branch for universe

[helm.git] / matita / contribs / LAMBDA-TYPES / LambdaDelta-1 / pr0 / fwd.ma

diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/fwd.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/fwd.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/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)).(insert_eq 
+T (TSort n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda 
+(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: 
+T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T 
+t (TSort n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H1) in 
+(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TSort n)) 
+t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
+t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let 
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in 
+(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: 
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: 
+(((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
+t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) 
+(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u 
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 
+t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda 
+(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2 
+v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2 
+t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
+(TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TSort n) H8) in (False_ind (eq T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind 
+b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to (eq T u2 
+u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda 
+(_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: 
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TSort 
+n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match 
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) 
+H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 
+H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
+(TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
+(Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind 
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee 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 t2 
+(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq 
+T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) 
+(TSort n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: 
+T).(match ee 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 t2 (THead (Flat Cast) u t1)) H4)))))))) y x 
+H0))) H))).
+
+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)).(insert_eq 
+T (TLRef n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda 
+(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: 
+T).((eq T t (TLRef n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T 
+t (TLRef n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TLRef n) H1) in 
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TLRef n)) 
+t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
+t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let 
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in 
+(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u: 
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: 
+(((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
+t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) 
+(TLRef n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u 
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2 
+t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda 
+(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2 
+v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda 
+(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2 
+t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) 
+(TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef n) H8) in (False_ind (eq T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind 
+b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda 
+(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (eq T u2 
+u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda 
+(_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_: 
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TLRef 
+n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match 
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | 
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) 
+H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1)) 
+H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 
+(TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
+(Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind 
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee 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 t2 
+(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq 
+T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1) 
+(TLRef n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee: 
+T).(match ee 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 t2 (THead (Flat Cast) u t1)) H4)))))))) y x 
+H0))) H))).
+
+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)).(insert_eq T (THead (Bind Abst) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (\lambda (y: 
+T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T 
+t (THead (Bind Abst) u1 t1)) \to (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))))))) (\lambda 
+(t: T).(\lambda (H1: (eq T t (THead (Bind Abst) u1 t1))).(let H2 \def 
+(f_equal T T (\lambda (e: T).e) t (THead (Bind Abst) u1 t1) H1) in (eq_ind_r 
+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)) t 
+H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda 
+(H2: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
+(THead (Bind Abst) u1 t1))).(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) H5) in ((let H7 \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) H5) in ((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 _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind 
+Abst) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind 
+Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead 
+(Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 
+(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def 
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T 
+u0 (\lambda (t: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: 
+T).(pr0 t u2)) H1 u1 H9) in (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)) H14 H12))))) k H10)))) 
+H7)) H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: 
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 
+t1)) \to (ex3_2 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))))))).(\lambda (H5: (eq T (THead (Flat Appl) 
+v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind 
+T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee 
+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) H5) in (False_ind (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: B).(\lambda 
+(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abst) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (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 (THead (Flat Appl) v1 (THead (Bind 
+b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee 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) H8) in (False_ind (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 (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) 
+H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 
+w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 
+t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee: 
+T).(match ee 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) H6) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abst) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b 
+Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(H3: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 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))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O 
+t0)) (THead (Bind Abst) u1 t1))).(let H5 \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) H4) in 
+((let H6 \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) H4) in ((let H7 \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) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abst)).(let H10 
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H9) in (let 
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abst) u1 t)) 
+\to (ex3_2 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 t t3)))))) H3 (lift (S O) O t0) H7) in (eq_ind T 
+(lift (S O) O t0) (\lambda (t: T).(ex3_2 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 t t3))))) (let H12 
+\def (match (H10 (refl_equal B Abst)) in False return (\lambda (_: 
+False).(ex3_2 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 (lift (S O) O t0) t3))))) with []) in H12) t1 
+H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: 
+(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 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))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u 
+t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u 
+t0) (\lambda (ee: T).(match ee 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) H3) in (False_ind (ex3_2 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)))) H4)))))))) y x H0))) H)))).
