X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fsubst0%2Ffwd.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fsubst0%2Ffwd.ma;h=4b387483e16e9e23b29af037f9440f983b9f2842;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma
new file mode 100644
index 000000000..4b387483e
--- /dev/null
+++ b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/subst0/fwd.ma
@@ -0,0 +1,710 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/subst0/defs.ma".
+
+include "LambdaDelta-1/lift/props.ma".
+
+theorem subst0_gen_sort:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
+i v (TSort n) x) \to (\forall (P: Prop).P)))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
+(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n) 
+(\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda 
+(H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda 
+(t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_: 
+T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def 
+(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (TSort n) H1) in (False_ind P H2))))) 
+(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: 
+nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) 
+\to P))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t) 
+(TSort n))).(let H4 \def (eq_ind T (THead k u1 t) (\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 P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: 
+T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 t1 
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda 
+(H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k 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 P H4))))))))))) (\lambda (v0: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 
+i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (k: 
+K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (s k i0) v0 t1 
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\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 P H6)))))))))))))) i v y x H0))) H)))))).
+
+theorem subst0_gen_lref:
+ \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 
+i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
+\def
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda 
+(H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0 
+i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v)))) 
+(\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0: 
+nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n)) 
+\to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0: 
+T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def 
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with 
+[(TSort _) \Rightarrow i0 | (TLRef n0) \Rightarrow n0 | (THead _ _ _) 
+\Rightarrow i0])) (TLRef i0) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0: 
+nat).(land (eq nat n n0) (eq T (lift (S n0) O v0) (lift (S n) O v0)))) (conj 
+(eq nat n n) (eq T (lift (S n) O v0) (lift (S n) O v0)) (refl_equal nat n) 
+(refl_equal T (lift (S n) O v0))) i0 H2))))) (\lambda (v0: T).(\lambda (u2: 
+T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 
+u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 
+(lift (S n) O v0)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T 
+(THead k u1 t) (TLRef n))).(let H4 \def (eq_ind T (THead k u1 t) (\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 (land (eq nat n i0) (eq T (THead k u2 
+t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0: 
+T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 
+(s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s 
+k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T 
+(THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k 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 (land (eq nat n i0) (eq T (THead k u 
+t2) (lift (S n) O v0))) H4))))))))))) (\lambda (v0: T).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1 
+u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2 
+(lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: 
+T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef 
+n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O v0)))))).(\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 (land (eq nat n i0) (eq T (THead k u2 
+t2) (lift (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))).
+
+theorem subst0_gen_head:
+ \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall 
+(x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T 
+(\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: 
+T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 
+u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))))
+\def
+ \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda 
+(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) 
+x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda 
+(_: T).(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: 
+T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) 
+(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 
+t2)))))) (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda 
+(n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k 
+u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda 
+(u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 
+t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1 
+t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef 
+i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee: 
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
+(THead k u1 t1) H1) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq T (lift (S 
+i0) O v0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T 
+(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u1 t2))) (\lambda (t2: 
+T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (lift (S i0) O v0) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) 
+v0 t1 t2))))) H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u0: 
+T).(\lambda (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq 
+T u0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 
+t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T 
+u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (t: T).(\lambda (k0: 
+K).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H4 \def 
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with 
+[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
+\Rightarrow k1])) (THead k0 u0 t) (THead k 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 u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _) 
