From 7f82161596bdfccc1179f5edcc0bfd76d34516b5 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Wed, 13 Sep 2006 15:13:47 +0000 Subject: [PATCH] - ok pr0 pr1 pr2 - added one problem (like problems-4) --- .../Level-1/LambdaDelta/pr0/dec.ma | 530 +++ .../Level-1/LambdaDelta/pr0/defs.ma | 42 + .../Level-1/LambdaDelta/pr0/fwd.ma | 1642 ++++++++ .../Level-1/LambdaDelta/pr0/pr0.ma | 2818 +++++++++++++ .../Level-1/LambdaDelta/pr0/props.ma | 1775 ++++++++ .../Level-1/LambdaDelta/pr0/subst1.ma | 95 + .../Level-1/LambdaDelta/pr1/defs.ma | 25 + .../Level-1/LambdaDelta/pr1/pr1.ma | 64 + .../Level-1/LambdaDelta/pr1/props.ma | 62 + .../Level-1/LambdaDelta/pr2/clen.ma | 182 + .../Level-1/LambdaDelta/pr2/defs.ma | 30 + .../Level-1/LambdaDelta/pr2/fwd.ma | 3634 +++++++++++++++++ .../Level-1/LambdaDelta/pr2/pr2.ma | 248 ++ .../Level-1/LambdaDelta/pr2/props.ma | 285 ++ .../Level-1/LambdaDelta/pr2/subst1.ma | 132 + .../Level-1/LambdaDelta/pr3/defs.ma | 25 + .../Level-1/LambdaDelta/tau1/cnt.ma | 88 + .../Level-1/LambdaDelta/tau1/defs.ma | 25 + .../Level-1/LambdaDelta/tau1/props.ma | 144 + .../Level-1/LambdaDelta/theory.ma | 38 + .../LAMBDA-TYPES/Level-1/problems-1.ma | 183 + 21 files changed, 12067 insertions(+) create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/dec.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/defs.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/pr0.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/props.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/subst1.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/defs.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/pr1.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/props.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/clen.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/defs.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/fwd.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/pr2.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/defs.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/cnt.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/defs.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/props.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/problems-1.ma diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/dec.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/dec.ma new file mode 100644 index 000000000..d95c98683 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/dec.ma @@ -0,0 +1,530 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/dec". + +include "pr0/fwd.ma". + +include "subst0/dec.ma". + +include "T/dec.ma". + +include "T/props.ma". + +theorem nf0_dec: + \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T +(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 t1 t2)))) +\def + \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to +(eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl +(\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T +(\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda +(t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n) +t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T +(TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl +(\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T +(\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda +(t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n) +t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T +(TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t: +T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T +(\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0 +t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall +(P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(match k in K return (\lambda +(k0: K).(or (\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t +t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) with [(Bind b) +\Rightarrow (match b in B return (\lambda (b0: B).(or (\forall (t2: T).((pr0 +(THead (Bind b0) t t0) t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind b0) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b0) t t0) t2))))) with [Abbr +\Rightarrow (or_intror (\forall (t2: T).((pr0 (THead (Bind Abbr) t t0) t2) +\to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T +(THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in (let +H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) O v)) +(eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind +Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 +(lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 (lift +(S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T (THead +(Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S O) O +x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) t2)) +(THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind Abbr) +t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let H5 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t2) +\Rightarrow t2])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O +x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t2: T).(subst0 O t t2 (lift (S +O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6 +P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3))) +(\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x) +(\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3) +\to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2) +t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O) +O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind +Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t +(lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S +O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2))) +H1)))) | Abst \Rightarrow (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t +t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead +(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda +(H2: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H3 \def H0 in +(or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: +T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) +(or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead +(Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t +t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) +t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 +t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to +(eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead +(Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Bind Abst) t t0) t2))) (\lambda (t2: T).(\lambda (H5: (pr0 (THead +(Bind Abst) t t0) t2)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T +t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) +t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead (Bind +Abst) x0 x1))).(\lambda (H7: (pr0 t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y +\def (H4 x1 H8) in (let H_y0 \def (H2 x0 H7) in (let H9 \def (eq_ind_r T x1 +(\lambda (t3: T).(pr0 t0 t3)) H8 t0 H_y) in (let H10 \def (eq_ind_r T x1 +(\lambda (t3: T).(eq T t2 (THead (Bind Abst) x0 t3))) H6 t0 H_y) in (let H11 +\def (eq_ind_r T x0 (\lambda (t3: T).(pr0 t t3)) H7 t H_y0) in (let H12 \def +(eq_ind_r T x0 (\lambda (t3: T).(eq T t2 (THead (Bind Abst) t3 t0))) H10 t +H_y0) in (eq_ind_r T (THead (Bind Abst) t t0) (\lambda (t3: T).(eq T (THead +(Bind Abst) t t0) t3)) (refl_equal T (THead (Bind Abst) t t0)) t2 +H12)))))))))))) (pr0_gen_abst t t0 t2 H5)))))) (\lambda (H4: (ex2 T (\lambda +(t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 +t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t +t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq +T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T +t0 x) \to (\forall (P: Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror +(\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind +Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) +\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) +t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) +(THead (Bind Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead +(Bind Abst) t x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | +(TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind +Abst) t t0) (THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x +(\lambda (t2: T).(pr0 t0 t2)) H6 t0 H8) in (let H10 \def (eq_ind_r T x +(\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H5 t0 H8) in +(H10 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind +Abst))))))) H4)) H3))) (\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda +(t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) +(or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead +(Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t +t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) +t t0) t2)))) (\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P: +Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead +(Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T +(\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind +Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x +t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) +\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Abst) t t0) +(THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t2: +T).(pr0 t t2)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t2: T).((eq T +t t2) \to (\forall (P0: Prop).P0))) H3 t H6) in (H8 (refl_equal T t) P)))))) +(pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) | Void +\Rightarrow (let H_x \def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T +(\lambda (v: T).(or (subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O +v)))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T +(THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind +Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Bind Void) t t0) t2)))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 +(lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 (lift +(S O) O x)) (eq T t0 (lift (S O) O x)) (or (\forall (t2: T).((pr0 (THead +(Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda +(H3: (subst0 O t t0 (lift (S O) O x))).(let H4 \def H in (or_ind (\forall +(t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) +\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: +T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) +(\lambda (H5: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H6 \def +H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T +(\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to +(eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead +(Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Bind Void) t t0) t2)))) (\lambda (H7: ((\forall (t2: T).((pr0 t0 t2) +\to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Bind Void) t +t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq +T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Bind Void) t t0) t2))) (\lambda (t2: T).(\lambda (H8: (pr0 +(THead (Bind Void) t t0) t2)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (pr0 t0 (lift +(S O) O t2)) (eq T (THead (Bind Void) t t0) t2) (\lambda (H9: (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Bind Void) t t0) +t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T t2 (THead (Bind +Void) x0 x1))).(\lambda (H11: (pr0 t x0)).(\lambda (H12: (pr0 t0 x1)).(let +H_y \def (H7 x1 H12) in (let H_y0 \def (H5 x0 H11) in (let H13 \def (eq_ind_r +T x1 (\lambda (t3: T).(pr0 t0 t3)) H12 t0 H_y) in (let H14 \def (eq_ind_r T +x1 (\lambda (t3: T).(eq T t2 (THead (Bind Void) x0 t3))) H10 t0 H_y) in (let +H15 \def (eq_ind_r T x0 (\lambda (t3: T).(pr0 t t3)) H11 t H_y0) in (let H16 +\def (eq_ind_r T x0 (\lambda (t3: T).(eq T t2 (THead (Bind Void) t3 t0))) H14 +t H_y0) in (eq_ind_r T (THead (Bind Void) t t0) (\lambda (t3: T).(eq T (THead +(Bind Void) t t0) t3)) (refl_equal T (THead (Bind Void) t t0)) t2 +H16)))))))))))) H9)) (\lambda (H9: (pr0 t0 (lift (S O) O t2))).(let H_y \def +(H7 (lift (S O) O t2) H9) in (let H10 \def (eq_ind T t0 (\lambda (t3: +T).(subst0 O t t3 (lift (S O) O x))) H3 (lift (S O) O t2) H_y) in (eq_ind_r T +(lift (S O) O t2) (\lambda (t3: T).(eq T (THead (Bind Void) t t3) t2)) +(subst0_gen_lift_false t2 t (lift (S O) O x) (S O) O O (le_n O) (eq_ind_r nat +(plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S +O)) (plus_comm O (S O))) H10 (eq T (THead (Bind Void) t (lift (S O) O t2)) +t2)) t0 H_y)))) (pr0_gen_void t t0 t2 H8)))))) (\lambda (H7: (ex2 T (\lambda +(t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 +t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) +(\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t +t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq +T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (x0: T).(\lambda (H8: (((eq +T t0 x0) \to (\forall (P: Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror +(\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind +Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) +\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) +t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) +(THead (Bind Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead +(Bind Void) t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda +(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 +| (TLRef _) \Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Bind +Void) t t0) (THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 +(\lambda (t2: T).(pr0 t0 t2)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 +(\lambda (t2: T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H8 t0 H11) in +(H13 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind +Void))))))) H7)) H6))) (\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda +(t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) +(or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead +(Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t +t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) +t t0) t2)))) (\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P: +Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0 +(THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T +(\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind +Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void) +x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) +\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Bind Void) t t0) +(THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t2: +T).(pr0 t t2)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t2: T).((eq +T t t2) \to (\forall (P0: Prop).P0))) H6 t H9) in (H11 (refl_equal T t) +P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4))) +(\lambda (H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda +(t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq T t2 t3))) (ex2 T (\lambda +(t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 t2 +t3))))) H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2: +T).(or (\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead +(Bind Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t +t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) +t t2) t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S +O) O x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S +O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t +(lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead +(Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y +(Bind Void) x t (S O) O H5 P))) (pr0_zeta Void not_void_abst x x (pr0_refl x) +t))) t0 H3))) H2))) H1)))]) | (Flat f) \Rightarrow (match f in F return +(\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0) t2) \to +(eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead +(Flat f0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Flat f0) t t0) t2))))) with [Appl \Rightarrow (let H_x \def (binder_dec t0) +in (let H1 \def H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: +T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: +B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to +(\forall (P: Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) +t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T +(THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda +(b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w +u))))))).(ex_3_ind B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq +T t0 (THead (Bind b) w u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t +t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq +T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: +T).(\lambda (x2: T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 +\def (eq_ind T t0 (\lambda (t2: T).(or (\forall (t3: T).((pr0 t2 t3) \to (eq +T t2 t3))) (ex2 T (\lambda (t3: T).((eq T t2 t3) \to (\forall (P: Prop).P))) +(\lambda (t3: T).(pr0 t2 t3))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r +T (THead (Bind x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead +(Flat Appl) t t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T +(\lambda (t3: T).((eq T (THead (Flat Appl) t t2) t3) \to (\forall (P: +Prop).P))) (\lambda (t3: T).(pr0 (THead (Flat Appl) t t2) t3))))) ((match x0 +in B return (\lambda (b: B).((or (\forall (t2: T).((pr0 (THead (Bind b) x1 +x2) t2) \to (eq T (THead (Bind b) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T +(THead (Bind b) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Bind b) x1 x2) t2)))) \to (or (\forall (t2: T).((pr0 (THead (Flat +Appl) t (THead (Bind b) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead +(Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t +(THead (Bind b) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Flat Appl) t (THead (Bind b) x1 x2)) t2)))))) with [Abbr \Rightarrow +(\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abbr) x1 x2) t2) \to (eq +T (THead (Bind Abbr) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind +Abbr) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Bind Abbr) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat +Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead +(Bind Abbr) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t +(THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex_intro2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) +\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead +(Bind Abbr) x1 x2)) t2)) (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) +O t) x2)) (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) +(THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: +Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t2) \Rightarrow +(match t2 in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False +| (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) +\Rightarrow False])])])) I (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S +O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Abbr not_abbr_abst t t +(pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) | Abst \Rightarrow +(\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abst) x1 x2) t2) \to (eq +T (THead (Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind +Abst) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead +(Bind Abst) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat +Appl) t (THead (Bind Abst) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead +(Bind Abst) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t +(THead (Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: +T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex_intro2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) +\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead +(Bind Abst) x1 x2)) t2)) (THead (Bind Abbr) t x2) (\lambda (H6: (eq T (THead +(Flat Appl) t (THead (Bind Abst) x1 x2)) (THead (Bind Abbr) t x2))).(\lambda +(P: Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Abst) x1 +x2)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) +\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) t x2) +H6) in (False_ind P H7)))) (pr0_beta x1 t t (pr0_refl t) x2 x2 (pr0_refl +x2))))) | Void \Rightarrow (\lambda (_: (or (\forall (t2: T).((pr0 (THead +(Bind Void) x1 x2) t2) \to (eq T (THead (Bind Void) x1 x2) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 x2) t2))))).(or_intror +(\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) +\to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) +\to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead +(Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat +Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda +(t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2)) (THead +(Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T +(THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead (Bind Void) x1 (THead +(Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T +(THead (Flat Appl) t (THead (Bind Void) x1 x2)) (\lambda (ee: T).(match ee in +T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ t2) \Rightarrow (match t2 in T return (\lambda +(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False +| (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind +Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7)))) +(pr0_upsilon Void not_void_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 +(pr0_refl x2)))))]) H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b: +B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to +(\forall (P: Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0 +t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead +(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda +(H4: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H5 \def H0 in +(or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: +T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) +(or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead +(Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t +t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) +t t0) t2)))) (\lambda (H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 +t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to +(eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead +(Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 +(THead (Flat Appl) t t0) t2))) (\lambda (t2: T).(\lambda (H7: (pr0 (THead +(Flat Appl) t t0) t2)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (ex4_4 T T T +T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) +(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 t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T +t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) +(\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t0 t3))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 +t3))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H9: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H10: (pr0 t +x0)).(\lambda (H11: (pr0 t0 x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def +(H4 x0 H10) in (let H12 \def (eq_ind_r T x1 (\lambda (t3: T).(pr0 t0 t3)) H11 +t0 H_y) in (let H13 \def (eq_ind_r T x1 (\lambda (t3: T).(eq T t2 (THead +(Flat Appl) x0 t3))) H9 t0 H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t3: +T).(pr0 t t3)) H10 t H_y0) in (let H15 \def (eq_ind_r T x0 (\lambda (t3: +T).(eq T t2 (THead (Flat Appl) t3 t0))) H13 t H_y0) in (eq_ind_r T (THead +(Flat Appl) t t0) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) +(refl_equal T (THead (Flat Appl) t t0)) t2 H15)))))))))))) H8)) (\lambda (H8: +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq T (THead (Flat Appl) t t0) t2) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda +(H9: (eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (H10: (eq T t2 (THead +(Bind Abbr) x2 x3))).(\lambda (_: (pr0 t x2)).(\lambda (_: (pr0 x1 +x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t3: T).(eq T (THead +(Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 (\lambda (t3: T).(\forall +(t4: T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind Abst) x0 x1) H9) in +(let H14 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w: +T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P: +Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in (eq_ind_r T (THead (Bind +Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind +Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind Abst) x0 x1) (pr0_refl +(THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t (THead (Bind Abst) x0 +x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 H10))))))))) H8)) (\lambda (H8: +(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 t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T +t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda +(_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3))))))))).(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 t0 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: +B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: +T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H10: (eq T +t0 (THead (Bind x0) x1 x2))).(\lambda (H11: (eq T t2 (THead (Bind x0) x4 +(THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (_: (pr0 t x3)).(\lambda +(_: (pr0 x1 x4)).(\lambda (_: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0) x4 +(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(eq T (THead (Flat +Appl) t t0) t3)) (let H15 \def (eq_ind T t0 (\lambda (t3: T).(\forall (t4: +T).((pr0 t3 t4) \to (eq T t3 t4)))) H6 (THead (Bind x0) x1 x2) H10) in (let +H16 \def (eq_ind T t0 (\lambda (t3: T).(\forall (b: B).(\forall (w: +T).(\forall (u: T).((eq T t3 (THead (Bind b) w u)) \to (\forall (P: +Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) in (eq_ind_r T (THead (Bind x0) +x1 x2) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind x0) x4 +(THead (Flat Appl) (lift (S O) O x3) x5)))) (H16 x0 x1 x2 (H15 (THead (Bind +x0) x1 x2) (pr0_refl (THead (Bind x0) x1 x2))) (eq T (THead (Flat Appl) t +(THead (Bind x0) x1 x2)) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O +x3) x5)))) t0 H10))) t2 H11))))))))))))) H8)) (pr0_gen_appl t t0 t2 H7)))))) +(\lambda (H6: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T +t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall +(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) +(\lambda (x: T).(\lambda (H7: (((eq T t0 x) \to (\forall (P: +Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 +(THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat +Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) t +x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) +\Rightarrow t0 | (THead _ _ t2) \Rightarrow t2])) (THead (Flat Appl) t t0) +(THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x (\lambda (t2: +T).(pr0 t0 t2)) H8 t0 H10) in (let H12 \def (eq_ind_r T x (\lambda (t2: +T).((eq T t0 t2) \to (\forall (P0: Prop).P0))) H7 t0 H10) in (H12 (refl_equal +T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat Appl))))))) H6)) H5))) +(\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T +t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall +(t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) +t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to +(\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) +(\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P: +Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead +(Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T +(\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: +Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat +Appl) x t0) (\lambda (H7: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) x +t0))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) +\Rightarrow t | (THead _ t2 _) \Rightarrow t2])) (THead (Flat Appl) t t0) +(THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t2: +T).(pr0 t t2)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t2: T).((eq +T t t2) \to (\forall (P0: Prop).P0))) H5 t H8) in (H10 (refl_equal T t) +P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3))) +H1))) | Cast \Rightarrow (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) +t t0) t2) \to (eq T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: +T).((eq T (THead (Flat Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda +(t2: T).(pr0 (THead (Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: +T).((eq T (THead (Flat Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda +(t2: T).(pr0 (THead (Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead +(Flat Cast) t t0) t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 +P))) (pr0_epsilon t0 t0 (pr0_refl t0) t)))])])))))) t1). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/defs.ma new file mode 100644 index 000000000..0047ef3e5 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/defs.ma @@ -0,0 +1,42 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/defs". + +include "subst0/defs.ma". + +inductive pr0: T \to (T \to Prop) \def +| pr0_refl: \forall (t: T).(pr0 t t) +| pr0_comp: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: +T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (k: K).(pr0 (THead k u1 t1) +(THead k u2 t2)))))))) +| pr0_beta: \forall (u: T).(\forall (v1: T).(\forall (v2: T).((pr0 v1 v2) \to +(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) u t1)) (THead (Bind Abbr) v2 t2)))))))) +| pr0_upsilon: \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: +T).(\forall (v2: T).((pr0 v1 v2) \to (\forall (u1: T).(\forall (u2: T).((pr0 +u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr0 (THead +(Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t2))))))))))))) +| pr0_delta: \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: +T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst0 O u2 t2 w) \to +(pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))) +| pr0_zeta: \forall (b: B).((not (eq B b Abst)) \to (\forall (t1: T).(\forall +(t2: T).((pr0 t1 t2) \to (\forall (u: T).(pr0 (THead (Bind b) u (lift (S O) O +t1)) t2)))))) +| pr0_epsilon: \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (u: +T).(pr0 (THead (Flat Cast) u t1) t2)))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma new file mode 100644 index 000000000..6f10158a3 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma @@ -0,0 +1,1642 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd". + +include "pr0/props.ma". + +theorem pr0_gen_sort: + \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n)))) +\def + \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(let H0 +\def (match H in pr0 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: +(pr0 t t0)).((eq T t (TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n))))))) +with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TSort n))).(\lambda +(H1: (eq T t x)).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t0 x) \to (eq T +x (TSort n)))) (\lambda (H2: (eq T (TSort n) x)).(eq_ind T (TSort n) (\lambda +(t0: T).(eq T t0 (TSort n))) (refl_equal T (TSort n)) x H2)) t (sym_eq T t +(TSort n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda +(H2: (eq T (THead k u1 t1) (TSort n))).(\lambda (H3: (eq T (THead k u2 t2) +x)).((let H4 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H2) in +(False_ind ((eq T (THead k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to +(eq T x (TSort n))))) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) +\Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u +t1)) (TSort n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 +\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TSort n) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 +v2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4)) H3 H0 H1))) | +(pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3) \Rightarrow (\lambda (H4: (eq T +(THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (TSort n))).(\lambda (H5: (eq T +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 +\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TSort n) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 +u1 u2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))))) H6)) H5 H0 H1 H2 H3))) | +(pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead +(Bind Abbr) u1 t1) (TSort n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) +x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match +e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) +H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u1 u2) \to +((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TSort n)))))) H5)) H4 H0 H1 +H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead +(Bind b) u (lift (S O) O t1)) (TSort n))).(\lambda (H3: (eq T t2 x)).((let H4 +\def (eq_ind T (THead (Bind b) u (lift (S O) O t1)) (\lambda (e: T).(match e +in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef +_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H2) in +(False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (eq T x +(TSort n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1 t2 H0 u) \Rightarrow +(\lambda (H1: (eq T (THead (Flat Cast) u t1) (TSort n))).(\lambda (H2: (eq T +t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TSort n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort +n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TSort n)) (refl_equal T x))))). + +theorem pr0_gen_lref: + \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n)))) +\def + \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(let H0 +\def (match H in pr0 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: +(pr0 t t0)).((eq T t (TLRef n)) \to ((eq T t0 x) \to (eq T x (TLRef n))))))) +with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TLRef n))).(\lambda +(H1: (eq T t x)).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t0 x) \to (eq T +x (TLRef n)))) (\lambda (H2: (eq T (TLRef n) x)).(eq_ind T (TLRef n) (\lambda +(t0: T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) x H2)) t (sym_eq T t +(TLRef n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda +(H2: (eq T (THead k u1 t1) (TLRef n))).(\lambda (H3: (eq T (THead k u2 t2) +x)).((let H4 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in +(False_ind ((eq T (THead k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to +(eq T x (TLRef n))))) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) +\Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u +t1)) (TLRef n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 +\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TLRef n) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 +v2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4)) H3 H0 H1))) | +(pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3) \Rightarrow (\lambda (H4: (eq T +(THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (TLRef n))).(\lambda (H5: (eq T +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 +\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TLRef n) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 +u1 u2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))))) H6)) H5 H0 H1 H2 H3))) | +(pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead +(Bind Abbr) u1 t1) (TLRef n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) +x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match +e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | +(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) +H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u1 u2) \to +((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TLRef n)))))) H5)) H4 H0 H1 +H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead +(Bind b) u (lift (S O) O t1)) (TLRef n))).(\lambda (H3: (eq T t2 x)).((let H4 +\def (eq_ind T (THead (Bind b) u (lift (S O) O t1)) (\lambda (e: T).(match e +in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef +_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in +(False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (eq T x +(TLRef n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1 t2 H0 u) \Rightarrow +(\lambda (H1: (eq T (THead (Flat Cast) u t1) (TLRef n))).(\lambda (H2: (eq T +t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I +(TLRef n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TLRef +n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TLRef n)) (refl_equal T x))))). + +theorem pr0_gen_abst: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 +t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind +Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr0 t1 t2))))))) +\def + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Bind Abst) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t: +T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abst) u1 +t1)) \to ((eq T t0 x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T +x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))))) with [(pr0_refl t) +\Rightarrow (\lambda (H0: (eq T t (THead (Bind Abst) u1 t1))).(\lambda (H1: +(eq T t x)).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t0: T).((eq T t0 x) +\to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) +u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr0 t1 t2)))))) (\lambda (H2: (eq T (THead (Bind Abst) +u1 t1) x)).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T +(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T +(THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 +t2))) u1 t1 (refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl +t1)) x H2)) t (sym_eq T t (THead (Bind Abst) u1 t1) H0) H1))) | (pr0_comp u0 +u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead +(Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) +\Rightarrow t])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H5 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) +\Rightarrow t])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H6 +\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) +\Rightarrow k0])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in (eq_ind K +(Bind Abst) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T +(THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda +(u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T +t0 t1) \to ((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 +t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind +Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3))))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T +t1 (\lambda (t: T).((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 u1 u2) \to +((pr0 t t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead +(Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) (\lambda (H9: (eq T (THead (Bind +Abst) u2 t2) x)).(eq_ind T (THead (Bind Abst) u2 t2) (\lambda (t: T).((pr0 u1 +u2) \to ((pr0 t1 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t +(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))) (\lambda (H10: (pr0 u1 +u2)).(\lambda (H11: (pr0 t1 t2)).(ex3_2_intro T T (\lambda (u3: T).(\lambda +(t3: T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda +(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 +t1 t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H10 H11))) x H9)) t0 +(sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Abst) +H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow +(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead +(Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let +H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda +(e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind Abst) u1 t1) H2) in (False_ind ((eq T +(THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 +t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind +b) u0 t0)) (THead (Bind Abst) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead +(Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abst) u1 t1) H4) in (False_ind ((eq T (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) +\to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda +(u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3))))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) +\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) +u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def +(eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void +\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Abst) u1 +t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to +((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) H5)) H4 +H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T +(THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1))).(\lambda +(H3: (eq T t2 x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: +((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) +\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in +lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t +with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match +(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 +t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in +lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) +\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 +t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | +(THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead +(Bind Abst) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e +in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) +\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead +(Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in (eq_ind B Abst +(\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 +x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T +(lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B Abst Abst)) \to ((pr0 +t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3))))))))) (\lambda (H8: (eq T (lift (S O) O t0) +t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B +Abst Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 +u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))))))) (\lambda (H9: (eq +T t2 x)).(eq_ind T x (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t0 t) +\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))))) (\lambda (H10: (not (eq +B Abst Abst))).(\lambda (_: (pr0 t0 x)).(False_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift +(S O) O t0) t3)))) (H10 (refl_equal B Abst))))) t2 (sym_eq T t2 x H9))) t1 +H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abst H6))) H5)) H4)) H3 H0 H1))) | +(pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u +t0) (THead (Bind Abst) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def +(eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abst) u1 t1) H1) in (False_ind ((eq T t2 x) \to +((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead +(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H3)) H2 H0)))]) in (H0 (refl_equal T +(THead (Bind Abst) u1 t1)) (refl_equal T x)))))). + +theorem pr0_gen_appl: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (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).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 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).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) +v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t2: T).(pr0 z1 t2)))))))))))) +\def + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Flat Appl) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t: +T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Flat Appl) u1 +t1)) \to ((eq T t0 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).(pr0 +u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 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).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x +(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 +v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))))) with [(pr0_refl +t) \Rightarrow (\lambda (H0: (eq T t (THead (Flat Appl) u1 t1))).(\lambda +(H1: (eq T t x)).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t0: T).((eq T +t0 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).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 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).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) +v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t2: T).(pr0 z1 t2))))))))))) (\lambda (H2: (eq T (THead (Flat Appl) u1 t1) +x)).(eq_ind T (THead (Flat Appl) u1 t1) (\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).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind +Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t2: T).(pr0 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).(eq T t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda +(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda +(u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 +t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) +u1 t1) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 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).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T (THead (Flat +Appl) u1 t1) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) +t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda +(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 +y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T +(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead +(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Flat Appl) +u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) t (sym_eq T t (THead (Flat Appl) +u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: +(eq T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead +k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 +| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) +H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 +| (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) +H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e in T return +(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | +(THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat Appl) u1 t1) +H2) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) +\to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) +v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 +u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 +(\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Flat Appl) u2 t2) x) \to +((pr0 t u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T +T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) +(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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x +(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: +T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 +v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H8: +(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Appl) u2 t2) +x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (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 t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))))))) (\lambda (H9: (eq T (THead (Flat Appl) u2 t2) +x)).(eq_ind T (THead (Flat Appl) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to +((pr0 t1 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t +(THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(t3: T).(eq T t (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t (THead (Bind b) +v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 +u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3)))))))))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: +(pr0 t1 t2)).(or3_intro0 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T +(THead (Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) +u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Flat +Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) +t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda +(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 +y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex3_2_intro T T +(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead +(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Appl) +u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) +k (sym_eq K k (Flat Appl) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 +t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind +Abst) u t0)) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead (Bind +Abbr) v2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u +t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) +\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat +Appl) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) +\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead +(Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H2) in (eq_ind T u1 (\lambda (t: +T).((eq T (THead (Bind Abst) u t0) t1) \to ((eq T (THead (Bind Abbr) v2 t2) +x) \to ((pr0 t v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (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 t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b) v3 (THead +(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H6: (eq T (THead (Bind Abst) u t0) +t1)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).((eq T (THead (Bind +Abbr) v2 t2) x) \to ((pr0 u1 v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (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 t (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b) v3 (THead +(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))))))) (\lambda (H7: (eq T (THead (Bind Abbr) v2 t2) +x)).(eq_ind T (THead (Bind Abbr) v2 t2) (\lambda (t: T).((pr0 u1 v2) \to +((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr0 (THead (Bind Abst) u t0) t3)))) (ex4_4 +T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq +T (THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (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 (THead (Bind +Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t +(THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 +v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))) (\lambda (H8: (pr0 +u1 v2)).(\lambda (H9: (pr0 t0 t2)).(or3_intro1 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 (THead (Bind Abst) u t0) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) +v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (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 +(THead (Bind Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda +(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) +(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3))))) u t0 v2 t2 (refl_equal T (THead (Bind Abst) u t0)) (refl_equal T +(THead (Bind Abbr) v2 t2)) H8 H9)))) x H7)) t1 H6)) v1 (sym_eq T v1 u1 H5))) +H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow +(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead +(Flat Appl) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead +(Bind b) u0 t0) | (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ +t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead +(Flat Appl) u1 t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) +\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead +(Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H4) in (eq_ind T u1 (\lambda (t: +T).((eq T (THead (Bind b) u0 t0) t1) \to ((eq T (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 t +v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (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).(eq T t1 (THead (Bind +b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v3 (THead +(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3)))))))))))))))) (\lambda (H8: (eq T (THead (Bind b) u0 t0) +t1)).(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).((eq T (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to +((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda +(u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (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).(eq T t (THead (Bind +b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v3 (THead +(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))))))))) (\lambda (H9: (eq T (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t2)) x)).(eq_ind T (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t2)) (\lambda (t: T).((not (eq B b +Abst)) \to ((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T +(\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u3 t3)))) +(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: +T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 +t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) +(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).(eq T (THead (Bind b) u0 t0) (THead (Bind b0) y1 +z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (v3: T).(\lambda (t3: T).(eq T t (THead (Bind b0) v3 (THead (Flat +Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3)))))))))))))) (\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 u1 +v2)).(\lambda (H12: (pr0 u0 u2)).(\lambda (H13: (pr0 t0 t2)).(or3_intro2 +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead +(Bind b) u0 t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) +(THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda +(_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (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).(eq T (THead (Bind b) +u0 t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) +v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 +u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3)))))))) (ex6_6_intro 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).(eq T (THead (Bind b) u0 +t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead +(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))) b u0 t0 v2 u2 t2 H10 (refl_equal T (THead (Bind b) +u0 t0)) (refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t2))) H11 H12 H13)))))) x H9)) t1 H8)) v1 (sym_eq T v1 u1 H7))) H6)) H5 H0 H1 +H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T +(THead (Bind Abbr) u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H4: (eq T +(THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 +t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 +t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to +((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or3 (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda +(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (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 t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 +u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) +(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind +T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat Appl) u1 t1) H2) in (False_ind ((eq T t2 x) \to +((not (eq B b Abst)) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (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).(eq T t1 (THead (Bind +b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v2 (THead +(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3)))))))))))) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) +\Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) +u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat +Cast) u t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: +F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead +(Flat Appl) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) +v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead +(Flat Appl) u1 t1)) (refl_equal T x)))))). + +theorem pr0_gen_cast: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 +t1) x) \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).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))))) +\def + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Flat Cast) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t: +T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Flat Cast) u1 +t1)) \to ((eq T t0 x) \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).(pr0 +u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))))))) +with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Flat Cast) u1 +t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda +(t0: T).((eq T t0 x) \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).(pr0 u1 +u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))) (\lambda +(H2: (eq T (THead (Flat Cast) u1 t1) x)).(eq_ind T (THead (Flat Cast) u1 t1) +(\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).(pr0 u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl +(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) +(THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 +t1)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat +Cast) u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 +(refl_equal T (THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) +t (sym_eq T t (THead (Flat Cast) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 +H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Flat Cast) u1 +t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) +(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H5 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) +(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H6 \def (f_equal T K +(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) +\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) +(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in (eq_ind K (Flat Cast) +(\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) +x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 x)))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda +(t: T).((eq T t0 t1) \to ((eq T (THead (Flat Cast) u2 t2) x) \to ((pr0 t u2) +\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x +(THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))))) (\lambda +(H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Cast) u2 +t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 x)))))) (\lambda (H9: (eq T (THead (Flat Cast) u2 t2) +x)).