+
+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)).(insert_eq T (THead (Flat Appl) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(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 (y: 
+T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T 
+t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(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)))))))))))) (\lambda (t: 
+T).(\lambda (H1: (eq T t (THead (Flat Appl) u1 t1))).(let H2 \def (f_equal T 
+T (\lambda (e: T).e) t (THead (Flat Appl) u1 t1) H1) in (eq_ind_r 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))) t H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 
+u2)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
+Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr0 u1 u3))))) (\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 
+(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O u3) t2))))))))) (\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 
+(t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 t1)) \to 
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 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 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)))))))))))).(\lambda (k: K).(\lambda (H5: (eq 
+T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(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) H5) in ((let H7 \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) H5) in ((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 _ _ t) \Rightarrow t])) 
+(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0 
+u1)).(\lambda (H10: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda 
+(k0: K).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 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 k0 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 k0 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)))))))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq 
+T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T 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 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 
+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))))))))))) H4 t1 H8) in (let 
+H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def 
+(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (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 u2 (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 u2 (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))))))))))) H2 u1 H9) in (let H14 \def (eq_ind 
+T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (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)) H14 H12)))))) k H10)))) H7)) 
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(H1: (pr0 v1 v2)).(\lambda (H2: (((eq T v1 (THead (Flat Appl) u1 t1)) \to 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (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 v2 (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 (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind 
+b) v3 (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 (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 
+t1)) \to (or3 (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 (v3: T).(\lambda (t3: T).(eq T 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)))))))))))).(\lambda (H5: (eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \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) H5) in ((let H7 \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) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T v1 
+(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (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 v2 (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 v2 (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))))))))))) H2 u1 H8) in (let H10 \def (eq_ind T v1 
+(\lambda (t: T).(pr0 t v2)) H1 u1 H8) in (let H11 \def (eq_ind_r T t1 
+(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (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 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 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 t (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T 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))))))))))) H4 (THead (Bind Abst) u t0) H7) in (let H12 
+\def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 t)) \to 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (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 v2 (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 v2 (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))))))))))) H9 (THead (Bind Abst) u t0) H7) in 
+(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(or3 (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 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) 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 t (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)))))))))) (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)) H10 H3)) t1 H7))))))) H6)))))))))))) (\lambda (b: B).(\lambda (H1: (not 
+(eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H2: (pr0 v1 
+v2)).(\lambda (H3: (((eq T v1 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (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 v2 (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 (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 (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind b0) v3 (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 (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t2: T).(pr0 z1 t2)))))))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda 
+(H4: (pr0 u0 u2)).(\lambda (H5: (((eq T u0 (THead (Flat Appl) u1 t1)) \to 
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) 
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (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 (t2: T).(eq T u2 (THead (Bind 
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\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 (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H6: (pr0 t0 t2)).(\lambda (H7: (((eq T t0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 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 t1 
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T 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)))))))))))).(\lambda (H8: (eq T (THead (Flat 
+Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \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) H8) in ((let H10 \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) 
+H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t: 
+T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T v2 (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 v2 (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 v2 (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))))))))))) H3 u1 H11) in (let H13 \def (eq_ind T v1 
+(\lambda (t: T).(pr0 t v2)) H2 u1 H11) in (let H14 \def (eq_ind_r T t1 
+(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (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 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 t (THead (Bind 
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T 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))))))))))) H7 (THead (Bind b) u0 t0) H10) in (let H15 \def 
+(eq_ind_r T t1 (\lambda (t: T).((eq T u0 (THead (Flat Appl) u1 t)) \to (or3 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (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 u2 (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 u2 (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))))))))))) H5 (THead (Bind b) u0 t0) H10) in 