+\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H6 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) 
+\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T 
+u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T 
+(\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3: 
+T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) 
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda 
+(t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead 
+k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k 
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind 
+T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda 
+(u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) 
+(ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 
+(s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 
+(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7) 
+in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1 
+H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 
+t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T 
+(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 
+t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) 
+(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T 
+(\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: 
+T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0 
+H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2: 
+T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0 
+t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: 
+T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) 
+(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 
+(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
+t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 
+t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda 
+(_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead 
+k1 _ _) \Rightarrow k1])) (THead k0 u t0) (THead k 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 u | (TLRef _) \Rightarrow u | (THead _ t _) 
+\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H6 \def 
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with 
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) 
+\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in (\lambda (H7: (eq T u 
+u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T 
+(\lambda (u2: T).(eq T (THead k0 t t2) (THead k u2 t1))) (\lambda (u2: 
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k0 t t2) 
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H9 \def (eq_ind T t0 
+(\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: 
+T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2))) 
+(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 
+(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0 
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
+t3))))))) H2 t1 H6) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s 
+k0 i0) v0 t t2)) H1 t1 H6) in (let H11 \def (eq_ind K k0 (\lambda (k1: 
+K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 
+(THead k u2 t1))) (\lambda (u2: T).(subst0 (s k1 i0) v0 u1 u2))) (ex2 T 
+(\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s 
+k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
+(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 
+t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s 
+k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T 
+(\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) (\lambda (u2: 
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u1 t2) 
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T 
+(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: 
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) 
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3: 
+T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) 
+v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5)) 
+H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda 
+(i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead 
+k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2 
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0: 
+T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4: 
+(((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead 
+k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda 
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) 
+v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 
+t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6 
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
+with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) 
+\Rightarrow k1])) (THead k0 u0 t0) (THead k 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 k0 u0 t0) (THead k 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 k0 u0 t0) (THead k u1 t1) H5) in (\lambda (H9: (eq T 
+u0 u1)).(\lambda (H10: (eq K k0 k)).(let H11 \def (eq_ind T t0 (\lambda (t: 
+T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead 
+k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda 
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0)) 
+v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3))) 
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H4 
+t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t 
+t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1 
+(THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T 
+t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))) 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 u3))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))))))) H11 k H10) in 
+(let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s k1 i0) v0 t1 t2)) H12 
+k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T (\lambda (u3: T).(eq T 
+(THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) 
+(ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u1 t3))) (\lambda 
+(t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda 
+(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k 
+i0) v0 t1 t3)))))) (let H15 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead 
+k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T u2 
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 \def (eq_ind T u0 
+(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in (or3_intro2 (ex2 T (\lambda 
+(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 
+v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3))) 
+(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: 
+T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda 