(eq_ind T (THead (Flat Cast) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to +((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t +(THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t))))) (\lambda (H10: +(pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) +(ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Cast) +u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T +(THead (Flat Cast) u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 +(sym_eq T u0 u1 H7))) k (sym_eq K k (Flat Cast) H6))) H5)) H4)) H3 H0 H1))) | +(pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat +Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: +(eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat +Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f +in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast +\Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H2) in (False_ind ((eq T +(THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 +T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (pr0 t1 x))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 +H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 +(THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq T +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 +\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) +\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow +True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H4) in +(False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 +t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead +(Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))))) H6)) H5 H0 H1 H2 +H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T +(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(\lambda (H4: (eq T +(THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 +t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 +t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to +((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 x)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) +\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead +(Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T +(THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow +False])])) I (THead (Flat Cast) u1 t1) H2) in (False_ind ((eq T t2 x) \to +((not (eq B b Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 x))))) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow +(\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u1 +t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | +(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) +u t0) (THead (Flat Cast) u1 t1) H1) in ((let H4 \def (f_equal T T (\lambda +(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u +| (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) +u t0) (THead (Flat Cast) u1 t1) H1) in (eq_ind T u1 (\lambda (_: T).((eq T t0 +t1) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 x)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: +T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 +x))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 t1 t) \to +(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))) (\lambda (H7: (pr0 t1 +x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead +(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x) H7)) t2 (sym_eq T t2 x +H6))) t0 (sym_eq T t0 t1 H5))) u (sym_eq T u u1 H4))) H3)) H2 H0)))]) in (H0 +(refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x)))))). + +theorem pr0_gen_abbr: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (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).(pr0 u1 u2))) (\lambda +(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) +(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x)))))) +\def + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Bind Abbr) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t: +T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abbr) u1 +t1)) \to ((eq T t0 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).(pr0 +u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda +(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S +O) O x)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead +(Bind Abbr) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Abbr) +u1 t1) (\lambda (t0: T).((eq T t0 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).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 +t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y +t2))))))) (pr0 t1 (lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Abbr) +u1 t1) x)).(eq_ind T (THead (Bind Abbr) u1 t1) (\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).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: +T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 +O u2 y t2))))))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda +(u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) +u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: +T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) +(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O (THead (Bind +Abbr) u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T +(THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 +t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y +t2)))))) u1 t1 (refl_equal T (THead (Bind Abbr) u1 t1)) (pr0_refl u1) +(or_introl (pr0 t1 t1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: +T).(subst0 O u1 y t1))) (pr0_refl t1)))) x H2)) t (sym_eq T t (THead (Bind +Abbr) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda +(H2: (eq T (THead k u0 t0) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T +(THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) +\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind +Abbr) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) +\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind +Abbr) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e in T +return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind +Abbr) u1 t1) H2) in (eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u1) \to +((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 +t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) +(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) +(\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to +((eq T (THead (Bind Abbr) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) +(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) +(\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind +Abbr) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda +(u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 +t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y +t3))))))) (pr0 t1 (lift (S O) O x))))))) (\lambda (H9: (eq T (THead (Bind +Abbr) u2 t2) x)).(eq_ind T (THead (Bind Abbr) u2 t2) (\lambda (t: T).((pr0 u1 +u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq +T t (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 +u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: +T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O +t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) +(THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) +(\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 +t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O (THead +(Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T +(THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 +t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y +t3)))))) u2 t2 (refl_equal T (THead (Bind Abbr) u2 t2)) H10 (or_introl (pr0 +t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y +t2))) H11))))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k +(sym_eq K k (Bind Abbr) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 +t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind +Abst) u t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T (THead (Bind +Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind +Abst) u t0)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 +t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) +\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 +t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)))))) +H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow +(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead +(Bind Abbr) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) +v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind +Abbr) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 +u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 +u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda +(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S +O) O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) +\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) +u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) +\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) in +((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t +_) \Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) +in (eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr) +u2 w) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or +(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 +t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) +(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) +(\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind +Abbr) u2 w) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to ((subst0 O u2 t2 w) \to +(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) +u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) +(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) +(\lambda (H8: (eq T (THead (Bind Abbr) u2 w) x)).(eq_ind T (THead (Bind Abbr) +u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) +\to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind +Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) +(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O t))))))) +(\lambda (H9: (pr0 u1 u2)).(\lambda (H10: (pr0 t1 t2)).(\lambda (H11: (subst0 +O u2 t2 w)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T +(THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 +t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y +t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro T T +(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind +Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: +T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) +(\lambda (y: T).(subst0 O u3 y t3)))))) u2 w (refl_equal T (THead (Bind Abbr) +u2 w)) H9 (or_intror (pr0 t1 w) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda +(y: T).(subst0 O u2 y w))) (ex_intro2 T (\lambda (y: T).(pr0 t1 y)) (\lambda +(y: T).(subst0 O u2 y w)) t2 H10 H11))))))) x H8)) t0 (sym_eq T t0 t1 H7))) +u0 (sym_eq T u0 u1 H6))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) +\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead +(Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T +\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T +\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) +\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 +t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | +(THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead +(Bind Abbr) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e +in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) +\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead +(Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in (eq_ind B Abbr +(\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 +x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 +t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y +t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind +T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not +(eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 +t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y +t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O +t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not +(eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t +t3) (ex2 T (\lambda (y: T).(pr0 t y)) (\lambda (y: T).(subst0 O u2 y +t3))))))) (pr0 t (lift (S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T +x (\lambda (t: T).((not (eq B Abbr Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: +T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y: T).(pr0 (lift (S O) O +t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 (lift (S O) O t0) (lift +(S O) O x)))))) (\lambda (_: (not (eq B Abbr Abst))).(\lambda (H11: (pr0 t0 +x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead +(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(u2: T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y: +T).(pr0 (lift (S O) O t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 +(lift (S O) O t0) (lift (S O) O x)) (pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq +T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abbr H6))) H5)) +H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T +(THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t1))).(\lambda (H2: (eq T t2 +x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e +in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef +_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return +(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abbr) u1 t1) H1) in (False_ind ((eq T t2 x) \to +((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 +t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x))))) +H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T +x)))))). + +theorem pr0_gen_void: + \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (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).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x)))))) +\def + \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead +(Bind Void) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t: +T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Void) u1 +t1)) \to ((eq T t0 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).(pr0 +u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) +O x)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind +Void) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Void) u1 t1) +(\lambda (t0: T).((eq T t0 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).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 +(lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Void) u1 t1) +x)).(eq_ind T (THead (Bind Void) u1 t1) (\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).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: +T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda +(u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) +u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O (THead (Bind Void) +u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead +(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 +(refl_equal T (THead (Bind Void) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) +t (sym_eq T t (THead (Bind Void) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 +H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Void) u1 +t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) +(THead k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H5 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) +(THead k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T K +(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) +\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) +(THead k u0 t0) (THead (Bind Void) u1 t1) H2) in (eq_ind K (Bind Void) +(\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) +x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T +u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Void) u2 t2) x) \to +((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda +(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: +T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 +(lift (S O) O x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: +T).((eq T (THead (Bind Void) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to +(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) +u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))) (\lambda +(H9: (eq T (THead (Bind Void) u2 t2) x)).(eq_ind T (THead (Bind Void) u2 t2) +(\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda +(u3: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 (lift (S O) O t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda +(H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: +T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T +(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead +(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void) +u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) +k (sym_eq K k (Bind Void) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 +t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind +Abst) u t0)) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T (THead (Bind +Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind +Abst) u t0)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 +t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) +\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)))))) +H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow +(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead +(Bind Void) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) +v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind +Void) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 +u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: +T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 +u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) +O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) +\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Void) +u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def +(eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: +B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void +\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Void) u1 +t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to +((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: +T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: +T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 (lift (S O) O x))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 +t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) +(THead (Bind Void) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T +\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T +\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) +\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 +t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | +(THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead +(Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e +in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) +\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead +(Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in (eq_ind B Void +(\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 +x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T +u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not +(eq B Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O t0) +t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B +Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift +(S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not +(eq B Void Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift +(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x)))))) (\lambda (_: +(not (eq B Void Abst))).(\lambda (H11: (pr0 t0 x)).(or_intror (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 (lift (S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x)) +(pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u +u1 H7))) b (sym_eq B b Void H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 +H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind +Void) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead +(Flat Cast) u t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ +_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 +t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))) H3)) H2 H0)))]) in (H0 +(refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x)))))). + +theorem pr0_gen_lift: + \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 +(lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda +(t2: T).(pr0 t1 t2))))))) +\def + \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t +x)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr0 t1 +t2))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat d (\lambda (n: +nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h n +t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t: T).(\forall +(x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h +x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t: T).(\lambda +(t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (ex2 +T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 +t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: +(eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h x1 +t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0)))))) (\lambda (u1: +T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0: +T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: +T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda +(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall +(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: +T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (k: +K).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead k u1 t2) +(lift h x1 x0))).(K_ind (\lambda (k0: K).((eq T (THead k0 u1 t2) (lift h x1 +x0)) \to (ex2 T (\lambda (t4: T).(eq T (THead k0 u2 t3) (lift h x1 t4))) +(\lambda (t4: T).(pr0 x0 t4))))) (\lambda (b: B).(\lambda (H6: (eq T (THead +(Bind b) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: +T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T +u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) +z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) +(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda +(H7: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 +x2))).(\lambda (H9: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) +x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) +(lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: +T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T +(\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: +T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 +(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h (S x1) +x4) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t) (lift +h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4)))) (ex2_ind T +(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 +T (\lambda (t4: T).(eq T (THead (Bind b) u2 (lift h (S x1) x4)) (lift h x1 +t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x5: +T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 +x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T +(THead (Bind b) t (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 +(THead (Bind b) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead +(Bind b) (lift h x1 x5) (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: +T).(pr0 (THead (Bind b) x2 x3) t4)) (THead (Bind b) x5 x4) (sym_eq T (lift h +x1 (THead (Bind b) x5 x4)) (THead (Bind b) (lift h x1 x5) (lift h (S x1) x4)) +(lift_bind b x5 x4 h x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Bind b))) u2 +H_x0)))) (H2 x2 x1 H8)) t3 H_x)))) (H4 x3 (S x1) H9)) x0 H7)))))) +(lift_gen_bind b u1 t2 x0 h x1 H6)))) (\lambda (f: F).(\lambda (H6: (eq T +(THead (Flat f) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: +T).(\lambda (z: T).(eq T x0 (THead (Flat f) y0 z)))) (\lambda (y0: +T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: +T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f) u2 +t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda +(x3: T).(\lambda (H7: (eq T x0 (THead (Flat f) x2 x3))).(\lambda (H8: (eq T +u1 (lift h x1 x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T +(THead (Flat f) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead +(Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T +(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 +T (\lambda (t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Flat f) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq +T t3 (lift h x1 x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4) +(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Flat f) u2 t) (lift h +x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4)))) (ex2_ind T +(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 +T (\lambda (t4: T).(eq T (THead (Flat f) u2 (lift h x1 x4)) (lift h x1 t4))) +(\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4))) (\lambda (x5: T).(\lambda +(H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 x5)).(eq_ind_r T +(lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Flat f) +t (lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 +x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Flat f) (lift h x1 x5) +(lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) +t4)) (THead (Flat f) x5 x4) (sym_eq T (lift h x1 (THead (Flat f) x5 x4)) +(THead (Flat f) (lift h x1 x5) (lift h x1 x4)) (lift_flat f x5 x4 h x1)) +(pr0_comp x2 x5 H11 x3 x4 H10 (Flat f))) u2 H_x0)))) (H2 x2 x1 H8)) t3 +H_x)))) (H4 x3 x1 H9)) x0 H7)))))) (lift_gen_flat f u1 t2 x0 h x1 H6)))) k +H5))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda +(_: (pr0 v1 v2)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T v1 +(lift h x1 x0)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x1 t2))) +(\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: +T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: +nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h +x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (x0: T).(\lambda (x1: +nat).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t2)) +(lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 +(THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h +x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (THead (Bind Abst) u t2) +(lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) +(lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda +(x3: T).(\lambda (H6: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H7: (eq +T v1 (lift h x1 x2))).(\lambda (H8: (eq T (THead (Bind Abst) u t2) (lift h x1 +x3))).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t: T).(ex2 T (\lambda +(t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: +T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 +(THead (Bind Abst) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h +x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 +T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Flat Appl) x2 x3) t4))) (\lambda (x4: T).(\lambda (x5: +T).(\lambda (H9: (eq T x3 (THead (Bind Abst) x4 x5))).(\lambda (_: (eq T u +(lift h x1 x4))).(\lambda (H11: (eq T t2 (lift h (S x1) x5))).(eq_ind_r T +(THead (Bind Abst) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T +(THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat +Appl) x2 t) t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) +(\lambda (t4: T).(pr0 x5 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind +Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 +(THead (Bind Abst) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 +(lift h (S x1) x6))).(\lambda (H12: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) +x6) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t) +(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind +Abst) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq T v2 (lift h x1 t4))) +(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind +Abbr) v2 (lift h (S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead +(Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))) (\lambda (x7: T).(\lambda +(H_x0: (eq T v2 (lift h x1 x7))).(\lambda (H13: (pr0 x2 x7)).(eq_ind_r T +(lift h x1 x7) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind +Abbr) t (lift h (S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead +(Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)))) (ex_intro2 T (\lambda (t4: +T).(eq T (THead (Bind Abbr) (lift h x1 x7) (lift h (S x1) x6)) (lift h x1 +t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) +t4)) (THead (Bind Abbr) x7 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x7 +x6)) (THead (Bind Abbr) (lift h x1 x7) (lift h (S x1) x6)) (lift_bind Abbr x7 +x6 h x1)) (pr0_beta x4 x2 x7 H13 x5 x6 H12)) v2 H_x0)))) (H2 x2 x1 H7)) t3 +H_x)))) (H4 x5 (S x1) H11)) x3 H9)))))) (lift_gen_bind Abst u t2 x3 h x1 H8)) +x0 H6)))))) (lift_gen_flat Appl v1 (THead (Bind Abst) u t2) x0 h x1 +H5)))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda +(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (H3: ((\forall +(x0: T).(\forall (x1: nat).((eq T v1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: +T).(eq T v2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda +(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H5: ((\forall +(x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: +T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda +(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H7: ((\forall +(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: +T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda +(x0: T).(\lambda (x1: nat).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead +(Bind b) u1 t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda +(z: T).(eq T x0 (THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: +T).(eq T v1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (THead +(Bind b) u1 t2) (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda +(t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H9: (eq T +x0 (THead (Flat Appl) x2 x3))).(\lambda (H10: (eq T v1 (lift h x1 +x2))).(\lambda (H11: (eq T (THead (Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r +T (THead (Flat Appl) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) +(\lambda (t4: T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0: T).(\lambda (z: +T).(eq T x3 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T +u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) +z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat +Appl) x2 x3) t4))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T x3 +(THead (Bind b) x4 x5))).(\lambda (H13: (eq T u1 (lift h x1 x4))).(\lambda +(H14: (eq T t2 (lift h (S x1) x5))).(eq_ind_r T (THead (Bind b) x4 x5) +(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead +(Flat Appl) x2 t) t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1) +t4))) (\lambda (t4: T).(pr0 x5 t4)) (ex2 T (\lambda (t4: T).(eq T (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) +(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4))) +(\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x6))).(\lambda (H15: +(pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6) (\lambda (t: T).(ex2 T (\lambda +(t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t)) +(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) +x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda +(t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda +(x7: T).(\lambda (H_x0: (eq T u2 (lift h x1 x7))).(\lambda (H16: (pr0 x4 +x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T +(THead (Bind b) t (THead (Flat Appl) (lift (S O) O v2) (lift h (S x1) x6))) +(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) +x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq T v2 (lift h x1 t4))) (\lambda +(t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 +x7) (THead (Flat Appl) (lift (S O) O v2) (lift h (S x1) x6))) (lift h x1 +t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) +t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T v2 (lift h x1 x8))).(\lambda +(H17: (pr0 x2 x8)).(eq_ind_r T (lift h x1 x8) (\lambda (t: T).(ex2 T (\lambda +(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O +t) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat +Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (plus (S O) x1) +(lift (S O) O x8)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind +b) (lift h x1 x7) (THead (Flat Appl) t (lift h (S x1) x6))) (lift h x1 t4))) +(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) +(eq_ind T (lift h (S x1) (THead (Flat Appl) (lift (S O) O x8) x6)) (\lambda +(t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) t) (lift +h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 +x5)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 +x7) (lift h (S x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift h x1 +t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) +t4)) (THead (Bind b) x7 (THead (Flat Appl) (lift (S O) O x8) x6)) (sym_eq T +(lift h x1 (THead (Bind b) x7 (THead (Flat Appl) (lift (S O) O x8) x6))) +(THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat Appl) (lift (S O) +O x8) x6))) (lift_bind b x7 (THead (Flat Appl) (lift (S O) O x8) x6) h x1)) +(pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5 x6 H15)) (THead (Flat Appl) (lift h +(S x1) (lift (S O) O x8)) (lift h (S x1) x6)) (lift_flat Appl (lift (S O) O +x8) x6 h (S x1))) (lift (S O) O (lift h x1 x8)) (lift_d x8 h (S O) x1 O +(le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2 H_x0)))) (H5 x4 x1 H13)) t3 +H_x)))) (H7 x5 (S x1) H14)) x3 H12)))))) (lift_gen_bind b u1 t2 x3 h x1 H11)) +x0 H9)))))) (lift_gen_flat Appl v1 (THead (Bind b) u1 t2) x0 h x1 +H8))))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 +u2)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 +x0)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: +T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 +t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 +x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: +T).(pr0 x0 t4)))))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t3 +w)).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H6: (eq T (THead (Bind +Abbr) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: +T).(eq T x0 (THead (Bind Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq +T u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S +x1) z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 +t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: +T).(\lambda (H7: (eq T x0 (THead (Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 +(lift h x1 x2))).(\lambda (H9: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T +(THead (Bind Abbr) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T +(THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) +(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: +T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift +h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda +(x4: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x4))).(\lambda (H10: (pr0 x3 +x4)).(let H11 \def (eq_ind T t3 (\lambda (t: T).(subst0 O u2 t w)) H5 (lift h +(S x1) x4) H_x) in (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4))) +(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind +Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) +t4))) (\lambda (x5: T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda +(H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda +(t4: T).(eq T (THead (Bind Abbr) t w) (lift h x1 t4))) (\lambda (t4: T).(pr0 +(THead (Bind Abbr) x2 x3) t4)))) (let H13 \def (eq_ind T u2 (\lambda (t: +T).(subst0 O t (lift h (S x1) x4) w)) H11 (lift h x1 x5) H_x0) in (let H14 +\def (refl_equal nat (S (plus O x1))) in (let H15 \def (eq_ind nat (S x1) +(\lambda (n: nat).(subst0 O (lift h x1 x5) (lift h n x4) w)) H13 (S (plus O +x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq T w (lift h (S (plus O x1)) +t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2 T (\lambda (t4: T).(eq T +(THead (Bind Abbr) (lift h x1 x5) w) (lift h x1 t4))) (\lambda (t4: T).(pr0 +(THead (Bind Abbr) x2 x3) t4))) (\lambda (x6: T).(\lambda (H16: (eq T w (lift +h (S (plus O x1)) x6))).(\lambda (H17: (subst0 O x5 x4 x6)).(eq_ind_r T (lift +h (S (plus O x1)) x6) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead +(Bind Abbr) (lift h x1 x5) t) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead +(Bind Abbr) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind +Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6)) (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)) (THead (Bind Abbr) x5 x6) (sym_eq +T (lift h x1 (THead (Bind Abbr) x5 x6)) (THead (Bind Abbr) (lift h x1 x5) +(lift h (S (plus O x1)) x6)) (lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta +x2 x5 H12 x3 x4 H10 x6 H17)) w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 +H15))))) u2 H_x0)))) (H2 x2 x1 H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) +(lift_gen_bind Abbr u1 t2 x0 h x1 H6))))))))))))))) (\lambda (b: B).(\lambda +(H1: (not (eq B b Abst))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 +t2 t3)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h +x1 x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: +T).(pr0 x0 t4)))))))).(\lambda (u: T).(\lambda (x0: T).(\lambda (x1: +nat).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t2)) (lift h x1 +x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind +b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 y0)))) +(\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2) (lift h (S x1) z)))) +(ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 +t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq T x0 (THead (Bind +b) x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H7: (eq T (lift +(S O) O t2) (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda +(t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: +T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S O) x1) (\lambda (n: +nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1)) (plus x1 (S O)) +(plus_comm x1 (S O))) in (let H9 \def (eq_ind nat (S x1) (\lambda (n: +nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O)) H8) in +(ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4: T).(eq +T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) +(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: T).(\lambda +(H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift h x1 +x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4: +T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t) +t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: +T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5: +T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4 +x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T +t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O +x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1 +t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5 +(refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4 +x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0 +H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda +(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall +(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: +T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: +T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast) +u t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T +x0 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift +h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T +(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) +(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast) +x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h +x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T +(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) +(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 +x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: +T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T +t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4) +(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda +(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: +T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat +Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_epsilon x3 x4 H7 x2)) +t3 H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1 +H3)))))))))) y x H0))))) H))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/pr0.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/pr0.ma new file mode 100644 index 000000000..f04a552db --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/pr0.ma @@ -0,0 +1,2818 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/pr0". + +include "pr0/fwd.ma". + +include "lift/tlt.ma". + +theorem pr0_confluence__pr0_cong_upsilon_refl: + \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: +T).((pr0 u0 u3) \to (\forall (t4: T).(\forall (t5: T).((pr0 t4 t5) \to +(\forall (u2: T).(\forall (v2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) +\to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4)) +t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O +v2) t5)) t))))))))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda +(u3: T).(\lambda (H0: (pr0 u0 u3)).(\lambda (t4: T).(\lambda (t5: T).(\lambda +(H1: (pr0 t4 t5)).(\lambda (u2: T).(\lambda (v2: T).(\lambda (x: T).(\lambda +(H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t: T).(pr0 +(THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t)) (\lambda (t: T).(pr0 (THead +(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead (Bind b) u3 +(THead (Flat Appl) (lift (S O) O x) t5)) (pr0_upsilon b H u2 x H2 u0 u3 H0 t4 +t5 H1) (pr0_comp u3 u3 (pr0_refl u3) (THead (Flat Appl) (lift (S O) O v2) t5) +(THead (Flat Appl) (lift (S O) O x) t5) (pr0_comp (lift (S O) O v2) (lift (S +O) O x) (pr0_lift v2 x H3 (S O) O) t5 t5 (pr0_refl t5) (Flat Appl)) (Bind +b))))))))))))))). + +theorem pr0_confluence__pr0_cong_upsilon_cong: + \forall (b: B).((not (eq B b Abst)) \to (\forall (u2: T).(\forall (v2: +T).(\forall (x: T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t2: T).(\forall +(t5: T).(\forall (x0: T).((pr0 t2 x0) \to ((pr0 t5 x0) \to (\forall (u5: +T).(\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to ((pr0 u3 x1) \to (ex2 T +(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) +(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) +t5)) t))))))))))))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u2: T).(\lambda +(v2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda (H1: (pr0 v2 +x)).(\lambda (t2: T).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H2: (pr0 t2 +x0)).(\lambda (H3: (pr0 t5 x0)).(\lambda (u5: T).(\lambda (u3: T).(\lambda +(x1: T).(\lambda (H4: (pr0 u5 x1)).(\lambda (H5: (pr0 u3 x1)).(ex_intro2 T +(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) +(\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) +t5)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x) x0)) +(pr0_upsilon b H u2 x H0 u5 x1 H4 t2 x0 H2) (pr0_comp u3 x1 H5 (THead (Flat +Appl) (lift (S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp +(lift (S O) O v2) (lift (S O) O x) (pr0_lift v2 x H1 (S O) O) t5 x0 H3 (Flat +Appl)) (Bind b))))))))))))))))))). + +theorem pr0_confluence__pr0_cong_upsilon_delta: + (not (eq B Abbr Abst)) \to (\forall (u5: T).(\forall (t2: T).(\forall (w: +T).((subst0 O u5 t2 w) \to (\forall (u2: T).(\forall (v2: T).(\forall (x: +T).((pr0 u2 x) \to ((pr0 v2 x) \to (\forall (t5: T).(\forall (x0: T).((pr0 t2 +x0) \to ((pr0 t5 x0) \to (\forall (u3: T).(\forall (x1: T).((pr0 u5 x1) \to +((pr0 u3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead +(Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead +(Flat Appl) (lift (S O) O v2) t5)) t)))))))))))))))))))) +\def + \lambda (H: (not (eq B Abbr Abst))).(\lambda (u5: T).(\lambda (t2: +T).(\lambda (w: T).(\lambda (H0: (subst0 O u5 t2 w)).(\lambda (u2: +T).(\lambda (v2: T).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: +(pr0 v2 x)).(\lambda (t5: T).(\lambda (x0: T).(\lambda (H3: (pr0 t2 +x0)).(\lambda (H4: (pr0 t5 x0)).(\lambda (u3: T).(\lambda (x1: T).(\lambda +(H5: (pr0 u5 x1)).(\lambda (H6: (pr0 u3 x1)).(or_ind (pr0 w x0) (ex2 T +(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x1 x0 w2))) (ex2 T +(\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) +(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O +v2) t5)) t))) (\lambda (H7: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 +(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 +(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) (THead +(Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x0)) (pr0_upsilon Abbr H +u2 x H1 u5 x1 H5 w x0 H7) (pr0_comp u3 x1 H6 (THead (Flat Appl) (lift (S O) O +v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) O v2) +(lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) (Bind +Abbr)))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: +T).(subst0 O x1 x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda +(w2: T).(subst0 O x1 x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) +u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 +(THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x2: T).(\lambda (H8: +(pr0 w x2)).(\lambda (H9: (subst0 O x1 x0 x2)).(ex_intro2 T (\lambda (t: +T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: +T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)) +(THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O x) x2)) (pr0_upsilon +Abbr H u2 x H1 u5 x1 H5 w x2 H8) (pr0_delta u3 x1 H6 (THead (Flat Appl) (lift +(S O) O v2) t5) (THead (Flat Appl) (lift (S O) O x) x0) (pr0_comp (lift (S O) +O v2) (lift (S O) O x) (pr0_lift v2 x H2 (S O) O) t5 x0 H4 (Flat Appl)) +(THead (Flat Appl) (lift (S O) O x) x2) (subst0_snd (Flat Appl) x1 x2 x0 O H9 +(lift (S O) O x))))))) H7)) (pr0_subst0 t2 x0 H3 u5 w O H0 x1 +H5))))))))))))))))))). + +theorem pr0_confluence__pr0_cong_upsilon_zeta: + \forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3: +T).((pr0 u0 u3) \to (\forall (u2: T).(\forall (v2: T).(\forall (x0: T).((pr0 +u2 x0) \to ((pr0 v2 x0) \to (\forall (x: T).(\forall (t3: T).(\forall (x1: +T).((pr0 x x1) \to ((pr0 t3 x1) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat +Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) +(lift (S O) O v2) (lift (S O) O x))) t))))))))))))))))) +\def + \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (u0: T).(\lambda +(u3: T).(\lambda (_: (pr0 u0 u3)).(\lambda (u2: T).(\lambda (v2: T).(\lambda +(x0: T).(\lambda (H1: (pr0 u2 x0)).(\lambda (H2: (pr0 v2 x0)).(\lambda (x: +T).(\lambda (t3: T).(\lambda (x1: T).(\lambda (H3: (pr0 x x1)).(\lambda (H4: +(pr0 t3 x1)).(eq_ind T (lift (S O) O (THead (Flat Appl) v2 x)) (\lambda (t: +T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: +T).(pr0 (THead (Bind b) u3 t) t0)))) (ex_intro2 T (\lambda (t: T).(pr0 (THead +(Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (lift (S O) O +(THead (Flat Appl) v2 x))) t)) (THead (Flat Appl) x0 x1) (pr0_comp u2 x0 H1 +t3 x1 H4 (Flat Appl)) (pr0_zeta b H (THead (Flat Appl) v2 x) (THead (Flat +Appl) x0 x1) (pr0_comp v2 x0 H2 x x1 H3 (Flat Appl)) u3)) (THead (Flat Appl) +(lift (S O) O v2) (lift (S O) O x)) (lift_flat Appl v2 x (S O) +O)))))))))))))))). + +theorem pr0_confluence__pr0_cong_delta: + \forall (u3: T).(\forall (t5: T).(\forall (w: T).((subst0 O u3 t5 w) \to +(\forall (u2: T).(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall +(t3: T).(\forall (x0: T).((pr0 t3 x0) \to ((pr0 t5 x0) \to (ex2 T (\lambda +(t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind +Abbr) u3 w) t)))))))))))))) +\def + \lambda (u3: T).(\lambda (t5: T).(\lambda (w: T).(\lambda (H: (subst0 O u3 +t5 w)).(\lambda (u2: T).(\lambda (x: T).(\lambda (H0: (pr0 u2 x)).(\lambda +(H1: (pr0 u3 x)).(\lambda (t3: T).(\lambda (x0: T).(\lambda (H2: (pr0 t3 +x0)).(\lambda (H3: (pr0 t5 x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: +T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: +T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) +u3 w) t))) (\lambda (H4: (pr0 w x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead +(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)) +(THead (Bind Abbr) x x0) (pr0_comp u2 x H0 t3 x0 H2 (Bind Abbr)) (pr0_comp u3 +x H1 w x0 H4 (Bind Abbr)))) (\lambda (H4: (ex2 T (\lambda (w2: T).(pr0 w w2)) +(\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w +w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead +(Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) +(\lambda (x1: T).(\lambda (H5: (pr0 w x1)).(\lambda (H6: (subst0 O x x0 +x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda +(t: T).(pr0 (THead (Bind Abbr) u3 w) t)) (THead (Bind Abbr) x x1) (pr0_delta +u2 x H0 t3 x0 H2 x1 H6) (pr0_comp u3 x H1 w x1 H5 (Bind Abbr)))))) H4)) +(pr0_subst0 t5 x0 H3 u3 w O H x H1))))))))))))). + +theorem pr0_confluence__pr0_upsilon_upsilon: + \forall (b: B).((not (eq B b Abst)) \to (\forall (v1: T).(\forall (v2: +T).(\forall (x0: T).((pr0 v1 x0) \to ((pr0 v2 x0) \to (\forall (u1: +T).(\forall (u2: T).(\forall (x1: T).((pr0 u1 x1) \to ((pr0 u2 x1) \to +(\forall (t1: T).(\forall (t2: T).(\forall (x2: T).((pr0 t1 x2) \to ((pr0 t2 +x2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) +(lift (S O) O v1) t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t2)) t))))))))))))))))))) +\def + \lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda +(v2: T).(\lambda (x0: T).(\lambda (H0: (pr0 v1 x0)).(\lambda (H1: (pr0 v2 +x0)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (x1: T).(\lambda (H2: (pr0 u1 +x1)).(\lambda (H3: (pr0 u2 x1)).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(x2: T).(\lambda (H4: (pr0 t1 x2)).(\lambda (H5: (pr0 t2 x2)).(ex_intro2 T +(\lambda (t: T).(pr0 (THead (Bind b) u1 (THead (Flat Appl) (lift (S O) O v1) +t1)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t2)) t)) (THead (Bind b) x1 (THead (Flat Appl) (lift (S O) O x0) +x2)) (pr0_comp u1 x1 H2 (THead (Flat Appl) (lift (S O) O v1) t1) (THead (Flat +Appl) (lift (S O) O x0) x2) (pr0_comp (lift (S O) O v1) (lift (S O) O x0) +(pr0_lift v1 x0 H0 (S O) O) t1 x2 H4 (Flat Appl)) (Bind b)) (pr0_comp u2 x1 +H3 (THead (Flat Appl) (lift (S O) O v2) t2) (THead (Flat Appl) (lift (S O) O +x0) x2) (pr0_comp (lift (S O) O v2) (lift (S O) O x0) (pr0_lift v2 x0 H1 (S +O) O) t2 x2 H5 (Flat Appl)) (Bind b))))))))))))))))))). + +theorem pr0_confluence__pr0_delta_delta: + \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to +(\forall (u3: T).(\forall (t5: T).(\forall (w0: T).((subst0 O u3 t5 w0) \to +(\forall (x: T).((pr0 u2 x) \to ((pr0 u3 x) \to (\forall (x0: T).((pr0 t3 x0) +\to ((pr0 t5 x0) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) +(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)))))))))))))))) +\def + \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 +t3 w)).(\lambda (u3: T).(\lambda (t5: T).(\lambda (w0: T).(\lambda (H0: +(subst0 O u3 t5 w0)).(\lambda (x: T).(\lambda (H1: (pr0 u2 x)).(\lambda (H2: +(pr0 u3 x)).(\lambda (x0: T).(\lambda (H3: (pr0 t3 x0)).(\lambda (H4: (pr0 t5 +x0)).(or_ind (pr0 w0 x0) (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: +T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) +t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H5: (pr0 w0 +x0)).(or_ind (pr0 w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: +T).(subst0 O x x0 w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) +t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H6: (pr0 w +x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda +(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x0) (pr0_comp +u2 x H1 w x0 H6 (Bind Abbr)) (pr0_comp u3 x H2 w0 x0 H5 (Bind Abbr)))) +(\lambda (H6: (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O +x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 +O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda +(t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: T).(\lambda (H7: +(pr0 w x1)).(\lambda (H8: (subst0 O x x0 x1)).(ex_intro2 T (\lambda (t: +T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) +u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x H1 w x1 H7 (Bind Abbr)) +(pr0_delta u3 x H2 w0 x0 H5 x1 H8))))) H6)) (pr0_subst0 t3 x0 H3 u2 w O H x +H1))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 w0 w2)) (\lambda (w2: +T).(subst0 O x x0 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 w0 w2)) (\lambda +(w2: T).(subst0 O x x0 w2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 +w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x1: +T).(\lambda (H6: (pr0 w0 x1)).(\lambda (H7: (subst0 O x x0 x1)).(or_ind (pr0 +w x0) (ex2 T (\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 +w2))) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: +T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H8: (pr0 w x0)).(ex_intro2 T +(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead +(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x0 H8 x1 +H7) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr)))) (\lambda (H8: (ex2 T (\lambda +(w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)))).(ex2_ind T +(\lambda (w2: T).(pr0 w w2)) (\lambda (w2: T).(subst0 O x x0 w2)) (ex2 T +(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead +(Bind Abbr) u3 w0) t))) (\lambda (x2: T).(\lambda (H9: (pr0 w x2)).(\lambda +(H10: (subst0 O x x0 x2)).(or4_ind (eq T x2 x1) (ex2 T (\lambda (t: +T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x x1 t))) (subst0 O x x2 x1) +(subst0 O x x1 x2) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) +(\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (H11: (eq T x2 +x1)).(let H12 \def (eq_ind T x2 (\lambda (t: T).(pr0 w t)) H9 x1 H11) in +(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: +T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x1) (pr0_comp u2 x +H1 w x1 H12 (Bind Abbr)) (pr0_comp u3 x H2 w0 x1 H6 (Bind Abbr))))) (\lambda +(H11: (ex2 T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: T).(subst0 O x +x1 t)))).(ex2_ind T (\lambda (t: T).(subst0 O x x2 t)) (\lambda (t: +T).(subst0 O x x1 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) +t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x3: +T).(\lambda (H12: (subst0 O x x2 x3)).(\lambda (H13: (subst0 O x x1 +x3)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda +(t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x3) (pr0_delta +u2 x H1 w x2 H9 x3 H12) (pr0_delta u3 x H2 w0 x1 H6 x3 H13))))) H11)) +(\lambda (H11: (subst0 O x x2 x1)).(ex_intro2 T (\lambda (t: T).(pr0 (THead +(Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t)) +(THead (Bind Abbr) x x1) (pr0_delta u2 x H1 w x2 H9 x1 H11) (pr0_comp u3 x H2 +w0 x1 H6 (Bind Abbr)))) (\lambda (H11: (subst0 O x x1 x2)).(ex_intro2 T +(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead +(Bind Abbr) u3 w0) t)) (THead (Bind Abbr) x x2) (pr0_comp u2 x H1 w x2 H9 +(Bind Abbr)) (pr0_delta u3 x H2 w0 x1 H6 x2 H11))) (subst0_confluence_eq x0 +x2 x O H10 x1 H7))))) H8)) (pr0_subst0 t3 x0 H3 u2 w O H x H1))))) H5)) +(pr0_subst0 t5 x0 H4 u3 w0 O H0 x H2))))))))))))))). + +theorem pr0_confluence__pr0_delta_epsilon: + \forall (u2: T).(\forall (t3: T).(\forall (w: T).((subst0 O u2 t3 w) \to +(\forall (t4: T).((pr0 (lift (S O) O t4) t3) \to (\forall (t2: T).(ex2 T +(\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 +t))))))))) +\def + \lambda (u2: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H: (subst0 O u2 +t3 w)).(\lambda (t4: T).(\lambda (H0: (pr0 (lift (S O) O t4) t3)).(\lambda +(t2: T).(ex2_ind T (\lambda (t5: T).(eq T t3 (lift (S O) O t5))) (\lambda +(t5: T).(pr0 t4 t5)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) +(\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H1: (eq T t3 (lift (S +O) O x))).(\lambda (_: (pr0 t4 x)).(let H3 \def (eq_ind T t3 (\lambda (t: +T).(subst0 O u2 t w)) H (lift (S O) O x) H1) in (subst0_gen_lift_false x u2 w +(S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) +(le_n (plus (S O) O)) (plus O (S O)) (plus_comm O (S O))) H3 (ex2 T (\lambda +(t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))))) +(pr0_gen_lift t4 t3 (S O) O H0)))))))). + +theorem pr0_confluence: + \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0 +t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))) +\def + \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to +(\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) +(\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall +(v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 +v t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2 +t3))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: +T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 in pr0 return (\lambda +(t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 t4)).((eq T t3 t) \to ((eq T t4 +t1) \to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 +t5)))))))) with [(pr0_refl t3) \Rightarrow (\lambda (H2: (eq T t3 +t)).(\lambda (H3: (eq T t3 t1)).(eq_ind T t (\lambda (t4: T).((eq T t4 t1) +\to (ex2 T (\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5))))) +(\lambda (H4: (eq T t t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t5: +T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t2 t5)))) (let H5 \def (match H1 in pr0 +return (\lambda (t4: T).(\lambda (t5: T).(\lambda (_: (pr0 t4 t5)).((eq T t4 +t) \to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: +T).(pr0 t2 t6)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H5: (eq T t4 +t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T t (\lambda (t5: T).