+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "basic_1/subst0/defs.ma".
-
-include "basic_1/lift/fwd.ma".
-
-implied rec lemma subst0_ind (P: (nat \to (T \to (T \to (T \to Prop))))) (f:
-(\forall (v: T).(\forall (i: nat).(P i v (TLRef i) (lift (S i) O v))))) (f0:
-(\forall (v: T).(\forall (u2: T).(\forall (u1: T).(\forall (i: nat).((subst0
-i v u1 u2) \to ((P i v u1 u2) \to (\forall (t: T).(\forall (k: K).(P i v
-(THead k u1 t) (THead k u2 t))))))))))) (f1: (\forall (k: K).(\forall (v:
-T).(\forall (t2: T).(\forall (t1: T).(\forall (i: nat).((subst0 (s k i) v t1
-t2) \to ((P (s k i) v t1 t2) \to (\forall (u: T).(P i v (THead k u t1) (THead
-k u t2))))))))))) (f2: (\forall (v: T).(\forall (u1: T).(\forall (u2:
-T).(\forall (i: nat).((subst0 i v u1 u2) \to ((P i v u1 u2) \to (\forall (k:
-K).(\forall (t1: T).(\forall (t2: T).((subst0 (s k i) v t1 t2) \to ((P (s k
-i) v t1 t2) \to (P i v (THead k u1 t1) (THead k u2 t2)))))))))))))) (n: nat)
-(t: T) (t0: T) (t1: T) (s0: subst0 n t t0 t1) on s0: P n t t0 t1 \def match
-s0 with [(subst0_lref v i) \Rightarrow (f v i) | (subst0_fst v u2 u1 i s1 t2
-k) \Rightarrow (f0 v u2 u1 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2 s1) t2
-k) | (subst0_snd k v t2 t3 i s1 u) \Rightarrow (f1 k v t2 t3 i s1
-((subst0_ind P f f0 f1 f2) (s k i) v t3 t2 s1) u) | (subst0_both v u1 u2 i s1
-k t2 t3 s2) \Rightarrow (f2 v u1 u2 i s1 ((subst0_ind P f f0 f1 f2) i v u1 u2
-s1) k t2 t3 s2 ((subst0_ind P f f0 f1 f2) (s k i) v t2 t3 s2))].
-
-lemma 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 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 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 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 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)))))).
-
-lemma 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 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 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 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 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))))).
-
-lemma 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 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 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 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 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 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 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 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 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
-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 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))))))).
-
-lemma 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)).
-
-lemma 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)) (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).
-
-lemma 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_O_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)).
-
-lemma subst0_gen_lift_rev_ge:
- \forall (t1: T).(\forall (v: T).(\forall (u2: T).(\forall (i: nat).(\forall
-(h: nat).(\forall (d: nat).((subst0 i v t1 (lift h d u2)) \to ((le (plus d h)
-i) \to (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1:
-T).(eq T t1 (lift h d u1)))))))))))
-\def
- \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (v: T).(\forall (u2:
-T).(\forall (i: nat).(\forall (h: nat).(\forall (d: nat).((subst0 i v t (lift
-h d u2)) \to ((le (plus d h) i) \to (ex2 T (\lambda (u1: T).(subst0 (minus i
-h) v u1 u2)) (\lambda (u1: T).(eq T t (lift h d u1)))))))))))) (\lambda (n:
-nat).(\lambda (v: T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H: (subst0 i v (TSort n) (lift h d
-u2))).(\lambda (_: (le (plus d h) i)).(subst0_gen_sort v (lift h d u2) i n H
-(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T
-(TSort n) (lift h d u1))))))))))))) (\lambda (n: nat).(\lambda (v:
-T).(\lambda (u2: T).(\lambda (i: nat).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H: (subst0 i v (TLRef n) (lift h d u2))).(\lambda (H0: (le
-(plus d h) i)).(land_ind (eq nat n i) (eq T (lift h d u2) (lift (S n) O v))
-(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T
-(TLRef n) (lift h d u1)))) (\lambda (H1: (eq nat n i)).(\lambda (H2: (eq T
-(lift h d u2) (lift (S n) O v))).(let H3 \def (eq_ind_r nat i (\lambda (n0:
-nat).(le (plus d h) n0)) H0 n H1) in (eq_ind nat n (\lambda (n0: nat).(ex2 T
-(\lambda (u1: T).(subst0 (minus n0 h) v u1 u2)) (\lambda (u1: T).(eq T (TLRef
-n) (lift h d u1))))) (eq_ind_r nat (plus (minus n h) h) (\lambda (n0:
-nat).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 u2)) (\lambda (u1:
-T).(eq T (TLRef n0) (lift h d u1))))) (eq_ind T (lift h d (TLRef (minus n
-h))) (\lambda (t: T).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 u2))
-(\lambda (u1: T).(eq T t (lift h d u1))))) (let H4 \def (eq_ind nat n
-(\lambda (n0: nat).(eq T (lift h d u2) (lift (S n0) O v))) H2 (plus h (minus
-n h)) (le_plus_minus h n (le_trans h (plus d h) n (le_plus_r d h) H3))) in
-(let H5 \def (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(eq T
-(lift h d u2) (lift n0 O v))) H4 (plus h (S (minus n h))) (plus_n_Sm h (minus
-n h))) in (let H6 \def (eq_ind_r T (lift (plus h (S (minus n h))) O v)
-(\lambda (t: T).(eq T (lift h d u2) t)) H5 (lift h d (lift (S (minus n h)) O
-v)) (lift_free v (S (minus n h)) h O d (le_S d (minus n h) (le_minus d n h
-H3)) (le_O_n d))) in (eq_ind_r T (lift (S (minus n h)) O v) (\lambda (t:
-T).(ex2 T (\lambda (u1: T).(subst0 (minus n h) v u1 t)) (\lambda (u1: T).(eq
-T (lift h d (TLRef (minus n h))) (lift h d u1))))) (ex_intro2 T (\lambda (u1:
-T).(subst0 (minus n h) v u1 (lift (S (minus n h)) O v))) (\lambda (u1: T).(eq
-T (lift h d (TLRef (minus n h))) (lift h d u1))) (TLRef (minus n h))
-(subst0_lref v (minus n h)) (refl_equal T (lift h d (TLRef (minus n h))))) u2
-(lift_inj u2 (lift (S (minus n h)) O v) h d H6))))) (TLRef (plus (minus n h)
-h)) (lift_lref_ge (minus n h) h d (le_minus d n h H3))) n (le_plus_minus_sym
-h n (le_trans h (plus d h) n (le_plus_r d h) H3))) i H1)))) (subst0_gen_lref
-v (lift h d u2) i n H)))))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H:
-((\forall (v: T).(\forall (u2: T).(\forall (i: nat).(\forall (h:
-nat).(\forall (d: nat).((subst0 i v t (lift h d u2)) \to ((le (plus d h) i)
-\to (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1:
-T).(eq T t (lift h d u1))))))))))))).(\lambda (t0: T).(\lambda (H0: ((\forall
-(v: T).(\forall (u2: T).(\forall (i: nat).(\forall (h: nat).(\forall (d:
-nat).((subst0 i v t0 (lift h d u2)) \to ((le (plus d h) i) \to (ex2 T
-(\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T t0
-(lift h d u1))))))))))))).(\lambda (v: T).(\lambda (u2: T).(\lambda (i:
-nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (subst0 i v (THead k t
-t0) (lift h d u2))).(\lambda (H2: (le (plus d h) i)).(or3_ind (ex2 T (\lambda
-(u3: T).(eq T (lift h d u2) (THead k u3 t0))) (\lambda (u3: T).(subst0 i v t
-u3))) (ex2 T (\lambda (t2: T).(eq T (lift h d u2) (THead k t t2))) (\lambda
-(t2: T).(subst0 (s k i) v t0 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T (lift h d u2) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_:
-T).(subst0 i v t u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t0
-t2)))) (ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 u2)) (\lambda (u1:
-T).(eq T (THead k t t0) (lift h d u1)))) (\lambda (H3: (ex2 T (\lambda (u3:
-T).(eq T (lift h d u2) (THead k u3 t0))) (\lambda (u3: T).(subst0 i v t
-u3)))).(ex2_ind T (\lambda (u3: T).(eq T (lift h d u2) (THead k u3 t0)))
-(\lambda (u3: T).(subst0 i v t u3)) (ex2 T (\lambda (u1: T).(subst0 (minus i
-h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda
-(x: T).(\lambda (H4: (eq T (lift h d u2) (THead k x t0))).(\lambda (H5:
-(subst0 i v t x)).(let H6 \def (sym_eq T (lift h d u2) (THead k x t0) H4) in
-(let H_x \def (lift_gen_head k x t0 u2 h d H6) in (let H7 \def H_x in
-(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T x (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T t0 (lift h (s k d) z)))) (ex2 T (\lambda (u1:
-T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift
-h d u1)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T u2 (THead k
-x0 x1))).(\lambda (H9: (eq T x (lift h d x0))).(\lambda (H10: (eq T t0 (lift
-h (s k d) x1))).(let H11 \def (eq_ind T x (\lambda (t2: T).(subst0 i v t t2))
-H5 (lift h d x0) H9) in (eq_ind_r T (THead k x0 x1) (\lambda (t2: T).(ex2 T
-(\lambda (u1: T).(subst0 (minus i h) v u1 t2)) (\lambda (u1: T).(eq T (THead
-k t t0) (lift h d u1))))) (eq_ind_r T (lift h (s k d) x1) (\lambda (t2:
-T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1)))
-(\lambda (u1: T).(eq T (THead k t t2) (lift h d u1))))) (let H_x0 \def (H v
-x0 i h d H11 H2) in (let H12 \def H_x0 in (ex2_ind T (\lambda (u1: T).(subst0
-(minus i h) v u1 x0)) (\lambda (u1: T).(eq T t (lift h d u1))) (ex2 T
-(\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1:
-T).(eq T (THead k t (lift h (s k d) x1)) (lift h d u1)))) (\lambda (x2:
-T).(\lambda (H13: (subst0 (minus i h) v x2 x0)).(\lambda (H14: (eq T t (lift
-h d x2))).(eq_ind_r T (lift h d x2) (\lambda (t2: T).(ex2 T (\lambda (u1:
-T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T (THead k
-t2 (lift h (s k d) x1)) (lift h d u1))))) (eq_ind T (lift h d (THead k x2
-x1)) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead
-k x0 x1))) (\lambda (u1: T).(eq T t2 (lift h d u1))))) (ex_intro2 T (\lambda
-(u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T
-(lift h d (THead k x2 x1)) (lift h d u1))) (THead k x2 x1) (subst0_fst v x0
-x2 (minus i h) H13 x1 k) (refl_equal T (lift h d (THead k x2 x1)))) (THead k
-(lift h d x2) (lift h (s k d) x1)) (lift_head k x2 x1 h d)) t H14)))) H12)))
-t0 H10) u2 H8))))))) H7))))))) H3)) (\lambda (H3: (ex2 T (\lambda (t2: T).(eq
-T (lift h d u2) (THead k t t2))) (\lambda (t2: T).(subst0 (s k i) v t0
-t2)))).(ex2_ind T (\lambda (t2: T).(eq T (lift h d u2) (THead k t t2)))
-(\lambda (t2: T).(subst0 (s k i) v t0 t2)) (ex2 T (\lambda (u1: T).(subst0
-(minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1))))
