X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2FLambdaDelta%2Fsubst0%2Ffwd.ma;h=622fbc20d9b97a09f90e3dd1bef19a36bb84e12b;hb=4216d3e6b09d1be8784ee9394f6c7f5aed4300a7;hp=ccaad15d26f0b459ddc1986c86fa8ec575fca135;hpb=9c4d497ef24a58d1a9ff907e1484e9b5093b726b;p=helm.git

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/fwd.ma
index ccaad15d2..622fbc20d 100644
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/fwd.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/fwd.ma
@@ -20,17 +20,196 @@ include "subst0/defs.ma".
 
 include "lift/props.ma".
 
-axiom subst0_gen_sort:
+theorem subst0_inv_coq:
+ \forall (i: nat).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall 
+(P: ((nat \to (T \to (T \to (T \to Prop)))))).((((subst0 i v t1 t2) \to 
+(\forall (v0: T).(\forall (i0: nat).((eq nat i0 i) \to ((eq T v0 v) \to ((eq 
+T (TLRef i0) t1) \to ((eq T (lift (S i0) O v0) t2) \to (P i v t1 t2))))))))) 
+\to ((((subst0 i v t1 t2) \to (\forall (v0: T).(\forall (u2: T).(\forall (u1: 
+T).(\forall (i0: nat).(\forall (t: T).(\forall (k: K).((eq nat i0 i) \to ((eq 
+T v0 v) \to ((eq T (THead k u1 t) t1) \to ((eq T (THead k u2 t) t2) \to 
+((subst0 i0 v0 u1 u2) \to (P i v t1 t2)))))))))))))) \to ((((subst0 i v t1 
+t2) \to (\forall (k: K).(\forall (v0: T).(\forall (t0: T).(\forall (t3: 
+T).(\forall (i0: nat).(\forall (u: T).((eq nat i0 i) \to ((eq T v0 v) \to 
+((eq T (THead k u t3) t1) \to ((eq T (THead k u t0) t2) \to ((subst0 (s k i0) 
+v0 t3 t0) \to (P i v t1 t2)))))))))))))) \to ((((subst0 i v t1 t2) \to 
+(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).(\forall (i0: 
+nat).(\forall (k: K).(\forall (t0: T).(\forall (t3: T).((eq nat i0 i) \to 
+((eq T v0 v) \to ((eq T (THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2) 
+\to ((subst0 i0 v0 u1 u2) \to ((subst0 (s k i0) v0 t0 t3) \to (P i v t1 
+t2)))))))))))))))) \to ((subst0 i v t1 t2) \to (P i v t1 t2))))))))))
+\def
+ \lambda (i: nat).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda 
+(P: ((nat \to (T \to (T \to (T \to Prop)))))).(\lambda (H: (((subst0 i v t1 
+t2) \to (\forall (v0: T).(\forall (i0: nat).((eq nat i0 i) \to ((eq T v0 v) 
+\to ((eq T (TLRef i0) t1) \to ((eq T (lift (S i0) O v0) t2) \to (P i v t1 