+(t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda 
+(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k 
+i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H16 H14)))) k0 
+H10)))))))) H7)) H6)))))))))))))) i v y x H0))) H))))))).
+
+theorem subst0_gen_lift_lt:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t1) 
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
+(t2: T).(subst0 i u t1 t2)))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: 
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift h d 
+u) (lift h (S (plus i d)) t) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h 
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2))))))))) (\lambda (n: 
+nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S (plus i d)) (TSort n)) 
+x)).(let H0 \def (eq_ind T (lift h (S (plus i d)) (TSort n)) (\lambda (t: 
+T).(subst0 i (lift h d u) t x)) H (TSort n) (lift_sort n h (S (plus i d)))) 
+in (subst0_gen_sort (lift h d u) x i n H0 (ex2 T (\lambda (t2: T).(eq T x 
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TSort n) 
+t2))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: nat).(\lambda 
+(h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i (lift h d u) (lift h (S 
+(plus i d)) (TLRef n)) x)).(lt_le_e n (S (plus i d)) (ex2 T (\lambda (t2: 
+T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef 
+n) t2))) (\lambda (H0: (lt n (S (plus i d)))).(let H1 \def (eq_ind T (lift h 
+(S (plus i d)) (TLRef n)) (\lambda (t: T).(subst0 i (lift h d u) t x)) H 
+(TLRef n) (lift_lref_lt n h (S (plus i d)) H0)) in (land_ind (eq nat n i) (eq 
+T x (lift (S n) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S 
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: 
+(eq nat n i)).(\lambda (H3: (eq T x (lift (S n) O (lift h d u)))).(eq_ind_r T 
+(lift (S n) O (lift h d u)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t 
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2)))) 
+(eq_ind_r nat i (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S n0) 
+O (lift h d u)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
+(TLRef n0) t2)))) (eq_ind T (lift h (plus (S i) d) (lift (S i) O u)) (\lambda 
+(t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h (S (plus i d)) t2))) (\lambda 
+(t2: T).(subst0 i u (TLRef i) t2)))) (ex_intro2 T (\lambda (t2: T).(eq T 
+(lift h (S (plus i d)) (lift (S i) O u)) (lift h (S (plus i d)) t2))) 
+(\lambda (t2: T).(subst0 i u (TLRef i) t2)) (lift (S i) O u) (refl_equal T 
+(lift h (S (plus i d)) (lift (S i) O u))) (subst0_lref u i)) (lift (S i) O 
+(lift h d u)) (lift_d u h (S i) d O (le_O_n d))) n H2) x H3))) 
+(subst0_gen_lref (lift h d u) x i n H1)))) (\lambda (H0: (le (S (plus i d)) 
+n)).(let H1 \def (eq_ind T (lift h (S (plus i d)) (TLRef n)) (\lambda (t: 
+T).(subst0 i (lift h d u) t x)) H (TLRef (plus n h)) (lift_lref_ge n h (S 
+(plus i d)) H0)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n 
+h)) O (lift h d u))) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
+t2))) (\lambda (t2: T).(subst0 i u (TLRef n) t2))) (\lambda (H2: (eq nat 
+(plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n h)) O (lift h d 
+u)))).(let H4 \def (eq_ind_r nat i (\lambda (n0: nat).(le (S (plus n0 d)) n)) 
+H0 (plus n h) H2) in (le_false n (plus (plus n h) d) (ex2 T (\lambda (t2: 
+T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (TLRef 
+n) t2))) (le_plus_trans n (plus n h) d (le_plus_l n h)) H4)))) 
+(subst0_gen_lref (lift h d u) x i (plus n h) H1))))))))))) (\lambda (k: 
+K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst0 i (lift h d u) (lift h (S (plus i d)) t) 
+x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda 
+(t2: T).(subst0 i u t t2)))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall 
+(x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i (lift 
+h d u) (lift h (S (plus i d)) t0) x) \to (ex2 T (\lambda (t2: T).(eq T x 
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t0 
+t2)))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H1: (subst0 i (lift h d u) (lift h (S (plus i d)) (THead k t 
+t0)) x)).(let H2 \def (eq_ind T (lift h (S (plus i d)) (THead k t t0)) 
+(\lambda (t2: T).(subst0 i (lift h d u) t2 x)) H1 (THead k (lift h (S (plus i 
+d)) t) (lift h (s k (S (plus i d))) t0)) (lift_head k t t0 h (S (plus i d)))) 
+in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k (S (plus 
+i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S (plus i d)) 
+t) u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) 
+t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i 
+d))) t0) t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k 
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S 
+(plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h 
+d u) (lift h (s k (S (plus i d))) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x 
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) 
+t2))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k 
+(S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h (S 
+(plus i d)) t) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h 
+(s k (S (plus i d))) t0)))) (\lambda (u2: T).(subst0 i (lift h d u) (lift h 
+(S (plus i d)) t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
+t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: 
+T).(\lambda (H4: (eq T x (THead k x0 (lift h (s k (S (plus i d))) 
+t0)))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) t) 
+x0)).(eq_ind_r T (THead k x0 (lift h (s k (S (plus i d))) t0)) (\lambda (t2: 
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda 
+(t3: T).(subst0 i u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T 
+x0 (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T 
+(\lambda (t2: T).(eq T (THead k x0 (lift h (s k (S (plus i d))) t0)) (lift h 
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) 
+(\lambda (x1: T).(\lambda (H6: (eq T x0 (lift h (S (plus i d)) x1))).(\lambda 
+(H7: (subst0 i u t x1)).(eq_ind_r T (lift h (S (plus i d)) x1) (\lambda (t2: 
+T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 (lift h (s k (S (plus i d))) 
+t0)) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) 
+t3)))) (eq_ind T (lift h (S (plus i d)) (THead k x1 t0)) (\lambda (t2: 
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda 
+(t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T 
+(lift h (S (plus i d)) (THead k x1 t0)) (lift h (S (plus i d)) t2))) (\lambda 
+(t2: T).(subst0 i u (THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h 
