TLRef n)) \to +(let TMP_100 \def (eq T t3 t) in (let TMP_103 \def (\lambda (d: C).(\lambda +(u: T).(\lambda (_: T).(let TMP_101 \def (Bind Abbr) in (let TMP_102 \def +(CHead d TMP_101 u) in (getl n c TMP_102)))))) in (let TMP_104 \def (\lambda +(d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_107 \def +(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_105 \def (S n) in +(let TMP_106 \def (lift TMP_105 O v) in (eq T t3 TMP_106)))))) in (let +TMP_108 \def (ex3_3 C T T TMP_103 TMP_104 TMP_107) in (or TMP_100 +TMP_108)))))))) in (let TMP_110 \def (S n) in (let TMP_111 \def (lift TMP_110 +O x1) in (let H10 \def (eq_ind T t2 TMP_109 H3 TMP_111 H9) in (let TMP_112 +\def (\lambda (t: T).(pr3 c t t3)) in (let TMP_113 \def (S n) in (let TMP_114 +\def (lift TMP_113 O x1) in (let H11 \def (eq_ind T t2 TMP_112 H2 TMP_114 H9) +in (let TMP_115 \def (S n) in (let TMP_116 \def (getl_drop Abbr c x0 x1 n H8) +in (let H12 \def (pr3_gen_lift c x1 t3 TMP_115 O H11 x0 TMP_116) in (let +TMP_119 \def (\lambda (t4: T).(let TMP_117 \def (S n) in (let TMP_118 \def +(lift TMP_117 O t4) in (eq T t3 TMP_118)))) in (let TMP_120 \def (\lambda +(t4: T).(pr3 x0 x1 t4)) in (let TMP_121 \def (TLRef n) in (let TMP_122 \def +(eq T t3 TMP_121) in (let TMP_125 \def (\lambda (d: C).(\lambda (u: +T).(\lambda (_: T).(let TMP_123 \def (Bind Abbr) in (let TMP_124 \def (CHead +d TMP_123 u) in (getl n c TMP_124)))))) in (let TMP_126 \def (\lambda (d: +C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) in (let TMP_129 \def +(\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let TMP_127 \def (S n) in +(let TMP_128 \def (lift TMP_127 O v) in (eq T t3 TMP_128)))))) in (let +TMP_130 \def (ex3_3 C T T TMP_125 TMP_126 TMP_129) in (let TMP_131 \def (or +TMP_122 TMP_130) in (let TMP_150 \def (\lambda (x2: T).(\lambda (H13: (eq T +t3 (lift (S n) O x2))).(\lambda (H14: (pr3 x0 x1 x2)).(let TMP_132 \def +(TLRef n) in (let TMP_133 \def (eq T t3 TMP_132) in (let TMP_136 \def +(\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(let TMP_134 \def (Bind Abbr) +in (let TMP_135 \def (CHead d TMP_134 u) in (getl n c TMP_135)))))) in (let +TMP_137 \def (\lambda (d: C).(\lambda (u: T).(\lambda (v: T).(pr3 d u v)))) +in (let TMP_140 \def (\lambda (_: C).(\lambda (_: T).(\lambda (v: T).(let +TMP_138 \def (S n) in (let TMP_139 \def (lift TMP_138 O v) in (eq T t3 +TMP_139)))))) in (let TMP_141 \def (ex3_3 C T T TMP_136 TMP_137 TMP_140) in +(let TMP_144 \def (\lambda (d: C).(\lambda (u: T).(\lambda (_: T).(let +TMP_142 \def (Bind Abbr) in (let TMP_143 \def (CHead d TMP_142 u) in (getl n +c TMP_143)))))) in (let TMP_145 \def (\lambda (d: C).(\lambda (u: T).(\lambda +(v: T).(pr3 d u v)))) in (let TMP_148 \def (\lambda (_: C).(\lambda (_: +T).(\lambda (v: T).(let TMP_146 \def (S n) in (let TMP_147 \def (lift TMP_146 +O v) in (eq T t3 TMP_147)))))) in (let TMP_149 \def (ex3_3_intro C T T +TMP_144 TMP_145 TMP_148 x0 x1 x2 H8 H14 H13) in (or_intror TMP_133 TMP_141 +TMP_149)))))))))))))) in (ex2_ind T TMP_119 TMP_120 TMP_131 TMP_150 +H12)))))))))))))))))))))))))) in (ex2_2_ind C T TMP_85 TMP_88 TMP_99 TMP_151 +H7)))))))))))) in (let TMP_153 \def (or_ind TMP_48 TMP_55 TMP_66 TMP_82 +TMP_152 H6) in (eq_ind_r T TMP_36 TMP_46 TMP_153 t1 +H4))))))))))))))))))))))))))))) in (pr3_ind c TMP_22 TMP_33 TMP_154 y x +H0)))))) in (insert_eq T TMP_1 TMP_2 TMP_12 TMP_155 H)))))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma index c7b2fb7aa..7ff6beaf4 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/iso.ma @@ -14,11 +14,11 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr3/fwd.ma". +include "basic_1/pr3/props.ma". -include "Basic-1/iso/props.ma". +include "basic_1/iso/props.ma". -include "Basic-1/tlist/props.ma". +include "basic_1/tlist/fwd.ma". theorem pr3_iso_appls_abbr: \forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c @@ -56,20 +56,19 @@ T).(\lambda (v: T).(pr3 d0 u v)))) (\lambda (_: C).(\lambda (_: T).(\lambda i) O x2) (\lambda (t: T).(pr3 c (lift (S i) O w) t)) (let H8 \def (eq_ind C (CHead d (Bind Abbr) w) (\lambda (c0: C).(getl i c c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) -H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c0 _ _) \Rightarrow -c0])) (CHead d (Bind Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d -(Bind Abbr) w) i H (CHead x0 (Bind Abbr) x1) H4)) in ((let H10 \def (f_equal -C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) w) (CHead -x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 (Bind -Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def (eq_ind_r T x1 -(\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) in (let H13 -\def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in (let H14 \def -(eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) w))) H12 d -H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w x2)) H13 d -H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 H15))))))) -H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: +H4)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e with [(CSort _) +\Rightarrow d | (CHead c0 _ _) \Rightarrow c0])) (CHead d (Bind Abbr) w) +(CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H (CHead x0 +(Bind Abbr) x1) H4)) in ((let H10 \def (f_equal C T (\lambda (e: C).(match e +with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind +Abbr) w) (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead d (Bind Abbr) w) i H +(CHead x0 (Bind Abbr) x1) H4)) in (\lambda (H11: (eq C d x0)).(let H12 \def +(eq_ind_r T x1 (\lambda (t: T).(getl i c (CHead x0 (Bind Abbr) t))) H8 w H10) +in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(pr3 x0 t x2)) H5 w H10) in +(let H14 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) +w))) H12 d H11) in (let H15 \def (eq_ind_r C x0 (\lambda (c0: C).(pr3 c0 w +x2)) H13 d H11) in (pr3_lift c d (S i) O (getl_drop Abbr c d w i H14) w x2 +H15))))))) H9))) u2 H6)))))))) H3)) H2))))) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H0: ((\forall (u2: T).((pr3 c (THeads (Flat Appl) t0 (TLRef i)) u2) \to ((((iso (THeads (Flat Appl) t0 (TLRef i)) u2) \to (\forall (P: Prop).P))) \to (pr3 c (THeads (Flat Appl) t0 (lift (S i) O w)) @@ -199,9 +198,6 @@ u2 (pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3)))))))) vs)))))). -(* COMMENTS -Initial nodes: 3759 -END *) theorem pr3_iso_appls_cast: \forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(let u1 @@ -362,9 +358,6 @@ O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))) vs)))). -(* COMMENTS -Initial nodes: 3297 -END *) theorem pr3_iso_appl_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: @@ -450,59 +443,44 @@ T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3)) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H10: (eq T (THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5))).(\lambda (H11: (pr3 c v2 x4)).(\lambda (H12: (pr3 (CHead c (Bind b) v2) -t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e in T return -(\lambda (_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow -Abst | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with -[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (THead (Bind Abst) -x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def (f_equal T T (\lambda -(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x0 -| (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind -Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) \Rightarrow t0])) -(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in (\lambda (H16: (eq T -x0 x4)).(\lambda (H17: (eq B Abst b)).(let H18 \def (eq_ind_r T x5 (\lambda -(t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H12 x1 H15) in (let H19 \def -(eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) H11 x0 H16) in (let H20 \def -(eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c (Bind b0) v2) t x1)) H18 Abst -H17) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H -Abst H17) in (eq_ind B Abst (\lambda (b0: B).(pr3 c (THead (Bind b0) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind Abbr) x2 x3))) (let H22 -\def (match (H21 (refl_equal B Abst)) in False return (\lambda (_: -False).(pr3 c (THead (Bind Abst) v2 (THead (Flat Appl) (lift (S O) O v1) t)) -(THead (Bind Abbr) x2 x3))) with []) in H22) b H17)))))))) H14)) H13))))))) -H9)) (\lambda (H9: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind -Abst) x0 x1)))).(pr3_t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O -x2) (lift (S O) O (THead (Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift -(S O) O v1) t) (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead -(Bind Abst) x0 x1))) (Bind b) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O -v1) (lift (S O) O x2) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop -(Bind b) O c c (drop_refl c) v2) v1 x2 H5) t (lift (S O) O (THead (Bind Abst) -x0 x1)) H9 Appl)) (THead (Bind Abbr) x2 x3) (eq_ind T (lift (S O) O (THead -(Flat Appl) x2 (THead (Bind Abst) x0 x1))) (\lambda (t0: T).(pr3 c (THead -(Bind b) v2 t0) (THead (Bind Abbr) x2 x3))) (pr3_sing c (THead (Bind Abbr) x2 -x1) (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) -x0 x1)))) (pr2_free c (THead (Bind b) v2 (lift (S O) O (THead (Flat Appl) x2 -(THead (Bind Abst) x0 x1)))) (THead (Bind Abbr) x2 x1) (pr0_zeta b H (THead -(Flat Appl) x2 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x1) (pr0_beta -x0 x2 x2 (pr0_refl x2) x1 x1 (pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) -(pr3_head_12 c x2 x2 (pr3_refl c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead -(Flat Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) -(lift_flat Appl x2 (THead (Bind Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) -H3)) (\lambda (H3: (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B -b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda -(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead -(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: -T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) -y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: -B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda -(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: -T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) -(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda -(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))))).(ex6_6_ind -B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +t x5)).(let H13 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _) +\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow +(match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) +(THead (Bind Abst) x0 x1) (THead (Bind b) x4 x5) H10) in ((let H14 \def +(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef +_) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) x0 +x1) (THead (Bind b) x4 x5) H10) in ((let H15 \def (f_equal T T (\lambda (e: +T).