+t2)))))))))).(\lambda (H0: (((subst0 i v t1 t2) \to (\forall (v0: T).(\forall 
+(u2: T).(\forall (u1: T).(\forall (i0: nat).(\forall (t: T).(\forall (k: 
+K).((eq nat i0 i) \to ((eq T v0 v) \to ((eq T (THead k u1 t) t1) \to ((eq T 
+(THead k u2 t) t2) \to ((subst0 i0 v0 u1 u2) \to (P i v t1 
+t2))))))))))))))).(\lambda (H1: (((subst0 i v t1 t2) \to (\forall (k: 
+K).(\forall (v0: T).(\forall (t0: T).(\forall (t3: T).(\forall (i0: 
+nat).(\forall (u: T).((eq nat i0 i) \to ((eq T v0 v) \to ((eq T (THead k u 
+t3) t1) \to ((eq T (THead k u t0) t2) \to ((subst0 (s k i0) v0 t3 t0) \to (P 
+i v t1 t2))))))))))))))).(\lambda (H2: (((subst0 i v t1 t2) \to (\forall (v0: 
+T).(\forall (u1: T).(\forall (u2: T).(\forall (i0: nat).(\forall (k: 
+K).(\forall (t0: T).(\forall (t3: T).((eq nat i0 i) \to ((eq T v0 v) \to ((eq 
+T (THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2) \to ((subst0 i0 v0 u1 
+u2) \to ((subst0 (s k i0) v0 t0 t3) \to (P i v t1 
+t2))))))))))))))))).(\lambda (H3: (subst0 i v t1 t2)).(let H4 \def (match H3 
+in subst0 return (\lambda (n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda 
+(t3: T).(\lambda (_: (subst0 n t t0 t3)).((eq nat n i) \to ((eq T t v) \to 
+((eq T t0 t1) \to ((eq T t3 t2) \to (P i v t1 t2)))))))))) with [(subst0_lref 
+v0 i0) \Rightarrow (\lambda (H4: (eq nat i0 i)).(\lambda (H5: (eq T v0 
+v)).(\lambda (H6: (eq T (TLRef i0) t1)).(\lambda (H7: (eq T (lift (S i0) O 
+v0) t2)).(H H3 v0 i0 H4 H5 H6 H7))))) | (subst0_fst v0 u2 u1 i0 H4 t k) 
+\Rightarrow (\lambda (H5: (eq nat i0 i)).(\lambda (H6: (eq T v0 v)).(\lambda 
+(H7: (eq T (THead k u1 t) t1)).(\lambda (H8: (eq T (THead k u2 t) t2)).(H0 H3 
+v0 u2 u1 i0 t k H5 H6 H7 H8 H4))))) | (subst0_snd k v0 t0 t3 i0 H4 u) 
+\Rightarrow (\lambda (H5: (eq nat i0 i)).(\lambda (H6: (eq T v0 v)).(\lambda 
+(H7: (eq T (THead k u t3) t1)).(\lambda (H8: (eq T (THead k u t0) t2)).(H1 H3 
+k v0 t0 t3 i0 u H5 H6 H7 H8 H4))))) | (subst0_both v0 u1 u2 i0 H4 k t0 t3 H5) 
+\Rightarrow (\lambda (H6: (eq nat i0 i)).(\lambda (H7: (eq T v0 v)).(\lambda 
+(H8: (eq T (THead k u1 t0) t1)).(\lambda (H9: (eq T (THead k u2 t3) t2)).(H2 
+H3 v0 u1 u2 i0 k t0 t3 H6 H7 H8 H9 H4 H5)))))]) in (H4 (refl_equal nat i) 
+(refl_equal T v) (refl_equal T t1) (refl_equal T t2)))))))))))).