+(S (plus i d)) (THead k x1 t0))) (subst0_fst u x1 t i H7 t0 k)) (THead k 
+(lift h (S (plus i d)) x1) (lift h (s k (S (plus i d))) t0)) (lift_head k x1 
+t0 h (S (plus i d)))) x0 H6)))) (H x0 i h d H5)) x H4)))) H3)) (\lambda (H3: 
+(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i d)) t) t2))) 
+(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) 
+t0) t2)))).(ex2_ind T (\lambda (t2: T).(eq T x (THead k (lift h (S (plus i 
+d)) t) t2))) (\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S 
+(plus i d))) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) 
+t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x0: 
+T).(\lambda (H4: (eq T x (THead k (lift h (S (plus i d)) t) x0))).(\lambda 
+(H5: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) 
+x0)).(eq_ind_r T (THead k (lift h (S (plus i d)) t) x0) (\lambda (t2: T).(ex2 
+T (\lambda (t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: 
+T).(subst0 i u (THead k t t0) t3)))) (let H6 \def (eq_ind nat (s k (S (plus i 
+d))) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x0)) H5 (S 
+(s k (plus i d))) (s_S k (plus i d))) in (let H7 \def (eq_ind nat (s k (plus 
+i d)) (\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x0)) 
+H6 (plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x0 
+(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) 
+(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) x0) (lift h 
+(S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2))) 
+(\lambda (x1: T).(\lambda (H8: (eq T x0 (lift h (S (plus (s k i) d)) 
+x1))).(\lambda (H9: (subst0 (s k i) u t0 x1)).(eq_ind_r T (lift h (S (plus (s 
+k i) d)) x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h 
+(S (plus i d)) t) t2) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i 
+u (THead k t t0) t3)))) (eq_ind nat (s k (plus i d)) (\lambda (n: nat).(ex2 T 
+(\lambda (t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h (S n) x1)) 
+(lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) 
+t2)))) (eq_ind nat (s k (S (plus i d))) (\lambda (n: nat).(ex2 T (\lambda 
+(t2: T).(eq T (THead k (lift h (S (plus i d)) t) (lift h n x1)) (lift h (S 
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind 
+T (lift h (S (plus i d)) (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda 
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u 
+(THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift h (S (plus i 
+d)) (THead k t x1)) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
+(THead k t t0) t2)) (THead k t x1) (refl_equal T (lift h (S (plus i d)) 
+(THead k t x1))) (subst0_snd k u x1 t0 i H9 t)) (THead k (lift h (S (plus i 
+d)) t) (lift h (s k (S (plus i d))) x1)) (lift_head k t x1 h (S (plus i d)))) 
+(S (s k (plus i d))) (s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x0 
+H8)))) (H0 x0 (s k i) h d H7)))) x H4)))) H3)) (\lambda (H3: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(subst0 i (lift h d u) (lift h (S (plus i d)) t) u2))) 
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) (lift h d u) (lift h (s k (S 
+(plus i d))) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq 
+T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i (lift h d 
+u) (lift h (S (plus i d)) t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
+(s k i) (lift h d u) (lift h (s k (S (plus i d))) t0) t2))) (ex2 T (\lambda 
+(t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
+(THead k t t0) t2))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T x 
+(THead k x0 x1))).(\lambda (H5: (subst0 i (lift h d u) (lift h (S (plus i d)) 
+t) x0)).(\lambda (H6: (subst0 (s k i) (lift h d u) (lift h (s k (S (plus i 
+d))) t0) x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda 
+(t3: T).(eq T t2 (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u 
+(THead k t t0) t3)))) (let H7 \def (eq_ind nat (s k (S (plus i d))) (\lambda 
+(n: nat).(subst0 (s k i) (lift h d u) (lift h n t0) x1)) H6 (S (s k (plus i 
+d))) (s_S k (plus i d))) in (let H8 \def (eq_ind nat (s k (plus i d)) 
+(\lambda (n: nat).(subst0 (s k i) (lift h d u) (lift h (S n) t0) x1)) H7 
+(plus (s k i) d) (s_plus k i d)) in (ex2_ind T (\lambda (t2: T).(eq T x1 
+(lift h (S (plus (s k i) d)) t2))) (\lambda (t2: T).(subst0 (s k i) u t0 t2)) 
+(ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h (S (plus i d)) t2))) 
+(\lambda (t2: T).(subst0 i u (THead k t t0) t2))) (\lambda (x2: T).(\lambda 
+(H9: (eq T x1 (lift h (S (plus (s k i) d)) x2))).(\lambda (H10: (subst0 (s k 
+i) u t0 x2)).(ex2_ind T (\lambda (t2: T).(eq T x0 (lift h (S (plus i d)) 
+t2))) (\lambda (t2: T).(subst0 i u t t2)) (ex2 T (\lambda (t2: T).(eq T 
+(THead k x0 x1) (lift h (S (plus i d)) t2))) (\lambda (t2: T).(subst0 i u 
+(THead k t t0) t2))) (\lambda (x3: T).(\lambda (H11: (eq T x0 (lift h (S 
+(plus i d)) x3))).(\lambda (H12: (subst0 i u t x3)).(eq_ind_r T (lift h (S 
+(plus i d)) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 
+x1) (lift h (S (plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) 
+t3)))) (eq_ind_r T (lift h (S (plus (s k i) d)) x2) (\lambda (t2: T).(ex2 T 
+(\lambda (t3: T).(eq T (THead k (lift h (S (plus i d)) x3) t2) (lift h (S 
+(plus i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (eq_ind 
+nat (s k (plus i d)) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k 
+(lift h (S (plus i d)) x3) (lift h (S n) x2)) (lift h (S (plus i d)) t2))) 
+(\lambda (t2: T).(subst0 i u (THead k t t0) t2)))) (eq_ind nat (s k (S (plus 
+i d))) (\lambda (n: nat).(ex2 T (\lambda (t2: T).(eq T (THead k (lift h (S 
+(plus i d)) x3) (lift h n x2)) (lift h (S (plus i d)) t2))) (\lambda (t2: 
+T).(subst0 i u (THead k t t0) t2)))) (eq_ind T (lift h (S (plus i d)) (THead 
+k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h (S (plus 
+i d)) t3))) (\lambda (t3: T).(subst0 i u (THead k t t0) t3)))) (ex_intro2 T 
+(\lambda (t2: T).(eq T (lift h (S (plus i d)) (THead k x3 x2)) (lift h (S 
+(plus i d)) t2))) (\lambda (t2: T).(subst0 i u (THead k t t0) t2)) (THead k 
+x3 x2) (refl_equal T (lift h (S (plus i d)) (THead k x3 x2))) (subst0_both u 
+t x3 i H12 k t0 x2 H10)) (THead k (lift h (S (plus i d)) x3) (lift h (s k (S 
+(plus i d))) x2)) (lift_head k x3 x2 h (S (plus i d)))) (S (s k (plus i d))) 
+(s_S k (plus i d))) (plus (s k i) d) (s_plus k i d)) x1 H9) x0 H11)))) (H x0 
+i h d H5))))) (H0 x1 (s k i) h d H8)))) x H4)))))) H3)) (subst0_gen_head k 
+(lift h d u) (lift h (S (plus i d)) t) (lift h (s k (S (plus i d))) t0) x i 
+H2))))))))))))) t1)).