(match e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | +(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind b) x4 +x5) H10) in (\lambda (H16: (eq T x0 x4)).(\lambda (H17: (eq B Abst b)).(let +H18 \def (eq_ind_r T x5 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) +H12 x1 H15) in (let H19 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c v2 t0)) +H11 x0 H16) in (let H20 \def (eq_ind_r B b (\lambda (b0: B).(pr3 (CHead c +(Bind b0) v2) t x1)) H18 Abst H17) in (let H21 \def (eq_ind_r B b (\lambda +(b0: B).(not (eq B b0 Abst))) H Abst H17) in (eq_ind B Abst (\lambda (b0: +B).(pr3 c (THead (Bind b0) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead +(Bind Abbr) x2 x3))) (let H22 \def (match (H21 (refl_equal B Abst)) in False +with []) in H22) b H17)))))))) H14)) H13))))))) H9)) (\lambda (H9: (pr3 +(CHead c (Bind b) v2) t (lift (S O) O (THead (Bind Abst) x0 x1)))).(pr3_t +(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O x2) (lift (S O) O (THead +(Bind Abst) x0 x1)))) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) +t)) c (pr3_head_2 c v2 (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat +Appl) (lift (S O) O x2) (lift (S O) O (THead (Bind Abst) x0 x1))) (Bind b) +(pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O x2) (pr3_lift +(CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c (drop_refl c) v2) +v1 x2 H5) t (lift (S O) O (THead (Bind Abst) x0 x1)) H9 Appl)) (THead (Bind +Abbr) x2 x3) (eq_ind T (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) +x0 x1))) (\lambda (t0: T).(pr3 c (THead (Bind b) v2 t0) (THead (Bind Abbr) x2 +x3))) (pr3_sing c (THead (Bind Abbr) x2 x1) (THead (Bind b) v2 (lift (S O) O +(THead (Flat Appl) x2 (THead (Bind Abst) x0 x1)))) (pr2_free c (THead (Bind +b) v2 (lift (S O) O (THead (Flat Appl) x2 (THead (Bind Abst) x0 x1)))) (THead +(Bind Abbr) x2 x1) (pr0_zeta b H (THead (Flat Appl) x2 (THead (Bind Abst) x0 +x1)) (THead (Bind Abbr) x2 x1) (pr0_beta x0 x2 x2 (pr0_refl x2) x1 x1 +(pr0_refl x1)) v2)) (THead (Bind Abbr) x2 x3) (pr3_head_12 c x2 x2 (pr3_refl +c x2) (Bind Abbr) x1 x3 (H7 Abbr x2))) (THead (Flat Appl) (lift (S O) O x2) +(lift (S O) O (THead (Bind Abst) x0 x1))) (lift_flat Appl x2 (THead (Bind +Abst) x0 x1) (S O) O)))) H8))) u2 H4))))))))) H3)) (\lambda (H3: (ex6_6 B T T +T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead (Bind b0) y1 z1)))))))) (\lambda @@ -513,54 +491,64 @@ O u3) z2)) u2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) -y2) z1 z2))))))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O -v1) t)) u2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: -T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 -Abst))).(\lambda (H5: (pr3 c (THead (Bind b) v2 t) (THead (Bind x0) x1 -x2))).(\lambda (H6: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) -O x4) x3)) u2)).(\lambda (H7: (pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 -x5)).(\lambda (H9: (pr3 (CHead c (Bind x0) x5) x2 x3)).(pr3_t (THead (Bind -x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) c (let H_x \def (pr3_gen_bind b H c v2 t -(THead (Bind x0) x1 x2) H5) in (let H10 \def H_x in (or_ind (ex3_2 T T -(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind -b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: -T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2)))) (pr3 (CHead c (Bind -b) v2) t (lift (S O) O (THead (Bind x0) x1 x2))) (pr3 c (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat -Appl) (lift (S O) O x4) x3))) (\lambda (H11: (ex3_2 T T (\lambda (u3: -T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind b) v2) t t2))))).(ex3_2_ind T T (\lambda (u3: -T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) u3 t2)))) -(\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda -(t2: T).(pr3 (CHead c (Bind b) v2) t t2))) (pr3 c (THead (Bind b) v2 (THead -(Flat Appl) (lift (S O) O v1) t)) (THead (Bind x0) x5 (THead (Flat Appl) -(lift (S O) O x4) x3))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H12: (eq -T (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7))).(\lambda (H13: (pr3 c v2 -x6)).(\lambda (H14: (pr3 (CHead c (Bind b) v2) t x7)).(let H15 \def (f_equal -T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) -\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) \Rightarrow (match -k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -((let H16 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 -_) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -((let H17 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: -T).T) with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ -t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in -(\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 \def -(eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 H17) -in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 H18) -in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) x5) x2 -x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq B b0 -Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind b) v2 -(THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead (Flat -Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 x5 H8) -(Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) (lift (S -O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift (S O) O -x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c c -(drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 +y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b0: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B +b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind b) v2 t) (THead +(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u3: T).(\lambda (y2: T).(pr3 c (THead (Bind b0) +y2 (THead (Flat Appl) (lift (S O) O u3) z2)) u2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr3 c v1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b0: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b0) y2) z1 z2))))))) (pr3 c +(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) u2) (\lambda (x0: +B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: +T).(\lambda (x5: T).(\lambda (H4: (not (eq B x0 Abst))).(\lambda (H5: (pr3 c +(THead (Bind b) v2 t) (THead (Bind x0) x1 x2))).(\lambda (H6: (pr3 c (THead +(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)).(\lambda (H7: +(pr3 c v1 x4)).(\lambda (H8: (pr3 c x1 x5)).(\lambda (H9: (pr3 (CHead c (Bind +x0) x5) x2 x3)).(pr3_t (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O +x4) x3)) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) c (let +H_x \def (pr3_gen_bind b H c v2 t (THead (Bind x0) x1 x2) H5) in (let H10 +\def H_x in (or_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead +(Bind x0) x1 x2) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: +T).(pr3 c v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) +v2) t t2)))) (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 +x2))) (pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) +(THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))) (\lambda (H11: +(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind x0) x1 x2) +(THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c v2 u3))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t +t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind +x0) x1 x2) (THead (Bind b) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr3 c +v2 u3))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b) v2) t t2))) +(pr3 c (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead +(Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4) x3))) (\lambda (x6: +T).(\lambda (x7: T).(\lambda (H12: (eq T (THead (Bind x0) x1 x2) (THead (Bind +b) x6 x7))).(\lambda (H13: (pr3 c v2 x6)).(\lambda (H14: (pr3 (CHead c (Bind +b) v2) t x7)).(let H15 \def (f_equal T B (\lambda (e: T).(match e with +[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) +\Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow +x0])])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 x7) H12) in ((let H16 \def +(f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x1 | (TLRef +_) \Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) +(THead (Bind b) x6 x7) H12) in ((let H17 \def (f_equal T T (\lambda (e: +T).(match e with [(TSort _) \Rightarrow x2 | (TLRef _) \Rightarrow x2 | +(THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind b) x6 +x7) H12) in (\lambda (H18: (eq T x1 x6)).(\lambda (H19: (eq B x0 b)).(let H20 +\def (eq_ind_r T x7 (\lambda (t0: T).(pr3 (CHead c (Bind b) v2) t t0)) H14 x2 +H17) in (let H21 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c v2 t0)) H13 x1 +H18) in (let H22 \def (eq_ind B x0 (\lambda (b0: B).(pr3 (CHead c (Bind b0) +x5) x2 x3)) H9 b H19) in (let H23 \def (eq_ind B x0 (\lambda (b0: B).(not (eq +B b0 Abst))) H4 b H19) in (eq_ind_r B b (\lambda (b0: B).(pr3 c (THead (Bind +b) v2 (THead (Flat Appl) (lift (S O) O v1) t)) (THead (Bind b0) x5 (THead +(Flat Appl) (lift (S O) O x4) x3)))) (pr3_head_21 c v2 x5 (pr3_t x1 v2 c H21 +x5 H8) (Bind b) (THead (Flat Appl) (lift (S O) O v1) t) (THead (Flat Appl) +(lift (S O) O x4) x3) (pr3_flat (CHead c (Bind b) v2) (lift (S O) O v1) (lift +(S O) O x4) (pr3_lift (CHead c (Bind b) v2) c (S O) O (drop_drop (Bind b) O c +c (drop_refl c) v2) v1 x4 H7) t x3 (pr3_t x2 t (CHead c (Bind b) v2) H20 x3 (pr3_pr3_pr3_t c v2 x1 H21 x2 x3 (Bind b) (pr3_pr3_pr3_t c x1 x5 H8 x2 x3 (Bind b) H22))) Appl)) x0 H19)))))))) H16)) H15))))))) H11)) (\lambda (H11: (pr3 (CHead c (Bind b) v2) t (lift (S O) O (THead (Bind x0) x1 x2)))).(pr3_t @@ -587,9 +575,6 @@ H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) O) O x4) Appl))) (THead (Flat Appl) (lift (S O) O x4) (lift (S O) O (THead (Bind x0) x1 x2))) (lift_flat Appl x4 (THead (Bind x0) x1 x2) (S O) O)))) H10))) u2 H6))))))))))))) H3)) H2)))))))))). -(* COMMENTS -Initial nodes: 4805 -END *) theorem pr3_iso_appls_appl_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (v: T).(\forall (u: @@ -758,9 +743,6 @@ Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H9 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H10 (lift (S O) O x4) Appl)) u2 H7)))))))))))))) H4)) H3))))))))) vs)))))). -(* COMMENTS -Initial nodes: 3571 -END *) theorem pr3_iso_appls_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (vs: TList).(\forall (u: @@ -841,9 +823,6 @@ t) t0))) (iso_head t1 t1 (THeads (Flat Appl) ts0 (THead (Flat Appl) t (THead (Flat Appl) (TApp (lifts (S O) O ts) (lift (S O) O t)) t0) (theads_tapp (Flat Appl) (lift (S O) O t) t0 (lifts (S O) O ts))) (lifts (S O) O (TApp ts t)) (lifts_tapp (S O) O t ts))))))))))) vs))). -(* COMMENTS -Initial nodes: 1681 -END *) theorem pr3_iso_beta: \forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 \def (THead (Flat @@ -915,16 +894,15 @@ u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (x4: T).