((eq T t5 t2) +\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))) +(\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t6: +T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))) (let H8 \def (eq_ind T t +(\lambda (t5: T).(eq T t4 t5)) H5 t2 H7) in (let H9 \def (eq_ind T t (\lambda +(t5: T).(eq T t5 t1)) H4 t2 H7) in (let H10 \def (eq_ind T t (\lambda (t5: +T).(eq T t3 t5)) H2 t2 H7) in (let H11 \def (eq_ind T t (\lambda (t5: +T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall +(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: +T).(pr0 t7 t8)))))))))) H t2 H7) in (let H12 \def (eq_ind T t2 (\lambda (t5: +T).(\forall (v: T).((tlt v t5) \to (\forall (t6: T).((pr0 v t6) \to (\forall +(t7: T).((pr0 v t7) \to (ex2 T (\lambda (t8: T).(pr0 t6 t8)) (\lambda (t8: +T).(pr0 t7 t8)))))))))) H11 t1 H9) in (eq_ind_r T t1 (\lambda (t5: T).(ex2 T +(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t5 t6)))) (let H13 \def +(eq_ind T t2 (\lambda (t5: T).(eq T t3 t5)) H10 t1 H9) in (ex_intro2 T +(\lambda (t5: T).(pr0 t1 t5)) (\lambda (t5: T).(pr0 t1 t5)) t1 (pr0_refl t1) +(pr0_refl t1))) t2 H9)))))) t (sym_eq T t t2 H7))) t4 (sym_eq T t4 t H5) +H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead +k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 +t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 +t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 +t7))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) +(\lambda (t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 u1 +u2)).(\lambda (H11: (pr0 t4 t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: +T).(eq T t6 t1)) H4 (THead k u1 t4) H7) in (eq_ind T (THead k u1 t4) (\lambda +(t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead k +u2 t5) t7)))) (let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 +(THead k u1 t4) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall +(v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: +T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 +t8 t9)))))))))) H (THead k u1 t4) H7) in (ex_intro2 T (\lambda (t6: T).(pr0 +(THead k u1 t4) t6)) (\lambda (t6: T).(pr0 (THead k u2 t5) t6)) (THead k u2 +t5) (pr0_comp u1 u2 H10 t4 t5 H11 k) (pr0_refl (THead k u2 t5))))) t1 H12)))) +t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 H6) \Rightarrow +(\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) +t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat +Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) +v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 +t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T (THead (Bind +Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t6: T).((pr0 +v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda +(t7: T).(pr0 t6 t7)))))) (\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 t4 +t5)).(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead +(Flat Appl) v1 (THead (Bind Abst) u t4)) H7) in (eq_ind T (THead (Flat Appl) +v1 (THead (Bind Abst) u t4)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t6 +t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t5) t7)))) (let H13 \def +(eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead +(Bind Abst) u t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: +T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall +(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: +T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H7) +in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind +Abst) u t4)) t6)) (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 t5) t6)) (THead +(Bind Abbr) v2 t5) (pr0_beta u v1 v2 H10 t4 t5 H11) (pr0_refl (THead (Bind +Abbr) v2 t5))))) t1 H12)))) t2 H9)) t H7 H8 H5 H6))) | (pr0_upsilon b H5 v1 +v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) +v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 +(THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 +v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 +t7)) (\lambda (t7: T).(pr0 t2 t7))))))))) (\lambda (H11: (eq T (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t6: T).((not (eq B +b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))))))) (\lambda +(H12: (not (eq B b Abst))).(\lambda (H13: (pr0 v1 v2)).(\lambda (H14: (pr0 u1 +u2)).(\lambda (H15: (pr0 t4 t5)).(let H16 \def (eq_ind_r T t (\lambda (t6: +T).(eq T t6 t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in +(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (t6: T).(ex2 +T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t5)) t7)))) (let H17 \def (eq_ind_r T t +(\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1 +t4)) H9) in (let H18 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: +T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v +t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 +t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in +(pr0_confluence__pr0_cong_upsilon_refl b H12 u1 u2 H14 t4 t5 H15 v1 v2 v2 H13 +(pr0_refl v2)))) t1 H16)))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta +u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 +t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead +(Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to +((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda +(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) (\lambda (H10: (eq T +(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda +(t6: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7))))))) (\lambda +(H11: (pr0 u1 u2)).(\lambda (H12: (pr0 t4 t5)).(\lambda (H13: (subst0 O u2 t5 +w)).(let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead +(Bind Abbr) u1 t4) H8) in (eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (t6: +T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 (THead (Bind +Abbr) u2 w) t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) +H2 (THead (Bind Abbr) u1 t4) H8) in (let H16 \def (eq_ind_r T t (\lambda (t6: +T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall +(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: +T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t4) H8) in (ex_intro2 T +(\lambda (t6: T).(pr0 (THead (Bind Abbr) u1 t4) t6)) (\lambda (t6: T).(pr0 +(THead (Bind Abbr) u2 w) t6)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H11 +t4 t5 H12 w H13) (pr0_refl (THead (Bind Abbr) u2 w))))) t1 H14))))) t2 H10)) +t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7: +(eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H8: (eq T t5 +t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 +t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))) (\lambda (H9: (eq T t5 +t2)).(eq_ind T t2 (\lambda (t6: T).((not (eq B b Abst)) \to ((pr0 t4 t6) \to +(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))) +(\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 t4 t2)).(let H12 \def +(eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead (Bind b) u (lift (S O) +O t4)) H7) in (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (t6: +T).(ex2 T (\lambda (t7: T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let +H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t3 t6)) H2 (THead (Bind b) u +(lift (S O) O t4)) H7) in (let H14 \def (eq_ind_r T t (\lambda (t6: +T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall +(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: +T).(pr0 t8 t9)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H7) in +(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t4)) t6)) +(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_zeta b H10 t4 t2 H11 u) (pr0_refl +t2)))) t1 H12)))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_epsilon t4 +t5 H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) +t)).(\lambda (H7: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda +(_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 +t7)) (\lambda (t7: T).(pr0 t2 t7)))))) (\lambda (H8: (eq T t5 t2)).(eq_ind T +t2 (\lambda (t6: T).((pr0 t4 t6) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) +(\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H9: (pr0 t4 t2)).(let H10 \def +(eq_ind_r T t (\lambda (t6: T).(eq T t6 t1)) H4 (THead (Flat Cast) u t4) H6) +in (eq_ind T (THead (Flat Cast) u t4) (\lambda (t6: T).(ex2 T (\lambda (t7: +T).(pr0 t6 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H11 \def (eq_ind_r T t +(\lambda (t6: T).(eq T t3 t6)) H2 (THead (Flat Cast) u t4) H6) in (let H12 +\def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall +(t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: +T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u +t4) H6) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Flat Cast) u t4) t6)) +(\lambda (t6: T).(pr0 t2 t6)) t2 (pr0_epsilon t4 t2 H9 u) (pr0_refl t2)))) t1 +H10))) t5 (sym_eq T t5 t2 H8))) t H6 H7 H5)))]) in (H5 (refl_equal T t) +(refl_equal T t2))) t (sym_eq T t t1 H4))) t3 (sym_eq T t3 t H2) H3))) | +(pr0_comp u1 u2 H2 t3 t4 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1 +t3) t)).(\lambda (H5: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u1 t3) +(\lambda (_: T).((eq T (THead k u2 t4) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) +\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) +(\lambda (H6: (eq T (THead k u2 t4) t1)).(eq_ind T (THead k u2 t4) (\lambda +(t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 +t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda +(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5: +T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 +t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 +t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 +t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T +(\lambda (t7: T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) +(let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead k u1 +t3) H4) in (eq_ind T (THead k u1 t3) (\lambda (t6: T).(ex2 T (\lambda (t7: +T).(pr0 (THead k u2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def +(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead k u1 t3) H4) in (let +H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to +(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T +(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead +k u1 t3) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead k u2 t4) t6)) +(\lambda (t6: T).(pr0 (THead k u1 t3) t6)) (THead k u2 t4) (pr0_refl (THead k +u2 t4)) (pr0_comp u1 u2 H7 t3 t4 H8 k)))) t2 H12)) t (sym_eq T t t2 H11))) t5 +(sym_eq T t5 t H9) H10))) | (pr0_comp u0 u3 H9 t5 t6 H10 k0) \Rightarrow +(\lambda (H11: (eq T (THead k0 u0 t5) t)).(\lambda (H12: (eq T (THead k0 u3 +t6) t2)).(eq_ind T (THead k0 u0 t5) (\lambda (_: T).((eq T (THead k0 u3 t6) +t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead +k0 u3 t6) t2)).(eq_ind T (THead k0 u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda +(t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t5 +t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) +H4 (THead k0 u0 t5) H11) in (let H17 \def (match H16 in eq return (\lambda +(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k0 u0 t5)) \to (ex2 T +(\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 (THead k0 u3 +t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead k u1 +t3) (THead k0 u0 t5))).(let H18 \def (f_equal T T (\lambda (e: T).(match e in +T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) +\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead k0 +u0 t5) H17) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) +\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0 +u0 t5) H17) in ((let H20 \def (f_equal T K (\lambda (e: T).(match e in T +return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 u0 +t5) H17) in (eq_ind K k0 (\lambda (k1: K).((eq T u1 u0) \to ((eq T t3 t5) \to +(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7)) (\lambda (t7: T).(pr0 +(THead k0 u3 t6) t7)))))) (\lambda (H21: (eq T u1 u0)).(eq_ind T u0 (\lambda +(_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead k0 u2 t4) t8)) +(\lambda (t8: T).(pr0 (THead k0 u3 t6) t8))))) (\lambda (H22: (eq T t3 +t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead k0 u2 +t4) t8)) (\lambda (t8: T).(pr0 (THead k0 u3 t6) t8)))) (let H23 \def +(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: +T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: +T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5) +H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H22) +in (let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u0 H21) in +(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T +(\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 +u3 t6) t7))) (\lambda (x: T).(\lambda (H26: (pr0 u2 x)).(\lambda (H27: (pr0 +u3 x)).(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) +(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 +(THead k0 u3 t6) t7))) (\lambda (x0: T).(\lambda (H28: (pr0 t4 x0)).(\lambda +(H29: (pr0 t6 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) +(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x x0) (pr0_comp u2 x +H26 t4 x0 H28 k0) (pr0_comp u3 x H27 t6 x0 H29 k0))))) (H23 t5 (tlt_head_dx +k0 u0 t5) t4 H24 t6 H15))))) (H23 u0 (tlt_head_sx k0 u0 t5) u2 H25 u3 +H14))))) t3 (sym_eq T t3 t5 H22))) u1 (sym_eq T u1 u0 H21))) k (sym_eq K k k0 +H20))) H19)) H18)))]) in (H17 (refl_equal T (THead k0 u0 t5))))))) t2 H13)) t +H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda +(H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) t)).(\lambda +(H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 +(THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) +t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead +(Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: +T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 +t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v1 +v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: +T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind Abst) u +t5)) H11) in (let H17 \def (match H16 in eq return (\lambda (t7: T).(\lambda +(_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind Abst) u t5))) +\to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 +(THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal \Rightarrow (\lambda +(H17: (eq T (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u +t5)))).(let H18 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 +| (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 +(THead (Bind Abst) u t5)) H17) in ((let H19 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | +(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) +(THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H17) in ((let H20 \def +(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with +[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) +\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u +t5)) H17) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u1 v1) \to ((eq T +t3 (THead (Bind Abst) u t5)) \to (ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 +t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))))) (\lambda +(H21: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq T t3 (THead (Bind Abst) +u t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 t4) t8)) +(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8))))) (\lambda (H22: (eq T +t3 (THead (Bind Abst) u t5))).(eq_ind T (THead (Bind Abst) u t5) (\lambda (_: +T).(ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 t4) t8)) (\lambda (t8: +T).(pr0 (THead (Bind Abbr) v2 t6) t8)))) (let H23 \def (eq_ind_r T t (\lambda +(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to +(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind +Abst) u t5)) H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) +H8 (THead (Bind Abst) u t5) H22) in (let H25 \def (match H24 in pr0 return +(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead +(Bind Abst) u t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead +(Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) +t9)))))))) with [(pr0_refl t7) \Rightarrow (\lambda (H25: (eq T t7 (THead +(Bind Abst) u t5))).(\lambda (H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst) +u t5) (\lambda (t8: T).((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead +(Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) +t9))))) (\lambda (H27: (eq T (THead (Bind Abst) u t5) t4)).(eq_ind T (THead +(Bind Abst) u t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat +Appl) u2 t8) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (let +H28 \def (eq_ind T u1 (\lambda (t8: T).(pr0 t8 u2)) H7 v1 H21) in (ex2_ind T +(\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda +(t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda +(t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H29: +(pr0 u2 x)).(\lambda (H30: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 +(THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 +(THead (Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H29 +t5 t6 H15) (pr0_comp v2 x H30 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H23 v1 +(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H28 v2 H14))) t4 +H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3 +H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead +(Bind Abst) u t5))).(\lambda (H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) +\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H30 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) +\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in ((let H31 +\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) +\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t5) H27) in (eq_ind K +(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t5) \to ((eq T (THead +k1 u3 t8) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: +T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind +Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda +(t9: T).((eq T t7 t5) \to ((eq T (THead (Bind Abst) u3 t8) t4) \to ((pr0 t9 +u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 +t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))))) +(\lambda (H33: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq T (THead +(Bind Abst) u3 t8) t4) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda +(t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead +(Bind Abbr) v2 t6) t10))))))) (\lambda (H34: (eq T (THead (Bind Abst) u3 t8) +t4)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to +((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) +t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda (_: +(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(let H37 \def (eq_ind T u1 (\lambda +(t9: T).(pr0 t9 u2)) H7 v1 H21) in (ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) +(\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) +u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) +v2 t6) t9))) (\lambda (x: T).(\lambda (H38: (pr0 u2 x)).(\lambda (H39: (pr0 +v2 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) +(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) +t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0: +T).(\lambda (H40: (pr0 t8 x0)).(\lambda (H41: (pr0 t6 x0)).(ex_intro2 T +(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9)) +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x x0) +(pr0_beta u3 u2 x H38 t8 x0 H40) (pr0_comp v2 x H39 t6 x0 H41 (Bind +Abbr)))))) (H23 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat Appl) +v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5) (tlt_head_dx +(Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15))))) (H23 v1 +(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H37 v2 H14))))) t4 +H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 u H32))) k0 (sym_eq K k0 +(Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | (pr0_beta u0 v0 v3 H25 t7 t8 +H26) \Rightarrow (\lambda (H27: (eq T (THead (Flat Appl) v0 (THead (Bind +Abst) u0 t7)) (THead (Bind Abst) u t5))).(\lambda (H28: (eq T (THead (Bind +Abbr) v3 t8) t4)).((let H29 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind +Abst) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) +\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) +H27) in (False_ind ((eq T (THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))) H29)) H28 H25 H26))) +| (pr0_upsilon b H25 v0 v3 H26 u0 u3 H27 t7 t8 H28) \Rightarrow (\lambda +(H29: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) +u t5))).(\lambda (H30: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S +O) O v3) t8)) t4)).((let H31 \def (eq_ind T (THead (Flat Appl) v0 (THead +(Bind b) u0 t7)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ +_) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t5) +H29) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O +v3) t8)) t4) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) H31)) H30 H25 H26 +H27 H28))) | (pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: +(eq T (THead (Bind Abbr) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H29: (eq +T (THead (Bind Abbr) u3 w) t4)).((let H30 \def (eq_ind T (THead (Bind Abbr) +u0 t7) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) +\Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind b) +\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow +True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) +\Rightarrow False])])) I (THead (Bind Abst) u t5) H28) in (False_ind ((eq T +(THead (Bind Abbr) u3 w) t4) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O +u3 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))) H30)) H29 H25 H26 +H27))) | (pr0_zeta b H25 t7 t8 H26 u0) \Rightarrow (\lambda (H27: (eq T +(THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t5))).(\lambda +(H28: (eq T t8 t4)).((let H29 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: +((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) +\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in +lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match +t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 +d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ +t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind +Abst) u t5) H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) +\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S +O) O t7)) (THead (Bind Abst) u t5) H27) in ((let H31 \def (f_equal T B +(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) +\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match +k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u +t5) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S +O) O t7) t5) \to ((eq T t8 t4) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to +(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Bind Abbr) v2 t6) t9))))))))) (\lambda (H32: (eq T u0 +u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t5) \to ((eq T t8 +t4) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: +T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind +Abbr) v2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S O) O t7) t5)).(eq_ind +T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B Abst Abst)) +\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) +t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10))))))) (\lambda +(H34: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not (eq B Abst Abst)) \to +((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) +t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda +(H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t4)).(let H37 \def (match +(H35 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda +(t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead +(Bind Abbr) v2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t4 H34))) t5 +H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25 +H26))) | (pr0_epsilon t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead +(Flat Cast) u0 t7) (THead (Bind Abst) u t5))).(\lambda (H27: (eq T t8 +t4)).((let H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in +K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind Abst) u t5) H26) in (False_ind ((eq T t8 +t4) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 +t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27 +H25)))]) in (H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T +t4))))) t3 (sym_eq T t3 (THead (Bind Abst) u t5) H22))) u1 (sym_eq T u1 v1 +H21))) k (sym_eq K k (Flat Appl) H20))) H19)) H18)))]) in (H17 (refl_equal T +(THead (Flat Appl) v1 (THead (Bind Abst) u t5)))))))) t2 H13)) t H11 H12 H9 +H10))) | (pr0_upsilon b H9 v1 v2 H10 u0 u3 H11 t5 t6 H12) \Rightarrow +(\lambda (H13: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) +t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O +v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) +(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O +v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda +(t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to +((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: +T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda +(H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1 v2)).(\lambda (H18: (pr0 u0 +u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: +T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b) u0 +t5)) H13) in (let H21 \def (match H20 in eq return (\lambda (t7: T).(\lambda +(_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind b) u0 t5))) \to +(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))))) with +[refl_equal \Rightarrow (\lambda (H21: (eq T (THead k u1 t3) (THead (Flat +Appl) v1 (THead (Bind b) u0 t5)))).(let H22 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | +(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) +(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H21) in ((let H23 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) +(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H21) in ((let +H24 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) +\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 +t5)) H21) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u1 v1) \to ((eq T +t3 (THead (Bind b) u0 t5)) \to (ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) +t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) +O v2) t6)) t7)))))) (\lambda (H25: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: +T).((eq T t3 (THead (Bind b) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Flat Appl) u2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat +Appl) (lift (S O) O v2) t6)) t8))))) (\lambda (H26: (eq T t3 (THead (Bind b) +u0 t5))).(eq_ind T (THead (Bind b) u0 t5) (\lambda (_: T).(ex2 T (\lambda +(t8: T).(pr0 (THead (Flat Appl) u2 t4) t8)) (\lambda (t8: T).(pr0 (THead +(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))) (let H27 \def +(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: +T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: +T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) +v1 (THead (Bind b) u0 t5)) H13) in (let H28 \def (eq_ind T t3 (\lambda (t7: +T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H26) in (let H29 \def (match H28 in +pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T +t7 (THead (Bind b) u0 t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 +(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) with [(pr0_refl t7) +\Rightarrow (\lambda (H29: (eq T t7 (THead (Bind b) u0 t5))).(\lambda (H30: +(eq T t7 t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t8: T).((eq T t8 t4) +\to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) +(\lambda (H31: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0 +t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) +t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) +O v2) t6)) t9)))) (let H32 \def (eq_ind T u1 (\lambda (t8: T).(pr0 t8 u2)) H7 +v1 H25) in (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 +t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 +t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift +(S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H33: (pr0 u2 x)).(\lambda +(H34: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 u0 u3 H18 t5 +t6 H19 u2 v2 x H33 H34)))) (H27 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind +b) u0 t5)) u2 H32 v2 H17))) t4 H31)) t7 (sym_eq T t7 (THead (Bind b) u0 t5) +H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow (\lambda (H31: +(eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: (eq T (THead +k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | (TLRef _) +\Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) (THead +(Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | (TLRef _) +\Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) (THead +(Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e: T).(match e +in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) +\Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) (THead +(Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 u0) \to +((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to ((pr0 t7 +t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda +(t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) +t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T +t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to ((pr0 t7 +t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) +(\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O +v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: +T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to +(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: +T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) +t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t4)).(eq_ind T (THead +(Bind b) u5 t8) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to (ex2 T +(\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10)))))) +(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(let H41 \def (eq_ind +T u1 (\lambda (t9: T).(pr0 t9 u2)) H7 v1 H25) in (ex2_ind T (\lambda (t9: +T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: T).(pr0 +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda +(x: T).(\lambda (H42: (pr0 u2 x)).(\lambda (H43: (pr0 v2 x)).(ex2_ind T +(\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda +(t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: +T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) +(\lambda (x0: T).(\lambda (H44: (pr0 t8 x0)).(\lambda (H45: (pr0 t6 +x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9)) +(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) +t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) +O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H46: (pr0 u5 x1)).(\lambda (H47: +(pr0 u3 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x H42 H43 t8 +t6 x0 H44 H45 u5 u3 x1 H46 H47)))) (H27 u0 (tlt_trans (THead (Bind b) u0 t5) +u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) u0 t5) +(tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 H18))))) (H27 +t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 (THead (Bind b) +u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind +b) u0 t5))) t8 H40 t6 H19))))) (H27 v1 (tlt_head_sx (Flat Appl) v1 (THead +(Bind b) u0 t5)) u2 H41 v2 H17))))) t4 H38)) t7 (sym_eq T t7 t5 H37))) u4 +(sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) H32 H29 +H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: (eq T +(THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0 +t5))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def +(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in +K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind b) u0 t5) H31) in (False_ind ((eq T +(THead (Bind Abbr) v3 t8) t4) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))) H33)) +H32 H29 H30))) | (pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32) +\Rightarrow (\lambda (H33: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 +t7)) (THead (Bind b) u0 t5))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead +(Flat Appl) (lift (S O) O v3) t8)) t4)).((let H35 \def (eq_ind T (THead (Flat +Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind b) u0 t5) H33) in (False_ind ((eq T (THead (Bind b0) +u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t4) \to ((not (eq B b0 Abst)) +\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: +T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) +u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))) H35)) H34 H29 H30 H31 +H32))) | (pr0_delta u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: (eq +T (THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5))).(\lambda (H33: (eq T +(THead (Bind Abbr) u5 w) t4)).((let H34 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | +(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind +Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H35 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) +(THead (Bind Abbr) u4 t7) (THead (Bind b) u0 t5) H32) in ((let H36 \def +(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with +[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) +\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) +(THead (Bind b) u0 t5) H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u0) +\to ((eq T t7 t5) \to ((eq T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u4 u5) +\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b0) u3 +(THead (Flat Appl) (lift (S O) O v2) t6)) t9)))))))))) (\lambda (H37: (eq T +u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T t7 t5) \to ((eq T (THead (Bind +Abbr) u5 w) t4) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to +(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: +T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) +t10))))))))) (\lambda (H38: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9: T).((eq +T (THead (Bind Abbr) u5 w) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to ((subst0 +O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) +(\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O +v2) t6)) t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w) +t4)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u0 u5) \to +((pr0 t5 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead +(Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t10))))))) (\lambda (H40: (pr0 u0 +u5)).(\lambda (H41: (pr0 t5 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43 +\def (eq_ind_r B b (\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat +Appl) v1 (THead (Bind b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to +(\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) +(\lambda (t11: T).(pr0 t10 t11)))))))))) H27 Abbr H36) in (let H44 \def +(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H26 Abbr +H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) +H16 Abbr H36) in (let H46 \def (eq_ind T u1 (\lambda (t9: T).(pr0 t9 u2)) H7 +v1 H25) in (ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 +t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 +w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift +(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H47: (pr0 u2 x)).(\lambda +(H48: (pr0 v2 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: +T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead +(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H49: (pr0 +t8 x0)).(\lambda (H50: (pr0 t6 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) +(\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) +u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 +(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda +(H51: (pr0 u5 x1)).(\lambda (H52: (pr0 u3 +x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x H47 H48 +t6 x0 H49 H50 u3 x1 H51 H52)))) (H43 u0 (tlt_trans (THead (Bind Abbr) u0 t5) +u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind Abbr) +u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40 u3 +H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat Appl) +v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5) (tlt_head_dx +(Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19))))) (H43 v1 +(tlt_head_sx (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) u2 H46 v2 H17))))))))) +t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 (sym_eq T u4 u0 H37))) b H36)) H35)) +H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) \Rightarrow (\lambda +(H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 +t5))).(\lambda (H32: (eq T t8 t4)).((let H33 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let +rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match t9 +with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match +(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 +u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 d) t10))]) +in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match +t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 +d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ +t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) +u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) +\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S +O) O t7)) (THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T B (\lambda +(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 +| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return +(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow +b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0 t5) H31) in +(eq_ind B b (\lambda (b1: B).((eq T u u0) \to ((eq T (lift (S O) O t7) t5) +\to ((eq T t8 t4) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))))))) +(\lambda (H36: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O +t7) t5) \to ((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 +T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10: +T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) +t10)))))))) (\lambda (H37: (eq T (lift (S O) O t7) t5)).(eq_ind T (lift (S O) +O t7) (\lambda (_: T).((eq T t8 t4) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) +\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda +(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) +t10))))))) (\lambda (H38: (eq T t8 t4)).(eq_ind T t4 (\lambda (t9: T).((not +(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead +(Flat Appl) u2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t10)))))) (\lambda (H39: (not (eq B b +Abst))).(\lambda (H40: (pr0 t7 t4)).(let H41 \def (eq_ind_r T t5 (\lambda +(t9: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 +t9))) \to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) +\to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 +t12)))))))))) H27 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5 +(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H26 (lift (S O) O t7) H37) +in (let H43 \def (eq_ind_r T t5 (\lambda (t9: T).(pr0 t9 t6)) H19 (lift (S O) +O t7) H37) in (ex2_ind T (\lambda (t9: T).(eq T t6 (lift (S O) O t9))) +(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) +u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift +(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H44: (eq T t6 (lift (S O) O +x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: +T).(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda +(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t9)) +t10)))) (let H46 \def (eq_ind T u1 (\lambda (t9: T).(pr0 t9 u2)) H7 v1 H25) +in (ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) +(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O +x))) t9))) (\lambda (x0: T).(\lambda (H47: (pr0 u2 x0)).(\lambda (H48: (pr0 +v2 x0)).(ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9)) +(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O +x))) t9))) (\lambda (x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t4 +x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x0 H47 H48 +x t4 x1 H49 H50)))) (H41 t7 (tlt_trans (THead (Bind b) u0 (lift (S O) O t7)) +t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O t7))) (lift_tlt_dx +(Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 (lift +(S O) O t7)))) x H45 t4 H40))))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead +(Bind b) u0 (lift (S O) O t7))) u2 H46 v2 H17))) t6 H44)))) (pr0_gen_lift t7 +t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u (sym_eq T u u0 +H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | (pr0_epsilon t7 +t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u t7) (THead +(Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def (eq_ind T +(THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return (\lambda (_: +T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | +(THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind +b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 t7 t8) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) H32)) +H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) (refl_equal T +t4))))) t3 (sym_eq T t3 (THead (Bind b) u0 t5) H26))) u1 (sym_eq T u1 v1 +H25))) k (sym_eq K k (Flat Appl) H24))) H23)) H22)))]) in (H21 (refl_equal T +(THead (Flat Appl) v1 (THead (Bind b) u0 t5)))))))))) t2 H15)) t H13 H14 H9 +H10 H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda +(H12: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind +Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T +(THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O +u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: +T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w) +t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to +((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead +k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0 +u3)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18 +\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead +(Bind Abbr) u0 t5) H12) in (let H19 \def (match H18 in eq return (\lambda +(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u0 t5)) \to +(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 +(THead (Bind Abbr) u3 w) t8)))))) with [refl_equal \Rightarrow (\lambda (H19: +(eq T (THead k u1 t3) (THead (Bind Abbr) u0 t5))).(let H20 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) +(THead k u1 t3) (THead (Bind Abbr) u0 t5) H19) in ((let H21 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) +(THead k u1 t3) (THead (Bind Abbr) u0 t5) H19) in ((let H22 \def (f_equal T K +(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) +\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) +(THead k u1 t3) (THead (Bind Abbr) u0 t5) H19) in (eq_ind K (Bind Abbr) +(\lambda (k0: K).((eq T u1 u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t7: +T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) +t7)))))) (\lambda (H23: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3 +t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 t4) t8)) (\lambda +(t8: T).(pr0 (THead (Bind Abbr) u3 w) t8))))) (\lambda (H24: (eq T t3 +t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind +Abbr) u2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w) t8)))) (let +H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to +(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T +(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H +(THead (Bind Abbr) u0 t5) H12) in (let H26 \def (eq_ind T t3 (\lambda (t7: +T).(pr0 t7 t4)) H8 t5 H24) in (let H27 \def (eq_ind T u1 (\lambda (t7: +T).(pr0 t7 u2)) H7 u0 H23) in (ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) +(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) +u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x: +T).(\lambda (H28: (pr0 u2 x)).(\lambda (H29: (pr0 u3 x)).(ex2_ind T (\lambda +(t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind +Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 t4 x0)).(\lambda (H31: +(pr0 t6 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w H17 u2 x H28 H29 t4 x0 +H30 H31)))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16))))) (H25 +u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H27 u3 H15))))) t3 (sym_eq T t3 t5 +H24))) u1 (sym_eq T u1 u0 H23))) k (sym_eq K k (Bind Abbr) H22))) H21)) +H20)))]) in (H19 (refl_equal T (THead (Bind Abbr) u0 t5)))))))) t2 H14)) t +H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u) \Rightarrow (\lambda +(H11: (eq T (THead (Bind b) u (lift (S O) O t5)) t)).(\lambda (H12: (eq T t6 +t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 +t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: +T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda +(H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to +((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda +(t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not (eq B b Abst))).(\lambda (H15: +(pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 +t3) t7)) H4 (THead (Bind b) u (lift (S O) O t5)) H11) in (let H17 \def (match +H16 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead +(Bind b) u (lift (S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 +t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow +(\lambda (H17: (eq T (THead k u1 t3) (THead (Bind b) u (lift (S O) O +t5)))).(let H18 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 +| (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Bind b) u (lift (S +O) O t5)) H17) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) +\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead +(Bind b) u (lift (S O) O t5)) H17) in ((let H20 \def (f_equal T K (\lambda +(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k +| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) +(THead (Bind b) u (lift (S O) O t5)) H17) in (eq_ind K (Bind b) (\lambda (k0: +K).((eq T u1 u) \to ((eq T t3 (lift (S O) O t5)) \to (ex2 T (\lambda (t7: +T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))))) (\lambda +(H21: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t3 (lift (S O) O t5)) +\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t4) t8)) (\lambda (t8: +T).(pr0 t2 t8))))) (\lambda (H22: (eq T t3 (lift (S O) O t5))).(eq_ind T +(lift (S O) O t5) (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind +b) u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H23 \def (eq_ind_r T t +(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) +\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b) u (lift (S O) O +t5)) H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 +(lift (S O) O t5) H22) in (ex2_ind T (\lambda (t7: T).(eq T t4 (lift (S O) O +t7))) (\lambda (t7: T).(pr0 t5 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind +b) u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: +(eq T t4 (lift (S O) O x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S +O) O x) (\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) +t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H27 \def (eq_ind T u1 (\lambda (t7: +T).(pr0 t7 u2)) H7 u H21) in (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda +(t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S +O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H28: +(pr0 x x0)).(\lambda (H29: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 +(THead (Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 +(pr0_zeta b H14 x x0 H28 u2) H29)))) (H23 t5 (lift_tlt_dx (Bind b) u t5 (S O) +O) x H26 t2 H15))) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) t3 (sym_eq +T t3 (lift (S O) O t5) H22))) u1 (sym_eq T u1 u H21))) k (sym_eq K k (Bind b) +H20))) H19)) H18)))]) in (H17 (refl_equal T (THead (Bind b) u (lift (S O) O +t5)))))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon t5 t6 +H9 u) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda +(H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T +t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) +(\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T t2 +(\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 +t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H13: (pr0 t5 t2)).(let +H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead +(Flat Cast) u t5) H10) in (let H15 \def (match H14 in eq return (\lambda (t7: +T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Cast) u t5)) \to (ex2 T +(\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) +with [refl_equal \Rightarrow (\lambda (H15: (eq T (THead k u1 t3) (THead +(Flat Cast) u t5))).(let H16 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) +\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead +(Flat Cast) u t5) H15) in ((let H17 \def (f_equal T T (\lambda (e: T).(match +e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) +\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead +(Flat Cast) u t5) H15) in ((let H18 \def (f_equal T K (\lambda (e: T).(match +e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat +Cast) u t5) H15) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 u) \to +((eq T t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda +(t7: T).(pr0 t2 t7)))))) (\lambda (H19: (eq T u1 u)).(eq_ind T u (\lambda (_: +T).((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t4) +t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H20: (eq T t3 t5)).(eq_ind T +t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t4) +t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H21 \def (eq_ind_r T t (\lambda +(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to +(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u t5) H10) in +(let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H20) in (let +H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H19) in (ex2_ind T +(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda +(t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) +(\lambda (x: T).(\lambda (H24: (pr0 t4 x)).(\lambda (H25: (pr0 t2 +x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) +(\lambda (t7: T).(pr0 t2 t7)) x (pr0_epsilon t4 x H24 u2) H25)))) (H21 t5 +(tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13))))) t3 (sym_eq T t3 t5 H20))) +u1 (sym_eq T u1 u H19))) k (sym_eq K k (Flat Cast) H18))) H17)) H16)))]) in +(H15 (refl_equal T (THead (Flat Cast) u t5)))))) t6 (sym_eq T t6 t2 H12))) t +H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5 +H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4 H3) \Rightarrow (\lambda (H4: (eq T +(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t)).(\lambda (H5: (eq T +(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind +Abst) u t3)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4) t1) \to ((pr0 +v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda +(t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T (THead (Bind Abbr) v2 t4) +t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: T).((pr0 v1 v2) \to +((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 +t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda (H8: (pr0 t3 t4)).(let H9 +\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_: +(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 +(THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with +[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5 +t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: +T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))))) +(\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: +T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let +H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead (Flat Appl) +v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead (Flat Appl) v1 (THead +(Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind +Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t +(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat Appl) v1 (THead (Bind Abst) u +t3)) H4) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: +T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v +t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 +t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in +(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 t4) t6)) (\lambda +(t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t6)) (THead +(Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4)) (pr0_beta u v1 v2 H7 +t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) | +(pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda (H11: (eq T (THead k u1 +t5) t)).(\lambda (H12: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) +(\lambda (_: T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) +\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: +T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6) t2)).(eq_ind T +(THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 +t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 +\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind +Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let H17 \def (match H16 in eq +return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u1 t5)) +\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: +T).(pr0 (THead k u2 t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H17: +(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5))).(let +H18 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow +(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat +Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H17) in ((let H19 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) +\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 +t5) H17) in ((let H20 \def (f_equal T K (\lambda (e: T).(match e in T return +(\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) +\Rightarrow (Flat Appl) | (THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) +v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H17) in (eq_ind K (Flat Appl) +(\lambda (k0: K).((eq T v1 u1) \to ((eq T (THead (Bind Abst) u t3) t5) \to +(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: +T).(pr0 (THead k0 u2 t6) t7)))))) (\lambda (H21: (eq T v1 u1)).(eq_ind T u1 +(\lambda (_: T).((eq T (THead (Bind Abst) u t3) t5) \to (ex2 T (\lambda (t8: +T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat +Appl) u2 t6) t8))))) (\lambda (H22: (eq T (THead (Bind Abst) u t3) +t5)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (_: T).(ex2 T (\lambda (t8: +T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat +Appl) u2 t6) t8)))) (let H23 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead +k0 u1 t5) t)) H11 (Flat Appl) H20) in (let H24 \def (eq_ind_r T t5 (\lambda +(t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) H22) in (let H25 \def +(match H24 in pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 +t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T t8 t6) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow (\lambda +(H25: (eq T t7 (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t7 +t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq T t8 t6) \to +(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H27: (eq T (THead (Bind +Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).(ex2 T +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 t8) t9)))) (let H28 \def (eq_ind_r T t5 (\lambda (t8: +T).(eq T (THead (Flat Appl) u1 t8) t)) H23 (THead (Bind Abst) u t3) H22) in +(let H29 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to +(\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T +(\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H +(THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H28) in (let H30 \def (eq_ind +T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H21) in (ex2_ind T (\lambda (t8: +T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda (t8: T).(pr0 +(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 +(THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H31: (pr0 v2 +x)).(\lambda (H32: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead +(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead +(Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H31 t4 t4 +(pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H32 t3 t4 H8))))) (H29 u1 +(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H30 u2 H14))))) t6 +H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H25) H26))) | (pr0_comp u0 u3 +H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead +(Bind Abst) u t3))).(\lambda (H28: (eq T (THead k0 u3 t8) t6)).((let H29 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9) +\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H27) in ((let H30 +\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _) +\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H27) in ((let H31 +\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) +\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H27) in (eq_ind K +(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t3) \to ((eq T (THead +k1 u3 t8) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: +T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat +Appl) u2 t6) t9))))))))) (\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda +(t9: T).((eq T t7 t3) \to ((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 t9 +u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 +t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))))) +(\lambda (H33: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead +(Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda +(t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead +(Flat Appl) u2 t6) t10))))))) (\lambda (H34: (eq T (THead (Bind Abst) u3 t8) +t6)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to +((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) +t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)))))) (\lambda (_: +(pr0 u u3)).(\lambda (H36: (pr0 t3 t8)).