+
+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).(subst0_inv_coq i v (TSort 
+n) x (\lambda (_: nat).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).P)))) 
+(\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0: T).(\lambda (i0: 
+nat).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: 
+(eq T (TLRef i0) (TSort n))).(\lambda (H4: (eq T (lift (S i0) O v0) x)).(let 
+H5 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (lift (S n0) O v0) x)) H4 i 
+H1) in (let H6 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (TLRef n0) (TSort 
+n))) H3 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(eq T (lift (S i) 
+O t) x)) H5 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v 
+(TSort n) t)) H0 (lift (S i) O v) H7) in (let H9 \def (eq_ind_r T x (\lambda 
+(t: T).(subst0 i v (TSort n) t)) H (lift (S i) O v) H7) in (let H10 \def 
+(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+(THead _ _ _) \Rightarrow False])) I (TSort n) H6) in (False_ind P 
+H10)))))))))))))) (\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0: 
+T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (t: 
+T).(\lambda (k: K).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 
+v)).(\lambda (H3: (eq T (THead k u1 t) (TSort n))).(\lambda (H4: (eq T (THead 
+k u2 t) x)).(\lambda (H5: (subst0 i0 v0 u1 u2)).(let H6 \def (eq_ind nat i0 
+(\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H5 i H1) in (let H7 \def (eq_ind T 
+v0 (\lambda (t0: T).(subst0 i t0 u1 u2)) H6 v H2) in (let H8 \def (eq_ind_r T 
+x (\lambda (t0: T).(subst0 i v (TSort n) t0)) H0 (THead k u2 t) H4) in (let 
+H9 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v (TSort n) t0)) H (THead k 
+u2 t) H4) in (let H10 \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 H10)))))))))))))))))) (\lambda (H0: (subst0 i v (TSort n) 
+x)).(\lambda (k: K).(\lambda (v0: T).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (i0: nat).(\lambda (u: T).(\lambda (H1: (eq nat i0 i)).(\lambda 
+(H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u t3) (TSort n))).(\lambda 
+(H4: (eq T (THead k u t0) x)).(\lambda (H5: (subst0 (s k i0) v0 t3 t0)).(let 
+H6 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t3 t0)) H5 i 
+H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i) t t3 t0)) H6 
+v H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TSort n) t)) 
+H0 (THead k u t0) H4) in (let H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i 
+v (TSort n) t)) H (THead k u t0) H4) in (let H10 \def (eq_ind T (THead k u 
+t3) (\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 H10)))))))))))))))))) 
+(\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0: T).(\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (k: K).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 
+v)).(\lambda (H4: (eq T (THead k u1 t0) (TSort n))).(\lambda (H5: (eq T 
+(THead k u2 t3) x)).(\lambda (H1: (subst0 i0 v0 u1 u2)).(\lambda (H6: (subst0 
+(s k i0) v0 t0 t3)).(let H7 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s 
+k n0) v0 t0 t3)) H6 i H2) in (let H8 \def (eq_ind nat i0 (\lambda (n0: 
+nat).(subst0 n0 v0 u1 u2)) H1 i H2) in (let H9 \def (eq_ind T v0 (\lambda (t: 
+T).(subst0 (s k i) t t0 t3)) H7 v H3) in (let H10 \def (eq_ind T v0 (\lambda 
+(t: T).(subst0 i t u1 u2)) H8 v H3) in (let H11 \def (eq_ind_r T x (\lambda 
+(t: T).(subst0 i v (TSort n) t)) H0 (THead k u2 t3) H5) in (let H12 \def 
+(eq_ind_r T x (\lambda (t: T).(subst0 i v (TSort n) t)) H (THead k u2 t3) H5) 
+in (let H13 \def (eq_ind T (THead k u1 t0) (\lambda (ee: T).(match ee in T 
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H4) in 
+(False_ind P H13)))))))))))))))))))))) H)))))).