(\lambda (x5: T).(\lambda (H8: (eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5))).(\lambda (H9: (pr3 c w x4)).(\lambda (H10: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x5))))).(let H11 \def (f_equal -T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) -(THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in ((let H12 \def -(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ _ t0) -\Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) x4 x5) H8) in -(\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 (\lambda (t0: -T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) H10 x1 -H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w t0)) H9 x0 -H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c +T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef _) +\Rightarrow x0 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst) x0 x1) +(THead (Bind Abst) x4 x5) H8) in ((let H12 \def (f_equal T T (\lambda (e: +T).(match e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | +(THead _ _ t0) \Rightarrow t0])) (THead (Bind Abst) x0 x1) (THead (Bind Abst) +x4 x5) H8) in (\lambda (H13: (eq T x0 x4)).(let H14 \def (eq_ind_r T x5 +(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t +t0)))) H10 x1 H12) in (let H15 \def (eq_ind_r T x4 (\lambda (t0: T).(pr3 c w +t0)) H9 x0 H13) in (pr3_t (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) v t) c (pr3_head_12 c v x2 H4 (Bind Abbr) t x3 (pr3_t x1 t (CHead c (Bind Abbr) x2) (H14 Abbr x2) x3 (H6 Abbr x2))) u2 H3))))) H11))))))) H7)))))))))) H2)) (\lambda (H2: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: @@ -964,30 +942,25 @@ u3))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (x6: T).(\lambda (x7: T).(\lambda (H10: (eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) x6 x7))).(\lambda (H11: (pr3 c w x6)).(\lambda (H12: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t x7))))).(let H13 \def -(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with -[(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead k _ _) -\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b) +(f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow x0 | (TLRef +_) \Rightarrow x0 | (THead k _ _) \Rightarrow (match k with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x0])])) (THead (Bind x0) x1 x2) (THead (Bind Abst) x6 x7) H10) in ((let H14 \def (f_equal T T (\lambda (e: T).(match -e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x1 | (TLRef _) -\Rightarrow x1 | (THead _ t0 _) \Rightarrow t0])) (THead (Bind x0) x1 x2) -(THead (Bind Abst) x6 x7) H10) in ((let H15 \def (f_equal T T (\lambda (e: -T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow x2 | -(TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) (THead (Bind x0) -x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq T x1 -x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 (\lambda -(t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t t0)))) -H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w t0)) -H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c (Bind -b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: +e with [(TSort _) \Rightarrow x1 | (TLRef _) \Rightarrow x1 | (THead _ t0 _) +\Rightarrow t0])) (THead (Bind x0) x1 x2) (THead (Bind Abst) x6 x7) H10) in +((let H15 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) +\Rightarrow x2 | (TLRef _) \Rightarrow x2 | (THead _ _ t0) \Rightarrow t0])) +(THead (Bind x0) x1 x2) (THead (Bind Abst) x6 x7) H10) in (\lambda (H16: (eq +T x1 x6)).(\lambda (H17: (eq B x0 Abst)).(let H18 \def (eq_ind_r T x7 +(\lambda (t0: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t +t0)))) H12 x2 H15) in (let H19 \def (eq_ind_r T x6 (\lambda (t0: T).(pr3 c w +t0)) H11 x1 H16) in (let H20 \def (eq_ind B x0 (\lambda (b: B).(pr3 (CHead c +(Bind b) x5) x2 x3)) H8 Abst H17) in (let H21 \def (eq_ind B x0 (\lambda (b: B).(pr3 c (THead (Bind b) x5 (THead (Flat Appl) (lift (S O) O x4) x3)) u2)) H5 Abst H17) in (let H22 \def (eq_ind B x0 (\lambda (b: B).(not (eq B b Abst))) H3 Abst H17) in (let H23 \def (match (H22 (refl_equal B Abst)) in -False return (\lambda (_: False).(pr3 c (THead (Bind Abbr) v t) u2)) with []) -in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) H1)))))))). -(* COMMENTS -Initial nodes: 2459 -END *) +False with []) in H23))))))))) H14)) H13))))))) H9)))))))))))))) H2)) +H1)))))))). theorem pr3_iso_appls_beta: \forall (us: TList).(\forall (v: T).(\forall (w: T).(\forall (t: T).(let u1 @@ -1149,7 +1122,4 @@ O) O x4) x3)) (THead (Bind x0) x1 (THead (Flat Appl) (lift (S O) O x4) x2)) c (pr3_head_12 c x1 x5 H8 (Bind x0) (THead (Flat Appl) (lift (S O) O x4) x2) (THead (Flat Appl) (lift (S O) O x4) x3) (pr3_thin_dx (CHead c (Bind x0) x5) x2 x3 H9 (lift (S O) O x4) Appl)) u2 H6)))))))))))))) H3)) H2)))))))))))) us). -(* COMMENTS -Initial nodes: 3345 -END *) diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma index e428daf37..3501531aa 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr1.ma @@ -14,21 +14,20 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr3/defs.ma". +include "basic_1/pr3/defs.ma". -include "Basic-1/pr1/defs.ma". +include "basic_1/pr1/fwd.ma". theorem pr3_pr1: \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1 t2)))) \def - \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda -(t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) (\lambda (t: -T).(\lambda (c: C).(pr3_refl c t))) (\lambda (t0: T).(\lambda (t3: -T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: (pr1 t0 -t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 t4)))).(\lambda (c: -C).(pr3_sing c t0 t3 (pr2_free c t3 t0 H0) t4 (H2 c))))))))) t1 t2 H))). -(* COMMENTS -Initial nodes: 95 -END *) + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(let TMP_1 \def +(\lambda (t: T).(\lambda (t0: T).(\forall (c: C).(pr3 c t t0)))) in (let +TMP_2 \def (\lambda (t: T).(\lambda (c: C).(pr3_refl c t))) in (let TMP_5 +\def (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda +(t4: T).(\lambda (_: (pr1 t0 t4)).(\lambda (H2: ((\forall (c: C).(pr3 c t0 +t4)))).(\lambda (c: C).(let TMP_3 \def (pr2_free c t3 t0 H0) in (let TMP_4 +\def (H2 c) in (pr3_sing c t0 t3 TMP_3 t4 TMP_4)))))))))) in (pr1_ind TMP_1 +TMP_2 TMP_5 t1 t2 H)))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma index 935850b1c..b50ab1cbb 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/pr3.ma @@ -14,9 +14,9 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr3/props.ma". +include "basic_1/pr3/props.ma". -include "Basic-1/pr2/pr2.ma". +include "basic_1/pr2/pr2.ma". theorem pr3_strip: \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall @@ -24,26 +24,32 @@ theorem pr3_strip: T).(pr3 c t2 t)))))))) \def \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 -t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 c t -t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 -t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t -t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 -t3)) t2 (pr3_pr2 c t t2 H0) (pr3_refl c t2))))) (\lambda (t2: T).(\lambda -(t3: T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 -t4)).(\lambda (H2: ((\forall (t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: -T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda -(H3: (pr2 c t3 t5)).(ex2_ind T (\lambda (t: T).(pr2 c t5 t)) (\lambda (t: -T).(pr2 c t2 t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c -t5 t))) (\lambda (x: T).(\lambda (H4: (pr2 c t5 x)).(\lambda (H5: (pr2 c t2 -x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) -(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda -(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T -(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_sing c -x t5 H4 x0 H7))))) (H2 x H5))))) (pr2_confluence c t3 t5 H3 t2 H0)))))))))) -t0 t1 H)))). -(* COMMENTS -Initial nodes: 375 -END *) +t1)).(let TMP_3 \def (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr2 +c t t3) \to (let TMP_1 \def (\lambda (t4: T).(pr3 c t2 t4)) in (let TMP_2 +\def (\lambda (t4: T).(pr3 c t3 t4)) in (ex2 T TMP_1 TMP_2))))))) in (let +TMP_8 \def (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let +TMP_4 \def (\lambda (t3: T).(pr3 c t t3)) in (let TMP_5 \def (\lambda (t3: +T).(pr3 c t2 t3)) in (let TMP_6 \def (pr3_pr2 c t t2 H0) in (let TMP_7 \def +(pr3_refl c t2) in (ex_intro2 T TMP_4 TMP_5 t2 TMP_6 TMP_7)))))))) in (let +TMP_26 \def (\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 +t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda (H2: ((\forall +(t5: T).((pr2 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: +T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr2 c t3 t5)).(let +TMP_9 \def (\lambda (t: T).(pr2 c t5 t)) in (let TMP_10 \def (\lambda (t: +T).(pr2 c t2 t)) in (let TMP_11 \def (\lambda (t: T).(pr3 c t4 t)) in (let +TMP_12 \def (\lambda (t: T).(pr3 c t5 t)) in (let TMP_13 \def (ex2 T TMP_11 +TMP_12) in (let TMP_24 \def (\lambda (x: T).(\lambda (H4: (pr2 c t5 +x)).(\lambda (H5: (pr2 c t2 x)).(let TMP_14 \def (\lambda (t: T).(pr3 c t4 +t)) in (let TMP_15 \def (\lambda (t: T).(pr3 c x t)) in (let TMP_16 \def +(\lambda (t: T).(pr3 c t4 t)) in (let TMP_17 \def (\lambda (t: T).(pr3 c t5 +t)) in (let TMP_18 \def (ex2 T TMP_16 TMP_17) in (let TMP_22 \def (\lambda +(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(let TMP_19 +\def (\lambda (t: T).(pr3 c t4 t)) in (let TMP_20 \def (\lambda (t: T).(pr3 c +t5 t)) in (let TMP_21 \def (pr3_sing c x t5 H4 x0 H7) in (ex_intro2 T TMP_19 +TMP_20 x0 H6 TMP_21))))))) in (let TMP_23 \def (H2 x H5) in (ex2_ind T TMP_14 +TMP_15 TMP_18 TMP_22 TMP_23))))))))))) in (let TMP_25 \def (pr2_confluence c +t3 t5 H3 t2 H0) in (ex2_ind T TMP_9 TMP_10 TMP_13 TMP_24 +TMP_25)))))))))))))))) in (pr3_ind c TMP_3 TMP_8 TMP_26 t0 t1 H))))))). theorem pr3_confluence: \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall @@ -51,24 +57,29 @@ theorem pr3_confluence: T).(pr3 c t2 t)))))))) \def \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr3 c t0 -t1)).(pr3_ind c (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 c t -t3) \to (ex2 T (\lambda (t4: T).(pr3 c t2 t4)) (\lambda (t4: T).(pr3 c t3 -t4))))))) (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t -t2)).(ex_intro2 T (\lambda (t3: T).(pr3 c t t3)) (\lambda (t3: T).(pr3 c t2 -t3)) t2 H0 (pr3_refl c t2))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda -(H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 t4)).(\lambda -(H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: T).(pr3 c t4 -t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda (H3: (pr3 c -t3 t5)).(ex2_ind T (\lambda (t: T).(pr3 c t5 t)) (\lambda (t: T).(pr3 c t2 -t)) (ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) -(\lambda (x: T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 -x)).