(let H37 \def (eq_ind_r T t5 (\lambda +(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H23 (THead (Bind Abst) u t3) H22) +in (let H38 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) +\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to +(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 +t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H37) in (let +H39 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H21) in (ex2_ind T +(\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda +(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead +(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x: T).(\lambda +(H40: (pr0 v2 x)).(\lambda (H41: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0 +t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead +(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead +(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H42: (pr0 t8 +x0)).(\lambda (H43: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead +(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead +(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H40 t4 x0 +H43 (Bind Abbr)) (pr0_beta u3 u2 x H41 t8 x0 H42))))) (H38 t3 (tlt_trans +(THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) +(tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst) +u t3))) t8 H36 t4 H8))))) (H38 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind +Abst) u t3)) v2 H39 u2 H14))))))) t6 H34)) t7 (sym_eq T t7 t3 H33))) u0 +(sym_eq T u0 u H32))) k0 (sym_eq K k0 (Bind Abst) H31))) H30)) H29)) H28 H25 +H26))) | (pr0_beta u0 v0 v3 H25 t7 t8 H26) \Rightarrow (\lambda (H27: (eq T +(THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u +t3))).(\lambda (H28: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H29 \def +(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in +K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T +(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 t6) t9)))))) H29)) H28 H25 H26))) | (pr0_upsilon b H25 +v0 v3 H26 u0 u3 H27 t7 t8 H28) \Rightarrow (\lambda (H29: (eq T (THead (Flat +Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H30: +(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6)).((let +H31 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in +K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind Abst) u t3) H29) in (False_ind ((eq T +(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not +(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 +(THead (Flat Appl) u2 t6) t9)))))))) H31)) H30 H25 H26 H27 H28))) | +(pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: (eq T (THead +(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H29: (eq T (THead +(Bind Abbr) u3 w) t6)).((let H30 \def (eq_ind T (THead (Bind Abbr) u0 t7) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow +(match k0 in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match +b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst +\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow +False])])) I (THead (Bind Abst) u t3) H28) in (False_ind ((eq T (THead (Bind +Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to +(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H30)) H29 H25 H26 H27))) | +(pr0_zeta b H25 t7 t8 H26 u0) \Rightarrow (\lambda (H27: (eq T (THead (Bind +b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T t8 +t6)).((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat +\to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u3 t10) +\Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 d) t10))]) in +lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T \def (match +t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0 +d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ +t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind +Abst) u t3) H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) +\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S +O) O t7)) (THead (Bind Abst) u t3) H27) in ((let H31 \def (f_equal T B +(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) +\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match +k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u +t3) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S +O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to +(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: +T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H32: (eq T u0 +u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8 +t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: +T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat +Appl) u2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S O) O t7) t3)).(eq_ind +T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq B Abst Abst)) +\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) +t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda +(H34: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to +((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4) +t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))) (\lambda +(H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H37 \def (match +(H35 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda +(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead +(Flat Appl) u2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t6 H34))) t3 +H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25 +H26))) | (pr0_epsilon t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead +(Flat Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8 +t6)).((let H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in +K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T t8 +t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 +t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H28)) H27 +H25)))]) in (H25 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T +t6))))) t5 H22)) v1 (sym_eq T v1 u1 H21))) k H20)) H19)) H18)))]) in (H17 +(refl_equal T (THead k u1 t5))))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta +u0 v0 v3 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 +(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6) +t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_: +T).((eq T (THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: +T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) +t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) +(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15: +(pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat +Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind +Abst) u0 t5)) H11) in (let H17 \def (match H16 in eq return (\lambda (t7: +T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v0 (THead (Bind +Abst) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) +t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))))) with +[refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v1 (THead +(Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)))).(let +H18 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) +\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow +t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 +(THead (Bind Abst) u0 t5)) H17) in ((let H19 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | +(TLRef _) \Rightarrow u | (THead _ _ t7) \Rightarrow (match t7 in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | +(THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u +t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H17) in ((let H20 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) +\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead +(Flat Appl) v0 (THead (Bind Abst) u0 t5)) H17) in (eq_ind T v0 (\lambda (_: +T).((eq T u u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) +t8)))))) (\lambda (H21: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3 +t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda +(t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8))))) (\lambda (H22: (eq T t3 +t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind +Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))) (let +H23 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to +(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T +(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H +(THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H11) in (let H24 \def +(eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H22) in (let H25 \def +(eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H7 v0 H20) in (ex2_ind T (\lambda +(t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7: +T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind +Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H26: (pr0 v2 x)).(\lambda (H27: +(pr0 v3 x)).(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 +t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda +(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda (x0: T).(\lambda (H28: +(pr0 t4 x0)).(\lambda (H29: (pr0 t6 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 +(THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 +t6) t7)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H26 t4 x0 H28 (Bind Abbr)) +(pr0_comp v3 x H27 t6 x0 H29 (Bind Abbr)))))) (H23 t5 (tlt_trans (THead (Bind +Abst) u0 t5) t5 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx +(Bind Abst) u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4 +H24 t6 H15))))) (H23 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 +t5)) v2 H25 v3 H14))))) t3 (sym_eq T t3 t5 H22))) u (sym_eq T u u0 H21))) v1 +(sym_eq T v1 v0 H20))) H19)) H18)))]) in (H17 (refl_equal T (THead (Flat +Appl) v0 (THead (Bind Abst) u0 t5)))))))) t2 H13)) t H11 H12 H9 H10))) | +(pr0_upsilon b H9 v0 v3 H10 u1 u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: +(eq T (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T +(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b +Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda +(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 +t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v3) t6)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v3) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to +((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind +Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq +B b Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: +(pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat +Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind +b) u1 t5)) H13) in (let H21 \def (match H20 in eq return (\lambda (t7: +T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v0 (THead (Bind b) +u1 t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) +(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) +t6)) t8)))))) with [refl_equal \Rightarrow (\lambda (H21: (eq T (THead (Flat +Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 +t5)))).(let H22 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 +| (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) +\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead +(Flat Appl) v0 (THead (Bind b) u1 t5)) H21) in ((let H23 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) \Rightarrow (match +t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) +\Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 +(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H21) +in ((let H24 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda +(_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | +(THead _ _ t7) \Rightarrow (match t7 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 (Flat Appl) v1 +(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H21) +in ((let H25 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead +_ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) +(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H21) in (eq_ind T v0 (\lambda +(_: T).((eq B Abst b) \to ((eq T u u1) \to ((eq T t3 t5) \to (ex2 T (\lambda +(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t8))))))) (\lambda +(H26: (eq B Abst b)).(eq_ind B Abst (\lambda (b0: B).((eq T u u1) \to ((eq T +t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) +(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O +v3) t6)) t7)))))) (\lambda (H27: (eq T u u1)).(eq_ind T u1 (\lambda (_: +T).((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) +t8)) (\lambda (t8: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S +O) O v3) t6)) t8))))) (\lambda (H28: (eq T t3 t5)).(eq_ind T t5 (\lambda (_: +T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: +T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) +t8)))) (let H29 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16 +Abst H26) in (let H30 \def (match (H29 (refl_equal B Abst)) in False return +(\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) +t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S +O) O v3) t6)) t7)))) with []) in H30)) t3 (sym_eq T t3 t5 H28))) u (sym_eq T +u u1 H27))) b H26)) v1 (sym_eq T v1 v0 H25))) H24)) H23)) H22)))]) in (H21 +(refl_equal T (THead (Flat Appl) v0 (THead (Bind b) u1 t5)))))))))) t2 H15)) +t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 H10 w H11) +\Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) t)).(\lambda (H13: +(eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u1 t5) +(\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to +((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: +(eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) +(\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: +T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5 +t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda +(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead +(Bind Abbr) u1 t5) H12) in (let H19 \def (match H18 in eq return (\lambda +(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u1 t5)) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: +T).(pr0 (THead (Bind Abbr) u2 w) t8)))))) with [refl_equal \Rightarrow +(\lambda (H19: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead +(Bind Abbr) u1 t5))).(let H20 \def (eq_ind T (THead (Flat Appl) v1 (THead +(Bind Abst) u t3)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ +_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 +t5) H19) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) +t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) H20)))]) in (H19 +(refl_equal T (THead (Bind Abbr) u1 t5)))))))) t2 H14)) t H12 H13 H9 H10 +H11))) | (pr0_zeta b H9 t5 t6 H10 u0) \Rightarrow (\lambda (H11: (eq T (THead +(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T +(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not +(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind +Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 +t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: +T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5 +t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 +(THead (Bind Abst) u t3)) t7)) H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) +in (let H17 \def (match H16 in eq return (\lambda (t7: T).(\lambda (_: (eq ? +? t7)).((eq T t7 (THead (Bind b) u0 (lift (S O) O t5))) \to (ex2 T (\lambda +(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) +with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v1 +(THead (Bind Abst) u t3)) (THead (Bind b) u0 (lift (S O) O t5)))).(let H18 +\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H17) in +(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) +(\lambda (t7: T).(pr0 t2 t7))) H18)))]) in (H17 (refl_equal T (THead (Bind b) +u0 (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | +(pr0_epsilon t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) +u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) +(\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: +T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) +(\lambda (H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: +T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t +(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) +H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def (match H14 in eq return +(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Cast) u0 +t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda +(t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H15: (eq T +(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Cast) u0 +t5))).(let H16 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl +\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u0 t5) +H15) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) +t7)) (\lambda (t7: T).(pr0 t2 t7))) H16)))]) in (H15 (refl_equal T (THead +(Flat Cast) u0 t5)))))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 +(refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | +(pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5) \Rightarrow (\lambda (H6: (eq T +(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t)).(\lambda (H7: (eq T (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T (THead +(Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (_: T).((eq T (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1) \to ((not (eq B b Abst)) \to +((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: +T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))))) (\lambda (H8: (eq T +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: +T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) +\to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) +(\lambda (H9: (not (eq B b Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: +(pr0 u1 u2)).(\lambda (H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0 +return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 +t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 +t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H13: (eq T t5 +t)).(\lambda (H14: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H15: (eq T t +t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 +t7)))) (let H16 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead +(Flat Appl) v1 (THead (Bind b) u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: +T).(pr0 t6 t7)))) (let H17 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) +H13 (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6) in (let H18 \def +(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: +T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: +T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) H6) in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead +(Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 +v2 v2 H10 (pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 +t H13) H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: +(eq T (THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T +(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) +\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) +(\lambda (H17: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda +(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: +T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 +t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def (match +H20 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead +k u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead k u3 t6) +t8)))))) with [refl_equal \Rightarrow (\lambda (H21: (eq T (THead (Flat Appl) +v1 (THead (Bind b) u1 t3)) (THead k u0 t5))).(let H22 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead (Bind b) u1 +t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) +u1 t3)) (THead k u0 t5) H21) in ((let H23 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | +(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat +Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H21) in ((let H24 \def +(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with +[(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | +(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind b) u1 +t3)) (THead k u0 t5) H21) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v1 +u0) \to ((eq T (THead (Bind b) u1 t3) t5) \to (ex2 T (\lambda (t7: T).(pr0 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda +(t7: T).(pr0 (THead k0 u3 t6) t7)))))) (\lambda (H25: (eq T v1 u0)).(eq_ind T +u0 (\lambda (_: T).((eq T (THead (Bind b) u1 t3) t5) \to (ex2 T (\lambda (t8: +T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) +(\lambda (t8: T).(pr0 (THead (Flat Appl) u3 t6) t8))))) (\lambda (H26: (eq T +(THead (Bind b) u1 t3) t5)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: +T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u3 t6) t8)))) +(let H27 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15 +(Flat Appl) H24) in (let H28 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 +t6)) H19 (THead (Bind b) u1 t3) H26) in (let H29 \def (match H28 in pr0 +return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 +(THead (Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda +(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7) +\Rightarrow (\lambda (H29: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H30: +(eq T t7 t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6) +\to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))) +(\lambda (H31: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1 +t3) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat +Appl) u3 t8) t9)))) (let H32 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T +(THead (Flat Appl) u0 t8) t)) H27 (THead (Bind b) u1 t3) H26) in (let H33 +\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall +(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda +(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead +(Flat Appl) u0 (THead (Bind b) u1 t3)) H32) in (let H34 \def (eq_ind T v1 +(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H25) in (ex2_ind T (\lambda (t8: T).(pr0 +v2 t8)) (\lambda (t8: T).(pr0 u3 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: +T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)) t8))) (\lambda (x: +T).(\lambda (H35: (pr0 v2 x)).(\lambda (H36: (pr0 u3 x)).(ex2_sym T (pr0 +(THead (Flat Appl) u3 (THead (Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b +H9 u1 u2 H11 t3 t4 H12 u3 v2 x H36 H35))))) (H33 u0 (tlt_head_sx (Flat Appl) +u0 (THead (Bind b) u1 t3)) v2 H34 u3 H18))))) t6 H31)) t7 (sym_eq T t7 (THead +(Bind b) u1 t3) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow +(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: +(eq T (THead k0 u5 t8) t6)).((let H33 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | +(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) +(THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | +(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) +(THead (Bind b) u1 t3) H31) in ((let H35 \def (f_equal T K (\lambda (e: +T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | +(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) +(THead (Bind b) u1 t3) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 +u1) \to ((eq T t7 t3) \to ((eq T (THead k1 u5 t8) t6) \to ((pr0 u4 u5) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 +t6) t9))))))))) (\lambda (H36: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9: +T).((eq T t7 t3) \to ((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to +((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat +Appl) u3 t6) t10)))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda +(t9: T).((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) +\to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) +t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead +(Bind b) u5 t8) (\lambda (t9: T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T +(\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t9) t10)))))) +(\lambda (H39: (pr0 u1 u5)).(\lambda (H40: (pr0 t3 t8)).(let H41 \def +(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H27 +(THead (Bind b) u1 t3) H26) in (let H42 \def (eq_ind_r T t (\lambda (t9: +T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v t10) \to +(\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) +(\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind +b) u1 t3)) H41) in (let H43 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) +H10 u0 H25) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 +u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 +(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H44: (pr0 v2 +x)).(\lambda (H45: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) +(\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead +(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H46: +(pr0 t8 x0)).(\lambda (H47: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 +t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 +(THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x1: T).(\lambda +(H48: (pr0 u5 x1)).(\lambda (H49: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat +Appl) u3 (THead (Bind b) u5 t8))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x +H45 H44 t8 t4 x0 H46 H47 u5 u2 x1 H48 H49))))) (H42 u1 (tlt_trans (THead +(Bind b) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx +(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H39 +u2 H11))))) (H42 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl) +u0 (THead (Bind b) u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat +Appl) u0 (THead (Bind b) u1 t3))) t8 H40 t4 H12))))) (H42 u0 (tlt_head_sx +(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H43 u3 H18))))))) t6 H38)) t7 +(sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) k0 (sym_eq K k0 (Bind b) +H35))) H34)) H33)) H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) +\Rightarrow (\lambda (H31: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u +t7)) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) +t6)).((let H33 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow +(match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False +| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H31) in (False_ind +((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to +(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))) +H33)) H32 H29 H30))) | (pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32) +\Rightarrow (\lambda (H33: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 +t7)) (THead (Bind b) u1 t3))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead +(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H35 \def (eq_ind T (THead (Flat +Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind b) u1 t3) H33) in (False_ind ((eq T (THead (Bind b0) +u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst)) +\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: +T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) +(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H35)) H34 H29 H30 +H31 H32))) | (pr0_delta u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32: +(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T +(THead (Bind Abbr) u5 w) t6)).((let H34 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | +(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind +Abbr) u4 t7) (THead (Bind b) u1 t3) H32) in ((let H35 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) +(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H32) in ((let H36 \def +(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with +[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _) +\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7) +(THead (Bind b) u1 t3) H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1) +\to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5) +\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0 +(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda +(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H37: (eq T u4 +u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind +Abbr) u5 w) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to +(ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) +t10))))))))) (\lambda (H38: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq +T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to ((subst0 +O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat +Appl) u3 t6) t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w) +t6)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u1 u5) \to +((pr0 t3 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead +(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: +T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H40: (pr0 u1 +u5)).(\lambda (H41: (pr0 t3 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43 +\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H26 +Abbr H36) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 +Abst))) H9 Abbr H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(eq T +(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)) H8 Abbr +H36) in (let H46 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat +Appl) u0 t9) t)) H27 (THead (Bind Abbr) u1 t3) H43) in (let H47 \def +(eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: +T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: +T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat +Appl) u0 (THead (Bind Abbr) u1 t3)) H46) in (let H48 \def (eq_ind T v1 +(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H25) in (ex2_ind T (\lambda (t9: T).(pr0 +v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead +(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: +T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) (\lambda (x: +T).(\lambda (H49: (pr0 v2 x)).(\lambda (H50: (pr0 u3 x)).(ex2_ind T (\lambda +(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: +T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) +(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) +(\lambda (x0: T).(\lambda (H51: (pr0 t8 x0)).(\lambda (H52: (pr0 t4 +x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9)) +(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead +(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H53: (pr0 u5 +x1)).(\lambda (H54: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead +(Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) +O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H44 u5 t8 w H42 u3 v2 x +H50 H49 t4 x0 H51 H52 u2 x1 H53 H54))))) (H47 u1 (tlt_trans (THead (Bind +Abbr) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_sx +(Bind Abbr) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) u5 +H40 u2 H11))))) (H47 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat +Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_dx (Bind Abbr) u1 t3) +(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H41 t4 H12))))) +(H47 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H48 u3 +H18))))))))))) t6 H39)) t7 (sym_eq T t7 t3 H38))) u4 (sym_eq T u4 u1 H37))) b +H36)) H35)) H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) +\Rightarrow (\lambda (H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead +(Bind b) u1 t3))).(\lambda (H32: (eq T t8 t6)).((let H33 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T +\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 +d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (TLRef _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T +\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0 +d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _ +t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) +u1 t3) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) +\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S +O) O t7)) (THead (Bind b) u1 t3) H31) in ((let H35 \def (f_equal T B (\lambda +(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0 +| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return +(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow +b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H31) in +(eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t7) t3) +\to ((eq T t8 t6) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T +(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))))) +(\lambda (H36: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O +t7) t3) \to ((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2 +T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))))))) +(\lambda (H37: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7) +(\lambda (_: T).((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to +(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) +t10))))))) (\lambda (H38: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not +(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: +T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H39: (not (eq B b +Abst))).(\lambda (H40: (pr0 t7 t6)).(let H41 \def (eq_ind_r T t3 (\lambda +(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H26 (lift (S O) O t7) H37) in (let +H42 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) +H27 (THead (Bind b) u1 (lift (S O) O t7)) H41) in (let H43 \def (eq_ind_r T t +(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v +t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 +t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead +(Bind b) u1 (lift (S O) O t7))) H42) in (let H44 \def (eq_ind_r T t3 (\lambda +(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H37) in (ex2_ind T (\lambda (t9: +T).(eq T t4 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda +(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x: +T).(\lambda (H45: (eq T t4 (lift (S O) O x))).(\lambda (H46: (pr0 t7 +x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10: +T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t9)) t10)) +(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H47 \def +(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H25) in (ex2_ind T (\lambda +(t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: +T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O +x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x0: +T).(\lambda (H48: (pr0 v2 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T +(\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda +(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S +O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda +(x1: T).(\lambda (H50: (pr0 x x1)).(\lambda (H51: (pr0 t6 x1)).(ex2_sym T +(pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta +b H39 u1 u2 H11 u3 v2 x0 H49 H48 x t6 x1 H50 H51))))) (H43 t7 (tlt_trans +(THead (Bind b) u1 (lift (S O) O t7)) t7 (THead (Flat Appl) u0 (THead (Bind +b) u1 (lift (S O) O t7))) (lift_tlt_dx (Bind b) u1 t7 (S O) O) (tlt_head_dx +(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H46 t6 H40))))) (H43 +u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H47 +u3 H18))) t4 H45)))) (pr0_gen_lift t7 t4 (S O) O H44)))))))) t8 (sym_eq T t8 +t6 H38))) t3 H37)) u (sym_eq T u u1 H36))) b0 (sym_eq B b0 b H35))) H34)) +H33)) H32 H29 H30))) | (pr0_epsilon t7 t8 H29 u) \Rightarrow (\lambda (H30: +(eq T (THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T +t8 t6)).((let H32 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in +K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind ((eq T t8 +t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead +(Flat Appl) u3 t6) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead +(Bind b) u1 t3)) (refl_equal T t6))))) t5 H26)) v1 (sym_eq T v1 u0 H25))) k +H24)) H23)) H22)))]) in (H21 (refl_equal T (THead k u0 t5))))))) t2 H17)) t +H15 H16 H13 H14))) | (pr0_beta u v0 v3 H13 t5 t6 H14) \Rightarrow (\lambda +(H15: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) t)).(\lambda +(H16: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Flat Appl) v0 +(THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t6) +t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: +T).(pr0 t2 t8))))))) (\lambda (H17: (eq T (THead (Bind Abbr) v3 t6) +t2)).(eq_ind T (THead (Bind Abbr) v3 t6) (\lambda (t7: T).((pr0 v0 v3) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda +(_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t +(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 +(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def (match +H20 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead +(Flat Appl) v0 (THead (Bind Abst) u t5))) \to (ex2 T (\lambda (t8: T).(pr0 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda +(t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))))) with [refl_equal \Rightarrow +(\lambda (H21: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead +(Flat Appl) v0 (THead (Bind Abst) u t5)))).(let H22 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match +t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) +\Rightarrow t3 | (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H21) +in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead +_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t8 _) \Rightarrow +t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 +(THead (Bind Abst) u t5)) H21) in ((let H24 \def (f_equal T B (\lambda (e: +T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | +(TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match t7 in T return +(\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind +b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])])) (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H21) +in ((let H25 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda +(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead +_ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) +(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H21) in (eq_ind T v0 (\lambda +(_: T).((eq B b Abst) \to ((eq T u1 u) \to ((eq T t3 t5) \to (ex2 T (\lambda +(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8))))))) (\lambda (H26: +(eq B b Abst)).(eq_ind B Abst (\lambda (b0: B).((eq T u1 u) \to ((eq T t3 t5) +\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) +t7)))))) (\lambda (H27: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t3 +t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) +t8))))) (\lambda (H28: (eq T t3 t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T +(\lambda (t8: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O +v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))) (let H29 +\def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall +(t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda +(t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat +Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H30 \def (eq_ind T t3 +(\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in (let H31 \def (eq_ind T u1 +(\lambda (t7: T).(pr0 t7 u2)) H11 u H27) in (let H32 \def (eq_ind B b +(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H33 \def (match +(H32 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda +(t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in +H33))))) t3 (sym_eq T t3 t5 H28))) u1 (sym_eq T u1 u H27))) b (sym_eq B b +Abst H26))) v1 (sym_eq T v1 v0 H25))) H24)) H23)) H22)))]) in (H21 +(refl_equal T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)))))))) t2 H17)) +t H15 H16 H13 H14))) | (pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) +\Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 +t5)) t)).(\lambda (H18: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S +O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) +(\lambda (_: T).((eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O +v3) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) +(\lambda (H19: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) +t6)) t2)).(eq_ind T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) +t6)) (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) +\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) +(\lambda (_: (not (eq B b0 Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: +(pr0 u0 u3)).(\lambda (H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda +(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead +(Flat Appl) v0 (THead (Bind b0) u0 t5)) H17) in (let H25 \def (match H24 in +eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat +Appl) v0 (THead (Bind b0) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: +T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) +t8)))))) with [refl_equal \Rightarrow (\lambda (H25: (eq T (THead (Flat Appl) +v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 +t5)))).(let H26 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 +| (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) +\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead +(Flat Appl) v0 (THead (Bind b0) u0 t5)) H25) in ((let H27 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match +t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) +\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H25) +in ((let H28 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda +(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ +_ t7) \Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _) +\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k +in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) +\Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead +(Flat Appl) v0 (THead (Bind b0) u0 t5)) H25) in ((let H29 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) +(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead +(Bind b0) u0 t5)) H25) in (eq_ind T v0 (\lambda (_: T).((eq B b b0) \to ((eq +T u1 u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 +(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t8))))))) +(\lambda (H30: (eq B b b0)).(eq_ind B b0 (\lambda (b1: B).((eq T u1 u0) \to +((eq T t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b1) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind +b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7)))))) (\lambda (H31: (eq +T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8: +T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) +(\lambda (t8: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O +v3) t6)) t8))))) (\lambda (H32: (eq T t3 t5)).(eq_ind T t5 (\lambda (_: +T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift +(S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b0) u3 (THead (Flat +Appl) (lift (S O) O v3) t6)) t8)))) (let H33 \def (eq_ind_r T t (\lambda (t7: +T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall +(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: +T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) +H17) in (let H34 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H32) +in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H31) in +(let H36 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0 H30) +in (let H37 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H10 v0 H29) in +(ex2_ind T (\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T +(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O +v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) +(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H38: (pr0 v2 +x)).(\lambda (H39: (pr0 v3 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) +(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 +(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda +(x0: T).(\lambda (H40: (pr0 u2 x0)).(\lambda (H41: (pr0 u3 x0)).(ex2_ind T +(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda +(t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) +O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H42: (pr0 t4 x1)).(\lambda (H43: +(pr0 t6 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H36 v2 v3 x H38 H39 u2 +u3 x0 H40 H41 t4 t6 x1 H42 H43)))) (H33 t5 (tlt_trans (THead (Bind b0) u0 t5) +t5 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0 +t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) t4 H34 t6 H23))))) +(H33 u0 (tlt_trans (THead (Bind b0) u0 t5) u0 (THead (Flat Appl) v0 (THead +(Bind b0) u0 t5)) (tlt_head_sx (Bind b0) u0 t5) (tlt_head_dx (Flat Appl) v0 +(THead (Bind b0) u0 t5))) u2 H35 u3 H22))))) (H33 v0 (tlt_head_sx (Flat Appl) +v0 (THead (Bind b0) u0 t5)) v2 H37 v3 H21))))))) t3 (sym_eq T t3 t5 H32))) u1 +(sym_eq T u1 u0 H31))) b (sym_eq B b b0 H30))) v1 (sym_eq T v1 v0 H29))) +H28)) H27)) H26)))]) in (H25 (refl_equal T (THead (Flat Appl) v0 (THead (Bind +b0) u0 t5)))))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3 +H13 t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0 +t5) t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead +(Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to +((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda +(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind +Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 +u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda +(t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 +t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def (eq_ind_r T t (\lambda +(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead +(Bind Abbr) u0 t5) H16) in (let H23 \def (match H22 in eq return (\lambda +(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u0 t5)) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w) t8)))))) +with [refl_equal \Rightarrow (\lambda (H23: (eq T (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (THead (Bind Abbr) u0 t5))).(let H24 \def (eq_ind T +(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (e: T).(match e in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind Abbr) u0 t5) H23) in (False_ind (ex2 T (\lambda (t7: +T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) +(\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) H24)))]) in (H23 +(refl_equal T (THead (Bind Abbr) u0 t5)))))))) t2 H18)) t H16 H17 H13 H14 +H15))) | (pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T +(THead (Bind b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 +t2)).(eq_ind T (THead (Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T +t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: +T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) +(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 +(\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda +(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 +Abst))).(\lambda (_: (pr0 t5 t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: +T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Bind +b0) u (lift (S O) O t5)) H15) in (let H21 \def (match H20 in eq return +(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind b0) u (lift +(S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) +with [refl_equal \Rightarrow (\lambda (H21: (eq T (THead (Flat Appl) v1 +(THead (Bind b) u1 t3)) (THead (Bind b0) u (lift (S O) O t5)))).(let H22 \def +(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) +\Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H21) in +(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H22)))]) in (H21 +(refl_equal T (THead (Bind b0) u (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 +H17))) t H15 H16 H13 H14))) | (pr0_epsilon t5 t6 H13 u) \Rightarrow (\lambda +(H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15: (eq T t6 t2)).(eq_ind +T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H16: (eq T t6 +t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: +T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) +(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H18 \def +(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 +t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in (let H19 \def (match H18 in eq +return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat +Cast) u t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) with +[refl_equal \Rightarrow (\lambda (H19: (eq T (THead (Flat Appl) v1 (THead +(Bind b) u1 t3)) (THead (Flat Cast) u t5))).(let H20 \def (eq_ind T (THead +(Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f +in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast +\Rightarrow False])])])) I (THead (Flat Cast) u t5) H19) in (False_ind (ex2 T +(\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H20)))]) in (H19 (refl_equal T +(THead (Flat Cast) u t5)))))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in +(H13 (refl_equal T t) (refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) +| (pr0_delta u1 u2 H2 t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead +(Bind Abbr) u1 t3) t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) +t1)).(eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind +Abbr) u2 w) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to +(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) +(\lambda (H7: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind +Abbr) u2 w) (\lambda (t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 +t4 w) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2 +t6))))))) (\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda +(H10: (subst0 O u2 t4 w)).(let H11 \def (match H1 in pr0 return (\lambda (t5: +T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: +T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5 +t)).(\lambda (H12: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: +T).(pr0 t2 t7))))) (\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_: +T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: +T).(pr0 t2 t7)))) (let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) +H13 (THead (Bind Abbr) u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3) +(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) +(\lambda (t7: T).(pr0 t6 t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: +T).(eq T t5 t6)) H11 (THead (Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r +T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v +t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) +(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in +(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda +(t6: T).(pr0 (THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w) +(pr0_refl (THead (Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2 +H14)) t (sym_eq T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3 +H11 t5 t6 H12 k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5) +t)).(\lambda (H14: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u0 t5) +(\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) +\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: +T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T +(THead k u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 +t8)))))) (\lambda (H16: (pr0 u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18 +\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 +(THead k u0 t5) H13) in (let H19 \def (match H18 in eq return (\lambda (t7: +T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u0 t5)) \to (ex2 T (\lambda +(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead k u3 +t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H19: (eq T (THead (Bind +Abbr) u1 t3) (THead k u0 t5))).(let H20 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | +(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind +Abbr) u1 t3) (THead k u0 t5) H19) in ((let H21 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | +(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind +Abbr) u1 t3) (THead k u0 t5) H19) in ((let H22 \def (f_equal T K (\lambda (e: +T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind +Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | (THead k0 _ _) \Rightarrow k0])) +(THead (Bind Abbr) u1 t3) (THead k u0 t5) H19) in (eq_ind K (Bind Abbr) +(\lambda (k0: K).((eq T u1 u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) +t7)))))) (\lambda (H23: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3 +t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda +(t8: T).(pr0 (THead (Bind Abbr) u3 t6) t8))))) (\lambda (H24: (eq T t3 +t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind +Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 t6) t8)))) (let +H25 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13 (Bind +Abbr) H22) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: +T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v +t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 +t10)))))))))) H (THead (Bind Abbr) u0 t5) H25) in (let H27 \def (eq_ind T t3 +(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H24) in (let H28 \def (eq_ind T u1 +(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (ex2_ind T (\lambda (t7: T).(pr0 +u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead +(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 t6) t7))) +(\lambda (x: T).(\lambda (H29: (pr0 u2 x)).(\lambda (H30: (pr0 u3 +x)).(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) +(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H31: (pr0 +t4 x0)).(\lambda (H32: (pr0 t6 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 +t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w +H10 u3 x H30 H29 t6 x0 H32 H31))))) (H26 t5 (tlt_head_dx (Bind Abbr) u0 t5) +t4 H27 t6 H17))))) (H26 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H28 u3 +H16)))))) t3 (sym_eq T t3 t5 H24))) u1 (sym_eq T u1 u0 H23))) k H22)) H21)) +H20)))]) in (H19 (refl_equal T (THead k u0 t5))))))) t2 H15)) t H13 H14 H11 +H12))) | (pr0_beta u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T +(THead (Flat Appl) v1 (THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T +(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind +Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 +v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) +u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead +(Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: +T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 +v2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: +T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind +Abst) u t5)) H13) in (let H19 \def (match H18 in eq return (\lambda (t7: +T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind +Abst) u t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) +(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal +\Rightarrow (\lambda (H19: (eq T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) +v1 (THead (Bind Abst) u t5)))).(let H20 \def (eq_ind T (THead (Bind Abbr) u1 +t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 +(THead (Bind Abst) u t5)) H19) in (False_ind (ex2 T (\lambda (t7: T).(pr0 +(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) +t7))) H20)))]) in (H19 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) +u t5)))))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 +u3 H13 t5 t6 H14) \Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 +(THead (Bind b) u0 t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 +(THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 +v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: +(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) +t2)).(eq_ind T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) +(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) +(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b +Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 +t5 t6)).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) +u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 +\def (match H22 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq +T t7 (THead (Flat Appl) v1 (THead (Bind b) u0 t5))) \to (ex2 T (\lambda (t8: +T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) +u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))))) with [refl_equal +\Rightarrow (\lambda (H23: (eq T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) +v1 (THead (Bind b) u0 t5)))).(let H24 \def (eq_ind T (THead (Bind Abbr) u1 +t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 +(THead (Bind b) u0 t5)) H23) in (False_ind (ex2 T (\lambda (t7: T).(pr0 +(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead +(Flat Appl) (lift (S O) O v2) t6)) t7))) H24)))]) in (H23 (refl_equal T +(THead (Flat Appl) v1 (THead (Bind b) u0 t5)))))))))) t2 H17)) t H15 H16 H11 +H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow (\lambda +(H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T (THead (Bind +Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T +(THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 +O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) +(\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead (Bind Abbr) +u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7: T).((pr0 u0 u3) +\to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 +(THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda +(H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: (subst0 O u3 t6 +w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 +t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 \def (match H20 in eq +return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind +Abbr) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) +(\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w0) t8)))))) with [refl_equal +\Rightarrow (\lambda (H21: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) +u0 t5))).(let H22 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 +| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind +Abbr) u0 t5) H21) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in +T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) +\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) +(THead (Bind Abbr) u0 t5) H21) in (eq_ind T u0 (\lambda (_: T).((eq T t3 t5) +\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: +T).