+(let H16 \def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 
+t)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T v2 (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 v2 (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 v2 (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))))))))))) H12 (THead (Bind b) u0 t0) H10) in 
+(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).(or3 (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 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 (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 t 
+(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)))))))))) (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 H1 (refl_equal T (THead (Bind b) u0 t0)) 
+(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))) 
+H13 H4 H6)) t1 H10)))))))) H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (u3: T).(\lambda 
+(t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\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 (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind 
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\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 (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 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 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)))))))))))).(\lambda (w: T).(\lambda (_: 
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead 
+(Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) 
+(\lambda (ee: T).(match ee 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) H6) in (False_ind 
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 
+w) (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 (Bind Abbr) u2 w) (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 (Bind Abbr) u2 w) (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))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not 
+(eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 t1)) \to (or3 (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 (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 t2 (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)))))))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead 
+(Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 \def 
+(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee 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) H4) in (False_ind (or3 (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 (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 t2 (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))))))))) H5)))))))))) (\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 
+t1)) \to (or3 (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)))))))))))).(\lambda (u: T).(\lambda (H3: (eq 
+T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(let H4 \def (eq_ind T 
+(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee 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) H3) in (False_ind (or3 (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))))))))) H4)))))))) y x H0))) H)))).
+
+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)).(insert_eq T (THead (Flat Cast) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))) 
+(\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda 
+(t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2)))) (pr0 t1 t0))))) (\lambda (t: T).(\lambda (H1: (eq T t (THead (Flat 
+Cast) u1 t1))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (THead (Flat 
+Cast) u1 t1) H1) in (eq_ind_r 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))) t H2)))) (\lambda 
+(u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 t2)).(\lambda 
+(H4: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) 
+(THead (Flat Cast) u1 t1))).(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) H5) in ((let H7 \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) H5) in ((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 _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat 
+Cast) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat 
+Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead k0 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 k0 u2 t2)))) (let H11 \def (eq_ind T t0 
+(\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(pr0 t1 t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0 
+(\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda 
+(t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: 
+T).(pr0 t u2)) H1 u1 H9) in (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)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+v2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (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))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee 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) H5) in (False_ind (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) 
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) v2 
+t2))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b 
+Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda 
+(_: (((eq T v1 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2)))) (pr0 t1 v2))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
+u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda 
+(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead 
+(Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9 
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: 
+T).(match ee 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) H8) in 
+(False_ind (or (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 Cast) u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: T).(\lambda 
+(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) 
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead 
+(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq 
+T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 
+t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T 
+(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(let H7 \def (eq_ind T 
+(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee 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) H6) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T (THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 
+t3)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H7))))))))))))) (\lambda (b: 
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 
+t1)) \to (or (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))))).(\lambda (u: 
+T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat 
+Cast) u1 t1))).(let H5 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) 
+(\lambda (ee: T).(match ee 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) H4) in (False_ind 
+(or (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)) H5)))))))))) (\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0 
+(THead (Flat Cast) u1 t1)) \to (or (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))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead 
+(Flat Cast) u1 t1))).(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) H3) in ((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 (Flat Cast) 
+u t0) (THead (Flat Cast) u1 t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def 
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or 
+(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)))) H2 t1 H5) in (let H8 \def 
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 t1 H5) in (or_intror (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) H8))))) H4)))))))) y x H0))) H)))).