(ex2_ind T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c x t)) -(ex2 T (\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))) (\lambda -(x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: (pr3 c x x0)).(ex_intro2 T -(\lambda (t: T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t)) x0 H6 (pr3_t x t5 -c H4 x0 H7))))) (H2 x H5))))) (pr3_strip c t3 t5 H3 t2 H0)))))))))) t0 t1 -H)))). -(* COMMENTS -Initial nodes: 367 -END *) +t1)).(let TMP_3 \def (\lambda (t: T).(\lambda (t2: T).(\forall (t3: T).((pr3 +c t t3) \to (let TMP_1 \def (\lambda (t4: T).(pr3 c t2 t4)) in (let TMP_2 +\def (\lambda (t4: T).(pr3 c t3 t4)) in (ex2 T TMP_1 TMP_2))))))) in (let +TMP_7 \def (\lambda (t: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t t2)).(let +TMP_4 \def (\lambda (t3: T).(pr3 c t t3)) in (let TMP_5 \def (\lambda (t3: +T).(pr3 c t2 t3)) in (let TMP_6 \def (pr3_refl c t2) in (ex_intro2 T TMP_4 +TMP_5 t2 H0 TMP_6))))))) in (let TMP_25 \def (\lambda (t2: T).(\lambda (t3: +T).(\lambda (H0: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (_: (pr3 c t2 +t4)).(\lambda (H2: ((\forall (t5: T).((pr3 c t2 t5) \to (ex2 T (\lambda (t: +T).(pr3 c t4 t)) (\lambda (t: T).(pr3 c t5 t))))))).(\lambda (t5: T).(\lambda +(H3: (pr3 c t3 t5)).(let TMP_8 \def (\lambda (t: T).(pr3 c t5 t)) in (let +TMP_9 \def (\lambda (t: T).(pr3 c t2 t)) in (let TMP_10 \def (\lambda (t: +T).(pr3 c t4 t)) in (let TMP_11 \def (\lambda (t: T).(pr3 c t5 t)) in (let +TMP_12 \def (ex2 T TMP_10 TMP_11) in (let TMP_23 \def (\lambda (x: +T).(\lambda (H4: (pr3 c t5 x)).(\lambda (H5: (pr3 c t2 x)).(let TMP_13 \def +(\lambda (t: T).(pr3 c t4 t)) in (let TMP_14 \def (\lambda (t: T).(pr3 c x +t)) in (let TMP_15 \def (\lambda (t: T).(pr3 c t4 t)) in (let TMP_16 \def +(\lambda (t: T).(pr3 c t5 t)) in (let TMP_17 \def (ex2 T TMP_15 TMP_16) in +(let TMP_21 \def (\lambda (x0: T).(\lambda (H6: (pr3 c t4 x0)).(\lambda (H7: +(pr3 c x x0)).(let TMP_18 \def (\lambda (t: T).(pr3 c t4 t)) in (let TMP_19 +\def (\lambda (t: T).(pr3 c t5 t)) in (let TMP_20 \def (pr3_t x t5 c H4 x0 +H7) in (ex_intro2 T TMP_18 TMP_19 x0 H6 TMP_20))))))) in (let TMP_22 \def (H2 +x H5) in (ex2_ind T TMP_13 TMP_14 TMP_17 TMP_21 TMP_22))))))))))) in (let +TMP_24 \def (pr3_strip c t3 t5 H3 t2 H0) in (ex2_ind T TMP_8 TMP_9 TMP_12 +TMP_23 TMP_24)))))))))))))))) in (pr3_ind c TMP_3 TMP_7 TMP_25 t0 t1 H))))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma index 777f5d85d..299b08b9a 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma @@ -14,11 +14,13 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr3/pr1.ma". +include "basic_1/pr3/fwd.ma". -include "Basic-1/pr2/props.ma". +include "basic_1/pr3/pr1.ma". -include "Basic-1/pr1/props.ma". +include "basic_1/pr2/props.ma". + +include "basic_1/pr1/props.ma". theorem clear_pr3_trans: \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to @@ -30,9 +32,6 @@ T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5: T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3 t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 107 -END *) theorem pr3_pr2: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c @@ -40,9 +39,6 @@ t1 t2)))) \def \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))). -(* COMMENTS -Initial nodes: 31 -END *) theorem pr3_t: \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall @@ -55,9 +51,6 @@ c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3: (pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 127 -END *) theorem pr3_thin_dx: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall @@ -72,9 +65,6 @@ T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 167 -END *) theorem pr3_head_1: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall @@ -89,9 +79,6 @@ T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t) (H2 k t)))))))))) u1 u2 H)))). -(* COMMENTS -Initial nodes: 167 -END *) theorem pr3_head_2: \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall @@ -106,9 +93,6 @@ T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0) (THead k u t4) H2))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 175 -END *) theorem pr3_head_21: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall @@ -119,9 +103,6 @@ c (THead k u1 t1) (THead k u2 t2))))))))) u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 (CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))). -(* COMMENTS -Initial nodes: 89 -END *) theorem pr3_head_12: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall @@ -132,9 +113,6 @@ c (THead k u1 t1) (THead k u2 t2))))))))) u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 (CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))). -(* COMMENTS -Initial nodes: 89 -END *) theorem pr3_cflat: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall @@ -148,9 +126,6 @@ T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c (Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 151 -END *) theorem pr3_flat: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall @@ -161,9 +136,6 @@ theorem pr3_flat: u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda (f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f u2))))))))). -(* COMMENTS -Initial nodes: 59 -END *) theorem pr3_pr0_pr2_t: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall @@ -189,28 +161,26 @@ c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda (H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \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) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 -(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) +(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead +c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) -u2) (CHead d (Bind Abbr) u) H9))) in ((let H12 \def (f_equal C T (\lambda (e: -C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | -(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) -u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind -b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda (H13: (eq B Abbr -b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0: -T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3 -(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 -t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t) -(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x -t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr) -u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind -Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) +u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 \def (f_equal C B (\lambda (e: +C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow +(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) +(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d +(Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) +H9))) in ((let H12 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) +\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) +(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 +(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda +(H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u +(\lambda (t0: T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda +(b0: B).(pr3 (CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: +T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind +Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda +(H17: (pr0 x t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c +(Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 +(CHead c (Bind Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) (pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f: F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O @@ -235,9 +205,6 @@ u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))). -(* COMMENTS -Initial nodes: 1557 -END *) theorem pr3_pr2_pr2_t: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall @@ -272,56 +239,52 @@ t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t) (\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 (Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda -(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 -| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind -b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 -\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) -with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K -return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) -\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) -(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind -Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) -u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let -H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in -(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind -T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i -O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda -(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let -H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O -i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n -u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7)) -(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda -(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0 -x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind -Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead -c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i) -(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1) -(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x -O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14)) -H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0 -(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 -(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0 -(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5 -H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0) -H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0 -(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4 -t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr) -u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl -(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1) -t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t) -(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta -(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind -Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4 -t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat -f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 -(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f) -c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10))))) -i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))). -(* COMMENTS -Initial nodes: 1697 -END *) +(e: C).