 
-axiom subst0_gen_lref:
+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)).(subst0_inv_coq i v (TLRef n) x (\lambda (n0: 
+nat).(\lambda (t: T).(\lambda (_: T).(\lambda (t1: T).(land (eq nat n n0) (eq 
+T t1 (lift (S n) O t))))))) (\lambda (H0: (subst0 i v (TLRef n) x)).(\lambda 
+(v0: T).(\lambda (i0: nat).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T 
+v0 v)).(\lambda (H3: (eq T (TLRef i0) (TLRef n))).(\lambda (H4: (eq T (lift 
+(S i0) O v0) x)).(let H5 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (lift 
+(S n0) O v0) x)) H4 i H1) in (let H6 \def (eq_ind nat i0 (\lambda (n0: 
+nat).(eq T (TLRef n0) (TLRef n))) H3 i H1) in (let H7 \def (eq_ind T v0 
+(\lambda (t: T).(eq T (lift (S i) O t) x)) H5 v H2) in (let H8 \def (eq_ind_r 
+T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H0 (lift (S i) O v) H7) in (let 
+H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H (lift (S i) 
+O v) H7) in (eq_ind T (lift (S i) O v) (\lambda (t: T).(land (eq nat n i) (eq 
+T t (lift (S n) O v)))) (let H10 \def (f_equal T nat (\lambda (e: T).(match e 
+in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0) 
+\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H6) in 
+(let H11 \def (eq_ind_r nat n (\lambda (n0: nat).(subst0 i v (TLRef n0) (lift 
+(S i) O v))) H8 i H10) in (let H12 \def (eq_ind_r nat n (\lambda (n0: 
+nat).(subst0 i v (TLRef n0) (lift (S i) O v))) H9 i H10) in (eq_ind nat i 
+(\lambda (n0: nat).(land (eq nat n0 i) (eq T (lift (S i) O v) (lift (S n0) O 
+v)))) (conj (eq nat i i) (eq T (lift (S i) O v) (lift (S i) O v)) (refl_equal 
+nat i) (refl_equal T (lift (S i) O v))) n H10)))) x H7))))))))))))) (\lambda 
+(H0: (subst0 i v (TLRef n) x)).(\lambda (v0: T).(\lambda (u2: T).(\lambda 
+(u1: T).(\lambda (i0: nat).(\lambda (t: T).(\lambda (k: K).(\lambda (H1: (eq 
+nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) 
+(TLRef n))).(\lambda (H4: (eq T (THead k u2 t) x)).(\lambda (H5: (subst0 i0 
+v0 u1 u2)).(let H6 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 
+u2)) H5 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t0: T).(subst0 i t0 u1 
+u2)) H6 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v 
+(TLRef n) t0)) H0 (THead k u2 t) H4) in (let H9 \def (eq_ind_r T x (\lambda 
+(t0: T).(subst0 i v (TLRef n) t0)) H (THead k u2 t) H4) in (eq_ind T (THead k 
+u2 t) (\lambda (t0: T).(land (eq nat n i) (eq T t0 (lift (S n) O v)))) (let 
+H10 \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 i) (eq T (THead k u2 t) (lift (S n) O v))) H10)) x 
+H4))))))))))))))))) (\lambda (H0: (subst0 i v (TLRef n) x)).(\lambda (k: 
+K).(\lambda (v0: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: 
+nat).(\lambda (u: T).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 
+v)).(\lambda (H3: (eq T (THead k u t3) (TLRef n))).(\lambda (H4: (eq T (THead 
+k u t0) x)).(\lambda (H5: (subst0 (s k i0) v0 t3 t0)).(let H6 \def (eq_ind 
+nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t3 t0)) H5 i H1) in (let H7 
+\def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i) t t3 t0)) H6 v H2) in (let 
+H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H0 (THead k u 
+t0) H4) in (let H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n) 
+t)) H (THead k u t0) H4) in (eq_ind T (THead k u t0) (\lambda (t: T).(land 
+(eq nat n i) (eq T t (lift (S n) O v)))) (let H10 \def (eq_ind T (THead k u 
+t3) (\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 i) (eq T 
+(THead k u t0) (lift (S n) O v))) H10)) x H4))))))))))))))))) (\lambda (H0: 
+(subst0 i v (TLRef n) x)).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (i0: nat).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq 
+T (THead k u1 t0) (TLRef n))).(\lambda (H5: (eq T (THead k u2 t3) 
+x)).(\lambda (H1: (subst0 i0 v0 u1 u2)).(\lambda (H6: (subst0 (s k i0) v0 t0 
+t3)).(let H7 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t0 
+t3)) H6 i H2) in (let H8 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 n0 v0 
+u1 u2)) H1 i H2) in (let H9 \def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i) 
+t t0 t3)) H7 v H3) in (let H10 \def (eq_ind T v0 (\lambda (t: T).(subst0 i t 
+u1 u2)) H8 v H3) in (let H11 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v 
+(TLRef n) t)) H0 (THead k u2 t3) H5) in (let H12 \def (eq_ind_r T x (\lambda 
+(t: T).(subst0 i v (TLRef n) t)) H (THead k u2 t3) H5) in (eq_ind T (THead k 
+u2 t3) (\lambda (t: T).(land (eq nat n i) (eq T t (lift (S n) O v)))) (let 
+H13 \def (eq_ind T (THead k u1 t0) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in 
+(False_ind (land (eq nat n i) (eq T (THead k u2 t3) (lift (S n) O v))) H13)) 
+x H5))))))))))))))))))))) H))))).