-(\lambda (x: T).(\lambda (H4: (eq T (lift h d u2) (THead k t x))).(\lambda
-(H5: (subst0 (s k i) v t0 x)).(let H6 \def (sym_eq T (lift h d u2) (THead k t
-x) H4) in (let H_x \def (lift_gen_head k t x u2 h d H6) in (let H7 \def H_x
-in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z))))
-(\lambda (y: T).(\lambda (_: T).(eq T t (lift h d y)))) (\lambda (_:
-T).(\lambda (z: T).(eq T x (lift h (s k d) z)))) (ex2 T (\lambda (u1:
-T).(subst0 (minus i h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift
-h d u1)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T u2 (THead k
-x0 x1))).(\lambda (H9: (eq T t (lift h d x0))).(\lambda (H10: (eq T x (lift h
-(s k d) x1))).(let H11 \def (eq_ind T x (\lambda (t2: T).(subst0 (s k i) v t0
-t2)) H5 (lift h (s k d) x1) H10) in (eq_ind_r T (THead k x0 x1) (\lambda (t2:
-T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 t2)) (\lambda (u1: T).(eq
-T (THead k t t0) (lift h d u1))))) (eq_ind_r T (lift h d x0) (\lambda (t2:
-T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x0 x1)))
-(\lambda (u1: T).(eq T (THead k t2 t0) (lift h d u1))))) (let H_y \def (H0 v
-x1 (s k i) h (s k d) H11) in (let H12 \def (eq_ind_r nat (plus (s k d) h)
-(\lambda (n: nat).((le n (s k i)) \to (ex2 T (\lambda (u1: T).(subst0 (minus
-(s k i) h) v u1 x1)) (\lambda (u1: T).(eq T t0 (lift h (s k d) u1)))))) H_y
-(s k (plus d h)) (s_plus k d h)) in (let H13 \def (eq_ind_r nat (minus (s k
-i) h) (\lambda (n: nat).((le (s k (plus d h)) (s k i)) \to (ex2 T (\lambda
-(u1: T).(subst0 n v u1 x1)) (\lambda (u1: T).(eq T t0 (lift h (s k d)
-u1)))))) H12 (s k (minus i h)) (s_minus k i h (le_trans h (plus d h) i
-(le_plus_r d h) H2))) in (let H14 \def (H13 (s_le k (plus d h) i H2)) in
-(ex2_ind T (\lambda (u1: T).(subst0 (s k (minus i h)) v u1 x1)) (\lambda (u1:
-T).(eq T t0 (lift h (s k d) u1))) (ex2 T (\lambda (u1: T).(subst0 (minus i h)
-v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T (THead k (lift h d x0) t0)
-(lift h d u1)))) (\lambda (x2: T).(\lambda (H15: (subst0 (s k (minus i h)) v
-x2 x1)).(\lambda (H16: (eq T t0 (lift h (s k d) x2))).(eq_ind_r T (lift h (s
-k d) x2) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1
-(THead k x0 x1))) (\lambda (u1: T).(eq T (THead k (lift h d x0) t2) (lift h d
-u1))))) (eq_ind T (lift h d (THead k x0 x2)) (\lambda (t2: T).(ex2 T (\lambda
-(u1: T).(subst0 (minus i h) v u1 (THead k x0 x1))) (\lambda (u1: T).(eq T t2
-(lift h d u1))))) (ex_intro2 T (\lambda (u1: T).(subst0 (minus i h) v u1
-(THead k x0 x1))) (\lambda (u1: T).(eq T (lift h d (THead k x0 x2)) (lift h d
-u1))) (THead k x0 x2) (subst0_snd k v x1 x2 (minus i h) H15 x0) (refl_equal T
-(lift h d (THead k x0 x2)))) (THead k (lift h d x0) (lift h (s k d) x2))
-(lift_head k x0 x2 h d)) t0 H16)))) H14))))) t H9) u2 H8))))))) H7)))))))
-H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (lift h
-d u2) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v t u3)))
-(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t0 t2))))).(ex3_2_ind T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T (lift h d u2) (THead k u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(subst0 i v t u3))) (\lambda (_: T).(\lambda
-(t2: T).(subst0 (s k i) v t0 t2))) (ex2 T (\lambda (u1: T).(subst0 (minus i
-h) v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda
-(x0: T).(\lambda (x1: T).(\lambda (H4: (eq T (lift h d u2) (THead k x0