+
+theorem subst0_gen_lift_false:
+ \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall 
+(d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst0 i u 
+(lift h d t) x) \to (\forall (P: Prop).P)))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (u: T).(\forall (x: 
+T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i 
+(plus d h)) \to ((subst0 i u (lift h d t0) x) \to (\forall (P: 
+Prop).P)))))))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (x: T).(\lambda 
+(h: nat).(\lambda (d: nat).(\lambda (i: nat).(\lambda (_: (le d i)).(\lambda 
+(_: (lt i (plus d h))).(\lambda (H1: (subst0 i u (lift h d (TSort n)) 
+x)).(\lambda (P: Prop).(let H2 \def (eq_ind T (lift h d (TSort n)) (\lambda 
+(t0: T).(subst0 i u t0 x)) H1 (TSort n) (lift_sort n h d)) in 
+(subst0_gen_sort u x i n H2 P)))))))))))) (\lambda (n: nat).(\lambda (u: 
+T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (i: 
+nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d h))).(\lambda (H1: 
+(subst0 i u (lift h d (TLRef n)) x)).(\lambda (P: Prop).(lt_le_e n d P 
+(\lambda (H2: (lt n d)).(let H3 \def (eq_ind T (lift h d (TLRef n)) (\lambda 
+(t0: T).(subst0 i u t0 x)) H1 (TLRef n) (lift_lref_lt n h d H2)) in (land_ind 
+(eq nat n i) (eq T x (lift (S n) O u)) P (\lambda (H4: (eq nat n i)).(\lambda 
+(_: (eq T x (lift (S n) O u))).(let H6 \def (eq_ind nat n (\lambda (n0: 
+nat).(lt n0 d)) H2 i H4) in (le_false d i P H H6)))) (subst0_gen_lref u x i n 
+H3)))) (\lambda (H2: (le d n)).(let H3 \def (eq_ind T (lift h d (TLRef n)) 
+(\lambda (t0: T).(subst0 i u t0 x)) H1 (TLRef (plus n h)) (lift_lref_ge n h d 
+H2)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S (plus n h)) O u)) P 
+(\lambda (H4: (eq nat (plus n h) i)).(\lambda (_: (eq T x (lift (S (plus n 
+h)) O u))).(let H6 \def (eq_ind_r nat i (\lambda (n0: nat).(lt n0 (plus d 
+h))) H0 (plus n h) H4) in (le_false d n P H2 (lt_le_S n d (simpl_lt_plus_r h 
+n d H6)))))) (subst0_gen_lref u x i (plus n h) H3))))))))))))))) (\lambda (k: 
+K).(\lambda (t0: T).(\lambda (H: ((\forall (u: T).(\forall (x: T).(\forall 
+(h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) 
+\to ((subst0 i u (lift h d t0) x) \to (\forall (P: 
+Prop).P))))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (u: T).(\forall 
+(x: T).(\forall (h: nat).(\forall (d: nat).(\forall (i: nat).((le d i) \to 
+((lt i (plus d h)) \to ((subst0 i u (lift h d t1) x) \to (\forall (P: 
+Prop).P))))))))))).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (i: nat).(\lambda (H1: (le d i)).(\lambda (H2: (lt i (plus 
+d h))).(\lambda (H3: (subst0 i u (lift h d (THead k t0 t1)) x)).(\lambda (P: 
+Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: 
+T).(subst0 i u t2 x)) H3 (THead k (lift h d t0) (lift h (s k d) t1)) 
+(lift_head k t0 t1 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead k 
+u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2))) 
+(ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: 
+T).(subst0 (s k i) u (lift h (s k d) t1) t2))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u (lift h d t0) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 
+(s k i) u (lift h (s k d) t1) t2)))) P (\lambda (H5: (ex2 T (\lambda (u2: 
+T).(eq T x (THead k u2 (lift h (s k d) t1)))) (\lambda (u2: T).(subst0 i u 
+(lift h d t0) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead k u2 (lift h 
+(s k d) t1)))) (\lambda (u2: T).(subst0 i u (lift h d t0) u2)) P (\lambda 
+(x0: T).(\lambda (_: (eq T x (THead k x0 (lift h (s k d) t1)))).(\lambda (H7: 
+(subst0 i u (lift h d t0) x0)).(H u x0 h d i H1 H2 H7 P)))) H5)) (\lambda 
+(H5: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t0) t2))) (\lambda 
+(t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2)))).(ex2_ind T (\lambda (t2: 
+T).(eq T x (THead k (lift h d t0) t2))) (\lambda (t2: T).(subst0 (s k i) u 
+(lift h (s k d) t1) t2)) P (\lambda (x0: T).(\lambda (_: (eq T x (THead k 
+(lift h d t0) x0))).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t1) 