(match e with [(CSort _) \Rightarrow d0 | (CHead c2 _ _) \Rightarrow +c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 +(CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 \def (f_equal C B (\lambda (e: +C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow +(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) +(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead +d0 (Bind Abbr) u0) t H12)) in ((let H15 \def (f_equal C T (\lambda (e: +C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) +(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead +d0 (Bind Abbr) u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq +C d0 c0)).(let H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) +H11 t H15) in (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 +t6)) (ex2_ind T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: +T).(subst0 (S (plus i O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) +(\lambda (x: T).(\lambda (H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S +(plus i O)) u x t6)).(let H21 \def (f_equal nat nat S (plus i O) i (sym_eq +nat i (plus i O) (plus_n_O i))) in (let H22 \def (eq_ind nat (S (plus i O)) +(\lambda (n: nat).(subst0 n u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda +(t7: T).(subst0 O t1 t5 t7)) (\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 +(Bind Abbr) t1) t4 t6) (\lambda (x0: T).(\lambda (H23: (subst0 O t1 t5 +x0)).(\lambda (H24: (pr0 x0 x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 +(pr2_delta (CHead c0 (Bind Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 +x0 H23) t6 (pr3_pr2 (CHead c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 +(Bind Abbr) t1) d u (S i) (getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 +t1 (clear_bind Abbr c0 t1) (CHead d (Bind Abbr) u) i H0) x0 x H24 t6 +H22)))))) (pr0_subst0_back t2 t5 x O H19 t1 H1))))))) (subst0_subst0 t5 t6 t +O H18 t2 u i H2)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: +(clear (CHead c0 (Flat f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 +(Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 O (getl_intro O c0 +(CHead d0 (Bind Abbr) u0) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead d0 +(Bind Abbr) u0) t H12)) t4 t5 H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k +t) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda (_: (((getl +i1 (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to +(pr3 (CHead c0 k t1) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) +(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind +(\lambda (k0: K).((getl (S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to +(pr3 (CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) +(CHead c0 (Bind b) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) +t1) t4 t6 (pr2_delta (CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) +i1 c0 (CHead d0 (Bind Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) +u0) t i1 H12) t1) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S +i1) (CHead c0 (Flat f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 +(Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) +(getl_gen_S (Flat f) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) +f t1)))) k H10))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c +u1 u2 H)))). theorem pr3_pr2_pr3_t: \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall @@ -338,9 +301,6 @@ T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_: \to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1 u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 u1 H3)))))))))) t1 t2 H)))))). -(* COMMENTS -Initial nodes: 199 -END *) theorem pr3_pr3_pr3_t: \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall @@ -357,9 +317,6 @@ t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3 (CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0: T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0 t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))). -(* COMMENTS -Initial nodes: 187 -END *) theorem pr3_lift: \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h @@ -374,9 +331,6 @@ d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2 H0)))))))). -(* COMMENTS -Initial nodes: 167 -END *) theorem pr3_eta: \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind @@ -397,11 +351,1254 @@ O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O) (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O)) (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u)) -(lift (S O) O u) (subst1_lift_S u O O (le_n O))) u (pr1_pr0 (THead (Bind +(lift (S O) O u) (subst1_lift_S u O O (le_O_n O))) u (pr1_pr0 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u (pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead (Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))). -(* COMMENTS -Initial nodes: 523 -END *) + +theorem pr3_gen_void: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c +(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 +(lift (S O) O x))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 +t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c +(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y +x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 +(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c +(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t +(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 +(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: +T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind +Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 +(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c +(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T +(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead +(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead +c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) +(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) +u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c +t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall +(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 +T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) +(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: +T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let +H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 +x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) +(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O +t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind +Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda +(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead +(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead +c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq +T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) +O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind +Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind +T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind +Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) +(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead +c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O +t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 +(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c +(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c +(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda +(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) +(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda +(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 +x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall +(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda +(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c +(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 +(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead +c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: +(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) +(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind +Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift +(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: +((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O +t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead +c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) +(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S +O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) +O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). + +theorem pr3_gen_abbr: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c +(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) +u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 +t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S +O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda +(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) +O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind +Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x +(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 +(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: +T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind +Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 +(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda +(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 +x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) +x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda +(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 +x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) +(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: +T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: +T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 +(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c +t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 +t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 +(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: +T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda +(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: +T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) +z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 +(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 +t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 +t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind +b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead +c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 +(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 +z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z +t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: +T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda +(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: +T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) +z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda +(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 +c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c +(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 +(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: +T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: +T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) +z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c +(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 +(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 +z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) +(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c +(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind +T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind +Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 +(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 +x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) +in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind +Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c +(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 +u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 +t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 +t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c +(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: +T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c +x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T +(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda +(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 +x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 +(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 +H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O +t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) +x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) +O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: +T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 +x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c +(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 +t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: +T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) +x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall +(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S +O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 +(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S +O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) +x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda +(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 +x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) +x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 +T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) +x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda +(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 +x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) +(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 +x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c +(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 +x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O +t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T +(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) +(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: +T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: +T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c +(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq +T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 +t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: +T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 +x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) +x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall +(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S +O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 +(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S +O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) +x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda +(x6: T).(\lambda (x7: T).(\lambda (H18: (eq T t4 (THead (Bind Abbr) x6 +x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) +x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 +T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) +x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S +O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda +(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 +(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 +x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 +H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind +Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 +(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) +(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O +t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 +(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) +x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 +(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing +(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 +(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) +H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda +(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) +x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing +(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) +(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c +(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))). + +theorem pr3_gen_appl: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c +(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 +c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 +t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda +(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 +t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 +(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 +u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda +(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T +T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: +T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall +(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 +T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) +(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t +(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda +(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead +(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind +Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) +(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat +Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 +c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 +(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 +x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda +(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 +t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat +Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 +z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 +(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c +t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 +t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 +(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq +T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead +(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 +T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 +t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat +Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) +t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 +c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 +x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 +\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t +(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 +c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 +(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: +T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 +(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda +(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 +t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 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z2))))))))) (\lambda +(x4: T).(\lambda (x5: T).(\lambda (H15: (eq T t4 (THead (Flat Appl) x4 +x5))).(\lambda (H16: (pr3 c x2 x4)).(\lambda (H17: (pr3 c x3 x5)).(eq_ind_r T +(THead (Flat Appl) x4 x5) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t5: T).(eq T t (THead (Flat Appl) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 +t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 +t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t5: 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T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: +(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda +(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b: +B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T +T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro +T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: +T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 +t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10 +(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T +T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 +z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b +Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift +(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 +(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 +c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: +T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3 +c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9 +(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2 +x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9) +x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro +B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) +x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 +(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) +(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind +Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind +Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 +(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq +T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 +x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 +x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c +(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 +\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t +(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T +T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 +c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 +(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: +T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 +z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat +Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c +(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind +Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) +x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind +B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: +B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 +y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: +T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) +(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) +u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: +T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) +t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 +c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) +y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda +(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: +T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T +x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 +(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 +x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) +x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 +(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 +(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) +x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead +(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) +(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall +(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T +T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 +c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind +b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda +(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 +(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let +H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 +(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda +(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind +Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) +z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) +(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda +(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 +z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat +Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c +(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) +t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: +B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: +T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) +y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) +H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) +x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))). + +theorem pr3_gen_bind: + \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: +T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 +t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind +b) u1) t1 (lift (S O) O x))))))))) +\def + \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall +(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind +b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x +(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 +(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B +Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: +T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def +(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) +u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T +T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) +t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 +c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 +t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind +Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda +(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) +t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x +(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 +(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S +O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead +(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 +H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S +O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x +(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 +(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: +(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: +T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 +\def (match (H (refl_equal B Abst)) in False with []) in H1))))))) (\lambda +(_: (not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: +T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 +\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall +(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c +(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) +u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda +(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) +u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) +u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x +(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead +c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 +c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 +t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 +x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall +(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda +(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 +(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S +O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead +(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 +H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 +(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 +c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 +t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b). diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma index 3bcbd4965..048fc9beb 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/subst1.ma @@ -14,9 +14,9 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr3/defs.ma". +include "basic_1/pr3/fwd.ma". -include "Basic-1/pr2/subst1.ma". +include "basic_1/pr2/subst1.ma". theorem pr3_subst1: \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c @@ -26,27 +26,33 @@ w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) \def \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: -T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 -w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1: -T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 -w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda -(t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5: -T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v -t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i -v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T -(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T -(\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2))) -(\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 -x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5 -w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 -w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v -t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 -i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c -e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))). -(* COMMENTS -Initial nodes: 425 -END *) +T).(\lambda (H0: (pr3 c t1 t2)).(let TMP_3 \def (\lambda (t: T).(\lambda (t0: +T).(\forall (w1: T).((subst1 i v t w1) \to (let TMP_1 \def (\lambda (w2: +T).(pr3 c w1 w2)) in (let TMP_2 \def (\lambda (w2: T).(subst1 i v t0 w2)) in +(ex2 T TMP_1 TMP_2))))))) in (let TMP_7 \def (\lambda (t: T).(\lambda (w1: +T).(\lambda (H1: (subst1 i v t w1)).(let TMP_4 \def (\lambda (w2: T).(pr3 c +w1 w2)) in (let TMP_5 \def (\lambda (w2: T).(subst1 i v t w2)) in (let TMP_6 +\def (pr3_refl c w1) in (ex_intro2 T TMP_4 TMP_5 w1 TMP_6 H1))))))) in (let +TMP_25 \def (\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 +t3)).(\lambda (t5: T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall +(w1: T).((subst1 i v t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) +(\lambda (w2: T).(subst1 i v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: +(subst1 i v t4 w1)).(let TMP_8 \def (\lambda (w2: T).(pr2 c w1 w2)) in (let +TMP_9 \def (\lambda (w2: T).(subst1 i v t3 w2)) in (let TMP_10 \def (\lambda +(w2: T).(pr3 c w1 w2)) in (let TMP_11 \def (\lambda (w2: T).(subst1 i v t5 +w2)) in (let TMP_12 \def (ex2 T TMP_10 TMP_11) in (let TMP_23 \def (\lambda +(x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3 x)).(let +TMP_13 \def (\lambda (w2: T).(pr3 c x w2)) in (let TMP_14 \def (\lambda (w2: +T).(subst1 i v t5 w2)) in (let TMP_15 \def (\lambda (w2: T).(pr3 c w1 w2)) in +(let TMP_16 \def (\lambda (w2: T).(subst1 i v t5 w2)) in (let TMP_17 \def +(ex2 T TMP_15 TMP_16) in (let TMP_21 \def (\lambda (x0: T).(\lambda (H7: (pr3 +c x x0)).(\lambda (H8: (subst1 i v t5 x0)).(let TMP_18 \def (\lambda (w2: +T).(pr3 c w1 w2)) in (let TMP_19 \def (\lambda (w2: T).(subst1 i v t5 w2)) in +(let TMP_20 \def (pr3_sing c x w1 H5 x0 H7) in (ex_intro2 T TMP_18 TMP_19 x0 +TMP_20 H8))))))) in (let TMP_22 \def (H3 x H6) in (ex2_ind T TMP_13 TMP_14 +TMP_17 TMP_21 TMP_22))))))))))) in (let TMP_24 \def (pr2_subst1 c e v i H t4 +t3 H1 w1 H4) in (ex2_ind T TMP_8 TMP_9 TMP_12 TMP_23 TMP_24)))))))))))))))) +in (pr3_ind c TMP_3 TMP_7 TMP_25 t1 t2 H0))))))))))). theorem pr3_gen_cabbr: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall @@ -57,39 +63,50 @@ a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T x1 x2)))))))))))))))) \def \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 -t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u: -T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: -C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall -(x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 -d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))) -(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda -(_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: -(csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 -a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d -x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2))) -(\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda -(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: -T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: -T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: -C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall -(x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: -T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 -x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda -(H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4: -(csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0 -a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d -x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda -(x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d -x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1 -d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda -(x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2)) -(ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: -T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O) -d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d -u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x -x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0 -e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))). -(* COMMENTS -Initial nodes: 731 -END *) +t2)).