 
-axiom subst0_gen_head:
+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 
@@ -38,25 +217,601 @@ 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)).(subst0_inv_coq i v (THead k u1 t1) x (\lambda (n: nat).(\lambda (t: 
+T).(\lambda (_: T).(\lambda (t2: T).(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 (H0: (subst0 i 
+v (THead k u1 t1) x)).(\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq 
+nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (TLRef i0) (THead k 
+u1 t1))).(\lambda (H4: (eq T (lift (S i0) O v0) x)).(let H5 \def (eq_ind nat 
+i0 (\lambda (n: nat).(eq T (lift (S n) O v0) x)) H4 i H1) in (let H6 \def 
+(eq_ind nat i0 (\lambda (n: nat).(eq T (TLRef n) (THead k u1 t1))) H3 i H1) 
+in (let H7 \def (eq_ind T v0 (\lambda (t: T).(eq T (lift (S i) O t) x)) H5 v 
+H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1) 
+t)) H0 (lift (S i) O v) H7) in (let H9 \def (eq_ind_r T x (\lambda (t: 
+T).(subst0 i v (THead k u1 t1) t)) H (lift (S i) O v) H7) in (eq_ind T (lift 
+(S i) O v) (\lambda (t: T).(or3 (ex2 T (\lambda (u2: T).(eq T t (THead k u2 
+t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T t 
+(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 t (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)))))) (let H10 \def (eq_ind T (TLRef i) (\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) H6) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq 
+T (lift (S i) O v) (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) 
+(ex2 T (\lambda (t2: T).(eq T (lift (S i) O v) (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 (lift (S i) O v) (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))))) H10)) x H7))))))))))))) (\lambda (H0: (subst0 i v (THead k u1 t1) 
+x)).(\lambda (v0: T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i0: 
+nat).(\lambda (t: T).(\lambda (k0: K).(\lambda (H1: (eq nat i0 i)).(\lambda 
+(H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 
+t1))).(\lambda (H4: (eq T (THead k0 u2 t) x)).(\lambda (H5: (subst0 i0 v0 u0 
+u2)).(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(subst0 n v0 u0 u2)) H5 i 
+H1) in (let H7 \def (eq_ind T v0 (\lambda (t0: T).(subst0 i t0 u0 u2)) H6 v 
+H2) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v (THead k u1 
+t1) t0)) H0 (THead k0 u2 t) H4) in (let H9 \def (eq_ind_r T x (\lambda (t0: 
+T).(subst0 i v (THead k u1 t1) t0)) H (THead k0 u2 t) H4) in (eq_ind T (THead 
+k0 u2 t) (\lambda (t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T t0 (THead k u3 
+t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T t0 
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T t0 (THead k u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s k i) v t1 t2)))))) (let H10 \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 H11 \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 H12 \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 (H13: (eq T u0 u1)).(\lambda (H14: (eq K k0 k)).(let H15 \def 
+(eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u2 t))) 
+H9 k H14) in (let H16 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k 
+u1 t1) (THead k1 u2 t))) H8 k H14) in (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 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) (THead 
+k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v 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 i v u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s k i) v t1 t2)))))) (let H17 \def (eq_ind T t (\lambda (t0: 
+T).(subst0 i v (THead k u1 t1) (THead k u2 t0))) H15 t1 H12) in (let H18 \def 
+(eq_ind T t (\lambda (t0: T).(subst0 i v (THead k u1 t1) (THead k u2 t0))) 
+H16 t1 H12) in (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 i v u1 
+u3))) (ex2 T (\lambda (t2: T).(eq T (THead k u2 t0) (THead k u1 t2))) 