+x0)).(H0 u x0 h (s k d) (s k i) (s_le k d i H1) (eq_ind nat (s k (plus d h)) 
+(\lambda (n: nat).(lt (s k i) n)) (lt_le_S (s k i) (s k (plus d h)) (s_lt k i 
+(plus d h) H2)) (plus (s k d) h) (s_plus k d h)) H7 P)))) H5)) (\lambda (H5: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))))).(ex3_2_ind 
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda 
+(u2: T).(\lambda (_: T).(subst0 i u (lift h d t0) u2))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t1) t2))) P (\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (_: (eq T x (THead k x0 x1))).(\lambda (H7: 
+(subst0 i u (lift h d t0) x0)).(\lambda (_: (subst0 (s k i) u (lift h (s k d) 
+t1) x1)).(H u x0 h d i H1 H2 H7 P)))))) H5)) (subst0_gen_head k u (lift h d 
+t0) (lift h (s k d) t1) x i H4))))))))))))))))) t).
+
+theorem subst0_gen_lift_ge:
+ \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst0 i u (lift h d t1) x) \to ((le (plus d h) 
+i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
+T).(subst0 (minus i h) u t1 t2))))))))))
+\def
+ \lambda (u: T).(\lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: 
+T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i u (lift h 
+d t) x) \to ((le (plus d h) i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d 
+t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)))))))))) (\lambda (n: 
+nat).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d: 
+nat).(\lambda (H: (subst0 i u (lift h d (TSort n)) x)).(\lambda (_: (le (plus 
+d h) i)).(let H1 \def (eq_ind T (lift h d (TSort n)) (\lambda (t: T).(subst0 
+i u t x)) H (TSort n) (lift_sort n h d)) in (subst0_gen_sort u x i n H1 (ex2 
+T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i 
+h) u (TSort n) t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (i: 
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (subst0 i u (lift h d 
+(TLRef n)) x)).(\lambda (H0: (le (plus d h) i)).(lt_le_e n d (ex2 T (\lambda 
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef 
+n) t2))) (\lambda (H1: (lt n d)).(let H2 \def (eq_ind T (lift h d (TLRef n)) 
+(\lambda (t: T).(subst0 i u t x)) H (TLRef n) (lift_lref_lt n h d H1)) in 
+(land_ind (eq nat n i) (eq T x (lift (S n) O u)) (ex2 T (\lambda (t2: T).(eq 
+T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (TLRef n) t2))) 
+(\lambda (H3: (eq nat n i)).(\lambda (_: (eq T x (lift (S n) O u))).(let H5 
+\def (eq_ind nat n (\lambda (n0: nat).(lt n0 d)) H1 i H3) in (le_false (plus 
+d h) i (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
+T).(subst0 (minus i h) u (TLRef n) t2))) H0 (le_plus_trans (S i) d h H5))))) 
+(subst0_gen_lref u x i n H2)))) (\lambda (H1: (le d n)).(let H2 \def (eq_ind 
+T (lift h d (TLRef n)) (\lambda (t: T).(subst0 i u t x)) H (TLRef (plus n h)) 
+(lift_lref_ge n h d H1)) in (land_ind (eq nat (plus n h) i) (eq T x (lift (S 
+(plus n h)) O u)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
+(t2: T).(subst0 (minus i h) u (TLRef n) t2))) (\lambda (H3: (eq nat (plus n 
+h) i)).(\lambda (H4: (eq T x (lift (S (plus n h)) O u))).(eq_ind nat (plus n 
+h) (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) 
+(\lambda (t2: T).(subst0 (minus n0 h) u (TLRef n) t2)))) (eq_ind_r T (lift (S 
+(plus n h)) O u) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h d 
+t2))) (\lambda (t2: T).(subst0 (minus (plus n h) h) u (TLRef n) t2)))) 
+(eq_ind_r nat n (\lambda (n0: nat).(ex2 T (\lambda (t2: T).(eq T (lift (S 
+(plus n h)) O u) (lift h d t2))) (\lambda (t2: T).(subst0 n0 u (TLRef n) 
+t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift (S (plus n h)) O u) (lift h 
+d t2))) (\lambda (t2: T).(subst0 n u (TLRef n) t2)) (lift (S n) O u) 
+(eq_ind_r T (lift (plus h (S n)) O u) (\lambda (t: T).(eq T (lift (S (plus n 
+h)) O u) t)) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(eq T (lift (S n0) O 
+u) (lift (plus h (S n)) O u))) (eq_ind_r nat (plus h (S n)) (\lambda (n0: 