-x1))).(\lambda (H5: (subst0 i v t x0)).(\lambda (H6: (subst0 (s k i) v t0
-x1)).(let H7 \def (sym_eq T (lift h d u2) (THead k x0 x1) H4) in (let H_x
-\def (lift_gen_head k x0 x1 u2 h d H7) in (let H8 \def H_x in (ex3_2_ind T T
-(\lambda (y: T).(\lambda (z: T).(eq T u2 (THead k y z)))) (\lambda (y:
-T).(\lambda (_: T).(eq T x0 (lift h d y)))) (\lambda (_: T).(\lambda (z:
-T).(eq T x1 (lift h (s k d) z)))) (ex2 T (\lambda (u1: T).(subst0 (minus i h)
-v u1 u2)) (\lambda (u1: T).(eq T (THead k t t0) (lift h d u1)))) (\lambda
-(x2: T).(\lambda (x3: T).(\lambda (H9: (eq T u2 (THead k x2 x3))).(\lambda
-(H10: (eq T x0 (lift h d x2))).(\lambda (H11: (eq T x1 (lift h (s k d)
-x3))).(let H12 \def (eq_ind T x1 (\lambda (t2: T).(subst0 (s k i) v t0 t2))
-H6 (lift h (s k d) x3) H11) in (let H13 \def (eq_ind T x0 (\lambda (t2:
-T).(subst0 i v t t2)) H5 (lift h d x2) H10) in (eq_ind_r T (THead k x2 x3)
-(\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1 t2))
-(\lambda (u1: T).(eq T (THead k t t0) (lift h d u1))))) (let H_x0 \def (H v
-x2 i h d H13 H2) in (let H14 \def H_x0 in (ex2_ind T (\lambda (u1: T).(subst0
-(minus i h) v u1 x2)) (\lambda (u1: T).(eq T t (lift h d u1))) (ex2 T
-(\lambda (u1: T).(subst0 (minus i h) v u1 (THead k x2 x3))) (\lambda (u1:
-T).(eq T (THead k t t0) (lift h d u1)))) (\lambda (x: T).(\lambda (H15:
-(subst0 (minus i h) v x x2)).(\lambda (H16: (eq T t (lift h d x))).(eq_ind_r
-T (lift h d x) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v
-u1 (THead k x2 x3))) (\lambda (u1: T).(eq T (THead k t2 t0) (lift h d u1)))))
-(let H_y \def (H0 v x3 (s k i) h (s k d) H12) in (let H17 \def (eq_ind_r nat
-(plus (s k d) h) (\lambda (n: nat).((le n (s k i)) \to (ex2 T (\lambda (u1:
-T).(subst0 (minus (s k i) h) v u1 x3)) (\lambda (u1: T).(eq T t0 (lift h (s k
-d) u1)))))) H_y (s k (plus d h)) (s_plus k d h)) in (let H18 \def (eq_ind_r
-nat (minus (s k i) h) (\lambda (n: nat).((le (s k (plus d h)) (s k i)) \to
-(ex2 T (\lambda (u1: T).(subst0 n v u1 x3)) (\lambda (u1: T).(eq T t0 (lift h
-(s k d) u1)))))) H17 (s k (minus i h)) (s_minus k i h (le_trans h (plus d h)
-i (le_plus_r d h) H2))) in (let H19 \def (H18 (s_le k (plus d h) i H2)) in
-(ex2_ind T (\lambda (u1: T).(subst0 (s k (minus i h)) v u1 x3)) (\lambda (u1:
-T).(eq T t0 (lift h (s k d) u1))) (ex2 T (\lambda (u1: T).(subst0 (minus i h)
-v u1 (THead k x2 x3))) (\lambda (u1: T).(eq T (THead k (lift h d x) t0) (lift
-h d u1)))) (\lambda (x4: T).(\lambda (H20: (subst0 (s k (minus i h)) v x4
-x3)).(\lambda (H21: (eq T t0 (lift h (s k d) x4))).(eq_ind_r T (lift h (s k
-d) x4) (\lambda (t2: T).(ex2 T (\lambda (u1: T).(subst0 (minus i h) v u1
-(THead k x2 x3))) (\lambda (u1: T).(eq T (THead k (lift h d x) t2) (lift h d
-u1))))) (eq_ind T (lift h d (THead k x x4)) (\lambda (t2: T).(ex2 T (\lambda
-(u1: T).(subst0 (minus i h) v u1 (THead k x2 x3))) (\lambda (u1: T).(eq T t2
-(lift h d u1))))) (ex_intro2 T (\lambda (u1: T).(subst0 (minus i h) v u1
-(THead k x2 x3))) (\lambda (u1: T).(eq T (lift h d (THead k x x4)) (lift h d
-u1))) (THead k x x4) (subst0_both v x x2 (minus i h) H15 k x4 x3 H20)
-(refl_equal T (lift h d (THead k x x4)))) (THead k (lift h d x) (lift h (s k
-d) x4)) (lift_head k x x4 h d)) t0 H21)))) H19))))) t H16)))) H14))) u2
-H9)))))))) H8))))))))) H3)) (subst0_gen_head k v t t0 (lift h d u2) i
-H1)))))))))))))) t1).
-
projects / helm.git / blobdiff