(pr0 (THead (Bind Abbr) u3 w0) t8))))) (\lambda (H24: (eq T t3 +t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind +Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w0) t8)))) (let +H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to +(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T +(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H +(THead (Bind Abbr) u0 t5) H14) in (let H26 \def (eq_ind T t3 (\lambda (t7: +T).(pr0 t7 t4)) H9 t5 H24) in (let H27 \def (eq_ind T u1 (\lambda (t7: +T).(pr0 t7 u2)) H8 u0 H23) in (ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) +(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) +u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x: +T).(\lambda (H28: (pr0 u2 x)).(\lambda (H29: (pr0 u3 x)).(ex2_ind T (\lambda +(t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind +Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 t4 x0)).(\lambda +(H31: (pr0 t6 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w H10 u3 t6 w0 H19 +x H28 H29 x0 H30 H31)))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 +H18))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H27 u3 H17))))) t3 +(sym_eq T t3 t5 H24))) u1 (sym_eq T u1 u0 H23))) H22)))]) in (H21 (refl_equal +T (THead (Bind Abbr) u0 t5)))))))) t2 H16)) t H14 H15 H11 H12 H13))) | +(pr0_zeta b H11 t5 t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b) +u (lift (S O) O t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind +b) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b +Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) +u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6 +t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: +T).(pr0 t2 t8)))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 +t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) +u1 t3) t7)) H5 (THead (Bind b) u (lift (S O) O t5)) H13) in (let H19 \def +(match H18 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 +(THead (Bind b) u (lift (S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead +(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal +\Rightarrow (\lambda (H19: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u +(lift (S O) O t5)))).(let H20 \def (f_equal T T (\lambda (e: T).(match e in T +return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) +\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) +(THead (Bind b) u (lift (S O) O t5)) H19) in ((let H21 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) +(THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H19) in ((let +H22 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) +with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0) +\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3) +(THead (Bind b) u (lift (S O) O t5)) H19) in (eq_ind B Abbr (\lambda (_: +B).((eq T u1 u) \to ((eq T t3 (lift (S O) O t5)) \to (ex2 T (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))) +(\lambda (H23: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t3 (lift (S O) +O t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) +(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H24: (eq T t3 (lift (S O) O +t5))).(eq_ind T (lift (S O) O t5) (\lambda (_: T).(ex2 T (\lambda (t8: +T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let +H25 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16 Abbr H22) +in (let H26 \def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u +(lift (S O) O t5)) t)) H13 Abbr H22) in (let H27 \def (eq_ind_r T t (\lambda +(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to +(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O +t5)) H26) in (let H28 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 +(lift (S O) O t5) H24) in (ex2_ind T (\lambda (t7: T).(eq T t4 (lift (S O) O +t7))) (\lambda (t7: T).(pr0 t5 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind +Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda +(H29: (eq T t4 (lift (S O) O x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def +(eq_ind T t4 (\lambda (t7: T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) +in (let H32 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H23) in +(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T +(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 +t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2 +x0)).(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H31 x (pr0_refl +(lift (S O) O x)) t2)))) (H27 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30 +t2 H17))))))) (pr0_gen_lift t5 t4 (S O) O H28)))))) t3 (sym_eq T t3 (lift (S +O) O t5) H24))) u1 (sym_eq T u1 u H23))) b H22)) H21)) H20)))]) in (H19 +(refl_equal T (THead (Bind b) u (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 +H15))) t H13 H14 H11 H12))) | (pr0_epsilon t5 t6 H11 u) \Rightarrow (\lambda +(H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13: (eq T t6 t2)).(eq_ind +T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to +(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: +T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: +T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) +t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H16 \def +(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead +(Flat Cast) u t5) H12) in (let H17 \def (match H16 in eq return (\lambda (t7: +T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Cast) u t5)) \to (ex2 T +(\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 +t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Bind Abbr) +u1 t3) (THead (Flat Cast) u t5))).(let H18 \def (eq_ind T (THead (Bind Abbr) +u1 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t5) +H17) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) +(\lambda (t7: T).(pr0 t2 t7))) H18)))]) in (H17 (refl_equal T (THead (Flat +Cast) u t5)))))) t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 +(refl_equal T t) (refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | +(pr0_zeta b H2 t3 t4 H3 u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u +(lift (S O) O t3)) t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) +u (lift (S O) O t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) +\to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: +T).(pr0 t2 t6))))))) (\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: +T).((not (eq B b Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 +t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b +Abst))).(\lambda (H8: (pr0 t3 t1)).(let H9 \def (match H1 in pr0 return +(\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to +((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 +t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 +t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) +(\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t +(\lambda (t6: T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4) +in (eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def +(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O) +O t3)) H4) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: +T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v +t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 +t9)))))))))) H (THead (Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T +(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift +(S O) O t3)) t6)) t1 (pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t +(sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 +H10 k) \Rightarrow (\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: +(eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T +(THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda +(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T +(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) +\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: +T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 +t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u +(lift (S O) O t3)) t7)) H4 (THead k u1 t5) H11) in (let H17 \def (match H16 +in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u1 +t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead k +u2 t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead +(Bind b) u (lift (S O) O t3)) (THead k u1 t5))).(let H18 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T +\def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d u0) (lref_map f (s k0 +d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) +\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T +\def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) +| (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d u0) (lref_map f (s k0 +d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ +t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5) +H17) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | +(THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead +k u1 t5) H17) in ((let H20 \def (f_equal T K (\lambda (e: T).(match e in T +return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef _) +\Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u +(lift (S O) O t3)) (THead k u1 t5) H17) in (eq_ind K (Bind b) (\lambda (k0: +K).((eq T u u1) \to ((eq T (lift (S O) O t3) t5) \to (ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))))) (\lambda +(H21: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t3) t5) +\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind +b) u2 t6) t8))))) (\lambda (H22: (eq T (lift (S O) O t3) t5)).(eq_ind T (lift +(S O) O t3) (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda +(t8: T).(pr0 (THead (Bind b) u2 t6) t8)))) (let H23 \def (eq_ind_r K k +(\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H20) in (let H24 +\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H22) +in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7: +T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 +(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift (S +O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda +(t7: T).(eq T (THead (Bind b) u1 t7) t)) H23 (lift (S O) O t3) H22) in (let +H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to +(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T +(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H +(THead (Bind b) u1 (lift (S O) O t3)) H27) in (eq_ind_r T (lift (S O) O x) +(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 +(THead (Bind b) u2 t7) t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) +(\lambda (t7: T).(pr0 t1 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda +(t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7))) (\lambda (x0: +T).(\lambda (H29: (pr0 x x0)).(\lambda (H30: (pr0 t1 x0)).(ex_intro2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift +(S O) O x)) t7)) x0 H30 (pr0_zeta b H7 x x0 H29 u2))))) (H28 t3 (lift_tlt_dx +(Bind b) u1 t3 (S O) O) x H26 t1 H8)) t6 H25)))))) (pr0_gen_lift t3 t6 (S O) +O H24)))) t5 H22)) u (sym_eq T u u1 H21))) k H20)) H19)) H18)))]) in (H17 +(refl_equal T (THead k u1 t5))))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta +u0 v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 +(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) +t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_: +T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to +(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) +(\lambda (H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind +Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: +(pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda +(t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) +v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (match H16 in eq return +(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 +(THead (Bind Abst) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda +(t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal \Rightarrow +(\lambda (H17: (eq T (THead (Bind b) u (lift (S O) O t3)) (THead (Flat Appl) +v1 (THead (Bind Abst) u0 t5)))).(let H18 \def (eq_ind T (THead (Bind b) u +(lift (S O) O t3)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ +_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 +(THead (Bind Abst) u0 t5)) H17) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 +t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H18)))]) in (H17 +(refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)))))))) t2 H13)) +t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2 H11 t5 t6 H12) +\Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 +t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S +O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) +(\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O +v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 +t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2 (THead (Flat Appl) +(lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2 (THead (Flat Appl) +(lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 v1 +v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 +t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0 +Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 +t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u +(lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) u1 t5)) +H13) in (let H21 \def (match H20 in eq return (\lambda (t7: T).(\lambda (_: +(eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind b0) u1 t5))) \to +(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind b0) +u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))))) with [refl_equal +\Rightarrow (\lambda (H21: (eq T (THead (Bind b) u (lift (S O) O t3)) (THead +(Flat Appl) v1 (THead (Bind b0) u1 t5)))).(let H22 \def (eq_ind T (THead +(Bind b) u (lift (S O) O t3)) (\lambda (e: T).(match e in T return (\lambda +(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False +| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with +[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat +Appl) v1 (THead (Bind b0) u1 t5)) H21) in (False_ind (ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) +(lift (S O) O v2) t6)) t7))) H22)))]) in (H21 (refl_equal T (THead (Flat +Appl) v1 (THead (Bind b0) u1 t5)))))))))) t2 H15)) t H13 H14 H9 H10 H11 +H12))) | (pr0_delta u1 u2 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq +T (THead (Bind Abbr) u1 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) +t2)).(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind +Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to +(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) +(\lambda (H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind +Abbr) u2 w) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 +t6 w) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 +t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H16: (pr0 t5 t6)).(\lambda +(H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T +(THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Bind Abbr) u1 t5) H12) +in (let H19 \def (match H18 in eq return (\lambda (t7: T).(\lambda (_: (eq ? +? t7)).((eq T t7 (THead (Bind Abbr) u1 t5)) \to (ex2 T (\lambda (t8: T).(pr0 +t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)))))) with +[refl_equal \Rightarrow (\lambda (H19: (eq T (THead (Bind b) u (lift (S O) O +t3)) (THead (Bind Abbr) u1 t5))).(let H20 \def (f_equal T T (\lambda (e: +T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let +rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 +with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match +(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 +t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in +lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow +((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match +t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef +(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | +(THead k u0 t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) +t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7) +\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 +t5) H19) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | +(THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead +(Bind Abbr) u1 t5) H19) in ((let H22 \def (f_equal T B (\lambda (e: T).(match +e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) +\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: +K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead +(Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 t5) H19) in (eq_ind B +Abbr (\lambda (_: B).((eq T u u1) \to ((eq T (lift (S O) O t3) t5) \to (ex2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) +t7)))))) (\lambda (H23: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T +(lift (S O) O t3) t5) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: +T).(pr0 (THead (Bind Abbr) u2 w) t8))))) (\lambda (H24: (eq T (lift (S O) O +t3) t5)).(eq_ind T (lift (S O) O t3) (\lambda (_: T).(ex2 T (\lambda (t8: +T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)))) (let +H25 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3) +H24) in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda +(t7: T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: +T).(pr0 (THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H26: (eq T +t6 (lift (S O) O x))).(\lambda (H27: (pr0 t3 x)).(let H28 \def (eq_ind_r T t5 +(\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3) +H24) in (let H29 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v +t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to +(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 +t10)))))))))) H (THead (Bind Abbr) u1 (lift (S O) O t3)) H28) in (let H30 +\def (eq_ind T t6 (\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) +H26) in (let H31 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 +Abbr H22) in (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 +t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind +Abbr) u2 w) t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 +t1 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) +(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H30 x (pr0_refl +(lift (S O) O x)) t1))))) (H29 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x +H27 t1 H8))))))))) (pr0_gen_lift t3 t6 (S O) O H25))) t5 H24)) u (sym_eq T u +u1 H23))) b (sym_eq B b Abbr H22))) H21)) H20)))]) in (H19 (refl_equal T +(THead (Bind Abbr) u1 t5)))))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta +b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda (H11: (eq T (THead (Bind b0) u0 +(lift (S O) O t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind +b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b0 +Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda +(t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda +(t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: +T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 +Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: +T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Bind b0) u0 +(lift (S O) O t5)) H11) in (let H17 \def (match H16 in eq return (\lambda +(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind b0) u0 (lift (S O) O +t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 +t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Bind b) u +(lift (S O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)))).(let H18 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: +T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) +\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false +\Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) +(lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O +t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) +(t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | +(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | +false \Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f +d u1) (lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S +O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O +t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H17) in ((let H19 \def (f_equal T +T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) +(THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) +H17) in ((let H20 \def (f_equal T B (\lambda (e: T).(match e in T return +(\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | +(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind +b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S +O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H17) in (eq_ind B b0 +(\lambda (_: B).((eq T u u0) \to ((eq T (lift (S O) O t3) (lift (S O) O t5)) +\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))) +(\lambda (H21: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O +t3) (lift (S O) O t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: +T).(pr0 t2 t8))))) (\lambda (H22: (eq T (lift (S O) O t3) (lift (S O) O +t5))).(eq_ind T (lift (S O) O t3) (\lambda (_: T).(ex2 T (\lambda (t8: +T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H23 \def (eq_ind_r T t +(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) +\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) +(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b0) u0 (lift (S O) O +t5)) H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H8 t5 +(lift_inj t3 t5 (S O) O H22)) in (let H25 \def (eq_ind B b (\lambda (b1: +B).(not (eq B b1 Abst))) H7 b0 H20) in (ex2_ind T (\lambda (t7: T).(pr0 t1 +t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) +(\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H26: (pr0 t1 +x)).(\lambda (H27: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) +(\lambda (t7: T).(pr0 t2 t7)) x H26 H27)))) (H23 t5 (lift_tlt_dx (Bind b0) u0 +t5 (S O) O) t1 H24 t2 H15))))) (lift (S O) O t5) H22)) u (sym_eq T u u0 +H21))) b (sym_eq B b b0 H20))) H19)) H18)))]) in (H17 (refl_equal T (THead +(Bind b0) u0 (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 +H10))) | (pr0_epsilon t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead +(Flat Cast) u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat +Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda +(H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T +(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: +(pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind +b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 +\def (match H14 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq +T t7 (THead (Flat Cast) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) +(\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H15: +(eq T (THead (Bind b) u (lift (S O) O t3)) (THead (Flat Cast) u0 t5))).(let +H16 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) +\Rightarrow False])])) I (THead (Flat Cast) u0 t5) H15) in (False_ind (ex2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H16)))]) in (H15 +(refl_equal T (THead (Flat Cast) u0 t5)))))) t6 (sym_eq T t6 t2 H12))) t H10 +H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t4 (sym_eq T t4 t1 +H6))) t H4 H5 H2 H3))) | (pr0_epsilon t3 t4 H2 u) \Rightarrow (\lambda (H3: +(eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: (eq T t4 t1)).(eq_ind T +(THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to ((pr0 t3 t4) \to +(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))) +(\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0 t3 t5) \to +(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))) +(\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 in pr0 return (\lambda (t5: +T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) +\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) +with [(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 t)).(\lambda (H8: (eq +T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H9: (eq T t +t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) +(\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t (\lambda (t6: +T).(eq T t6 t2)) H9 (THead (Flat Cast) u t3) H3) in (eq_ind T (THead (Flat +Cast) u t3) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda +(t7: T).(pr0 t6 t7)))) (let H11 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 +t6)) H7 (THead (Flat Cast) u t3) H3) in (let H12 \def (eq_ind_r T t (\lambda +(t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to +(\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) +(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Cast) u t3) H3) in +(ex_intro2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 (THead (Flat +Cast) u t3) t6)) t1 (pr0_refl t1) (pr0_epsilon t3 t1 H6 u)))) t2 H10)) t +(sym_eq T t t2 H9))) t5 (sym_eq T t5 t H7) H8))) | (pr0_comp u1 u2 H7 t5 t6 +H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t5) t)).(\lambda (H10: (eq +T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T +(THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda +(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T +(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) +\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: +T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: (pr0 t5 +t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u +t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (match H14 in eq return +(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u1 t5)) \to +(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead k u2 t6) +t8)))))) with [refl_equal \Rightarrow (\lambda (H15: (eq T (THead (Flat Cast) +u t3) (THead k u1 t5))).(let H16 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) +\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3) +(THead k u1 t5) H15) in ((let H17 \def (f_equal T T (\lambda (e: T).(match e +in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) +\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3) +(THead k u1 t5) H15) in ((let H18 \def (f_equal T K (\lambda (e: T).(match e +in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat Cast) | +(TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _) \Rightarrow k0])) (THead +(Flat Cast) u t3) (THead k u1 t5) H15) in (eq_ind K (Flat Cast) (\lambda (k0: +K).((eq T u u1) \to ((eq T t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) +(\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))))) (\lambda (H19: (eq T u +u1)).(eq_ind T u1 (\lambda (_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8: +T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t6) t8))))) +(\lambda (H20: (eq T t3 t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda +(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t6) t8)))) +(let H21 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H9 +(Flat Cast) H18) in (let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: +T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v +t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 +t10)))))))))) H (THead (Flat Cast) u1 t5) H21) in (let H23 \def (eq_ind T t3 +(\lambda (t7: T).(pr0 t7 t1)) H6 t5 H20) in (ex2_ind T (\lambda (t7: T).(pr0 +t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) +(\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) t7))) (\lambda (x: +T).(\lambda (H24: (pr0 t1 x)).(\lambda (H25: (pr0 t6 x)).(ex_intro2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) +t7)) x H24 (pr0_epsilon t6 x H25 u2))))) (H22 t5 (tlt_head_dx (Flat Cast) u1 +t5) t1 H23 t6 H13))))) t3 (sym_eq T t3 t5 H20))) u (sym_eq T u u1 H19))) k +H18)) H17)) H16)))]) in (H15 (refl_equal T (THead k u1 t5))))))) t2 H11)) t +H9 H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9: +(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq +T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind +Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 +v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda +(t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6) +t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to +((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 +t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def +(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead +(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (match H14 in +eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat +Appl) v1 (THead (Bind Abst) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) +(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal +\Rightarrow (\lambda (H15: (eq T (THead (Flat Cast) u t3) (THead (Flat Appl) +v1 (THead (Bind Abst) u0 t5)))).(let H16 \def (eq_ind T (THead (Flat Cast) u +t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort +_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) +\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) +\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: +F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead +(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H15) in (False_ind (ex2 T (\lambda +(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) +H16)))]) in (H15 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0 +t5)))))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 +t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead +(Bind b) u1 t5)) t)).(\lambda (H12: (eq T (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead +(Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to +((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) +(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b +Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda +(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not +(eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda +(_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead +(Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) H11) +in (let H19 \def (match H18 in eq return (\lambda (t7: T).(\lambda (_: (eq ? +? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind b) u1 t5))) \to (ex2 T +(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t6)) t8)))))) with [refl_equal \Rightarrow +(\lambda (H19: (eq T (THead (Flat Cast) u t3) (THead (Flat Appl) v1 (THead +(Bind b) u1 t5)))).(let H20 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda +(e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K +return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) +\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow +False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 (THead (Bind b) +u1 t5)) H19) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: +T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) +H20)))]) in (H19 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u1 +t5)))))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta u1 u2 H7 t5 t6 +H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u1 t5) +t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind +Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 +u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 +t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T (THead (Bind +Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1 +u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 +t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda +(_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H16 \def (eq_ind_r T t +(\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind Abbr) u1 +t5) H10) in (let H17 \def (match H16 in eq return (\lambda (t7: T).(\lambda +(_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u1 t5)) \to (ex2 T (\lambda +(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)))))) +with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Flat Cast) u t3) +(THead (Bind Abbr) u1 t5))).(let H18 \def (eq_ind T (THead (Flat Cast) u t3) +(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t5) H17) in (False_ind +(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) +u2 w) t7))) H18)))]) in (H17 (refl_equal T (THead (Bind Abbr) u1 t5)))))))) +t2 H12)) t H10 H11 H7 H8 H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow +(\lambda (H9: (eq T (THead (Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: +(eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: +T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda +(H11: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to +((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 +t2 t8)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let +H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) +H3 (THead (Bind b) u0 (lift (S O) O t5)) H9) in (let H15 \def (match H14 in +eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind +b) u0 (lift (S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda +(t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H15: (eq T +(THead (Flat Cast) u t3) (THead (Bind b) u0 (lift (S O) O t5)))).(let H16 +\def (eq_ind T (THead (Flat Cast) u t3) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda +(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow +True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H15) in (False_ind (ex2 T +(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H16)))]) in (H15 +(refl_equal T (THead (Bind b) u0 (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 +H11))) t H9 H10 H7 H8))) | (pr0_epsilon t5 t6 H7 u0) \Rightarrow (\lambda +(H8: (eq T (THead (Flat Cast) u0 t5) t)).(\lambda (H9: (eq T t6 t2)).(eq_ind +T (THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) +\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) +(\lambda (H10: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to +(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) +(\lambda (H11: (pr0 t5 t2)).(let H12 \def (eq_ind_r T t (\lambda (t7: T).(eq +T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Cast) u0 t5) H8) in (let H13 +\def (match H12 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq +T t7 (THead (Flat Cast) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) +(\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H13: +(eq T (THead (Flat Cast) u t3) (THead (Flat Cast) u0 t5))).(let H14 \def +(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with +[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) +\Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) u0 t5) H13) in +((let H15 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: +T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 +_) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) u0 t5) H13) +in (eq_ind T u0 (\lambda (_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8: +T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H16: (eq T t3 +t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) +(\lambda (t8: T).(pr0 t2 t8)))) (let H17 \def (eq_ind_r T t (\lambda (t7: +T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall +(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: +T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u0 t5) H8) in (let H18 \def +(eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H16) in (ex2_ind T (\lambda +(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: +T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H19: +(pr0 t1 x)).(\lambda (H20: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 +t7)) (\lambda (t7: T).(pr0 t2 t7)) x H19 H20)))) (H17 t5 (tlt_head_dx (Flat +Cast) u0 t5) t1 H18 t2 H11)))) t3 (sym_eq T t3 t5 H16))) u (sym_eq T u u0 +H15))) H14)))]) in (H13 (refl_equal T (THead (Flat Cast) u0 t5)))))) t6 +(sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t) (refl_equal T +t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2 (refl_equal T t) +(refl_equal T t1))))))))) t0). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/props.ma new file mode 100644 index 000000000..8aff570af --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/props.ma @@ -0,0 +1,1775 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/props". + +include "pr0/defs.ma". + +include "subst0/subst0.ma". + +theorem pr0_lift: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall +(d: nat).(pr0 (lift h d t1) (lift h d t2)))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t) +(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d: +nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda +(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 +(lift h d u1) (lift h d u2)))))).(\lambda (t0: T).(\lambda (t3: T).(\lambda +(_: (pr0 t0 t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 +(lift h d t0) (lift h d t3)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda +(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t: +T).(pr0 t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2) +(lift h (s k d) t3)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k +d) t0)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) +t0) (lift h (s k d) t3) (H3 h (s k d)) k) (lift h d (THead k u2 t3)) +(lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) (lift_head k u1 t0 h +d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: +(pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h +d v1) (lift h d v2)))))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 +t0 t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0) +(lift h d t3)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead +(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u +t0))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t3)))) (eq_ind_r +T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s +(Flat Appl) d)) t0)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) +(lift h d (THead (Bind Abbr) v2 t3)))) (eq_ind_r T (THead (Bind Abbr) (lift h +d v2) (lift h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) +(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s +(Bind Abst) (s (Flat Appl) d)) t0))) t)) (pr0_beta (lift h (s (Flat Appl) d) +u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl) +d)) t0) (lift h (s (Bind Abbr) d) t3) (H3 h (s (Bind Abbr) d))) (lift h d +(THead (Bind Abbr) v2 t3)) (lift_head (Bind Abbr) v2 t3 h d)) (lift h (s +(Flat Appl) d) (THead (Bind Abst) u t0)) (lift_head (Bind Abst) u t0 h (s +(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t0))) +(lift_head (Flat Appl) v1 (THead (Bind Abst) u t0) h d))))))))))))) (\lambda +(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d: +nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d: +nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t0: T).(\lambda (t3: +T).(\lambda (_: (pr0 t0 t3)).(\lambda (H6: ((\forall (h: nat).(\forall (d: +nat).(pr0 (lift h d t0) (lift h d t3)))))).(\lambda (h: nat).(\lambda (d: +nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) +(THead (Bind b) u1 t0))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t3))))) (eq_ind_r T (THead (Bind b) +(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t0)) +(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))))) (eq_ind_r T (THead +(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O) +O v2) t3))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead +(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) +t0))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O +v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t3)) (\lambda (t: T).(pr0 (THead +(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift +h (s (Bind b) (s (Flat Appl) d)) t0))) (THead (Bind b) (lift h d u2) t))) +(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h +d v1) (THead (Bind b) (lift h d u1) (lift h n t0))) (THead (Bind b) (lift h d +u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t3))))) +(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat +Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d) +t0))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) +d) t3))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d +u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t0) (lift h (plus (S O) d) +t3) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d +v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b) +d) (THead (Flat Appl) (lift (S O) O v2) t3)) (lift_head (Flat Appl) (lift (S +O) O v2) t3 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t3))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t3) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t0)) +(lift_head (Bind b) u1 t0 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) +v1 (THead (Bind b) u1 t0))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t0) +h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 +u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) +(lift h d u2)))))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 t0 +t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0) +(lift h d t3)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t3 w)).(\lambda +(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift +h (s (Bind Abbr) d) t0)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) +u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr) +d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind +Abbr) d) t0)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S +d) t0) (lift h (S d) t3) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in +(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) +(lift h d' t3) (lift h d' w))) (subst0_lift_lt t3 w u2 O H4 (S d) (lt_le_S O +(S d) (le_lt_n_Sm O d (le_O_n d))) h) d (eq_ind nat d (\lambda (n: nat).(eq +nat n d)) (refl_equal nat d) (minus d O) (minus_n_O d))))) (lift h d (THead +(Bind Abbr) u2 w)) (lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind +Abbr) u1 t0)) (lift_head (Bind Abbr) u1 t0 h d)))))))))))))) (\lambda (b: +B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t3: +T).(\lambda (_: (pr0 t0 t3)).(\lambda (H2: ((\forall (h: nat).(\forall (d: +nat).(pr0 (lift h d t0) (lift h d t3)))))).(\lambda (u: T).(\lambda (h: +nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s +(Bind b) d) (lift (S O) O t0))) (\lambda (t: T).(pr0 t (lift h d t3))) +(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Bind b) (lift h d +u) (lift h n (lift (S O) O t0))) (lift h d t3))) (eq_ind_r T (lift (S O) O +(lift h d t0)) (\lambda (t: T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d +t3))) (pr0_zeta b H0 (lift h d t0) (lift h d t3) (H2 h d) (lift h d u)) (lift +h (plus (S O) d) (lift (S O) O t0)) (lift_d t0 h (S O) d O (le_O_n d))) (S d) +(refl_equal nat (S d))) (lift h d (THead (Bind b) u (lift (S O) O t0))) +(lift_head (Bind b) u (lift (S O) O t0) h d))))))))))) (\lambda (t0: +T).(\lambda (t3: T).(\lambda (_: (pr0 t0 t3)).(\lambda (H1: ((\forall (h: +nat).(\forall (d: nat).(pr0 (lift h d t0) (lift h d t3)))))).(\lambda (u: +T).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Flat Cast) (lift h +d u) (lift h (s (Flat Cast) d) t0)) (\lambda (t: T).(pr0 t (lift h d t3))) +(pr0_epsilon (lift h (s (Flat Cast) d) t0) (lift h d t3) (H1 h d) (lift h d +u)) (lift h d (THead (Flat Cast) u t0)) (lift_head (Flat Cast) u t0 h +d))))))))) t1 t2 H))). + +theorem pr0_subst0_back: + \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 +i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: +T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) +\def + \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda +(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T +(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3))))))))) +(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 +v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: +T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) +(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u0: T).(\lambda +(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: +((\forall (u3: T).((pr0 u3 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t)) +(\lambda (t: T).(pr0 t u0))))))).(\lambda (t: T).(\lambda (k: K).(\lambda +(u3: T).(\lambda (H2: (pr0 u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u3 +u1 t0)) (\lambda (t0: T).(pr0 t0 u0)) (ex2 T (\lambda (t0: T).(subst0 i0 u3 +(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u0 t)))) (\lambda (x: +T).(\lambda (H3: (subst0 i0 u3 u1 x)).(\lambda (H4: (pr0 x u0)).(ex_intro2 T +(\lambda (t0: T).(subst0 i0 u3 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 +(THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i0 H3 t k) (pr0_comp x u0 +H4 t t (pr0_refl t) k))))) (H1 u3 H2)))))))))))) (\lambda (k: K).(\lambda (v: +T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (_: (subst0 +(s k i0) v t3 t0)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T +(\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t +t0))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind +T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t t0)) (ex2 +T (\lambda (t: T).(subst0 i0 u1 (THead k u t3) t)) (\lambda (t: T).(pr0 t +(THead k u t0)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t3 +x)).(\lambda (H4: (pr0 x t0)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 +(THead k u t3) t)) (\lambda (t: T).(pr0 t (THead k u t0))) (THead k u x) +(subst0_snd k u1 x t3 i0 H3 u) (pr0_comp u u (pr0_refl u) x t0 H4 k))))) (H1 +u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda +(i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: ((\forall (u3: +T).((pr0 u3 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t)) (\lambda (t: +T).(pr0 t u0))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: +T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (u3: +T).((pr0 u3 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda +(t: T).(pr0 t t3))))))).(\lambda (u3: T).(\lambda (H4: (pr0 u3 v)).(ex2_ind T +(\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda (t: T).(pr0 t t3)) (ex2 T +(\lambda (t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 t +(THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u3 t0 +x)).(\lambda (H6: (pr0 x t3)).(ex2_ind T (\lambda (t: T).(subst0 i0 u3 u1 t)) +(\lambda (t: T).(pr0 t u0)) (ex2 T (\lambda (t: T).(subst0 i0 u3 (THead k u1 +t0) t)) (\lambda (t: T).(pr0 t (THead k u0 t3)))) (\lambda (x0: T).(\lambda +(H7: (subst0 i0 u3 u1 x0)).(\lambda (H8: (pr0 x0 u0)).(ex_intro2 T (\lambda +(t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 t (THead k u0 +t3))) (THead k x0 x) (subst0_both u3 u1 x0 i0 H7 k t0 x H5) (pr0_comp x0 u0 +H8 x t3 H6 k))))) (H1 u3 H4))))) (H3 u3 H4))))))))))))))) i u2 t1 t2 H))))). + +theorem pr0_subst0_fwd: + \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 +i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: +T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) +\def + \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda +(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T +(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4))))))))) +(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v +u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: +T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) +(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u0: T).(\lambda +(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: +((\forall (u3: T).((pr0 v u3) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t)) +(\lambda (t: T).(pr0 u0 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda +(u3: T).(\lambda (H2: (pr0 v u3)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u3 +u1 t0)) (\lambda (t0: T).(pr0 u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u3 +(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u0 t) t0))) (\lambda (x: +T).(\lambda (H3: (subst0 i0 u3 u1 x)).(\lambda (H4: (pr0 u0 x)).(ex_intro2 T +(\lambda (t0: T).(subst0 i0 u3 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 +(THead k u0 t) t0)) (THead k x t) (subst0_fst u3 x u1 i0 H3 t k) (pr0_comp u0 +x H4 t t (pr0_refl t) k))))) (H1 u3 H2)))))))))))) (\lambda (k: K).(\lambda +(v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (_: +(subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to +(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t0 +t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind +T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 +T (\lambda (t: T).(subst0 i0 u1 (THead k u t3) t)) (\lambda (t: T).(pr0 +(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t3 +x)).(\lambda (H4: (pr0 t0 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 +(THead k u t3) t)) (\lambda (t: T).(pr0 (THead k u t0) t)) (THead k u x) +(subst0_snd k u1 x t3 i0 H3 u) (pr0_comp u u (pr0_refl u) t0 x H4 k))))) (H1 +u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda +(i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: ((\forall (u3: +T).((pr0 v u3) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t)) (\lambda (t: +T).(pr0 u0 t))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: +T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (u3: +T).((pr0 v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda +(t: T).(pr0 t3 t))))))).(\lambda (u3: T).(\lambda (H4: (pr0 v u3)).(ex2_ind T +(\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T +(\lambda (t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 (THead +k u0 t3) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u3 t0 +x)).(\lambda (H6: (pr0 t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u3 u1 t)) +(\lambda (t: T).(pr0 u0 t)) (ex2 T (\lambda (t: T).(subst0 i0 u3 (THead k u1 +t0) t)) (\lambda (t: T).(pr0 (THead k u0 t3) t))) (\lambda (x0: T).(\lambda +(H7: (subst0 i0 u3 u1 x0)).(\lambda (H8: (pr0 u0 x0)).(ex_intro2 T (\lambda +(t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 (THead k u0 t3) +t)) (THead k x0 x) (subst0_both u3 u1 x0 i0 H7 k t0 x H5) (pr0_comp u0 x0 H8 +t3 x H6 k))))) (H1 u3 H4))))) (H3 u3 H4))))))))))))))) i u2 t1 t2 H))))). + +theorem pr0_subst0: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall +(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 +v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t2 w2)))))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 +t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0 +w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: +nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1 +v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd +v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: +(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 +u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2 +w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 +t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 +t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: +nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2: +T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1 +(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: +T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) +(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) +(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4)) +(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k +u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3 +t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq +T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 +(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 +(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3) +(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t +w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2) +(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) +(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead +k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda +(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 +(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T +(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind +T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 +(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3) +w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: +T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror +(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x +t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 +(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k) +(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7)))) +H6)) (\lambda (H6: (ex2 T (\lambda (t5: T).(eq T w1 (THead k u1 t5))) +(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq +T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0 +w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 +(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k +u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind +(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s k +i) v2 t4 w2))) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) +w2)))) (\lambda (H9: (pr0 x t4)).(or_introl (pr0 (THead k u1 x) (THead k u2 +t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2))) (pr0_comp u1 u2 H0 x t4 H9 k))) +(\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s +k i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: +T).(subst0 (s k i) v2 t4 w2)) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 +(THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x x0)).(\lambda +(H11: (subst0 (s k i) v2 t4 x0)).(or_intror (pr0 (THead k u1 x) (THead k u2 +t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 +(THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) (THead +k u2 x0) (pr0_comp u1 u2 H0 x x0 H10 k) (subst0_snd k v2 x0 t4 i H11 u2)))))) +H9)) (H3 v1 x (s k i) H8 v2 H5)) w1 H7)))) H6)) (\lambda (H6: (ex3_2 T T +(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s k i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda +(t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 +i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) +(or (pr0 w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda +(w2: T).(subst0 i v2 (THead k u2 t4) w2)))) 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+x5) (s (Bind b) i) (subst0_lift_ge_s v2 x5 v3 i H25 O (le_O_n i) b) (Flat +Appl) t4 x H19))))))) H23)) (H2 v0 x0 i H11 v3 H8))))) H20)) (H4 v0 x2 (s +(Flat Appl) i) H15 v3 H8))))) H17)) (H6 v0 x3 (s (Bind b) (s (Flat Appl) i)) +H16 v3 H8)) x1 H14) w1 H10)))))) H13)) (subst0_gen_head (Bind b) v0 u1 t3 x1 +(s (Flat Appl) i) H12))))))) H9)) (subst0_gen_head (Flat Appl) v0 v1 (THead +(Bind b) u1 t3) w1 i H7)))))))))))))))))))))) (\lambda (u1: T).(\lambda (u2: +T).(\lambda (H0: (pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: +T).(\forall (i: nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) +\to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 u2 w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda +(H2: (pr0 t3 t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall +(i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 +w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 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x1) (pr0_delta u1 u2 +H0 x x0 H11 x1 H13) (subst0_snd (Bind Abbr) v2 x1 w i H14 u2)))))) +(subst0_confluence_neq t4 x0 v2 (s (Bind Abbr) i) H12 w u2 O H4 (sym_not_eq +nat O (S i) (O_S i))))))) H10)) (H3 v1 x (s (Bind Abbr) i) H9 v2 H6)) w1 +H8)))) H7)) (\lambda (H7: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T +w1 (THead (Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 +u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 +t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead +(Bind Abbr) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) +(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abbr) i) v1 t3 t5))) (or +(pr0 w1 (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) +(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H8: (eq T w1 (THead (Bind Abbr) x0 +x1))).(\lambda (H9: (subst0 i v1 u1 x0)).(\lambda (H10: (subst0 (s (Bind +Abbr) i) 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(eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w +x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind +Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x +H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) +(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 +H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: +T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 +w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind +Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: +(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind +Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead +(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 +w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 +x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind +Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x +x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0 +(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead +(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) +x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)) +(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd +(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind +Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14: +(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 +u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 +x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O +x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind +Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 +x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal +nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 +\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S +i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) +(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead +(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) +w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda +(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: +(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind +Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 +H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 +(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) +(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S +i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i +H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7)) +(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b: +B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1: +T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) +\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda +(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift +(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T +(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda +(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) +u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 +t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or +(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i +v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b) +u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T +(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda +(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 +w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: +(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u +x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0 +t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda +(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: +T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda +(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b) +i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 +w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: +(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift +(S O) O t3) x)).