+(\lambda (t2: T).(subst0 (s k i) v 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 i v u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s k i) v t1 t2)))))) (let H19 \def (eq_ind T u0 (\lambda (t0: 
+T).(subst0 i v t0 u2)) H7 u1 H13) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq 
+T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) 
+(ex2 T (\lambda (t2: T).(eq T (THead k u2 t1) (THead k u1 t2))) (\lambda (t2: 
+T).(subst0 (s k i) v 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 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 
+t2)))) (ex_intro2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i v u1 u3)) u2 (refl_equal T (THead k u2 t1)) H19))) 
+t H12))) k0 H14)))))) H11)) H10)) x H4))))))))))))))))) (\lambda (H0: (subst0 
+i v (THead k u1 t1) x)).(\lambda (k0: K).(\lambda (v0: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (u: T).(\lambda (H1: (eq nat 
+i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u t3) (THead 
+k u1 t1))).(\lambda (H4: (eq T (THead k0 u t0) x)).(\lambda (H5: (subst0 (s 
+k0 i0) v0 t3 t0)).(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(subst0 (s k0 
+n) v0 t3 t0)) H5 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(subst0 
+(s k0 i) t t3 t0)) H6 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t: 
+T).(subst0 i v (THead k u1 t1) t)) H0 (THead k0 u t0) H4) in (let H9 \def 
+(eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1) t)) H (THead k0 u 
+t0) H4) in (eq_ind T (THead k0 u t0) (\lambda (t: T).(or3 (ex2 T (\lambda 
+(u2: T).(eq T t (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 
+T (\lambda (t2: T).(eq T t (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 t (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)))))) (let H10 \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 t3) (THead k u1 t1) H3) in ((let H11 \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 t3) 
+(THead k u1 t1) H3) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e 
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) 
+\Rightarrow t3 | (THead _ _ t) \Rightarrow t])) (THead k0 u t3) (THead k u1 
+t1) H3) in (\lambda (H13: (eq T u u1)).(\lambda (H14: (eq K k0 k)).(let H15 
+\def (eq_ind T u (\lambda (t: T).(subst0 i v (THead k u1 t1) (THead k0 t 
+t0))) H9 u1 H13) in (let H16 \def (eq_ind T u (\lambda (t: T).(subst0 i v 
+(THead k u1 t1) (THead k0 t t0))) H8 u1 H13) in (eq_ind_r T u1 (\lambda (t: 
+T).(or3 (ex2 T (\lambda (u2: T).(eq T (THead k0 t t0) (THead k u2 t1))) 
+(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k0 
+t t0) (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 (THead k0 t t0) (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)))))) (let H17 \def (eq_ind T t3 
+(\lambda (t: T).(subst0 (s k0 i) v t t0)) H7 t1 H12) in (let H18 \def (eq_ind 
+K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u1 t0))) H15 k 
+H14) in (let H19 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1 
+t1) (THead k1 u1 t0))) H16 k H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: 
+K).(subst0 (s k1 i) v t1 t0)) H17 k H14) in (eq_ind_r K k (\lambda (k1: 
+K).(or3 (ex2 T (\lambda (u2: T).(eq T (THead k1 u1 t0) (THead k u2 t1))) 
+(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k1 
+u1 t0) (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 (THead k1 u1 t0) (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)))))) (or3_intro1 (ex2 T (\lambda 
+(u2: T).(eq T (THead k u1 t0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i v 
+u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k u1 t0) (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 (THead k u1 t0) (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)))) (ex_intro2 T (\lambda (t2: T).(eq T (THead k 
+u1 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2)) t0 
+(refl_equal T (THead k u1 t0)) H20)) k0 H14))))) u H13)))))) H11)) H10)) x 
+H4))))))))))))))))) (\lambda (H0: (subst0 i v (THead k u1 t1) x)).(\lambda 