+nat).(eq T (lift n0 O u) (lift (plus h (S n)) O u))) (refl_equal T (lift 
+(plus h (S n)) O u)) (S (plus h n)) (plus_n_Sm h n)) (plus n h) (plus_sym n 
+h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans_plus_r O d 
+(plus O (S n)) (le_plus_plus O O d (S n) (le_n O) (le_S d n H1))) (le_O_n 
+d))) (subst0_lref u n)) (minus (plus n h) h) (minus_plus_r n h)) x H4) i 
+H3))) (subst0_gen_lref u x i (plus n h) H2)))))))))))) (\lambda (k: 
+K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (i: nat).(\forall 
+(h: nat).(\forall (d: nat).((subst0 i u (lift h d t) x) \to ((le (plus d h) 
+i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: 
+T).(subst0 (minus i h) u t t2))))))))))).(\lambda (t0: T).(\lambda (H0: 
+((\forall (x: T).(\forall (i: nat).(\forall (h: nat).(\forall (d: 
+nat).((subst0 i u (lift h d t0) x) \to ((le (plus d h) i) \to (ex2 T (\lambda 
+(t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t0 
+t2))))))))))).(\lambda (x: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda 
+(d: nat).(\lambda (H1: (subst0 i u (lift h d (THead k t t0)) x)).(\lambda 
+(H2: (le (plus d h) i)).(let H3 \def (eq_ind T (lift h d (THead k t t0)) 
+(\lambda (t2: T).(subst0 i u t2 x)) H1 (THead k (lift h d t) (lift h (s k d) 
+t0)) (lift_head k t t0 h d)) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x 
+(THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u (lift h d t) 
+u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda 
+(t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2))) (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s 
+k i) u (lift h (s k d) t0) t2)))) (ex2 T (\lambda (t2: T).(eq T x (lift h d 
+t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda 
+(H4: (ex2 T (\lambda (u2: T).(eq T x (THead k u2 (lift h (s k d) t0)))) 
+(\lambda (u2: T).(subst0 i u (lift h d t) u2)))).(ex2_ind T (\lambda (u2: 
+T).(eq T x (THead k u2 (lift h (s k d) t0)))) (\lambda (u2: T).(subst0 i u 
+(lift h d t) u2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda 
+(H5: (eq T x (THead k x0 (lift h (s k d) t0)))).(\lambda (H6: (subst0 i u 
+(lift h d t) x0)).(eq_ind_r T (THead k x0 (lift h (s k d) t0)) (\lambda (t2: 
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 
+(minus i h) u (THead k t t0) t3)))) (ex2_ind T (\lambda (t2: T).(eq T x0 
+(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t t2)) (ex2 T (\lambda 
+(t2: T).(eq T (THead k x0 (lift h (s k d) t0)) (lift h d t2))) (\lambda (t2: 
+T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x1: T).(\lambda (H7: 
+(eq T x0 (lift h d x1))).(\lambda (H8: (subst0 (minus i h) u t x1)).(eq_ind_r 
+T (lift h d x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k t2 
+(lift h (s k d) t0)) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u 
+(THead k t t0) t3)))) (eq_ind T (lift h d (THead k x1 t0)) (\lambda (t2: 
+T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 
+(minus i h) u (THead k t t0) t3)))) (ex_intro2 T (\lambda (t2: T).(eq T (lift 
+h d (THead k x1 t0)) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u 
+(THead k t t0) t2)) (THead k x1 t0) (refl_equal T (lift h d (THead k x1 t0))) 
+(subst0_fst u x1 t (minus i h) H8 t0 k)) (THead k (lift h d x1) (lift h (s k 
+d) t0)) (lift_head k x1 t0 h d)) x0 H7)))) (H x0 i h d H6 H2)) x H5)))) H4)) 
+(\lambda (H4: (ex2 T (\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) 
+(\lambda (t2: T).(subst0 (s k i) u (lift h (s k d) t0) t2)))).(ex2_ind T 
+(\lambda (t2: T).(eq T x (THead k (lift h d t) t2))) (\lambda (t2: T).(subst0 
+(s k i) u (lift h (s k d) t0) t2)) (ex2 T (\lambda (t2: T).(eq T x (lift h d 
+t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda 
+(x0: T).(\lambda (H5: (eq T x (THead k (lift h d t) x0))).(\lambda (H6: 
+(subst0 (s k i) u (lift h (s k d) t0) x0)).(eq_ind_r T (THead k (lift h d t) 
+x0) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d t3))) 
+(\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (ex2_ind T 