(ex2_ind T (\lambda (t5: T).(eq T x (lift (S O) O t5))) +(\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1 +t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda +(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(eq_ind_r T (THead (Bind +b) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) +(\lambda (w2: T).(subst0 i v2 t4 w2))))) (eq_ind_r T (lift (S O) O x0) +(\lambda (t: T).(or (pr0 (THead (Bind b) u t) t4) (ex2 T (\lambda (w2: +T).(pr0 (THead (Bind b) u t) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) +(let H10 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n v1 t3 +x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: T).(pr0 +x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind b) u +(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift +(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H11: (pr0 +x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H11 u))) (\lambda (H11: (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 +t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda +(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H12: (pr0 x0 +x1)).(\lambda (H13: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u +(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift +(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H12 u) H13))))) H11)) (H2 v1 +x0 i H10 v2 H4))) x H8) w1 H6)))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) +(S O) O H7 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) +(le_lt_n_Sm O i (le_O_n i)))))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda +(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda +(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T +(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0 +x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i) +v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O +t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or +(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i +v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O +x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(eq_ind_r +T (THead (Bind b) x0 x1) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (eq_ind_r T (lift (S +O) O x) (\lambda (t: T).(or (pr0 (THead (Bind b) x0 t) t4) (ex2 T (\lambda +(w2: T).(pr0 (THead (Bind b) x0 t) w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n +v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 T (\lambda (w2: +T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind +b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 +(lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H12: +(pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H12 x0))) (\lambda (H12: (ex2 T +(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))).(ex2_ind +T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) (or (pr0 +(THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead +(Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) +(\lambda (x2: T).(\lambda (H13: (pr0 x x2)).(\lambda (H14: (subst0 i v2 t4 +x2)).(or_intror (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda +(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind b) +x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) x2 (pr0_zeta +b H0 x x2 H13 x0) H14))))) H12)) (H2 v1 x i H11 v2 H4))) x1 H9) w1 H6)))) +(subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S O) O H8 (le_S_n (S O) (S i) +(lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n +i)))))))))))) H5)) (subst0_gen_head (Bind b) v1 u (lift (S O) O t3) w1 i +H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 +t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 +t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: +nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) w1)).(\lambda (v2: +T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u2: T).(eq T w1 +(THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u u2))) (ex2 T +(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 +(s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat +Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) +(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: (ex2 T (\lambda (u2: +T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u +u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) +(\lambda (u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: +T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: +T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda (_: (subst0 i +v1 u x)).(eq_ind_r T (THead (Flat Cast) x t3) (\lambda (t: T).(or (pr0 t t4) +(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) +(or_introl (pr0 (THead (Flat Cast) x t3) t4) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Cast) x t3) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) +(pr0_epsilon t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t5: +T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat +Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) +u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5)) (or (pr0 w1 t4) +(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) +(\lambda (x: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) u x))).(\lambda +(H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T (THead (Flat Cast) u x) +(\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda +(w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: +T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or +(pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat +Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H7: (pr0 x +t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) +(pr0_epsilon x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 x w2)) +(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda +(w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or +(pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat +Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: +T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4 +x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) +(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)) x0 (pr0_epsilon x x0 H8 u) H9))))) H7)) (H1 v1 x (s +(Flat Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda +(u2: T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: +T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0 +x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast) +i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0 +t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda +(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0 +x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda +(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0 +(THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) +x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_epsilon x1 t4 H8 +x0))) (\lambda (H8: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: +T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 +w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or (pr0 (THead +(Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) +w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: +(pr0 x1 x)).(\lambda (H10: (subst0 (s (Flat Cast) i) v2 t4 x)).(or_intror +(pr0 (THead (Flat Cast) x0 x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat +Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 t4 w2)) x (pr0_epsilon x1 x H9 x0) H10))))) H8)) (H1 v1 x1 (s (Flat +Cast) i) H7 v2 H3)) w1 H5)))))) H4)) (subst0_gen_head (Flat Cast) v1 u t3 w1 +i H2))))))))))))) t1 t2 H))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/subst1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/subst1.ma new file mode 100644 index 000000000..ec55d79e1 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/subst1.ma @@ -0,0 +1,95 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/subst1". + +include "pr0/props.ma". + +include "subst1/defs.ma". + +theorem pr0_delta1: + \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall +(t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead +(Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w))))))))) +\def + \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1: +(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind +Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind +Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H +t1 t2 H0 t0 H2))) w H1)))))))). + +theorem pr0_subst1_back: + \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 +i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) +\def + \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda +(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: +T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda +(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2 +T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1 +(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 +i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda +(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda +(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x +H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))). + +theorem pr0_subst1_fwd: + \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 +i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) +\def + \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda +(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: +T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda +(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2 +T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1 +(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 +i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda +(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda +(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t: +T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x +H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))). + +theorem pr0_subst1: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall +(w1: T).(\forall (i: nat).((subst1 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 +v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2 +w2))))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1: +T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1 +w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to +(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)))))) +(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0 +t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2)))) +(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda +(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) +(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) +(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2 +T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3 +(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2)) +(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0 +w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0 +w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4: +(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2: +T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i +v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/defs.ma new file mode 100644 index 000000000..2b5d1c3c0 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/defs.ma @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/defs". + +include "pr0/defs.ma". + +inductive pr1: T \to (T \to Prop) \def +| pr1_r: \forall (t: T).(pr1 t t) +| pr1_u: \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: +T).((pr1 t2 t3) \to (pr1 t1 t3))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/pr1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/pr1.ma new file mode 100644 index 000000000..d0efa9702 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/pr1.ma @@ -0,0 +1,64 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/pr1". + +include "pr1/props.ma". + +include "pr0/pr0.ma". + +theorem pr1_strip: + \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0 +t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) +\def + \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda +(t: T).(\lambda (t2: T).(\forall (t3: T).((pr0 t t3) \to (ex2 T (\lambda (t4: +T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda +(t2: T).(\lambda (H0: (pr0 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) +(\lambda (t3: T).(pr1 t2 t3)) t2 (pr1_pr0 t t2 H0) (pr1_r t2))))) (\lambda +(t2: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda +(_: (pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr0 t2 t5) \to (ex2 T +(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: +T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda +(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 +t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 +x)).(ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 x t)) (ex2 T +(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x0: +T).(\lambda (H6: (pr1 t4 x0)).(\lambda (H7: (pr1 x x0)).(ex_intro2 T (\lambda +(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 H6 (pr1_u x t5 H4 x0 +H7))))) (H2 x H5))))) (pr0_confluence t3 t5 H3 t2 H0)))))))))) t0 t1 H))). + +theorem pr1_confluence: + \forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0 +t2) \to (ex2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t))))))) +\def + \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr1 t0 t1)).(pr1_ind (\lambda +(t: T).(\lambda (t2: T).(\forall (t3: T).((pr1 t t3) \to (ex2 T (\lambda (t4: +T).(pr1 t2 t4)) (\lambda (t4: T).(pr1 t3 t4))))))) (\lambda (t: T).(\lambda +(t2: T).(\lambda (H0: (pr1 t t2)).(ex_intro2 T (\lambda (t3: T).(pr1 t t3)) +(\lambda (t3: T).(pr1 t2 t3)) t2 H0 (pr1_r t2))))) (\lambda (t2: T).(\lambda +(t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_: (pr1 t2 +t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda (t: +T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda +(H3: (pr1 t3 t5)).(ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 +t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) +(\lambda (x: T).(\lambda (H4: (pr1 t5 x)).(\lambda (H5: (pr1 t2 x)).(ex2_ind +T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 x t)) (ex2 T (\lambda (t: +T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x0: T).(\lambda (H6: +(pr1 t4 x0)).(\lambda (H7: (pr1 x x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 +t)) (\lambda (t: T).(pr1 t5 t)) x0 H6 (pr1_t x t5 H4 x0 H7))))) (H2 x H5))))) +(pr1_strip t3 t5 H3 t2 H0)))))))))) t0 t1 H))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/props.ma new file mode 100644 index 000000000..7b7fa2331 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/props.ma @@ -0,0 +1,62 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr1/props". + +include "pr1/defs.ma". + +theorem pr1_pr0: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr1_u t2 t1 H t2 +(pr1_r t2)))). + +theorem pr1_t: + \forall (t2: T).(\forall (t1: T).((pr1 t1 t2) \to (\forall (t3: T).((pr1 t2 +t3) \to (pr1 t1 t3))))) +\def + \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (t3: T).((pr1 t0 t3) \to (pr1 t t3))))) +(\lambda (t: T).(\lambda (t3: T).(\lambda (H0: (pr1 t t3)).H0))) (\lambda +(t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda +(_: (pr1 t0 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t4 t5) \to (pr1 t0 +t5))))).(\lambda (t5: T).(\lambda (H3: (pr1 t4 t5)).(pr1_u t0 t3 H0 t5 (H2 t5 +H3)))))))))) t1 t2 H))). + +theorem pr1_head_1: + \forall (u1: T).(\forall (u2: T).((pr1 u1 u2) \to (\forall (t: T).(\forall +(k: K).(pr1 (THead k u1 t) (THead k u2 t)))))) +\def + \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr1 u1 u2)).(\lambda (t: +T).(\lambda (k: K).(pr1_ind (\lambda (t0: T).(\lambda (t1: T).(pr1 (THead k +t0 t) (THead k t1 t)))) (\lambda (t0: T).(pr1_r (THead k t0 t))) (\lambda +(t2: T).(\lambda (t1: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t3: T).(\lambda +(_: (pr1 t2 t3)).(\lambda (H2: (pr1 (THead k t2 t) (THead k t3 t))).(pr1_u +(THead k t2 t) (THead k t1 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k) (THead k +t3 t) H2))))))) u1 u2 H))))). + +theorem pr1_head_2: + \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (u: T).(\forall +(k: K).(pr1 (THead k u t1) (THead k u t2)))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(\lambda (u: +T).(\lambda (k: K).(pr1_ind (\lambda (t: T).(\lambda (t0: T).(pr1 (THead k u +t) (THead k u t0)))) (\lambda (t: T).(pr1_r (THead k u t))) (\lambda (t0: +T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t0)).(\lambda (t4: T).(\lambda (_: +(pr1 t0 t4)).(\lambda (H2: (pr1 (THead k u t0) (THead k u t4))).(pr1_u (THead +k u t0) (THead k u t3) (pr0_comp u u (pr0_refl u) t3 t0 H0 k) (THead k u t4) +H2))))))) t1 t2 H))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/clen.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/clen.ma new file mode 100644 index 000000000..00dda8542 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/clen.ma @@ -0,0 +1,182 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/clen". + +include "pr2/props.ma". + +include "clen/getl.ma". + +theorem pr2_gen_ctail: + \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall +(t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_: +T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 +(clen c) u t t2))))))))) +\def + \lambda (k: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (H: (pr2 (CTail k u c) t1 t2)).(insert_eq C (CTail k u c) +(\lambda (c0: C).(pr2 c0 t1 t2)) (or (pr2 c t1 t2) (ex3 T (\lambda (_: T).(eq +K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 (clen +c) u t t2)))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda +(c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CTail k u c)) \to (or +(pr2 c t t0) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t3: +T).(pr0 t t3)) (\lambda (t3: T).(subst0 (clen c) u t3 t0)))))))) (\lambda +(c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda +(_: (eq C c0 (CTail k u c))).(or_introl (pr2 c t3 t4) (ex3 T (\lambda (_: +T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(subst0 +(clen c) u t t4))) (pr2_free c t3 t4 H1))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind +Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 +t4)).(\lambda (t: T).(\lambda (H3: (subst0 i u0 t4 t)).(\lambda (H4: (eq C c0 +(CTail k u c))).(let H5 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead +d (Bind Abbr) u0))) H1 (CTail k u c) H4) in (let H_x \def (getl_gen_tail k +Abbr u u0 d c i H5) in (let H6 \def H_x in (or_ind (ex2 C (\lambda (e: C).(eq +C d (CTail k u e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0)))) +(ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k +(Bind Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort +n)))) (or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda +(t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda +(H7: (ex2 C (\lambda (e: C).(eq C d (CTail k u e))) (\lambda (e: C).(getl i c +(CHead e (Bind Abbr) u0))))).(ex2_ind C (\lambda (e: C).(eq C d (CTail k u +e))) (\lambda (e: C).(getl i c (CHead e (Bind Abbr) u0))) (or (pr2 c t3 t) +(ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) +(\lambda (t0: T).(subst0 (clen c) u t0 t)))) (\lambda (x: C).(\lambda (_: (eq +C d (CTail k u x))).(\lambda (H9: (getl i c (CHead x (Bind Abbr) +u0))).(or_introl (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) +(\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) +(pr2_delta c x u0 i H9 t3 t4 H2 t H3))))) H7)) (\lambda (H7: (ex4 nat +(\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind +Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: nat).(eq C d (CSort +n))))).(ex4_ind nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: +nat).(eq K k (Bind Abbr))) (\lambda (_: nat).(eq T u u0)) (\lambda (n: +nat).(eq C d (CSort n))) (or (pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k +(Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) +u t0 t)))) (\lambda (x0: nat).(\lambda (H8: (eq nat i (clen c))).(\lambda +(H9: (eq K k (Bind Abbr))).(\lambda (H10: (eq T u u0)).(\lambda (_: (eq C d +(CSort x0))).(let H12 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u0 t4 +t)) H3 (clen c) H8) in (let H13 \def (eq_ind_r T u0 (\lambda (t0: T).(subst0 +(clen c) t0 t4 t)) H12 u H10) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or +(pr2 c t3 t) (ex3 T (\lambda (_: T).(eq K k0 (Bind Abbr))) (\lambda (t0: +T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))))) (or_intror (pr2 +c t3 t) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: +T).(pr0 t3 t0)) (\lambda (t0: T).(subst0 (clen c) u t0 t))) (ex3_intro T +(\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 t3 t0)) +(\lambda (t0: T).(subst0 (clen c) u t0 t)) t4 (refl_equal K (Bind Abbr)) H2 +H13)) k H9)))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))). + +theorem pr2_gen_cbind: + \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall +(t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) +(THead (Bind b) v t2))))))) +\def + \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(let H0 \def (match H +in pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: +(pr2 c0 t t0)).((eq C c0 (CHead c (Bind b) v)) \to ((eq T t t1) \to ((eq T t0 +t2) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))))) with +[(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Bind b) +v))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead +c (Bind b) v) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 +t3) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))) (\lambda (H4: +(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to +(pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))) (\lambda (H5: (eq T t3 +t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) v +t1) (THead (Bind b) v t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead +(Bind b) v t1) (THead (Bind b) v t2) (pr0_comp v v (pr0_refl v) t1 t2 H6 +(Bind b)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 +(CHead c (Bind b) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2) +\Rightarrow (\lambda (H3: (eq C c0 (CHead c (Bind b) v))).(\lambda (H4: (eq T +t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) v) (\lambda +(c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) +u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead (Bind b) v t1) +(THead (Bind b) v t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 +(\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Bind b) v) (CHead d +(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead +(Bind b) v t1) (THead (Bind b) v t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind +T t2 (\lambda (t4: T).((getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u)) +\to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c (THead (Bind b) v t1) +(THead (Bind b) v t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) v) (CHead +d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 +t2)).(let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) v i H8) in (let +H11 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d (Bind Abbr) u) +(CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda +(j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead (Bind b) v t1) +(THead (Bind b) v t2)) (\lambda (H12: (land (eq nat i O) (eq C (CHead d (Bind +Abbr) u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C (CHead d (Bind +Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v +t2)) (\lambda (H13: (eq nat i O)).(\lambda (H14: (eq C (CHead d (Bind Abbr) +u) (CHead c (Bind b) v))).(let H15 \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) v) H14) in ((let +H16 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) +with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K +return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) +\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H14) in +((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead +d (Bind Abbr) u) (CHead c (Bind b) v) H14) in (\lambda (H18: (eq B Abbr +b)).(\lambda (_: (eq C d c)).(let H20 \def (eq_ind nat i (\lambda (n: +nat).(subst0 n u t3 t2)) H10 O H13) in (let H21 \def (eq_ind T u (\lambda +(t4: T).(subst0 O t4 t3 t2)) H20 v H17) in (eq_ind B Abbr (\lambda (b0: +B).(pr2 c (THead (Bind b0) v t1) (THead (Bind b0) v t2))) (pr2_free c (THead +(Bind Abbr) v t1) (THead (Bind Abbr) v t2) (pr0_delta v v (pr0_refl v) t1 t3 +H9 t2 H21)) b H18)))))) H16)) H15)))) H12)) (\lambda (H12: (ex2 nat (\lambda +(j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) +u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: +nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) v t1) (THead +(Bind b) v t2)) (\lambda (x: nat).(\lambda (H13: (eq nat i (S x))).(\lambda +(H14: (getl x c (CHead d (Bind Abbr) u))).(let H15 \def (f_equal nat nat +(\lambda (e: nat).e) i (S x) H13) in (let H16 \def (eq_ind nat i (\lambda (n: +nat).(subst0 n u t3 t2)) H10 (S x) H15) in (pr2_head_2 c v t1 t2 (Bind b) +(pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead c (Bind +b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H14) t1 t3 H9 t2 +H16))))))) H12)) H11)))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 +(sym_eq C c0 (CHead c (Bind b) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal +C (CHead c (Bind b) v)) (refl_equal T t1) (refl_equal T t2)))))))). + +theorem pr2_gen_cflat: + \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall +(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2)))))) +\def + \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(let H0 \def (match H +in pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: +(pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) v)) \to ((eq T t t1) \to ((eq T t0 +t2) \to (pr2 c t1 t2)))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow +(\lambda (H1: (eq C c0 (CHead c (Flat f) v))).(\lambda (H2: (eq T t0 +t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (_: +C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c t1 t2))))) +(\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to +((pr0 t t3) \to (pr2 c t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 +(\lambda (t: T).((pr0 t1 t) \to (pr2 c t1 t2))) (\lambda (H6: (pr0 t1 +t2)).(pr2_free c t1 t2 H6)) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) +c0 (sym_eq C c0 (CHead c (Flat f) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i +H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) +v))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead +c (Flat f) v) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 +(CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c +t1 t2))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T +t t2) \to ((getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t4 +t3) \to ((subst0 i u t3 t) \to (pr2 c t1 t2)))))) (\lambda (H7: (eq T t +t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) v) (CHead d +(Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c t1 +t2))))) (\lambda (H8: (getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) +u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 t2)).(let H_y +\def (getl_gen_flat f c (CHead d (Bind Abbr) u) v i H8) in (pr2_delta c d u i +H_y t1 t3 H9 t2 H10))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 +(sym_eq C c0 (CHead c (Flat f) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal +C (CHead c (Flat f) v)) (refl_equal T t1) (refl_equal T t2)))))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/defs.ma new file mode 100644 index 000000000..8466ceef0 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/defs.ma @@ -0,0 +1,30 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/defs". + +include "pr0/defs.ma". + +include "getl/defs.ma". + +inductive pr2: C \to (T \to (T \to Prop)) \def +| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to +(pr2 c t1 t2)))) +| pr2_delta: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: +nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: +T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to (pr2 c t1 +t)))))))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/fwd.ma new file mode 100644 index 000000000..222786926 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/fwd.ma @@ -0,0 +1,3634 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/fwd". + +include "pr2/defs.ma". + +include "pr0/fwd.ma". + +include "getl/drop.ma". + +include "getl/clear.ma". + +theorem pr2_gen_sort: + \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to +(eq T x (TSort n))))) +\def + \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort +n) x)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t +(TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n))))))))) with [(pr2_free c0 +t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 +(TSort n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 +(TSort n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort n)))))) +(\lambda (H4: (eq T t1 (TSort n))).(eq_ind T (TSort n) (\lambda (t: T).((eq T +t2 x) \to ((pr0 t t2) \to (eq T x (TSort n))))) (\lambda (H5: (eq T t2 +x)).(eq_ind T x (\lambda (t: T).((pr0 (TSort n) t) \to (eq T x (TSort n)))) +(\lambda (H6: (pr0 (TSort n) x)).(let H7 \def (eq_ind T x (\lambda (t: +T).(pr2 c (TSort n) t)) H (TSort n) (pr0_gen_sort x n H6)) in (eq_ind_r T +(TSort n) (\lambda (t: T).(eq T t (TSort n))) (refl_equal T (TSort n)) x +(pr0_gen_sort x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1 (TSort n) +H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t +H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 (TSort +n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1 (TSort +n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) +\to ((subst0 i u t2 t) \to (eq T x (TSort n)))))))) (\lambda (H6: (eq T t1 +(TSort n))).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t x) \to ((getl i c +(CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (eq T x +(TSort n))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: T).((getl +i c (CHead d (Bind Abbr) u)) \to ((pr0 (TSort n) t2) \to ((subst0 i u t2 t0) +\to (eq T x (TSort n)))))) (\lambda (_: (getl i c (CHead d (Bind Abbr) +u))).(\lambda (H9: (pr0 (TSort n) t2)).(\lambda (H10: (subst0 i u t2 x)).(let +H11 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 x)) H10 (TSort n) +(pr0_gen_sort t2 n H9)) in (subst0_gen_sort u x i n H11 (eq T x (TSort +n))))))) t (sym_eq T t x H7))) t1 (sym_eq T t1 (TSort n) H6))) c0 (sym_eq C +c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (TSort +n)) (refl_equal T x)))))). + +theorem pr2_gen_lref: + \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to +(or (eq T x (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 x (lift (S +n) O u))))))))) +\def + \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef +n) x)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t +(TLRef n)) \to ((eq T t0 x) \to (or (eq T x (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 x (lift (S n) O u))))))))))))) with [(pr2_free c0 t1 +t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TLRef +n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TLRef +n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (or (eq T x (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 x (lift (S n) O u)))))))))) (\lambda (H4: (eq T +t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t: T).((eq T t2 x) \to ((pr0 t +t2) \to (or (eq T x (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 +x (lift (S n) O u))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda +(t: T).((pr0 (TLRef n) t) \to (or (eq T x (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 x (lift (S n) O u)))))))) (\lambda (H6: (pr0 (TLRef +n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TLRef n) t)) H (TLRef +n) (pr0_gen_lref x n H6)) in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T +t (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 t (lift (S n) O +u))))))) (or_introl (eq T (TLRef n) (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 (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef +n))) x (pr0_gen_lref x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1 +(TLRef n) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 +t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 +(TLRef n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1 +(TLRef n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 +t1 t2) \to ((subst0 i u t2 t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda +(d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: +C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))))))))) (\lambda (H6: (eq T +t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t x) \to ((getl i +c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or +(eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c +(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift +(S n) O u0))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: +T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (TLRef n) t2) \to ((subst0 i +u t2 t0) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: +T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: +T).(eq T x (lift (S n) O u0)))))))))) (\lambda (H8: (getl i c (CHead d (Bind +Abbr) u))).(\lambda (H9: (pr0 (TLRef n) t2)).(\lambda (H10: (subst0 i u t2 +x)).(let H11 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 x)) H10 (TLRef +n) (pr0_gen_lref t2 n H9)) in (and_ind (eq nat n i) (eq T x (lift (S n) O u)) +(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c +(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift +(S n) O u0)))))) (\lambda (H12: (eq nat n i)).(\lambda (H13: (eq T x (lift (S +n) O u))).(let H14 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead +d (Bind Abbr) u))) H8 n H12) in (let H15 \def (eq_ind T x (\lambda (t0: +T).(pr2 c (TLRef n) t0)) H (lift (S n) O u) H13) in (eq_ind_r T (lift (S n) O +u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) (ex2_2 C T (\lambda (d0: +C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: +C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (or_intror (eq T (lift +(S n) O u) (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c +(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S +n) O u) (lift (S n) O u0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: +T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: +T).(eq T (lift (S n) O u) (lift (S n) O u0)))) d u H14 (refl_equal T (lift (S +n) O u)))) x H13))))) (subst0_gen_lref u x i n H11)))))) t (sym_eq T t x +H7))) t1 (sym_eq T t1 (TLRef n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 +H2))))]) in (H0 (refl_equal C c) (refl_equal T (TLRef n)) (refl_equal T +x)))))). + +theorem pr2_gen_abst: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c +(THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 t2)))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(let H0 \def (match H in pr2 return +(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t +t0)).((eq C c0 c) \to ((eq T t (THead (Bind Abst) u1 t1)) \to ((eq T t0 x) +\to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) +u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +t1 t2))))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq +C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq +T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abst) u1 t1)) \to +((eq T t2 x) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 t3)))))))))) (\lambda (H4: (eq T t0 (THead +(Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t: T).((eq +T t2 x) \to ((pr0 t t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) t1 t3))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x +(\lambda (t: T).((pr0 (THead (Bind Abst) u1 t1) t) \to (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))))) (\lambda (H6: +(pr0 (THead (Bind Abst) u1 t1) x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind Abst) x0 +x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def +(eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead +(Bind Abst) x0 x1) H7) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t: +T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead +(Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind +Abst) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free +(CHead c (Bind b) u) t1 x1 H9)))) x H7))))))) (pr0_gen_abst u1 t1 x H6))) t2 +(sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Abst) u1 t1) H4))) c0 +(sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) +\Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind +Abst) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T +t0 (THead (Bind Abst) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind +Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))))))))) (\lambda +(H6: (eq T t0 (THead (Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) +(\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to +((pr0 t3 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda +(t4: T).(eq T x (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))))))) (\lambda (H7: (eq T t +x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to +((pr0 (THead (Bind Abst) u1 t1) t2) \to ((subst0 i u t2 t3) \to (ex3_2 T T +(\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abst) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 +t4)))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: +(pr0 (THead (Bind Abst) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 +x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: +T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (pr0 u1 +x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t3: +T).(subst0 i u t3 x)) H10 (THead (Bind Abst) x0 x1) H11) in (or3_ind (ex2 T +(\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 +i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3))) +(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Bind Abst) i) u x1 t3)))) (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (H15: (ex2 T (\lambda +(u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 +u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) +(\lambda (u2: T).(subst0 i u x0 u2)) (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda +(H16: (eq T x (THead (Bind Abst) x2 x1))).(\lambda (H17: (subst0 i u x0 +x2)).(let H18 \def (eq_ind T x (\lambda (t3: T).(pr2 c (THead (Bind Abst) u1 +t1) t3)) H (THead (Bind Abst) x2 x1) H16) in (eq_ind_r T (THead (Bind Abst) +x2 x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 +(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead +c (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x1 +(refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c d u i H8 u1 x0 H12 x2 +H17) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 +H13)))) x H16))))) H15)) (\lambda (H15: (ex2 T (\lambda (t3: T).(eq T x +(THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 +t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3))) +(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: +T).(\lambda (H16: (eq T x (THead (Bind Abst) x0 x2))).(\lambda (H17: (subst0 +(s (Bind Abst) i) u x1 x2)).(let H18 \def (eq_ind T x (\lambda (t3: T).(pr2 c +(THead (Bind Abst) u1 t1) t3)) H (THead (Bind Abst) x0 x2) H16) in (eq_ind_r +T (THead (Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))) (ex3_2_intro +T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead +(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead +c (Bind b) u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) +(pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c +(Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) +t1 x1 H13 x2 H17)))) x H16))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Bind Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: +T).(\lambda (H16: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (H17: (subst0 +i u x0 x2)).(\lambda (H18: (subst0 (s (Bind Abst) i) u x1 x3)).(let H19 \def +(eq_ind T x (\lambda (t3: T).(pr2 c (THead (Bind Abst) u1 t1) t3)) H (THead +(Bind Abst) x2 x3) H16) in (eq_ind_r T (THead (Bind Abst) x2 x3) (\lambda +(t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind +Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x3 +(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c d u i H8 u1 x0 H12 x2 +H17) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S +i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3 +H18)))) x H16))))))) H15)) (subst0_gen_head (Bind Abst) u x0 x1 x i +H14)))))))) (pr0_gen_abst u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T +t0 (THead (Bind Abst) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) +in (H0 (refl_equal C c) (refl_equal T (THead (Bind Abst) u1 t1)) (refl_equal +T x))))))). + +theorem pr2_gen_cast: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c +(THead (Flat Cast) u1 t1) x) \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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c +t1 x)))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(let H0 \def (match H in pr2 return +(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t +t0)).((eq C c0 c) \to ((eq T t (THead (Flat Cast) u1 t1)) \to ((eq T t0 x) +\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).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c t1 x))))))))) with [(pr2_free c0 +t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 +(THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda +(_: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) +\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat +Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))) (\lambda (H4: (eq T t0 +(THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t: +T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c +t1 x))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead +(Flat Cast) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))) +(\lambda (H6: (pr0 (THead (Flat Cast) u1 t1) x)).(or_ind (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 x) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H7: +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T +T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H8: (eq T x (THead (Flat Cast) x0 x1))).(\lambda (H9: (pr0 u1 +x0)).(\lambda (H10: (pr0 t1 x1)).(let H11 \def (eq_ind T x (\lambda (t: +T).(pr2 c (THead (Flat Cast) u1 t1) t)) H (THead (Flat Cast) x0 x1) H8) in +(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 +t3)))) (pr2 c t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 +t3)))) (pr2 c t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Cast) x0 +x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8))))))) H7)) (\lambda +(H7: (pr0 t1 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) +(pr2_free c t1 x H7))) (pr0_gen_cast u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 +(sym_eq T t0 (THead (Flat Cast) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 +H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq +C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq +T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to +((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to +((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))))) +(\lambda (H6: (eq T t0 (THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat +Cast) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) +u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 +t4)))) (pr2 c t1 x))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: +T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Flat Cast) u1 t1) +t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: +T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (pr2 c t1 +x)))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 +(THead (Flat Cast) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 +t3)))) (pr2 c t1 x)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind +T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: +T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Flat Cast) x0 +x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def +(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Flat Cast) x0 +x1) H12) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 +x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x +(THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 +t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H16: (ex2 T (\lambda (u2: +T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 +u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1))) +(\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c +t1 x)) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 +x1))).(\lambda (H18: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 +t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T +x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x1 H17 (pr2_delta c +d u i H8 u1 x0 H13 x2 H18) (pr2_free c t1 x1 H14)))))) H16)) (\lambda (H16: +(ex2 T (\lambda (t3: T).(eq T x (THead (Flat Cast) x0 t3))) (\lambda (t3: +T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x +(THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 +t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat +Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda +(H17: (eq T x (THead (Flat Cast) x0 x2))).(\lambda (H18: (subst0 (s (Flat +Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 H17 (pr2_free c u1 x0 H13) +(pr2_delta c d u i H8 t1 x1 H14 x2 H18)))))) H16)) (\lambda (H16: (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (x3: +T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 x3))).(\lambda (H18: (subst0 +i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Cast) i) u x1 x3)).(or_introl +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 +t3))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8 +t1 x1 H14 x3 H19)))))))) H16)) (subst0_gen_head (Flat Cast) u x0 x1 x i +H15)))))))) H11)) (\lambda (H11: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 +t3)))) (pr2 c t1 x) (pr2_delta c d u i H8 t1 t2 H11 x H10))) (pr0_gen_cast u1 +t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Cast) u1 t1) +H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) +(refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x))))))). + +theorem pr2_gen_csort: + \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) +\to (pr0 t1 t2)))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort +n) t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c: C).(\lambda (t: +T).(\lambda (t0: T).(\lambda (_: (pr2 c t t0)).((eq C c (CSort n)) \to ((eq T +t t1) \to ((eq T t0 t2) \to (pr0 t1 t2)))))))) with [(pr2_free c t0 t3 H0) +\Rightarrow (\lambda (H1: (eq C c (CSort n))).(\lambda (H2: (eq T t0 +t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CSort n) (\lambda (_: C).((eq T +t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr0 t1 t2))))) (\lambda (H4: +(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to +(pr0 t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 +t1 t) \to (pr0 t1 t2))) (\lambda (H6: (pr0 t1 t2)).H6) t3 (sym_eq T t3 t2 +H5))) t0 (sym_eq T t0 t1 H4))) c (sym_eq C c (CSort n) H1) H2 H3 H0)))) | +(pr2_delta c d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c (CSort +n))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CSort +n) (\lambda (c0: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c0 (CHead d +(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr0 t1 t2))))))) +(\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to +((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u +t3 t) \to (pr0 t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda +(t4: T).((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to +((subst0 i u t3 t4) \to (pr0 t1 t2))))) (\lambda (H8: (getl i (CSort n) +(CHead d (Bind Abbr) u))).(\lambda (_: (pr0 t1 t3)).(\lambda (_: (subst0 i u +t3 t2)).(getl_gen_sort n i (CHead d (Bind Abbr) u) H8 (pr0 t1 t2))))) t +(sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c (sym_eq C c (CSort n) H3) H4 +H5 H0 H1 H2))))]) in (H0 (refl_equal C (CSort n)) (refl_equal T t1) +(refl_equal T t2)))))). + +theorem pr2_gen_appl: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T +T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall +(b: B).(\forall (u: T).(pr2 (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).(eq +T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(let H0 \def (match H in pr2 return +(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t +t0)).((eq C c0 c) \to ((eq T t (THead (Flat Appl) u1 t1)) \to ((eq T t0 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).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: +T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr2 (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).(eq T t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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)))))))))))))))) with [(pr2_free c0 +t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 +(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda +(_: C).((eq T t0 (THead (Flat Appl) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) +\to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))))))) (\lambda (H4: (eq T t0 +(THead (Flat Appl) u1 t1))).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t: +T).((eq T t2 x) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T +T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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)))))))))))) (\lambda +(H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Flat Appl) u1 t1) +t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))))) (\lambda (H6: (pr0 (THead +(Flat Appl) u1 t1) x)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 +u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) +(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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x +(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 +v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) z1 t3)))))))) (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 t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 +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))))))))) (\lambda (H7: (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T x (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 u1 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))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T x +(THead (Flat Appl) x0 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H10: (pr0 t1 +x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) z1 t3)))))))) (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 t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 +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_intro0 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Flat Appl) x0 x1) (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 u1 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)))))))) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 +x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8)))))) H7)) (\lambda +(H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) z1 t3)))))))) (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 t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 +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))))))))) (\lambda (x0: T).(\lambda +(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t1 (THead (Bind +Abst) x0 x1))).(\lambda (H9: (eq T x (THead (Bind Abbr) x2 x3))).(\lambda +(H10: (pr0 u1 x2)).(\lambda (H11: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) +x2 x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 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)))))))))) (eq_ind_r +T (THead (Bind Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (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 t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind +Abbr) x2 x3) (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 u1 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_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +z1 t3)))))))) (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 (THead (Bind Abst) x0 x1) (THead (Bind b) y1 +z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: +T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x2 x3) (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 u1 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)))))))) (ex4_4_intro +T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq +T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) +x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T +(THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free +c u1 x2 H10) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) +x1 x3 H11))))) t1 H8) x H9))))))))) 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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) +v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 +u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3))))))))).(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 t1 (THead (Bind b) +y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat +Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 +t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda +(x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: +T).(\lambda (x5: T).(\lambda (H8: (not (eq B x0 Abst))).(\lambda (H9: (eq T +t1 (THead (Bind x0) x1 x2))).(\lambda (H10: (eq T x (THead (Bind x0) x4 +(THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (H11: (pr0 u1 +x3)).(\lambda (H12: (pr0 x1 x4)).(\lambda (H13: (pr0 x2 x5)).(eq_ind_r T +(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t: +T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (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 u1 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)))))))))) (eq_ind_r T (THead (Bind +x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) z1 t3)))))))) (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 t (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead +(Flat Appl) (lift (S O) O x3) x5)) (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 u1 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_intro2 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) +O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead +(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) z1 t3)))))))) (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 (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (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 u1 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_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).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) +(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 u1 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))))))) +x0 x1 x2 x5 x3 x4 H8 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T +(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c u1 +x3 H11) (pr2_free c x1 x4 H12) (pr2_free (CHead c (Bind x0) x4) x2 x5 H13))) +t1 H9) x H10))))))))))))) H7)) (pr0_gen_appl u1 t1 x H6))) t2 (sym_eq T t2 x +H5))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H4))) c0 (sym_eq C c0 c H1) +H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda +(H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(\lambda +(H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Appl) u1 +t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 +t2) \to ((subst0 i u t2 t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x +(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 u1 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))))))))))))))) (\lambda (H6: (eq T t0 (THead (Flat Appl) u1 t1))).(eq_ind +T (THead (Flat Appl) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c +(CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: +T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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)))))))))))))) (\lambda (H7: (eq T t +x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to +((pr0 (THead (Flat Appl) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or3 (ex3_2 T +T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x +(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c (Bind b) u0) z1 t4)))))))) (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 t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 +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))))))))))))) (\lambda (H8: (getl i c +(CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Flat Appl) u1 t1) +t2)).(\lambda (H10: (subst0 i u t2 x)).