(let TMP_5 \def (\lambda (t: T).(\lambda (t0: T).(\forall (e: +C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to +(\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) +\to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (let TMP_3 \def +(\lambda (x2: T).(let TMP_1 \def (S O) in (let TMP_2 \def (lift TMP_1 d x2) +in (subst1 d u t0 TMP_2)))) in (let TMP_4 \def (\lambda (x2: T).(pr3 a x1 +x2)) in (ex2 T TMP_3 TMP_4))))))))))))))) in (let TMP_11 \def (\lambda (t: +T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda (_: (getl d c +(CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_: (csubst1 d u c +a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0 a)).(\lambda (x1: +T).(\lambda (H3: (subst1 d u t (lift (S O) d x1))).(let TMP_8 \def (\lambda +(x2: T).(let TMP_6 \def (S O) in (let TMP_7 \def (lift TMP_6 d x2) in (subst1 +d u t TMP_7)))) in (let TMP_9 \def (\lambda (x2: T).(pr3 a x1 x2)) in (let +TMP_10 \def (pr3_refl a x1) in (ex_intro2 T TMP_8 TMP_9 x1 H3 +TMP_10))))))))))))))) in (let TMP_39 \def (\lambda (t0: T).(\lambda (t3: +T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 +t4)).(\lambda (H2: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl +d c (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to +(\forall (a: C).((drop (S O) d a0 a) \to (\forall (x1: T).((subst1 d u t0 +(lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d +x2))) (\lambda (x2: T).(pr3 a x1 x2))))))))))))))).(\lambda (e: C).(\lambda +(u: T).(\lambda (d: nat).(\lambda (H3: (getl d c (CHead e (Bind Abbr) +u))).(\lambda (a0: C).(\lambda (H4: (csubst1 d u c a0)).(\lambda (a: +C).(\lambda (H5: (drop (S O) d a0 a)).(\lambda (x1: T).(\lambda (H6: (subst1 +d u t3 (lift (S O) d x1))).(let TMP_14 \def (\lambda (x2: T).(let TMP_12 \def +(S O) in (let TMP_13 \def (lift TMP_12 d x2) in (subst1 d u t0 TMP_13)))) in +(let TMP_15 \def (\lambda (x2: T).(pr2 a x1 x2)) in (let TMP_18 \def (\lambda +(x2: T).(let TMP_16 \def (S O) in (let TMP_17 \def (lift TMP_16 d x2) in +(subst1 d u t4 TMP_17)))) in (let TMP_19 \def (\lambda (x2: T).(pr3 a x1 x2)) +in (let TMP_20 \def (ex2 T TMP_18 TMP_19) in (let TMP_37 \def (\lambda (x: +T).(\lambda (H7: (subst1 d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 +x)).(let TMP_23 \def (\lambda (x2: T).(let TMP_21 \def (S O) in (let TMP_22 +\def (lift TMP_21 d x2) in (subst1 d u t4 TMP_22)))) in (let TMP_24 \def +(\lambda (x2: T).(pr3 a x x2)) in (let TMP_27 \def (\lambda (x2: T).(let +TMP_25 \def (S O) in (let TMP_26 \def (lift TMP_25 d x2) in (subst1 d u t4 +TMP_26)))) in (let TMP_28 \def (\lambda (x2: T).(pr3 a x1 x2)) in (let TMP_29 +\def (ex2 T TMP_27 TMP_28) in (let TMP_35 \def (\lambda (x0: T).(\lambda (H9: +(subst1 d u t4 (lift (S O) d x0))).(\lambda (H10: (pr3 a x x0)).(let TMP_32 +\def (\lambda (x2: T).(let TMP_30 \def (S O) in (let TMP_31 \def (lift TMP_30 +d x2) in (subst1 d u t4 TMP_31)))) in (let TMP_33 \def (\lambda (x2: T).(pr3 +a x1 x2)) in (let TMP_34 \def (pr3_sing a x x1 H8 x0 H10) in (ex_intro2 T +TMP_32 TMP_33 x0 H9 TMP_34))))))) in (let TMP_36 \def (H2 e u d H3 a0 H4 a H5 +x H7) in (ex2_ind T TMP_23 TMP_24 TMP_29 TMP_35 TMP_36))))))))))) in (let +TMP_38 \def (pr2_gen_cabbr c t3 t0 H0 e u d H3 a0 H4 a H5 x1 H6) in (ex2_ind +T TMP_14 TMP_15 TMP_20 TMP_37 TMP_38)))))))))))))))))))))))) in (pr3_ind c +TMP_5 TMP_11 TMP_39 t1 t2 H))))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma index 052f4e5b2..c7f6669d5 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/wcpr0.ma @@ -14,53 +14,70 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/pr3/props.ma". +include "basic_1/pr3/props.ma". -include "Basic-1/wcpr0/getl.ma". +include "basic_1/wcpr0/getl.ma". theorem pr3_wcpr0_t: \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2)))))) \def - \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind + \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(let TMP_1 \def (\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0 -t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3: -C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: -T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: -T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: -T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1) -(\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t: -T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda -(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead -c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 -(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0)) -(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7: -(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq -C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c: -C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_: -(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0 -k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: -T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr) -u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda -(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k -u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind -Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2: -C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3)))) -(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3: -T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1: -T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) -x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T -(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3 -(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6 -x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta -(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t -(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1 -H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 -H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7))) -H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2 -H2)))))))))))))) c2 c1 H))). -(* COMMENTS -Initial nodes: 799 -END *) +t1 t2) \to (pr3 c t1 t2)))))) in (let TMP_2 \def (\lambda (c: C).(\lambda +(t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).H0)))) in (let TMP_57 +\def (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda +(_: ((\forall (t1: T).(\forall (t2: T).((pr3 c3 t1 t2) \to (pr3 c0 t1 +t2)))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 +u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H3: (pr3 +(CHead c3 k u2) t1 t2)).(let TMP_3 \def (CHead c3 k u1) in (let TMP_5 \def +(\lambda (t: T).(\lambda (t0: T).(let TMP_4 \def (CHead c0 k u1) in (pr3 +TMP_4 t t0)))) in (let TMP_7 \def (\lambda (t: T).(let TMP_6 \def (CHead c0 k +u1) in (pr3_refl TMP_6 t))) in (let TMP_54 \def (\lambda (t0: T).(\lambda +(t3: T).(\lambda (H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda +(_: (pr3 (CHead c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 +t4)).(let TMP_8 \def (CHead c0 k u1) in (let TMP_9 \def (CHead c3 k u1) in +(let TMP_10 \def (\lambda (c: C).(pr2 c t3 t0)) in (let TMP_12 \def (\lambda +(_: C).(let TMP_11 \def (CHead c0 k u1) in (pr3 TMP_11 t3 t0))) in (let +TMP_52 \def (\lambda (y: C).(\lambda (H7: (pr2 y t3 t0)).(let TMP_14 \def +(\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq C c (CHead c3 k u1)) +\to (let TMP_13 \def (CHead c0 k u1) in (pr3 TMP_13 t t5)))))) in (let TMP_18 +\def (\lambda (c: C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 +t6)).(\lambda (_: (eq C c (CHead c3 k u1))).(let TMP_15 \def (CHead c0 k u1) +in (let TMP_16 \def (CHead c0 k u1) in (let TMP_17 \def (pr2_free TMP_16 t5 +t6 H8) in (pr3_pr2 TMP_15 t5 t6 TMP_17))))))))) in (let TMP_51 \def (\lambda +(c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H8: (getl +i c (CHead d (Bind Abbr) u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: +(pr0 t5 t6)).(\lambda (t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: +(eq C c (CHead c3 k u1))).(let TMP_21 \def (\lambda (c4: C).(let TMP_19 \def +(Bind Abbr) in (let TMP_20 \def (CHead d TMP_19 u) in (getl i c4 TMP_20)))) +in (let TMP_22 \def (CHead c3 k u1) in (let H12 \def (eq_ind C c TMP_21 H8 +TMP_22 H11) in (let TMP_26 \def (\lambda (e2: C).(\lambda (u3: T).(let TMP_23 +\def (CHead c0 k u1) in (let TMP_24 \def (Bind Abbr) in (let TMP_25 \def +(CHead e2 TMP_24 u3) in (getl i TMP_23 TMP_25)))))) in (let TMP_27 \def +(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) in (let TMP_28 \def (\lambda +(_: C).(\lambda (u3: T).(pr0 u3 u))) in (let TMP_29 \def (CHead c0 k u1) in +(let TMP_30 \def (pr3 TMP_29 t5 t) in (let TMP_44 \def (\lambda (x0: +C).(\lambda (x1: T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind +Abbr) x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(let TMP_31 +\def (\lambda (t7: T).(subst0 i x1 t6 t7)) in (let TMP_32 \def (\lambda (t7: +T).(pr0 t7 t)) in (let TMP_33 \def (CHead c0 k u1) in (let TMP_34 \def (pr3 +TMP_33 t5 t) in (let TMP_42 \def (\lambda (x: T).(\lambda (H16: (subst0 i x1 +t6 x)).(\lambda (H17: (pr0 x t)).(let TMP_35 \def (CHead c0 k u1) in (let +TMP_36 \def (CHead c0 k u1) in (let TMP_37 \def (pr2_delta TMP_36 x0 x1 i H13 +t5 t6 H9 x H16) in (let TMP_38 \def (CHead c0 k u1) in (let TMP_39 \def +(CHead c0 k u1) in (let TMP_40 \def (pr2_free TMP_39 x t H17) in (let TMP_41 +\def (pr3_pr2 TMP_38 x t TMP_40) in (pr3_sing TMP_35 x t5 TMP_37 t +TMP_41))))))))))) in (let TMP_43 \def (pr0_subst0_back u t6 t i H10 x1 H15) +in (ex2_ind T TMP_31 TMP_32 TMP_34 TMP_42 TMP_43)))))))))))) in (let TMP_45 +\def (CHead c3 k u1) in (let TMP_46 \def (CHead c0 k u1) in (let TMP_47 \def +(pr0_refl u1) in (let TMP_48 \def (wcpr0_comp c0 c3 H0 u1 u1 TMP_47 k) in +(let TMP_49 \def (Bind Abbr) in (let TMP_50 \def (wcpr0_getl_back TMP_45 +TMP_46 TMP_48 i d u TMP_49 H12) in (ex3_2_ind C T TMP_26 TMP_27 TMP_28 TMP_30 +TMP_44 TMP_50))))))))))))))))))))))))))) in (pr2_ind TMP_14 TMP_18 TMP_51 y +t3 t0 H7)))))) in (let TMP_53 \def (insert_eq C TMP_9 TMP_10 TMP_12 TMP_52 +H4) in (pr3_t t0 t3 TMP_8 TMP_53 t4 H6))))))))))))) in (let TMP_55 \def +(pr2_free c3 u1 u2 H2) in (let TMP_56 \def (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 +TMP_55) in (pr3_ind TMP_3 TMP_5 TMP_7 TMP_54 t1 t2 TMP_56)))))))))))))))))) +in (wcpr0_ind TMP_1 TMP_2 TMP_57 c2 c1 H)))))). -- 2.39.2