+(\lambda (t2: T).(eq T x0 (lift h (s k d) t2))) (\lambda (t2: T).(subst0 
+(minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T (THead k (lift h d 
+t) x0) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) 
+t2))) (\lambda (x1: T).(\lambda (H7: (eq T x0 (lift h (s k d) x1))).(\lambda 
+(H8: (subst0 (minus (s k i) h) u t0 x1)).(eq_ind_r T (lift h (s k d) x1) 
+(\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead k (lift h d t) t2) 
+(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) 
+(eq_ind T (lift h d (THead k t x1)) (\lambda (t2: T).(ex2 T (\lambda (t3: 
+T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t 
+t0) t3)))) (let H9 \def (eq_ind_r nat (minus (s k i) h) (\lambda (n: 
+nat).(subst0 n u t0 x1)) H8 (s k (minus i h)) (s_minus k i h (le_trans_plus_r 
+d h i H2))) in (ex_intro2 T (\lambda (t2: T).(eq T (lift h d (THead k t x1)) 
+(lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead k t t0) t2)) 
+(THead k t x1) (refl_equal T (lift h d (THead k t x1))) (subst0_snd k u x1 t0 
+(minus i h) H9 t))) (THead k (lift h d t) (lift h (s k d) x1)) (lift_head k t 
+x1 h d)) x0 H7)))) (H0 x0 (s k i) h (s k d) H6 (eq_ind nat (s k (plus d h)) 
+(\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i H2) (plus (s k d) h) 
+(s_plus k d h)))) x H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(subst0 i u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s 
+k i) u (lift h (s k d) t0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i 
+u (lift h d t) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) u (lift 
+h (s k d) t0) t2))) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda 
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x0: T).(\lambda 
+(x1: T).(\lambda (H5: (eq T x (THead k x0 x1))).(\lambda (H6: (subst0 i u 
+(lift h d t) x0)).(\lambda (H7: (subst0 (s k i) u (lift h (s k d) t0) 
+x1)).(eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq 
+T t2 (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) 
+t3)))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h (s k d) t2))) (\lambda 
+(t2: T).(subst0 (minus (s k i) h) u t0 t2)) (ex2 T (\lambda (t2: T).(eq T 
+(THead k x0 x1) (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u (THead 
+k t t0) t2))) (\lambda (x2: T).(\lambda (H8: (eq T x1 (lift h (s k d) 
+x2))).(\lambda (H9: (subst0 (minus (s k i) h) u t0 x2)).(ex2_ind T (\lambda 
+(t2: T).(eq T x0 (lift h d t2))) (\lambda (t2: T).(subst0 (minus i h) u t 
+t2)) (ex2 T (\lambda (t2: T).(eq T (THead k x0 x1) (lift h d t2))) (\lambda 
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2))) (\lambda (x3: T).(\lambda 
+(H10: (eq T x0 (lift h d x3))).(\lambda (H11: (subst0 (minus i h) u t 
+x3)).(eq_ind_r T (lift h d x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T 
+(THead k t2 x1) (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead 
+k t t0) t3)))) (eq_ind_r T (lift h (s k d) x2) (\lambda (t2: T).(ex2 T 
+(\lambda (t3: T).(eq T (THead k (lift h d x3) t2) (lift h d t3))) (\lambda 
+(t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (eq_ind T (lift h d 
+(THead k x3 x2)) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h d 
+t3))) (\lambda (t3: T).(subst0 (minus i h) u (THead k t t0) t3)))) (let H12 
+\def (eq_ind_r nat (minus (s k i) h) (\lambda (n: nat).(subst0 n u t0 x2)) H9 
+(s k (minus i h)) (s_minus k i h (le_trans_plus_r d h i H2))) in (ex_intro2 T 
+(\lambda (t2: T).(eq T (lift h d (THead k x3 x2)) (lift h d t2))) (\lambda 
+(t2: T).(subst0 (minus i h) u (THead k t t0) t2)) (THead k x3 x2) (refl_equal 
+T (lift h d (THead k x3 x2))) (subst0_both u t x3 (minus i h) H11 k t0 x2 
+H12))) (THead k (lift h d x3) (lift h (s k d) x2)) (lift_head k x3 x2 h d)) 
+x1 H8) x0 H10)))) (H x0 i h d H6 H2))))) (H0 x1 (s k i) h (s k d) H7 (eq_ind 
+nat (s k (plus d h)) (\lambda (n: nat).(le n (s k i))) (s_le k (plus d h) i 
+H2) (plus (s k d) h) (s_plus k d h)))) x H5)))))) H4)) (subst0_gen_head k u 
+(lift h d t) (lift h (s k d) t0) x i H3)))))))))))))) t1)).
+