(or3_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3)))))) (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 t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead +(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 +u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda +(t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x +(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 u1 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))))))))) +(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda +(_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda (x0: T).(\lambda +(x1: T).(\lambda (H12: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H13: +(pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def (eq_ind T t2 (\lambda +(t3: T).(subst0 i u t3 x)) H10 (THead (Flat Appl) x0 x1) H12) in (or3_ind +(ex2 T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: +T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Flat Appl) x0 +t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c (Bind b) u0) z1 t3)))))))) (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 t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 +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))))))))) (\lambda (H16: (ex2 T +(\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 +i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) +(\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda (x2: T).(\lambda (H17: (eq T x +(THead (Flat Appl) x2 x1))).(\lambda (H18: (subst0 i u x0 x2)).(eq_ind_r T +(THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 +t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind +Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t4)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T t3 (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 u1 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_intro0 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Flat Appl) x2 x1) (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 u1 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)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 +t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c d u i H8 u1 +x0 H13 x2 H18) (pr2_free c t1 x1 H14))) x H17)))) H16)) (\lambda (H16: (ex2 T +(\lambda (t3: T).(eq T x (THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 +(s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead +(Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3)) +(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda (x2: T).(\lambda +(H17: (eq T x (THead (Flat Appl) x0 x2))).(\lambda (H18: (subst0 (s (Flat +Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3: +T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat +Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: +T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (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 u1 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_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat +Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T +T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr) +u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (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 u1 +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)))))))) (ex3_2_intro T T (\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) x0 +x2)) (pr2_free c u1 x0 H13) (pr2_delta c d u i H8 t1 x1 H14 x2 H18))) x +H17)))) H16)) (\lambda (H16: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u +x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 +t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) +(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda (x2: T).(\lambda +(x3: T).(\lambda (H17: (eq T x (THead (Flat Appl) x2 x3))).(\lambda (H18: +(subst0 i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Appl) i) u x1 +x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 +(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c (Bind b) u0) z1 t4)))))))) (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 t1 (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (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 u1 +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_intro0 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda +(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (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 u1 +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)))))))) (ex3_2_intro T T (\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) x2 +x3)) (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8 t1 x1 H14 +x3 H19))) x H17)))))) H16)) (subst0_gen_head (Flat Appl) u x0 x1 x i +H15)))))))) H11)) (\lambda (H11: (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda +(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 t1 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda (x0: T).(\lambda +(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t1 (THead +(Bind Abst) x0 x1))).(\lambda (H13: (eq T t2 (THead (Bind Abbr) x2 +x3))).(\lambda (H14: (pr0 u1 x2)).(\lambda (H15: (pr0 x1 x3)).(let H16 \def +(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Bind Abbr) x2 +x3) H13) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr2 c t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: +T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 t3 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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_ind (ex2 T (\lambda +(u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 +u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3))) (\lambda +(t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda (H17: (ex2 T (\lambda (u2: T).(eq T x (THead +(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T +(\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 +i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 (THead (Bind Abst) x0 x1) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda (x4: T).(\lambda +(H18: (eq T x (THead (Bind Abbr) x4 x3))).(\lambda (H19: (subst0 i u x2 +x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) +x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) +(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 (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t3 (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 u1 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_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 (THead (Bind Abst) x0 x1) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind +Abbr) x4 x3) (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 u1 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)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1)) +(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c d u i H8 u1 x2 H14 x4 +H19) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) x1 x3 +H15))))) x H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x +(THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 +t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3))) +(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda (x4: T).(\lambda (H18: (eq T x (THead (Bind Abbr) +x2 x4))).(\lambda (H19: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T +(THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c +(THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t3 (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 u1 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_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 (THead (Bind Abst) x0 x1) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind +Abbr) x2 x4) (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 u1 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)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1)) +(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c u1 x2 H14) (\lambda (b: +B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) +(getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d +(Bind Abbr) u) i H8) x1 x3 H15 x4 H19))))) x H18)))) H17)) (\lambda (H17: +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) +x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) +(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 (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H18: (eq T x +(THead (Bind Abbr) x4 x5))).(\lambda (H19: (subst0 i u x2 x4)).(\lambda (H20: +(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5) +(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 +(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) x0 x1) t4)))) +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead +(Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c (Bind b) u0) z1 t4)))))))) (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 (THead (Bind +Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +t3 (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 u1 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_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) +(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 (THead (Bind Abst) x0 x1) (THead +(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind +Abbr) x4 x5) (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 u1 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)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1)) +(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c d u i H8 u1 x2 H14 x4 +H19) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S +i) (getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d +(Bind Abbr) u) i H8) x1 x3 H15 x5 H20))))) x H18)))))) H17)) (subst0_gen_head +(Bind Abbr) u x2 x3 x i H16)) t1 H12)))))))))) H11)) (\lambda (H11: (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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T +t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: +T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 +v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(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 t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda +(_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead +(Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda +(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: +T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) +(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 +t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 t1 (THead (Bind b) y1 z1)))))))) +(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda +(u2: T).(\lambda (y2: T).(eq T x (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 u1 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))))))))) (\lambda (x0: B).(\lambda (x1: +T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: +T).(\lambda (H12: (not (eq B x0 Abst))).(\lambda (H13: (eq T t1 (THead (Bind +x0) x1 x2))).(\lambda (H14: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) +(lift (S O) O x3) x5)))).(\lambda (H15: (pr0 u1 x3)).(\lambda (H16: (pr0 x1 +x4)).(\lambda (H17: (pr0 x2 x5)).(let H18 \def (eq_ind T t2 (\lambda (t3: +T).(subst0 i u t3 x)) H10 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) +O x3) x5)) H14) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr2 c t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: +T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 t3 +(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: +T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 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_ind (ex2 T (\lambda +(u2: T).(eq T x (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) +x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T x +(THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead +(Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) +i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T x (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 u1 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))))))))) (\lambda (H19: (ex2 T (\lambda (u2: T).(eq T x (THead +(Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: +T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind x0) +u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u +x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 +T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq +T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (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 u1 +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))))))))) (\lambda (x6: T).(\lambda +(H20: (eq T x (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) +x5)))).(\lambda (H21: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 +(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t3 (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 u1 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_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 +T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq +T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) +(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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) +O x3) x5)) (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 u1 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_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).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead +(Flat Appl) (lift (S O) O x3) x5)) (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 u1 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))))))) x0 x1 x2 x5 x3 x6 H12 (refl_equal T (THead +(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift +(S O) O x3) x5))) (pr2_free c u1 x3 H15) (pr2_delta c d u i H8 x1 x4 H16 x6 +H21) (pr2_free (CHead c (Bind x0) x6) x2 x5 H17))) x H20)))) H19)) (\lambda +(H19: (ex2 T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda (t3: +T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) +t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda +(t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) +t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead +(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: +T).(\lambda (_: T).(\lambda 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+(Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead (Flat +Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s +(Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq +T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (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 u1 +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))))))))) (\lambda (x7: T).(\lambda +(H23: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H24: +(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat +Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 +t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 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T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead +(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c (Bind b) u0) z1 t3)))))))) (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 (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (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 u1 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_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).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) +(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda +(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 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(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s +(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: +T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x4 x6) (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 u1 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))))))))) (\lambda (x7: T).(\lambda (x8: +T).(\lambda (H23: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H24: (subst0 +(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H25: (subst0 (s (Flat +Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8) +(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T +(THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c +(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (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 u1 +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)))))))))) (ex2_ind T (\lambda (t3: +T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) +i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c (Bind b) u0) z1 t3)))))))) (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 (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (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 u1 +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))))))))) (\lambda (x9: T).(\lambda +(H26: (eq T x7 (lift (S O) O x9))).(\lambda (H27: (subst0 (minus (s (Bind x0) +i) (S O)) u x3 x9)).(let H28 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) +(\lambda (n: nat).(subst0 n u x3 x9)) H27 i (s_arith1 x0 i)) in (eq_ind_r T +(lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat +Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) +(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 u1 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_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) +(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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) +O x9) x8)) (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 u1 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_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).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead +(Flat Appl) (lift (S O) O x9) x8)) (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 u1 u2))))))) +(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 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+T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) +O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 +u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat +Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) +x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T x (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 u1 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))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x +(THead (Bind x0) x6 x7))).(\lambda (H21: (subst0 i u x4 x6)).(\lambda (H22: +(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) +x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T +(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T t3 (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 u1 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_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat +Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) +u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3) +t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) +O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind +x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (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 u1 +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))))))))) (\lambda (H23: (ex2 T +(\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: +T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda +(u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s +(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x6 x7) (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 u1 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))))))))) (\lambda (x8: T).(\lambda (H24: (eq T x7 +(THead (Flat Appl) x8 x5))).(\lambda (H25: (subst0 (s (Bind x0) i) u (lift (S +O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3 +(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3) +(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 +T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq +T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x6 t3) (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 u1 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)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8 +(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u +x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind +x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c +(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind +Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead +(Flat Appl) x8 x5)) (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 u1 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))))))))) (\lambda (x9: T).(\lambda (H26: (eq T x8 (lift (S O) O +x9))).(\lambda (H27: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H28 +\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u +x3 x9)) H27 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3: +T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c +(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 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+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 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_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 +T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq +T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: 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T).(\lambda (y2: T).(pr2 +(CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H12 (refl_equal T (THead +(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift +(S O) O x9) x5))) (pr2_delta c d u i H8 u1 x3 H15 x9 H28) (pr2_delta c d u i +H8 x1 x4 H16 x6 H21) (pr2_free (CHead c (Bind x0) x6) x2 x5 H17))) x8 +H26))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H25 (le_S_n (S +O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n +i))))))) x7 H24)))) H23)) (\lambda (H23: (ex2 T (\lambda (t3: T).(eq T x7 +(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat +Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 +(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat +Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda +(u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 +c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda +(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead +(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 +t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) +(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 u1 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))))))))) +(\lambda (x8: T).(\lambda (H24: (eq T x7 (THead (Flat Appl) (lift (S O) O x3) +x8))).(\lambda (H25: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 +x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3: +T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c +u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) +t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: +T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 +z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: +T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (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 (_: 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(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).(eq T +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) +(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 u1 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))))))) +x0 x1 x2 x8 x3 x6 H12 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T +(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c u1 +x3 H15) (pr2_delta c d u i H8 x1 x4 H16 x6 H21) (pr2_delta (CHead c (Bind x0) +x6) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 +c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 H17 x8 H25))) x7 H24)))) H23)) +(\lambda (H23: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead +(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) +i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s +(Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda +(_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) +(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 +x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) +(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: +T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: +T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (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 +(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: +B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda +(y2: T).(eq T (THead (Bind x0) x6 x7) (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 u1 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))))))))) (\lambda (x8: T).(\lambda (x9: +T).(\lambda (H24: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H25: (subst0 +(s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H26: (subst0 (s (Flat +Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9) +(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T +(THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c +(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind +Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda +(t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda +(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (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 u1 +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)))))))))) (ex2_ind T (\lambda (t3: +T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) +i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: +T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 +x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda +(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) +(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: +T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 +(CHead c (Bind b) u0) z1 t3)))))))) (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 (THead (Bind +x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: +T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (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 u1 +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))))))))) (\lambda (x10: T).(\lambda +(H27: (eq T x8 (lift (S O) O x10))).(\lambda (H28: (subst0 (minus (s (Bind +x0) i) (S O)) u x3 x10)).(let H29 \def (eq_ind nat (minus (s (Bind x0) i) (S +O)) (\lambda (n: nat).(subst0 n u x3 x10)) H28 i (s_arith1 x0 i)) in +(eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: +T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) +(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 +T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq +T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) +x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) +(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) +(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 u1 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_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat +Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T +(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T +(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: +T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) +x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3)))))) +(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: +T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) +(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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda +(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: +T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) +O x10) x9)) (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 u1 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_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).(eq T (THead (Bind x0) x1 x2) (THead (Bind +b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda +(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead +(Flat Appl) (lift (S O) O x10) x9)) (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 u1 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))))))) x0 x1 x2 x9 x10 x6 H12 (refl_equal T +(THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) +(lift (S O) O x10) x9))) (pr2_delta c d u i H8 u1 x3 H15 x10 H29) (pr2_delta +c d u i H8 x1 x4 H16 x6 H21) (pr2_delta (CHead c (Bind x0) x6) d u (S i) +(getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 c x6) (CHead d +(Bind Abbr) u) i H8) x2 x5 H17 x9 H26))) x8 H27))))) (subst0_gen_lift_ge u x3 +x8 (s (Bind x0) i) (S O) O H25 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) +(lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x7 H24)))))) H23)) +(subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s (Bind x0) i) H22)) +x H20)))))) H19)) (subst0_gen_head (Bind x0) u x4 (THead (Flat Appl) (lift (S +O) O x3) x5) x i H18)) t1 H13)))))))))))))) H11)) (pr0_gen_appl u1 t1 t2 +H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H6))) +c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) +(refl_equal T (THead (Flat Appl) u1 t1)) (refl_equal T x))))))). + +theorem pr2_gen_abbr: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: +T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T +T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) +(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(let H0 \def (match H in pr2 return +(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t +t0)).((eq C c0 c) \to ((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq T t0 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).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind +b) u) t1 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead +c (Bind Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 +(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) +(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) +(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O +x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 +c)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 +x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abbr) u1 t1)) \to ((eq +T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) +(\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: +T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: +T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c +(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c +(Bind b) u) t1 (lift (S O) O x))))))))) (\lambda (H4: (eq T t0 (THead (Bind +Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t: T).((eq T t2 +x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T +x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) +(\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: +T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: +T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c +(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c +(Bind b) u) t1 (lift (S O) O x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x +(\lambda (t: T).((pr0 (THead (Bind Abbr) u1 t1) t) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 +t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind +Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c +(Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda +(_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))) +(\lambda (H6: (pr0 (THead (Bind Abbr) u1 t1) x)).(or_ind (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 +t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y +t3))))))) (pr0 t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: +T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T +T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) +(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H7: (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: +T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 +O u2 y t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 +t1 y)) (\lambda (y: T).(subst0 O u2 y t3)))))) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T +(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 +t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) +u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda +(x0: T).(\lambda (x1: T).(\lambda (H8: (eq T x (THead (Bind Abbr) x0 +x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T +(\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y x1))))).(or_ind +(pr0 t1 x1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y +x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind +b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead +c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 +(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) +(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) +(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O 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(S O) O x))))) (\lambda (x3: T).(\lambda (H18: (eq T x +(THead (Bind Abbr) x3 x1))).(\lambda (H19: (subst0 i u x0 x3)).(ex2_ind T +(\lambda (t3: T).(subst0 O u1 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind +b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 +(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: +T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 +y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z +t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 +(lift (S O) O x))))) (\lambda (x4: T).(\lambda (_: (subst0 O u1 x2 +x4)).(\lambda (_: (pr0 x4 x1)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) +t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) +u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind +b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 +(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: +T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 +y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z +t3))))))) x3 x1 H18 (pr2_delta c d u i H8 u1 x0 H13 x3 H19) (or3_intro1 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 x1))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) +t1 x1))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) +u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x1)))) (ex_intro2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) +t1 x1)) x0 H13 (pr2_delta (CHead c (Bind Abbr) x0) c x0 O (getl_refl Abbr c +x0) t1 x2 H14 x1 H15)))))))) (pr0_subst0_back x0 x2 x1 O H15 u1 H13))))) +H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x0 +t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3)))).(ex2_ind T +(\lambda (t3: T).(eq T x (THead (Bind Abbr) x0 t3))) (\lambda (t3: T).(subst0 +(s (Bind Abbr) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 +u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T +(\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) +(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: +T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x3: +T).(\lambda (H18: (eq T x (THead (Bind Abbr) x0 x3))).(\lambda (H19: (subst0 +(s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 x2 t3)) +(\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 +u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T +(\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) +(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: +T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x4: +T).(\lambda (H20: (subst0 O u1 x2 x4)).(\lambda (H21: (pr0 x4 x1)).(or_introl +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind +b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 +(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: +T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 +y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z +t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 +(lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T +x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 +u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: +T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) +(\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda +(y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: +T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c +(Bind Abbr) u1) z t3))))))) x0 x3 H18 (pr2_free c u1 x0 H13) (or3_intro2 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 x3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) +t1 x3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) +u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x3)))) (ex3_2_intro T T +(\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) +(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c (Bind Abbr) u1) z x3))) x4 x1 (pr2_delta (CHead c (Bind +Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 x2 H14 x4 H20) H21 (pr2_delta +(CHead c (Bind Abbr) u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind +Abbr) u) H8 u1) x1 x1 (pr0_refl x1) x3 H19)))))))) (pr0_subst0_back x0 x2 x1 +O H15 u1 H13))))) H17)) (\lambda (H17: (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind +Abbr) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T +x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u +x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 +t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind +b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 +(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: +T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 +y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z +t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 +(lift (S O) O x))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H18: (eq T x +(THead (Bind Abbr) x3 x4))).(\lambda (H19: (subst0 i u x0 x3)).(\lambda (H20: +(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 +x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) +t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) +u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) +(\lambda (x5: T).(\lambda (H21: (subst0 O u1 x2 x5)).(\lambda (H22: (pr0 x5 +x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead +(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 +(CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda +(u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: +T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: +T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c +(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c +(Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T +(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) +t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) +u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: +T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x3 x4 H18 +(pr2_delta c d u i H8 u1 x0 H13 x3 H19) (or3_intro2 (\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 +u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 x4))) (ex3_2 T T +(\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) +(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: +T).(pr2 (CHead c (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y: +T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: +T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c +(Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O +(getl_refl Abbr c u1) t1 x2 H14 x5 H21) H22 (pr2_delta (CHead c (Bind Abbr) +u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) x1 x1 +(pr0_refl x1) x4 H20)))))))) (pr0_subst0_back x0 x2 x1 O H15 u1 H13))))))) +H17)) (subst0_gen_head (Bind Abbr) u x0 x1 x i H16)))))) H_x0)) H_x)))))) +H11)) (\lambda (H11: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 +t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c +(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 +(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) +(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) +(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O +x)))) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u +(S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O +t2) H11 (lift (S O) O x) (subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) +(pr0_gen_abbr u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead +(Bind Abbr) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 +(refl_equal C c) (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T +x))))))). + +theorem pr2_gen_void: + \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda +(H: (pr2 c (THead (Bind Void) u1 t1) x)).(let H0 \def (match H in pr2 return +(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t +t0)).((eq C c0 c) \to ((eq T t (THead (Bind Void) u1 t1)) \to ((eq T t0 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).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +t1 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 +(lift (S O) O x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda +(H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 t1))).(\lambda +(H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Void) u1 +t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: +B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) +(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind +Void) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) +(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O +x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead +(Bind Void) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) t1 (lift (S O) O x))))))) (\lambda (H6: (pr0 (THead +(Bind Void) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 +u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) +O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) +t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 +(lift (S O) O x))))) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 +u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead +c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind +b) u) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: +(eq T x (THead (Bind Void) x0 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H10: +(pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) (\lambda (t: T).(or (ex3_2 +T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) +(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O +t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead +(Bind Void) x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind Void) x0 x1))))) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Void) +x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c +u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind +Void) x0 x1)) (pr2_free c u1 x0 H9) (\lambda (b: B).(\lambda (u: T).(pr2_free +(CHead c (Bind b) u) t1 x1 H10))))) x H8)))))) H7)) (\lambda (H7: (pr0 t1 +(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: +T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 +c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: +T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 +(CHead c (Bind b) u) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u: +T).(pr2_free (CHead c (Bind b) u) t1 (lift (S O) O x) H7))))) (pr0_gen_void +u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Void) u1 +t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 +H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead +(Bind Void) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: +C).((eq T t0 (THead (Bind Void) u1 t1)) \to ((eq T t x) \to ((getl i c1 +(CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or +(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 +t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) +t1 (lift (S O) O x))))))))))) (\lambda (H6: (eq T t0 (THead (Bind Void) u1 +t1))).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t3: T).((eq T t x) \to +((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) +\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind +Void) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) t1 t4)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) +t1 (lift (S O) O x)))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda +(t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Bind Void) u1 +t1) t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t4: T).(eq T x (THead (Bind Void) u2 t4)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) t1 t4)))))) (\forall (b: B).(\forall (u0: +T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))))))) (\lambda (H8: (getl +i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Bind Void) u1 t1) +t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 +t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: +T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (H11: (ex3_2 T +T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: +T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Bind Void) +x0 x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 +\def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Bind Void) +x0 x1) H12) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Void) u2 +x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x +(THead (Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 +t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 +t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift +(S O) O x))))) (\lambda (H16: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind +Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda +(u2: T).(eq T x (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 +u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind +Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) +t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Bind +Void) x2 x1))).(\lambda (H18: (subst0 i u x0 x2)).(or_introl (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 +t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift +(S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x +(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) +(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead +c (Bind b) u0) t1 t3))))) x2 x1 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) +(\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 +H14)))))))) H16)) (\lambda (H16: (ex2 T (\lambda (t3: T).(eq T x (THead (Bind +Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 +t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Void) x0 t3))) +(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda +(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) +(\lambda (x2: T).(\lambda (H17: (eq T x (THead (Bind Void) x0 x2))).(\lambda +(H18: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) t1 t3))))) x0 x2 H17 (pr2_free c u1 x0 H13) (\lambda (b: B).(\lambda (u0: +T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead +d (Bind Abbr) u) H8 u0) t1 x1 H14 x2 H18)))))))) H16)) (\lambda (H16: (ex3_2 +T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: +T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T +(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) +(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda +(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 +t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift +(S O) O x))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq T x +(THead (Bind Void) x2 x3))).(\lambda (H18: (subst0 i u x0 x2)).(\lambda (H19: +(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: +T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: +T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall +(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: +B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) +(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) +u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: +T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) +u0) t1 t3))))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (\lambda (b: +B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head +(Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H14 x3 H19)))))))))) H16)) +(subst0_gen_head (Bind Void) u x0 x1 x i H15)))))))) H11)) (\lambda (H11: +(pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda +(t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: +T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall +(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: +T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda +(u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c +(CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H11 (lift (S O) O x) +(subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) (pr0_gen_void u1 t1 t2 +H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H6))) +c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) +(refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x))))))). + +theorem pr2_gen_lift: + \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall +(d: nat).((pr2 c (lift h d t1) x) \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).(pr2 e t1 +t2)))))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda +(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(\lambda (e: C).(\lambda (H0: +(drop h d c e)).(let H1 \def (match H in pr2 return (\lambda (c0: C).(\lambda +(t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t +(lift h d t1)) \to ((eq T t0 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d +t2))) (\lambda (t2: T).(pr2 e t1 t2)))))))))) with [(pr2_free c0 t0 t2 H1) +\Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t0 (lift h d +t1))).(\lambda (H4: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (lift +h d t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t3: T).(eq T +x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))) (\lambda (H5: (eq T t0 +(lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t: T).((eq T t2 x) \to +((pr0 t t2) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: +T).(pr2 e t1 t3)))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t: +T).((pr0 (lift h d t1) t) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) +(\lambda (t3: T).(pr2 e t1 t3))))) (\lambda (H7: (pr0 (lift h d t1) +x)).(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr0 +t1 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 +e t1 t3))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift h d x0))).(\lambda +(H9: (pr0 t1 x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda +(t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 +T (\lambda (t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 +e t1 t3)) x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H9)) x H8)))) +(pr0_gen_lift t1 x h d H7))) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 (lift h +d t1) H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 +t2 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0 +(lift h d t1))).(\lambda (H6: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T +t0 (lift h d t1)) \to ((eq T t x) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) +\to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex2 T (\lambda (t3: T).(eq T x +(lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))))) (\lambda (H7: (eq T t0 +(lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t3: T).((eq T t x) \to +((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) +\to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 e +t1 t4)))))))) (\lambda (H8: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i +c (CHead d0 (Bind Abbr) u)) \to ((pr0 (lift h d t1) t2) \to ((subst0 i u t2 +t3) \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 +e t1 t4))))))) (\lambda (H9: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda +(H10: (pr0 (lift h d t1) t2)).(\lambda (H11: (subst0 i u t2 x)).(ex2_ind T +(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 +T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) +(\lambda (x0: T).(\lambda (H12: (eq T t2 (lift h d x0))).(\lambda (H13: (pr0 +t1 x0)).(let H14 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H11 +(lift h d x0) H12) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T x (lift h d +t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H15: (lt i d)).(let H16 \def +(eq_ind nat d (\lambda (n: nat).(drop h n c e)) H0 (S (plus i (minus d (S +i)))) (lt_plus_minus i d H15)) in (let H17 \def (eq_ind nat d (\lambda (n: +nat).(subst0 i u (lift h n x0) x)) H14 (S (plus i (minus d (S i)))) +(lt_plus_minus i d H15)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: +C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0: +C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: +C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T x (lift h d +t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2: +C).(\lambda (H18: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H19: (getl +i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0 +x2)).(let H21 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus +i (minus d (S i)))) x0) x)) H17 (lift h (minus d (S i)) x1) H18) in (ex2_ind +T (\lambda (t3: T).(eq T x (lift h (S (plus i (minus d (S i)))) t3))) +(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h +d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H22: (eq +T x (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H23: (subst0 i x1 x0 +x3)).(let H24 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: +nat).(eq T x (lift h n x3))) H22 d (lt_plus_minus i d H15)) in (ex_intro2 T +(\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 +H24 (pr2_delta e x2 x1 i H19 t1 x0 H13 x3 H23)))))) (subst0_gen_lift_lt x1 x0 +x i h (minus d (S i)) H21)))))))) (getl_drop_conf_lt Abbr c d0 u i H9 e h +(minus d (S i)) H16))))) (\lambda (H15: (le d i)).(lt_le_e i (plus d h) (ex2 +T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) +(\lambda (H16: (lt i (plus d h))).(subst0_gen_lift_false x0 u x h d i H15 H16 +H14 (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e +t1 t3))))) (\lambda (H16: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq +T x (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T +(\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) +(\lambda (x1: T).(\lambda (H17: (eq T x (lift h d x1))).(\lambda (H18: +(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T x (lift h d +t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H17 (pr2_delta e d0 u (minus i h) +(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H9 e h d H0 H16) t1 x0 H13 x1 +H18))))) (subst0_gen_lift_ge u x0 x i h d H14 H16)))))))))) (pr0_gen_lift t1 +t2 h d H10))))) t (sym_eq T t x H8))) t0 (sym_eq T t0 (lift h d t1) H7))) c0 +(sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T +(lift h d t1)) (refl_equal T x)))))))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/pr2.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/pr2.ma new file mode 100644 index 000000000..d56d40f53 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/pr2.ma @@ -0,0 +1,248 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/pr2". + +include "pr2/defs.ma". + +include "pr0/pr0.ma". + +include "getl/props.ma". + +theorem pr2_confluence__pr2_free_free: + \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 +t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t)))))))) +\def + \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (pr0 t0 t1)).(\lambda (H0: (pr0 t0 t2)).(ex2_ind T (\lambda (t: T).(pr0 +t2 t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) +(\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H1: (pr0 t2 +x)).(\lambda (H2: (pr0 t1 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) +(\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H2) (pr2_free c t2 x H1))))) +(pr0_confluence t0 t2 H0 t1 H))))))). + +theorem pr2_confluence__pr2_free_delta: + \forall (c: C).(\forall (d: C).(\forall (t0: T).(\forall (t1: T).(\forall +(t2: T).(\forall (t4: T).(\forall (u: T).(\forall (i: nat).((pr0 t0 t1) \to +((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t2) +\to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 +t)))))))))))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (t4: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (pr0 +t0 t1)).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H1: (pr0 +t0 t4)).(\lambda (H2: (subst0 i u t4 t2)).(ex2_ind T (\lambda (t: T).(pr0 t4 +t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda +(t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H3: (pr0 t4 x)).(\lambda (H4: +(pr0 t1 x)).(or_ind (pr0 t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda +(w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t))) (\lambda (H5: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 +c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x (pr2_free c t1 x H4) (pr2_free c t2 +x H5))) (\lambda (H5: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: +T).(subst0 i u x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda +(w2: T).(subst0 i u x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t))) (\lambda (x0: T).(\lambda (H6: (pr0 t2 x0)).(\lambda (H7: +(subst0 i u x x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t)) x0 (pr2_delta c d u i H0 t1 x H4 x0 H7) (pr2_free c t2 x0 +H6))))) H5)) (pr0_subst0 t4 x H3 u t2 i H2 u (pr0_refl u)))))) +(pr0_confluence t0 t4 H1 t1 H))))))))))))). + +theorem pr2_confluence__pr2_delta_delta: + \forall (c: C).(\forall (d: C).(\forall (d0: C).(\forall (t0: T).(\forall +(t1: T).(\forall (t2: T).(\forall (t3: T).(\forall (t4: T).(\forall (u: +T).(\forall (u0: T).(\forall (i: nat).(\forall (i0: nat).((getl i c (CHead d +(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t1) \to ((getl i0 c +(CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to +(ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 +t)))))))))))))))))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (d0: C).(\lambda (t0: T).(\lambda +(t1: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (u: +T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (i0: nat).(\lambda (H: (getl i +c (CHead d (Bind Abbr) u))).(\lambda (H0: (pr0 t0 t3)).(\lambda (H1: (subst0 +i u t3 t1)).(\lambda (H2: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda +(H3: (pr0 t0 t4)).