+(v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (k0: 
+K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2: (eq nat i0 i)).(\lambda 
+(H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k0 u0 t0) (THead k u1 
+t1))).(\lambda (H5: (eq T (THead k0 u2 t3) x)).(\lambda (H1: (subst0 i0 v0 u0 
+u2)).(\lambda (H6: (subst0 (s k0 i0) v0 t0 t3)).(let H7 \def (eq_ind nat i0 
+(\lambda (n: nat).(subst0 (s k0 n) v0 t0 t3)) H6 i H2) in (let H8 \def 
+(eq_ind nat i0 (\lambda (n: nat).(subst0 n v0 u0 u2)) H1 i H2) in (let H9 
+\def (eq_ind T v0 (\lambda (t: T).(subst0 (s k0 i) t t0 t3)) H7 v H3) in (let 
+H10 \def (eq_ind T v0 (\lambda (t: T).(subst0 i t u0 u2)) H8 v H3) in (let 
+H11 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1) t)) H0 
+(THead k0 u2 t3) H5) in (let H12 \def (eq_ind_r T x (\lambda (t: T).(subst0 i 
+v (THead k u1 t1) t)) H (THead k0 u2 t3) H5) in (eq_ind T (THead k0 u2 t3) 
+(\lambda (t: T).(or3 (ex2 T (\lambda (u3: T).(eq T t (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T t (THead 
+k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda 
+(u3: T).(\lambda (t2: T).(eq T t (THead k u3 t2)))) (\lambda (u3: T).(\lambda 
+(_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) 
+v t1 t2)))))) (let H13 \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) H4) in 
+((let H14 \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) H4) in ((let H15 \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) H4) in (\lambda (H16: (eq T 
+u0 u1)).(\lambda (H17: (eq K k0 k)).(let H18 \def (eq_ind T t0 (\lambda (t: 
+T).(subst0 (s k0 i) v t t3)) H9 t1 H15) in (let H19 \def (eq_ind K k0 
+(\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u2 t3))) H12 k H17) in 
+(let H20 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1) 
+(THead k1 u2 t3))) H11 k H17) in (let H21 \def (eq_ind K k0 (\lambda (k1: 
+K).(subst0 (s k1 i) v t1 t3)) H18 k H17) in (eq_ind_r K k (\lambda (k1: 
+K).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k1 u2 t3) (THead k u3 t1))) 
+(\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 
+u2 t3) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T 
+T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t3) (THead k u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) (let H22 \def (eq_ind T u0 
+(\lambda (t: T).(subst0 i v t u2)) H10 u1 H16) in (or3_intro2 (ex2 T (\lambda 
+(u3: T).(eq T (THead k u2 t3) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v 
+u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k u2 t3) (THead k u1 t2))) 
+(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: 
+T).(\lambda (t2: T).(eq T (THead k u2 t3) (THead k u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(subst0 (s k i) v t1 t2)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda 
+(t2: T).(eq T (THead k u2 t3) (THead k u3 t2)))) (\lambda (u3: T).(\lambda 
+(_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) 
+v t1 t2))) u2 t3 (refl_equal T (THead k u2 t3)) H22 H21))) k0 H17)))))))) 
+H14)) H13)) x H5))))))))))))))))))))) H))))))).
 
-axiom subst0_gen_lift_lt:
+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 (and_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 (and_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)).
 
-axiom subst0_gen_lift_false:
+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 (and_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 (and_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).
 
-axiom subst0_gen_lift_ge:
+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 
+(and_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 (and_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_comm n 
+h)) (lift h d (lift (S n) O u)) (lift_free u (S n) h O d (le_trans d (S n) 
+(plus O (S n)) (le_S d n H1) (le_n (plus O (S n)))) (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)).