+
+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)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(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 (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: 
+T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: 
+T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: 
+T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T 
+T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r 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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
+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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind 
+Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t 
+H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda 
+(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 
+t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 
+t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind 
+Abbr) u1 t1))).(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) 
+H5) in ((let H7 \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) 
+H5) in ((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 _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) 
+H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind 
+Abbr))).(eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead k0 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 (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let 
+H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to 
+(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4 
+t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in 
+(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2 
+u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in 
+(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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind 
+Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7)) 
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: 
+(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O 
+v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
+t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def 
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: 
+T).(match ee 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) H5) in (False_ind (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) 
+(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 (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S 
+O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: 
+(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 
+v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead 
+(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq 
+T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S 
+O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T 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 (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq 
+T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 
+t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) 
+(\lambda (ee: T).(match ee 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) H8) in (False_ind 
+(or (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 (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: 
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) (\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 
+(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T 
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 
+t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: 
+(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq 
+T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \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) H6) in ((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 _ _ t) 
+\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in 
+(\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda (t: T).((eq T 
+t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T 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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 
+t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T t0 (\lambda (t: 
+T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).((eq T 
+t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T u2 (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 
+t1 (lift (S O) O u2))))) H2 u1 H9) in (let H13 \def (eq_ind T u0 (\lambda (t: 
+T).(pr0 t u2)) H1 u1 H9) in (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 (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O 
+u3 y0 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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind 
+Abbr) u2 w)) H13 (or_intror (pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1 
+y0)) (\lambda (y0: T).(subst0 O u2 y0 w)) t2 H11 H5)))))))))) H7))))))))))))) 
+(\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda 
+(t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind 
+Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(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 (y0: 
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S 
+O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S 
+O) O t0)) (THead (Bind Abbr) u1 t1))).(let H5 \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) H4) in 
+((let H6 \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) H4) in ((let H7 \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) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10 
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let 
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t)) 
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3 
+(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2)))) 
+(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (y0: 
+T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) 
+(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1 
+H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_: 
+(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O 
+t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead 
+(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda 
+(ee: T).(match ee 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) H3) in (False_ind 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (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 (y0: T).(pr0 t1 y0)) 
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2))) 
+H4)))))))) y x H0))) H)))).
+
+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)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t: 
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) 
+O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: 
+T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda 
+(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda 
+(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r 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))) t H2)))) (\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1 
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T 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 t2)))))).(\lambda (k: K).(\lambda 
+(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(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) H5) in ((let H7 \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) H5) in ((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 _ _ t) \Rightarrow t])) 
+(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0 
+u1)).(\lambda (H10: (eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda 
+(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 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 k0 u2 t2))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t 
+(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: 
+T).(eq T 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 t2))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: 
+T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T 
+t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T u2 (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 u2))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: 
+T).(pr0 t u2)) H1 u1 H9) in (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)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 
+(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T v2 (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 v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (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)))))).(\lambda (H5: (eq T (THead 
+(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6 
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: 
+T).(match ee 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) H5) in (False_ind (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 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 (THead 
+(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B 
+b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 
+v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (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 v2)))))).(\lambda (u0: T).(\lambda 
+(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) 
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead 
+(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O 
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda 
+(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T 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 t2)))))).(\lambda (H8: (eq T (THead (Flat Appl) 
+v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T 
+(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee 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) H8) in (False_ind (or (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 (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 b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))))) 
+H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0 
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda 
+(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) 
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T 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 
+t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T 
+(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T 
+(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee 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) H6) in (False_ind (or 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) 
+(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 Abbr) u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B 
+b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 
+t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (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)))))).(\lambda (u: T).(\lambda 
+(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 
+t1))).(let H5 \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) H4) in ((let H6 \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) H4) in ((let H7 \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) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Void)).(let H10 
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Void H9) in (let 
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Void) u1 t)) 
+\to (or (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 t t3)))) (pr0 t (lift (S O) O t2))))) H3 (lift (S O) 
+O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or (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 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror (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 (lift 
+(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 
+H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: 
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 
+t1)) \to (or (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)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead 
+(Bind Void) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda 
+(ee: T).(match ee 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) H3) in (False_ind 
+(or (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))) H4)))))))) y x 
+H0))) H)))).
+
+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)) (\lambda (_: T).(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_sym x1 (S O))) in (let H9 \def (eq_ind nat (S x1) 
+(\lambda (n: nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O)) 
+H8) in (ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4: 
+T).(eq T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 
+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_tau 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))))).
+