(\lambda (H4: (subst0 i0 u0 t4 t2)).(ex2_ind T (\lambda (t: +T).(pr0 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr2 c t1 +t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t4 +x)).(\lambda (H6: (pr0 t3 x)).(or_ind (pr0 t1 x) (ex2 T (\lambda (w2: T).(pr0 +t1 w2)) (\lambda (w2: T).(subst0 i u x w2))) (ex2 T (\lambda (t: T).(pr2 c t1 +t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H7: (pr0 t1 x)).(or_ind (pr0 t2 +x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x +w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) +(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda +(t: T).(pr2 c t2 t)) x (pr2_free c t1 x H7) (pr2_free c t2 x H8))) (\lambda +(H8: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 +u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) +(\lambda (x0: T).(\lambda (H9: (pr0 t2 x0)).(\lambda (H10: (subst0 i0 u0 x +x0)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) +x0 (pr2_delta c d0 u0 i0 H2 t1 x H7 x0 H10) (pr2_free c t2 x0 H9))))) H8)) +(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))) (\lambda (H7: (ex2 T +(\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)))).(ex2_ind +T (\lambda (w2: T).(pr0 t1 w2)) (\lambda (w2: T).(subst0 i u x w2)) (ex2 T +(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x0: +T).(\lambda (H8: (pr0 t1 x0)).(\lambda (H9: (subst0 i u x x0)).(or_ind (pr0 +t2 x) (ex2 T (\lambda (w2: T).(pr0 t2 w2)) (\lambda (w2: T).(subst0 i0 u0 x +w2))) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) +(\lambda (H10: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) +(\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 x0 H8) (pr2_delta c d u i H +t2 x H10 x0 H9))) (\lambda (H10: (ex2 T (\lambda (w2: T).(pr0 t2 w2)) +(\lambda (w2: T).(subst0 i0 u0 x w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t2 +w2)) (\lambda (w2: T).(subst0 i0 u0 x w2)) (ex2 T (\lambda (t: T).(pr2 c t1 +t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (x1: T).(\lambda (H11: (pr0 t2 +x1)).(\lambda (H12: (subst0 i0 u0 x x1)).(neq_eq_e i i0 (ex2 T (\lambda (t: +T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H13: (not (eq nat i +i0))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 +i0 u0 x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 +t))) (\lambda (x2: T).(\lambda (H14: (subst0 i u x1 x2)).(\lambda (H15: +(subst0 i0 u0 x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t)) x2 (pr2_delta c d0 u0 i0 H2 t1 x0 H8 x2 H15) (pr2_delta c d +u i H t2 x1 H11 x2 H14))))) (subst0_confluence_neq x x1 u0 i0 H12 x0 u i H9 +(sym_not_eq nat i i0 H13)))) (\lambda (H13: (eq nat i i0)).(let H14 \def +(eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u0 x x1)) H12 i H13) in (let H15 +\def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d0 (Bind Abbr) u0))) +H2 i H13) in (let H16 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c0: +C).(getl i c c0)) H (CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind +Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (let H17 \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) u) +(CHead d0 (Bind Abbr) u0) (getl_mono c (CHead d (Bind Abbr) u) i H (CHead d0 +(Bind Abbr) u0) H15)) in ((let H18 \def (f_equal C T (\lambda (e: C).(match e +in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) +\Rightarrow t])) (CHead d (Bind Abbr) u) (CHead d0 (Bind Abbr) u0) (getl_mono +c (CHead d (Bind Abbr) u) i H (CHead d0 (Bind Abbr) u0) H15)) in (\lambda +(H19: (eq C d d0)).(let H20 \def (eq_ind_r T u0 (\lambda (t: T).(subst0 i t x +x1)) H14 u H18) in (let H21 \def (eq_ind_r T u0 (\lambda (t: T).(getl i c +(CHead d0 (Bind Abbr) t))) H16 u H18) in (let H22 \def (eq_ind_r C d0 +(\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) u))) H21 d H19) in (or4_ind +(eq T x1 x0) (ex2 T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: +T).(subst0 i u x0 t))) (subst0 i u x1 x0) (subst0 i u x0 x1) (ex2 T (\lambda +(t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) (\lambda (H23: (eq T x1 +x0)).(let H24 \def (eq_ind T x1 (\lambda (t: T).(pr0 t2 t)) H11 x0 H23) in +(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 +(pr2_free c t1 x0 H8) (pr2_free c t2 x0 H24)))) (\lambda (H23: (ex2 T +(\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i u x0 +t)))).(ex2_ind T (\lambda (t: T).(subst0 i u x1 t)) (\lambda (t: T).(subst0 i +u x0 t)) (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))) +(\lambda (x2: T).(\lambda (H24: (subst0 i u x1 x2)).(\lambda (H25: (subst0 i +u x0 x2)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c +t2 t)) x2 (pr2_delta c d u i H22 t1 x0 H8 x2 H25) (pr2_delta c d u i H22 t2 +x1 H11 x2 H24))))) H23)) (\lambda (H23: (subst0 i u x1 x0)).(ex_intro2 T +(\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t)) x0 (pr2_free c t1 +x0 H8) (pr2_delta c d u i H22 t2 x1 H11 x0 H23))) (\lambda (H23: (subst0 i u +x0 x1)).(ex_intro2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 +t)) x1 (pr2_delta c d u i H22 t1 x0 H8 x1 H23) (pr2_free c t2 x1 H11))) +(subst0_confluence_eq x x1 u i H20 x0 H9))))))) H17)))))))))) H10)) +(pr0_subst0 t4 x H5 u0 t2 i0 H4 u0 (pr0_refl u0)))))) H7)) (pr0_subst0 t3 x +H6 u t1 i H1 u (pr0_refl u)))))) (pr0_confluence t0 t4 H3 t3 +H0))))))))))))))))))). + +theorem pr2_confluence: + \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall +(t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: +T).(pr2 c t2 t)))))))) +\def + \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 +t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H in +pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).(\lambda (_: +(pr2 c0 t t3)).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T +(\lambda (t4: T).(pr2 c t1 t4)) (\lambda (t4: T).(pr2 c t2 t4)))))))))) with +[(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: +(eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T +t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c +t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind +T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t5: +T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5)))))) (\lambda (H6: (eq T t4 +t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t5: +T).(pr2 c t1 t5)) (\lambda (t5: T).(pr2 c t2 t5))))) (\lambda (H7: (pr0 t0 +t1)).(let H8 \def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t: +T).(\lambda (t5: T).(\lambda (_: (pr2 c1 t t5)).((eq C c1 c) \to ((eq T t t0) +\to ((eq T t5 t2) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda (t6: +T).(pr2 c t2 t6)))))))))) with [(pr2_free c1 t5 t6 H8) \Rightarrow (\lambda +(H9: (eq C c1 c)).(\lambda (H10: (eq T t5 t0)).(\lambda (H11: (eq T t6 +t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 +t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 +t))))))) (\lambda (H12: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 +t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: +T).(pr2 c t2 t7)))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: +T).((pr0 t0 t) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: +T).(pr2 c t2 t7))))) (\lambda (H14: (pr0 t0 +t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H14)) t6 (sym_eq T t6 t2 +H13))) t5 (sym_eq T t5 t0 H12))) c1 (sym_eq C c1 c H9) H10 H11 H8)))) | +(pr2_delta c1 d u i H8 t5 t6 H9 t H10) \Rightarrow (\lambda (H11: (eq C c1 +c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t t2)).(eq_ind C c +(\lambda (c2: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c2 (CHead d +(Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda +(t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))))) (\lambda (H14: +(eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t t2) \to ((getl i c +(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (ex2 T +(\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8)))))))) +(\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (t7: T).((getl i c (CHead d +(Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t7) \to (ex2 T (\lambda +(t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 t8))))))) (\lambda (H16: +(getl i c (CHead d (Bind Abbr) u))).(\lambda (H17: (pr0 t0 t6)).(\lambda +(H18: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i +H7 H16 H17 H18)))) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 +(sym_eq C c1 c H11) H12 H13 H8 H9 H10))))]) in (H8 (refl_equal C c) +(refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T +t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4 +H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3 +t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0) +\to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) +\to ((subst0 i u t4 t) \to (ex2 T (\lambda (t5: T).(pr2 c t1 t5)) (\lambda +(t5: T).(pr2 c t2 t5))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 +(\lambda (t5: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to +((pr0 t5 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t6: T).(pr2 c t1 +t6)) (\lambda (t6: T).(pr2 c t2 t6)))))))) (\lambda (H8: (eq T t t1)).(eq_ind +T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) +\to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t6: T).(pr2 c t1 t6)) (\lambda +(t6: T).(pr2 c t2 t6))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr) +u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12 +\def (match H0 in pr2 return (\lambda (c1: C).(\lambda (t5: T).(\lambda (t6: +T).(\lambda (_: (pr2 c1 t5 t6)).((eq C c1 c) \to ((eq T t5 t0) \to ((eq T t6 +t2) \to (ex2 T (\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 +t7)))))))))) with [(pr2_free c1 t5 t6 H12) \Rightarrow (\lambda (H13: (eq C +c1 c)).(\lambda (H14: (eq T t5 t0)).(\lambda (H15: (eq T t6 t2)).(eq_ind C c +(\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T +(\lambda (t7: T).(pr2 c t1 t7)) (\lambda (t7: T).(pr2 c t2 t7))))))) (\lambda +(H16: (eq T t5 t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t2) \to ((pr0 t7 +t6) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 +t8)))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t0 +t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 +t8))))) (\lambda (H18: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1) +(pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H18 H9 H10 H11))) t6 +(sym_eq T t6 t2 H17))) t5 (sym_eq T t5 t0 H16))) c1 (sym_eq C c1 c H13) H14 +H15 H12)))) | (pr2_delta c1 d0 u0 i0 H12 t5 t6 H13 t7 H14) \Rightarrow +(\lambda (H15: (eq C c1 c)).(\lambda (H16: (eq T t5 t0)).(\lambda (H17: (eq T +t7 t2)).(eq_ind C c (\lambda (c2: C).((eq T t5 t0) \to ((eq T t7 t2) \to +((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 +t6 t7) \to (ex2 T (\lambda (t8: T).(pr2 c t1 t8)) (\lambda (t8: T).(pr2 c t2 +t8))))))))) (\lambda (H18: (eq T t5 t0)).(eq_ind T t0 (\lambda (t8: T).((eq T +t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t8 t6) \to +((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda +(t9: T).(pr2 c t2 t9)))))))) (\lambda (H19: (eq T t7 t2)).(eq_ind T t2 +(\lambda (t8: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to +((subst0 i0 u0 t6 t8) \to (ex2 T (\lambda (t9: T).(pr2 c t1 t9)) (\lambda +(t9: T).(pr2 c t2 t9))))))) (\lambda (H20: (getl i0 c (CHead d0 (Bind Abbr) +u0))).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 i0 u0 t6 +t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 +H11 H20 H21 H22)))) t7 (sym_eq T t7 t2 H19))) t5 (sym_eq T t5 t0 H18))) c1 +(sym_eq C c1 c H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C c) +(refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T +t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C +c) (refl_equal T t0) (refl_equal T t1)))))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props.ma new file mode 100644 index 000000000..fca780fee --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props.ma @@ -0,0 +1,285 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/props". + +include "pr2/defs.ma". + +include "pr0/props.ma". + +include "getl/drop.ma". + +include "getl/clear.ma". + +theorem pr2_thin_dx: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(u: T).(\forall (f: F).(pr2 c (THead (Flat f) u t1) (THead (Flat f) u +t2))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(\lambda (u: T).(\lambda (f: F).(pr2_ind (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).(pr2 c0 (THead (Flat f) u t) (THead (Flat f) u t0))))) +(\lambda (c0: C).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr0 t0 +t3)).(pr2_free c0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u +(pr0_refl u) t0 t3 H0 (Flat f))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind +Abbr) u0))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr0 t0 +t3)).(\lambda (t: T).(\lambda (H2: (subst0 i u0 t3 t)).(pr2_delta c0 d u0 i +H0 (THead (Flat f) u t0) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t0 +t3 H1 (Flat f)) (THead (Flat f) u t) (subst0_snd (Flat f) u0 t t3 i H2 +u)))))))))))) c t1 t2 H)))))). + +theorem pr2_head_1: + \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall +(k: K).(\forall (t: T).(pr2 c (THead k u1 t) (THead k u2 t))))))) +\def + \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 +u2)).(\lambda (k: K).(\lambda (t: T).(pr2_ind (\lambda (c0: C).(\lambda (t0: +T).(\lambda (t1: T).(pr2 c0 (THead k t0 t) (THead k t1 t))))) (\lambda (c0: +C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(pr2_free c0 +(THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H0 t t (pr0_refl t) k)))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 +t0)).(pr2_delta c0 d u i H0 (THead k t1 t) (THead k t2 t) (pr0_comp t1 t2 H1 +t t (pr0_refl t) k) (THead k t0 t) (subst0_fst u t0 t2 i H2 t k)))))))))))) c +u1 u2 H)))))). + +theorem pr2_head_2: + \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall +(k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u +t2))))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(k: K).(K_ind (\lambda (k0: K).((pr2 (CHead c k0 u) t1 t2) \to (pr2 c (THead +k0 u t1) (THead k0 u t2)))) (\lambda (b: B).(\lambda (H: (pr2 (CHead c (Bind +b) u) t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda +(t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 (CHead c (Bind +b) u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) u t1) +(THead (Bind b) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow +(\lambda (H1: (eq C c0 (CHead c (Bind b) u))).(\lambda (H2: (eq T t0 +t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (_: +C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind +b) u t1) (THead (Bind b) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 +(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) u +t1) (THead (Bind b) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 +(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) +u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) u t1) (THead +(Bind b) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Bind b)))) t3 (sym_eq T +t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) u) H1) +H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda +(H3: (eq C c0 (CHead c (Bind b) u))).(\lambda (H4: (eq T t0 t1)).(\lambda +(H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (c1: C).((eq T t0 +t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0 +t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) +u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T +t t2) \to ((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 +t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind +b) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: +T).((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3) +\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u +t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) +u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).((match +i in nat return (\lambda (n: nat).((getl n (CHead c (Bind b) u) (CHead d +(Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) +(THead (Bind b) u t2))))) with [O \Rightarrow (\lambda (H11: (getl O (CHead c +(Bind b) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 O u0 t3 +t2)).(let H13 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) +(CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d +(Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) +H11))) 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 d (Bind Abbr) u0) +(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u +(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in ((let H15 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u0 | (CHead _ _ t4) \Rightarrow t4])) (CHead d +(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) +u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in +(\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H18 \def (eq_ind +T u0 (\lambda (t4: T).(subst0 O t4 t3 t2)) H12 u H15) in (eq_ind B Abbr +(\lambda (b0: B).(pr2 c (THead (Bind b0) u t1) (THead (Bind b0) u t2))) +(pr2_free c (THead (Bind Abbr) u t1) (THead (Bind Abbr) u t2) (pr0_delta u u +(pr0_refl u) t1 t3 H9 t2 H18)) b H16))))) H14)) H13)))) | (S n) \Rightarrow +(\lambda (H11: (getl (S n) (CHead c (Bind b) u) (CHead d (Bind Abbr) +u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0 (r (Bind b) +n) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H11) (THead (Bind b) u +t1) (THead (Bind b) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 (Bind b)) +(THead (Bind b) u t2) (subst0_snd (Bind b) u0 t2 t3 (r (Bind b) n) H12 +u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq +C c0 (CHead c (Bind b) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C +(CHead c (Bind b) u)) (refl_equal T t1) (refl_equal T t2))))) (\lambda (f: +F).(\lambda (H: (pr2 (CHead c (Flat f) u) t1 t2)).(let H0 \def (match H in +pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: +(pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) u)) \to ((eq T t t1) \to ((eq T t0 +t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))))) with +[(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Flat f) +u))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead +c (Flat f) u) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 +t3) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H4: +(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to +(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))) (\lambda (H5: (eq T t3 +t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Flat f) u +t1) (THead (Flat f) u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead +(Flat f) u t1) (THead (Flat f) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 +(Flat f)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 +(CHead c (Flat f) u) H1) H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t +H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) u))).(\lambda (H4: +(eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) u) +(\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d +(Bind Abbr) u0)) \to ((pr0 t0 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead +(Flat f) u t1) (THead (Flat f) u t2)))))))) (\lambda (H6: (eq T t0 +t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Flat +f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t4 t3) \to ((subst0 i u0 t3 t) \to +(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))) (\lambda (H7: (eq T +t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) u) (CHead d +(Bind Abbr) u0)) \to ((pr0 t1 t3) \to ((subst0 i u0 t3 t4) \to (pr2 c (THead +(Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H8: (getl i (CHead c +(Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda +(H10: (subst0 i u0 t3 t2)).((match i in nat return (\lambda (n: nat).((getl n +(CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to +(pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))) with [O \Rightarrow +(\lambda (H11: (getl O (CHead c (Flat f) u) (CHead d (Bind Abbr) +u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(pr2_delta c d u0 O (getl_intro O c +(CHead d (Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind +Abbr) u0) u (getl_gen_O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H11))) +(THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 +H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 O H12 u)))) +| (S n) \Rightarrow (\lambda (H11: (getl (S n) (CHead c (Flat f) u) (CHead d +(Bind Abbr) u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0 +(r (Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H11) +(THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 +H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 (r (Flat f) +n) H12 u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 +(sym_eq C c0 (CHead c (Flat f) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal +C (CHead c (Flat f) u)) (refl_equal T t1) (refl_equal T t2))))) k))))). + +theorem clear_pr2_trans: + \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to +(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2)))))) +\def + \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 +t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def (match H in +pr2 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 +c t t0)).((eq C c c2) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c1 t1 +t2)))))))) with [(pr2_free c t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c +c2)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c2 +(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c1 +t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 +t2) \to ((pr0 t t3) \to (pr2 c1 t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind +T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c1 t1 t2))) (\lambda (H7: (pr0 t1 +t2)).(pr2_free c1 t1 t2 H7)) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 +H5))) c (sym_eq C c c2 H2) H3 H4 H1)))) | (pr2_delta c d u i H1 t0 t3 H2 t +H3) \Rightarrow (\lambda (H4: (eq C c c2)).(\lambda (H5: (eq T t0 +t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c2 (\lambda (c0: C).((eq T t0 t1) +\to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) +\to ((subst0 i u t3 t) \to (pr2 c1 t1 t2))))))) (\lambda (H7: (eq T t0 +t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i c2 (CHead d +(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1 +t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i c2 +(CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c1 +t1 t2))))) (\lambda (H9: (getl i c2 (CHead d (Bind Abbr) u))).(\lambda (H10: +(pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(pr2_delta c1 d u i +(clear_getl_trans i c2 (CHead d (Bind Abbr) u) H9 c1 H0) t1 t3 H10 t2 H11)))) +t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c (sym_eq C c c2 H4) H5 H6 H1 +H2 H3))))]) in (H1 (refl_equal C c2) (refl_equal T t1) (refl_equal T +t2)))))))). + +theorem pr2_cflat: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2)))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (f: +F).(\forall (v: T).(pr2 (CHead c0 (Flat f) v) t t0)))))) (\lambda (c0: +C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (f: +F).(\lambda (v: T).(pr2_free (CHead c0 (Flat f) v) t3 t4 H0))))))) (\lambda +(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl +i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda +(H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda +(f: F).(\lambda (v: T).(pr2_delta (CHead c0 (Flat f) v) d u i (getl_flat c0 +(CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))))) c t1 t2 H)))). + +theorem pr2_ctail: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall +(k: K).(\forall (u: T).(pr2 (CTail k u c) t1 t2)))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(\lambda (k: K).(\lambda (u: T).(pr2_ind (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).(pr2 (CTail k u c0) t t0)))) (\lambda (c0: C).(\lambda +(t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free (CTail k u c0) +t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: +nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: +T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: +(subst0 i u0 t4 t)).(pr2_delta (CTail k u c0) (CTail k u d) u0 i (getl_ctail +Abbr c0 d u0 i H0 k u) t3 t4 H1 t H2))))))))))) c t1 t2 H)))))). + +theorem pr2_lift: + \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h +d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift +h d t1) (lift h d t2))))))))) +\def + \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 +t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0: C).(\lambda (t: +T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 e) \to ((eq T t t1) +\to ((eq T t0 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) with +[(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 e)).(\lambda (H3: +(eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C e (\lambda (_: C).((eq T +t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (lift h d t1) (lift h d +t2)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 +t2) \to ((pr0 t t3) \to (pr2 c (lift h d t1) (lift h d t2))))) (\lambda (H6: +(eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (lift h d +t1) (lift h d t2)))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c (lift h d t1) +(lift h d t2) (pr0_lift t1 t2 H7 h d))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T +t0 t1 H5))) c0 (sym_eq C c0 e H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 +t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 e)).(\lambda (H5: (eq T t0 +t1)).(\lambda (H6: (eq T t t2)).(eq_ind C e (\lambda (c1: C).((eq T t0 t1) +\to ((eq T t t2) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t3) +\to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2)))))))) (\lambda +(H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i e +(CHead d0 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c +(lift h d t1) (lift h d t2))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 +(\lambda (t4: T).((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t1 t3) \to +((subst0 i u t3 t4) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H9: +(getl i e (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda +(H11: (subst0 i u t3 t2)).(lt_le_e i d (pr2 c (lift h d t1) (lift h d t2)) +(\lambda (H12: (lt i d)).(let H13 \def (drop_getl_trans_le i d (le_S_n i d +(le_S (S i) d H12)) c e h H (CHead d0 (Bind Abbr) u) H9) in (ex3_2_ind C C +(\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda +(e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear +e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t1) (lift h d t2)) (\lambda +(x0: C).(\lambda (x1: C).(\lambda (H14: (drop i O c x0)).(\lambda (H15: (drop +h (minus d i) x0 x1)).(\lambda (H16: (clear x1 (CHead d0 (Bind Abbr) +u))).(let H17 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0 +x1)) H15 (S (minus d (S i))) (minus_x_Sy d i H12)) in (let H18 \def +(drop_clear_S x1 x0 h (minus d (S i)) H17 Abbr d0 u H16) in (ex2_ind C +(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i)) +u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t1) +(lift h d t2)) (\lambda (x: C).(\lambda (H19: (clear x0 (CHead x (Bind Abbr) +(lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x +d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x +(Bind Abbr) (lift h (minus d (S i)) u)) x0 H14 H19) (lift h d t1) (lift h d +t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) (subst0_lift_lt t3 t2 u i H11 d +H12 h))))) H18)))))))) H13))) (\lambda (H12: (le d i)).(pr2_delta c d0 u +(plus i h) (drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H9 H12) +(lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) +(subst0_lift_ge t3 t2 u i h H11 d H12))))))) t (sym_eq T t t2 H8))) t0 +(sym_eq T t0 t1 H7))) c0 (sym_eq C c0 e H4) H5 H6 H1 H2 H3))))]) in (H1 +(refl_equal C e) (refl_equal T t1) (refl_equal T t2)))))))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma new file mode 100644 index 000000000..012d5b0ee --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma @@ -0,0 +1,132 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1". + +include "pr2/defs.ma". + +include "pr0/subst1.ma". + +include "csubst1/defs.ma". + +include "subst1/subst1.ma". + +include "getl/props.ma". + +theorem pr2_delta1: + \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c +(CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) +\to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t)))))))))) +\def + \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda +(H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: +T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2 +t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0) +(\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2 +H0 t0 H2))) t H1)))))))))). + +theorem pr2_subst1: + \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c +(CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) +\to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c +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: (pr2 c t1 t2)).(let H1 \def (match H0 in pr2 return (\lambda +(c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C +c0 c) \to ((eq T t t1) \to ((eq T t0 t2) \to (\forall (w1: T).((subst1 i v t1 +w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v +t2 w2)))))))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq +C c0 c)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c +(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (\forall +(w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) +(\lambda (w2: T).(subst1 i v t2 w2))))))))) (\lambda (H5: (eq T t0 +t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (\forall +(w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) +(\lambda (w2: T).(subst1 i v t2 w2)))))))) (\lambda (H6: (eq T t3 +t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (\forall (w1: T).((subst1 i +v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 +i v t2 w2))))))) (\lambda (H7: (pr0 t1 t2)).(\lambda (w1: T).(\lambda (H8: +(subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst1 i v t2 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: +T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H9: (pr0 w1 x)).(\lambda +(H10: (subst1 i v t2 x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) +(\lambda (w2: T).(subst1 i v t2 w2)) x (pr2_free c w1 x H9) H10)))) +(pr0_subst1 t1 t2 H7 v w1 i H8 v (pr0_refl v)))))) t3 (sym_eq T t3 t2 H6))) +t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d +u i0 H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: +(eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c (\lambda (c1: C).((eq T +t0 t1) \to ((eq T t t2) \to ((getl i0 c1 (CHead d (Bind Abbr) u)) \to ((pr0 +t0 t3) \to ((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to +(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 +w2))))))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq +T t t2) \to ((getl i0 c (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to +((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T +(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))) +(\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i0 c (CHead d +(Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i0 u t3 t4) \to (\forall (w1: +T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda +(w2: T).(subst1 i v t2 w2))))))))) (\lambda (H9: (getl i0 c (CHead d (Bind +Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i0 u t3 +t2)).(\lambda (w1: T).(\lambda (H12: (subst1 i v t1 w1)).(ex2_ind T (\lambda +(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T (\lambda +(w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x: +T).(\lambda (H13: (pr0 w1 x)).(\lambda (H14: (subst1 i v t3 x)).(neq_eq_e i +i0 (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 +w2))) (\lambda (H15: (not (eq nat i i0))).(ex2_ind T (\lambda (t4: T).(subst1 +i v t2 t4)) (\lambda (t4: T).(subst1 i0 u x t4)) (ex2 T (\lambda (w2: T).(pr2 +c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda +(H16: (subst1 i v t2 x0)).(\lambda (H17: (subst1 i0 u x x0)).(ex_intro2 T +(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 +(pr2_delta1 c d u i0 H9 w1 x H13 x0 H17) H16)))) (subst1_confluence_neq t3 t2 +u i0 (subst1_single i0 u t3 t2 H11) x v i H14 (sym_not_eq nat i i0 H15)))) +(\lambda (H15: (eq nat i i0)).(let H16 \def (eq_ind_r nat i0 (\lambda (n: +nat).(subst0 n u t3 t2)) H11 i H15) in (let H17 \def (eq_ind_r nat i0 +(\lambda (n: nat).(getl n c (CHead d (Bind Abbr) u))) H9 i H15) in (let H18 +\def (eq_ind C (CHead e (Bind Abbr) v) (\lambda (c1: C).(getl i c c1)) H +(CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d +(Bind Abbr) u) H17)) in (let H19 \def (f_equal C C (\lambda (e0: C).(match e0 +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _) +\Rightarrow c1])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono +c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H17)) in ((let H20 \def +(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow v | (CHead _ _ t4) \Rightarrow t4])) (CHead e (Bind +Abbr) v) (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H +(CHead d (Bind Abbr) u) H17)) in (\lambda (H21: (eq C e d)).(let H22 \def +(eq_ind_r T u (\lambda (t4: T).(getl i c (CHead d (Bind Abbr) t4))) H18 v +H20) in (let H23 \def (eq_ind_r T u (\lambda (t4: T).(subst0 i t4 t3 t2)) H16 +v H20) in (let H24 \def (eq_ind_r C d (\lambda (c1: C).(getl i c (CHead c1 +(Bind Abbr) v))) H22 e H21) in (ex2_ind T (\lambda (t4: T).(subst1 i v t2 +t4)) (\lambda (t4: T).(subst1 i v x t4)) (ex2 T (\lambda (w2: T).(pr2 c w1 +w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda (H25: +(subst1 i v t2 x0)).(\lambda (H26: (subst1 i v x x0)).(ex_intro2 T (\lambda +(w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 (pr2_delta1 c +e v i H24 w1 x H13 x0 H26) H25)))) (subst1_confluence_eq t3 t2 v i +(subst1_single i v t3 t2 H23) x H14))))))) H19)))))))))) (pr0_subst1 t1 t3 +H10 v w1 i H12 v (pr0_refl v)))))))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 +t1 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) +(refl_equal T t1) (refl_equal T t2)))))))))). + +axiom pr2_gen_cabbr: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\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 t1 (lift (S O) d x1)) \to (ex2 T +(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a +x1 x2)))))))))))))))) +. + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/defs.ma new file mode 100644 index 000000000..69266bbb1 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/defs.ma @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/defs". + +include "pr2/defs.ma". + +inductive pr3 (c:C): T \to (T \to Prop) \def +| pr3_refl: \forall (t: T).(pr3 c t t) +| pr3_sing: \forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall (t3: +T).((pr3 c t2 t3) \to (pr3 c t1 t3))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/cnt.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/cnt.ma new file mode 100644 index 000000000..9dccd4da0 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/cnt.ma @@ -0,0 +1,88 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/cnt". + +include "tau1/props.ma". + +include "cnt/props.ma". + +theorem tau1_cnt: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c +t1 t) \to (ex2 T (\lambda (t2: T).(tau1 g c t1 t2)) (\lambda (t2: T).(cnt +t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: +(tau0 g c t1 t)).(tau0_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_: +T).(ex2 T (\lambda (t3: T).(tau1 g c0 t0 t3)) (\lambda (t3: T).(cnt t3)))))) +(\lambda (c0: C).(\lambda (n: nat).(ex_intro2 T (\lambda (t2: T).(tau1 g c0 +(TSort n) t2)) (\lambda (t2: T).(cnt t2)) (TSort (next g n)) (tau1_tau0 g c0 +(TSort n) (TSort (next g n)) (tau0_sort g c0 n)) (cnt_sort (next g n))))) +(\lambda (c0: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda +(H0: (getl i c0 (CHead d (Bind Abbr) v))).(\lambda (w: T).(\lambda (_: (tau0 +g d v w)).(\lambda (H2: (ex2 T (\lambda (t2: T).(tau1 g d v t2)) (\lambda +(t2: T).(cnt t2)))).(let H3 \def H2 in (ex2_ind T (\lambda (t2: T).(tau1 g d +v t2)) (\lambda (t2: T).(cnt t2)) (ex2 T (\lambda (t2: T).(tau1 g c0 (TLRef +i) t2)) (\lambda (t2: T).(cnt t2))) (\lambda (x: T).(\lambda (H4: (tau1 g d v +x)).(\lambda (H5: (cnt x)).(ex_intro2 T (\lambda (t2: T).(tau1 g c0 (TLRef i) +t2)) (\lambda (t2: T).(cnt t2)) (lift (S i) O x) (tau1_abbr g c0 d v i H0 x +H4) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind +Abst) v))).(\lambda (w: T).(\lambda (H1: (tau0 g d v w)).(\lambda (H2: (ex2 T +(\lambda (t2: T).(tau1 g d v t2)) (\lambda (t2: T).(cnt t2)))).(let H3 \def +H2 in (ex2_ind T (\lambda (t2: T).(tau1 g d v t2)) (\lambda (t2: T).(cnt t2)) +(ex2 T (\lambda (t2: T).(tau1 g c0 (TLRef i) t2)) (\lambda (t2: T).(cnt t2))) +(\lambda (x: T).(\lambda (H4: (tau1 g d v x)).(\lambda (H5: (cnt +x)).(ex_intro2 T (\lambda (t2: T).(tau1 g c0 (TLRef i) t2)) (\lambda (t2: +T).(cnt t2)) (lift (S i) O x) (tau1_trans g c0 (TLRef i) (lift (S i) O v) +(tau1_tau0 g c0 (TLRef i) (lift (S i) O v) (tau0_abst g c0 d v i H0 w H1)) +(lift (S i) O x) (tau1_lift g d v x H4 c0 (S i) O (getl_drop Abst c0 d v i +H0))) (cnt_lift x H5 (S i) O))))) H3)))))))))) (\lambda (b: B).(\lambda (c0: +C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g +(CHead c0 (Bind b) v) t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: T).(tau1 g +(CHead c0 (Bind b) v) t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in +(ex2_ind T (\lambda (t4: T).(tau1 g (CHead c0 (Bind b) v) t2 t4)) (\lambda +(t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Bind b) v t2) +t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (tau1 g (CHead +c0 (Bind b) v) t2 x)).(\lambda (H4: (cnt x)).(ex_intro2 T (\lambda (t4: +T).(tau1 g c0 (THead (Bind b) v t2) t4)) (\lambda (t4: T).(cnt t4)) (THead +(Bind b) v x) (tau1_bind g b c0 v t2 x H3) (cnt_head x H4 (Bind b) v))))) +H2))))))))) (\lambda (c0: C).(\lambda (v: T).(\lambda (t2: T).(\lambda (t3: +T).(\lambda (_: (tau0 g c0 t2 t3)).(\lambda (H1: (ex2 T (\lambda (t4: +T).(tau1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)))).(let H2 \def H1 in +(ex2_ind T (\lambda (t4: T).(tau1 g c0 t2 t4)) (\lambda (t4: T).(cnt t4)) +(ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Appl) v t2) t4)) (\lambda +(t4: T).(cnt t4))) (\lambda (x: T).(\lambda (H3: (tau1 g c0 t2 x)).(\lambda +(H4: (cnt x)).(ex_intro2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Appl) v +t2) t4)) (\lambda (t4: T).(cnt t4)) (THead (Flat Appl) v x) (tau1_appl g c0 v +t2 x H3) (cnt_head x H4 (Flat Appl) v))))) H2)))))))) (\lambda (c0: +C).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (tau0 g c0 v1 +v2)).(\lambda (_: (ex2 T (\lambda (t2: T).(tau1 g c0 v1 t2)) (\lambda (t2: +T).(cnt t2)))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (tau0 g c0 t2 +t3)).(\lambda (H3: (ex2 T (\lambda (t4: T).(tau1 g c0 t2 t4)) (\lambda (t4: +T).(cnt t4)))).(let H4 \def H3 in (ex2_ind T (\lambda (t4: T).(tau1 g c0 t2 +t4)) (\lambda (t4: T).(cnt t4)) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead +(Flat Cast) v1 t2) t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x: T).(\lambda +(H5: (tau1 g c0 t2 x)).(\lambda (H6: (cnt x)).(let H_x \def (tau1_cast2 g c0 +t2 x H5 v1 v2 H0) in (let H7 \def H_x in (ex2_ind T (\lambda (v3: T).(tau1 g +c0 v1 v3)) (\lambda (v3: T).(tau1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat +Cast) v3 x))) (ex2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Cast) v1 t2) +t4)) (\lambda (t4: T).(cnt t4))) (\lambda (x0: T).(\lambda (_: (tau1 g c0 v1 +x0)).(\lambda (H9: (tau1 g c0 (THead (Flat Cast) v1 t2) (THead (Flat Cast) x0 +x))).(ex_intro2 T (\lambda (t4: T).(tau1 g c0 (THead (Flat Cast) v1 t2) t4)) +(\lambda (t4: T).(cnt t4)) (THead (Flat Cast) x0 x) H9 (cnt_head x H6 (Flat +Cast) x0))))) H7)))))) H4))))))))))) c t1 t H))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/defs.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/defs.ma new file mode 100644 index 000000000..52f32692d --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/defs.ma @@ -0,0 +1,25 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/defs". + +include "tau0/defs.ma". + +inductive tau1 (g:G) (c:C) (t1:T): T \to Prop \def +| tau1_tau0: \forall (t2: T).((tau0 g c t1 t2) \to (tau1 g c t1 t2)) +| tau1_sing: \forall (t: T).((tau1 g c t1 t) \to (\forall (t2: T).((tau0 g c +t t2) \to (tau1 g c t1 t2)))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/props.ma new file mode 100644 index 000000000..2ea9bd7f8 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/props.ma @@ -0,0 +1,144 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/tau1/props". + +include "tau1/defs.ma". + +include "tau0/props.ma". + +theorem tau1_trans: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c +t1 t) \to (\forall (t2: T).((tau1 g c t t2) \to (tau1 g c t1 t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: +(tau1 g c t1 t)).(\lambda (t2: T).(\lambda (H0: (tau1 g c t t2)).(tau1_ind g +c t (\lambda (t0: T).(tau1 g c t1 t0)) (\lambda (t3: T).(\lambda (H1: (tau0 g +c t t3)).(tau1_sing g c t1 t H t3 H1))) (\lambda (t0: T).(\lambda (_: (tau1 g +c t t0)).(\lambda (H2: (tau1 g c t1 t0)).(\lambda (t3: T).(\lambda (H3: (tau0 +g c t0 t3)).(tau1_sing g c t1 t0 H2 t3 H3)))))) t2 H0))))))). + +theorem tau1_bind: + \forall (g: G).(\forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: +T).(\forall (t2: T).((tau1 g (CHead c (Bind b) v) t1 t2) \to (tau1 g c (THead +(Bind b) v t1) (THead (Bind b) v t2)))))))) +\def + \lambda (g: G).(\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: +T).(\lambda (t2: T).(\lambda (H: (tau1 g (CHead c (Bind b) v) t1 +t2)).(tau1_ind g (CHead c (Bind b) v) t1 (\lambda (t: T).(tau1 g c (THead +(Bind b) v t1) (THead (Bind b) v t))) (\lambda (t3: T).(\lambda (H0: (tau0 g +(CHead c (Bind b) v) t1 t3)).(tau1_tau0 g c (THead (Bind b) v t1) (THead +(Bind b) v t3) (tau0_bind g b c v t1 t3 H0)))) (\lambda (t: T).(\lambda (_: +(tau1 g (CHead c (Bind b) v) t1 t)).(\lambda (H1: (tau1 g c (THead (Bind b) v +t1) (THead (Bind b) v t))).(\lambda (t3: T).(\lambda (H2: (tau0 g (CHead c +(Bind b) v) t t3)).(tau1_sing g c (THead (Bind b) v t1) (THead (Bind b) v t) +H1 (THead (Bind b) v t3) (tau0_bind g b c v t t3 H2))))))) t2 H))))))). + +theorem tau1_appl: + \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall +(t2: T).((tau1 g c t1 t2) \to (tau1 g c (THead (Flat Appl) v t1) (THead (Flat +Appl) v t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda +(t2: T).(\lambda (H: (tau1 g c t1 t2)).(tau1_ind g c t1 (\lambda (t: T).(tau1 +g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t))) (\lambda (t3: +T).(\lambda (H0: (tau0 g c t1 t3)).(tau1_tau0 g c (THead (Flat Appl) v t1) +(THead (Flat Appl) v t3) (tau0_appl g c v t1 t3 H0)))) (\lambda (t: +T).(\lambda (_: (tau1 g c t1 t)).(\lambda (H1: (tau1 g c (THead (Flat Appl) v +t1) (THead (Flat Appl) v t))).(\lambda (t3: T).(\lambda (H2: (tau0 g c t +t3)).(tau1_sing g c (THead (Flat Appl) v t1) (THead (Flat Appl) v t) H1 +(THead (Flat Appl) v t3) (tau0_appl g c v t t3 H2))))))) t2 H)))))). + +theorem tau1_lift: + \forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau1 g e +t1 t2) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c +e) \to (tau1 g c (lift h d t1) (lift h d t2)))))))))) +\def + \lambda (g: G).(\lambda (e: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (tau1 g e t1 t2)).(tau1_ind g e t1 (\lambda (t: T).(\forall (c: +C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (tau1 g c (lift h +d t1) (lift h d t))))))) (\lambda (t3: T).(\lambda (H0: (tau0 g e t1 +t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop +h d c e)).(tau1_tau0 g c (lift h d t1) (lift h d t3) (tau0_lift g e t1 t3 H0 +c h d H1)))))))) (\lambda (t: T).(\lambda (_: (tau1 g e t1 t)).(\lambda (H1: +((\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to +(tau1 g c (lift h d t1) (lift h d t)))))))).(\lambda (t3: T).(\lambda (H2: +(tau0 g e t t3)).(\lambda (c: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda +(H3: (drop h d c e)).(tau1_sing g c (lift h d t1) (lift h d t) (H1 c h d H3) +(lift h d t3) (tau0_lift g e t t3 H2 c h d H3))))))))))) t2 H))))). + +theorem tau1_correct: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c +t1 t) \to (ex T (\lambda (t2: T).(tau0 g c t t2))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: +(tau1 g c t1 t)).(tau1_ind g c t1 (\lambda (t0: T).(ex T (\lambda (t2: +T).(tau0 g c t0 t2)))) (\lambda (t2: T).(\lambda (H0: (tau0 g c t1 +t2)).(tau0_correct g c t1 t2 H0))) (\lambda (t0: T).(\lambda (_: (tau1 g c t1 +t0)).(\lambda (_: (ex T (\lambda (t2: T).(tau0 g c t0 t2)))).(\lambda (t2: +T).(\lambda (H2: (tau0 g c t0 t2)).(tau0_correct g c t0 t2 H2)))))) t H))))). + +theorem tau1_abbr: + \forall (g: G).(\forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: +nat).((getl i c (CHead d (Bind Abbr) v)) \to (\forall (w: T).((tau1 g d v w) +\to (tau1 g c (TLRef i) (lift (S i) O w))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: +nat).(\lambda (H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (w: +T).(\lambda (H0: (tau1 g d v w)).(tau1_ind g d v (\lambda (t: T).(tau1 g c +(TLRef i) (lift (S i) O t))) (\lambda (t2: T).(\lambda (H1: (tau0 g d v +t2)).(tau1_tau0 g c (TLRef i) (lift (S i) O t2) (tau0_abbr g c d v i H t2 +H1)))) (\lambda (t: T).(\lambda (_: (tau1 g d v t)).(\lambda (H2: (tau1 g c +(TLRef i) (lift (S i) O t))).(\lambda (t2: T).(\lambda (H3: (tau0 g d t +t2)).(tau1_sing g c (TLRef i) (lift (S i) O t) H2 (lift (S i) O t2) +(tau0_lift g d t t2 H3 c (S i) O (getl_drop Abbr c d v i H)))))))) w +H0)))))))). + +theorem tau1_cast2: + \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((tau1 g c +t1 t2) \to (\forall (v1: T).(\forall (v2: T).((tau0 g c v1 v2) \to (ex2 T +(\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat +Cast) v1 t1) (THead (Flat Cast) v3 t2))))))))))) +\def + \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda +(H: (tau1 g c t1 t2)).(tau1_ind g c t1 (\lambda (t: T).(\forall (v1: +T).(\forall (v2: T).((tau0 g c v1 v2) \to (ex2 T (\lambda (v3: T).(tau1 g c +v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat +Cast) v3 t)))))))) (\lambda (t3: T).(\lambda (H0: (tau0 g c t1 t3)).(\lambda +(v1: T).(\lambda (v2: T).(\lambda (H1: (tau0 g c v1 v2)).(ex_intro2 T +(\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat +Cast) v1 t1) (THead (Flat Cast) v3 t3))) v2 (tau1_tau0 g c v1 v2 H1) +(tau1_tau0 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v2 t3) (tau0_cast +g c v1 v2 H1 t1 t3 H0)))))))) (\lambda (t: T).(\lambda (_: (tau1 g c t1 +t)).(\lambda (H1: ((\forall (v1: T).(\forall (v2: T).((tau0 g c v1 v2) \to +(ex2 T (\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead +(Flat Cast) v1 t1) (THead (Flat Cast) v3 t))))))))).(\lambda (t3: T).(\lambda +(H2: (tau0 g c t t3)).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H3: (tau0 g +c v1 v2)).(let H_x \def (H1 v1 v2 H3) in (let H4 \def H_x in (ex2_ind T +(\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat +Cast) v1 t1) (THead (Flat Cast) v3 t))) (ex2 T (\lambda (v3: T).(tau1 g c v1 +v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) +v3 t3)))) (\lambda (x: T).(\lambda (H5: (tau1 g c v1 x)).(\lambda (H6: (tau1 +g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) x t))).(let H_x0 \def +(tau1_correct g c v1 x H5) in (let H7 \def H_x0 in (ex_ind T (\lambda (t4: +T).(tau0 g c x t4)) (ex2 T (\lambda (v3: T).(tau1 g c v1 v3)) (\lambda (v3: +T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat Cast) v3 t3)))) (\lambda +(x0: T).(\lambda (H8: (tau0 g c x x0)).(ex_intro2 T (\lambda (v3: T).(tau1 g +c v1 v3)) (\lambda (v3: T).(tau1 g c (THead (Flat Cast) v1 t1) (THead (Flat +Cast) v3 t3))) x0 (tau1_sing g c v1 x H5 x0 H8) (tau1_sing g c (THead (Flat +Cast) v1 t1) (THead (Flat Cast) x t) H6 (THead (Flat Cast) x0 t3) (tau0_cast +g c x x0 H8 t t3 H2))))) H7)))))) H4))))))))))) t2 H))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma index 423565e49..673c65b47 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/theory.ma @@ -164,6 +164,12 @@ include "tau0/defs.ma". include "tau0/props.ma". +include "tau1/defs.ma". + +include "tau1/props.ma". + +include "tau1/cnt.ma". + include "A/defs.ma". include "asucc/defs.ma". @@ -208,3 +214,35 @@ include "arity/cimp.ma". include "arity/aprem.ma". +include "pr0/defs.ma". + +include "pr0/fwd.ma". + +include "pr0/props.ma". + +include "pr0/pr0.ma". + +include "pr0/subst1.ma". + +include "pr0/dec.ma". + +include "pr1/defs.ma". + +include "pr1/props.ma". + +include "pr1/pr1.ma". + +include "pr2/defs.ma". + +include "pr2/fwd.ma". + +include "pr2/props.ma". + +include "pr2/clen.ma". + +include "pr2/pr2.ma". + +include "pr2/subst1.ma". + +include "pr3/defs.ma". + diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/problems-1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/problems-1.ma new file mode 100644 index 000000000..32e613ebb --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Level-1/problems-1.ma @@ -0,0 +1,183 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* Problematic objects for disambiguation/typechecking ********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems". + +include "LambdaDelta/theory.ma". + +theorem pr2_gen_cabbr: + \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\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 t1 (lift (S O) d x1)) \to (ex2 T +(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a +x1 x2)))))))))))))))) +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e: +C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to +(\forall (a0: C).((csubst1 d u c0 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).(pr2 a x1 +x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: +nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: +C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) +d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d +x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2: +T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d +x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0 +(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda +(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T +(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a +x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda +(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t)) +H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4 +(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0 +H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S +O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind +Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 +t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e: +C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e +(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 +a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1: +T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda +(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2)) +(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: +T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1) +x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x +(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2: +T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) +(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10: +(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0)) +H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1 +d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12: +(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: +T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 +t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: +T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) +d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d +(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3: +T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) +(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr) +u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i) +(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0 +(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr) +d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda +(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_: +C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2: +C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0 +t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4: +T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr) +x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1 +(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl +i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0 +(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i)))) +(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: +C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0: +C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: +C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0 +u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6: +T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i)) +x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop +(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0: +T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6) +H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i)) +x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S +i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0 +x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28: +(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda +(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind +nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S +O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in +(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S +i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9: +T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9))) +(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S +i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8 +(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift +(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat +(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S +i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let +H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n: +nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12)) +in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10))) +(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O) +d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32)))))))) +(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30)))))) +(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S +i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i +H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12 +c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i +(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12)))) +(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n: +nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def +(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15 +\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in +(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind +Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2: +T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let +H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) +H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead +e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ +_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) +(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in +((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) +(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind +Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d +e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind +Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: +T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r +T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u +(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) +(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1: +C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda +(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0) +t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: +T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t +x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind +T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0) +(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) +(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i +(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i +x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10)))))) +(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0) +H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T +(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S +O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) +(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0 +u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T +(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1 +(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O) +d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq +T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0 +x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 +(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t +(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u +(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 +(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d +(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: +nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S O)))) x1 x0 H10 x3 +H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S +O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S +O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift +(S O) d0 x0) u0 d0 H11 (sym_not_equal nat d0 i (lt_neq d0 i H12)))))))))) +(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0 +x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))). + -- 2.39.2