X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fty3%2Fsubst1.ma;h=9c1ecb7116c38296480c1e98a1fef431276fa326;hb=89519c7b52e06304a94019dd528925300380cdc0;hp=60e08dbe125c378b34ab3148b6b6f7468c0f9faf;hpb=81cb773cbc402fc74752fb69a436b25be49489ee;p=helm.git

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma
index 60e08dbe1..9c1ecb711 100644
--- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1.ma
@@ -14,19 +14,11 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/subst1".
+include "LambdaDelta-1/ty3/props.ma".
 
-include "ty3/props.ma".
+include "LambdaDelta-1/pc3/subst1.ma".
 
-include "pc3/subst1.ma".
-
-include "pc3/fwd.ma".
-
-include "csubst1/getl.ma".
-
-include "csubst1/fwd.ma".
-
-include "getl/getl.ma".
+include "LambdaDelta-1/getl/getl.ma".
 
 theorem ty3_gen_cabbr:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
@@ -242,10 +234,10 @@ nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) (ty3_abbr g (minus n (S
 O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) u) a0 (csubst1_getl_ge d0 
 n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0) a 
 (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 
-(plus d0 (S O)) (plus_comm d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O 
-d0 n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_comm (S O) (minus 
-n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n 
-(S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) 
+(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 
+n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S 
+O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S 
+O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) 
 H6))))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
 C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) 
 u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: 
@@ -379,10 +371,10 @@ nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O u))) (ty3_abst g (minus n (S
 O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abst) u) a0 (csubst1_getl_ge d0 
 n (le_S_n d0 n (le_S (S d0) n H6)) c0 a0 u0 H4 (CHead d (Bind Abst) u) H0) a 
 (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le n0 n)) H6 
-(plus d0 (S O)) (plus_comm d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O 
-d0 n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_comm (S O) (minus 
-n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n 
-(S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) 
+(plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n (le_lt_trans O d0 
+n (le_O_n d0) H6)))) (plus (S O) (minus n (S O))) (plus_sym (S O) (minus n (S 
+O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus O (minus n (S 
+O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) 
 H6))))))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: 
 T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: ((\forall (e: C).(\forall (u0: 
 T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: 
@@ -397,191 +389,169 @@ nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)) \to (\forall
 (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 t3 
 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t4 (lift 
 (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) 
-t4 t0)).(\lambda (H5: ((\forall (e: C).(\forall (u0: T).(\forall (d: 
-nat).((getl d (CHead c0 (Bind b) u) (CHead e (Bind Abbr) u0)) \to (\forall 
-(a0: C).((csubst1 d u0 (CHead c0 (Bind b) u) a0) \to (\forall (a: C).((drop 
-(S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 t4 
-(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 t0 (lift 
-(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
 y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda 
-(H6: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H7: 
-(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H8: (drop (S O) d a0 a)).(let 
-H9 \def (H1 e u0 d H6 a0 H7 a H8) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
+(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: 
+(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let 
+H7 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
 (_: T).(subst1 d u0 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
 T).(subst1 d u0 t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead 
 (Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 
 d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: 
+(y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: 
 (subst1 d u0 u (lift (S O) d x0))).(\lambda (_: (subst1 d u0 t (lift (S O) d 
-x1))).(\lambda (H12: (ty3 g a x0 x1)).(let H13 \def (H5 e u0 (S d) (getl_head 
-(Bind b) d c0 (CHead e (Bind Abbr) u0) H6 u) (CHead a0 (Bind b) (lift (S O) d 
-x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H10 c0 a0 H7) (CHead a (Bind b) 
-x0) (drop_skip_bind (S O) d a0 a H8 b x0)) in (ex3_2_ind T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 (S d) u0 t0 (lift (S O) (S d) y2)))) (\lambda 
+x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H3 e u0 (S d) (getl_head 
+(Bind b) d c0 (CHead e (Bind Abbr) u0) H4 u) (CHead a0 (Bind b) (lift (S O) d 
+x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H8 c0 a0 H5) (CHead a (Bind b) 
+x0) (drop_skip_bind (S O) d a0 a H6 b x0)) in (ex3_2_ind T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S O) (S d) y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S O) (S d) y2)))) (\lambda 
 (y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T 
 (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S 
 O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u 
 t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (subst1 (S d) u0 t4 (lift (S 
-O) (S d) x2))).(\lambda (_: (subst1 (S d) u0 t0 (lift (S O) (S d) 
-x3))).(\lambda (H16: (ty3 g (CHead a (Bind b) x0) x2 x3)).(let H17 \def (H3 e 
-u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind Abbr) u0) H6 u) (CHead a0 
-(Bind b) (lift (S O) d x0)) (csubst1_bind b d u0 u (lift (S O) d x0) H10 c0 
-a0 H7) (CHead a (Bind b) x0) (drop_skip_bind (S O) d a0 a H8 b x0)) in 
-(ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 (S d) u0 t3 (lift (S 
-O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 (S d) u0 t4 (lift (S 
-O) (S d) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) 
-x0) y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead 
-(Bind b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 
-d u0 (THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda 
-(y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H18: 
-(subst1 (S d) u0 t3 (lift (S O) (S d) x4))).(\lambda (H19: (subst1 (S d) u0 
-t4 (lift (S O) (S d) x5))).(\lambda (H20: (ty3 g (CHead a (Bind b) x0) x4 
-x5)).(let H21 \def (eq_ind T x5 (\lambda (t5: T).(ty3 g (CHead a (Bind b) x0) 
-x4 t5)) H20 x2 (subst1_confluence_lift t4 x5 u0 (S d) H19 x2 H14)) in 
-(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind 
-b) u t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 
-(THead (Bind b) u t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: 
-T).(ty3 g a y1 y2))) (THead (Bind b) x0 x4) (THead (Bind b) x0 x2) (eq_ind_r 
-T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x4)) (\lambda (t5: 
-T).(subst1 d u0 (THead (Bind b) u t3) t5)) (subst1_head u0 u (lift (S O) d 
-x0) d H10 (Bind b) t3 (lift (S O) (S d) x4) H18) (lift (S O) d (THead (Bind 
-b) x0 x4)) (lift_bind b x0 x4 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S 
-O) d x0) (lift (S O) (S d) x2)) (\lambda (t5: T).(subst1 d u0 (THead (Bind b) 
-u t4) t5)) (subst1_head u0 u (lift (S O) d x0) d H10 (Bind b) t4 (lift (S O) 
-(S d) x2) H14) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b x0 x2 (S O) 
-d)) (ty3_bind g a x0 x1 H12 b x4 x2 H21 x3 H16)))))))) H17))))))) H13))))))) 
-H9))))))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: 
-T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall (u0: 
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 (S d) u0 t3 (lift (S 
+O) (S d) x2))).(\lambda (H13: (subst1 (S d) u0 t4 (lift (S O) (S d) 
+x3))).(\lambda (H14: (ty3 g (CHead a (Bind b) x0) x2 x3)).(ex3_2_intro T T 
+(\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Bind b) u t3) (lift (S 
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind b) u 
+t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (eq_ind_r T (THead (Bind b) 
+(lift (S O) d x0) (lift (S O) (S d) x2)) (\lambda (t0: T).(subst1 d u0 (THead 
+(Bind b) u t3) t0)) (subst1_head u0 u (lift (S O) d x0) d H8 (Bind b) t3 
+(lift (S O) (S d) x2) H12) (lift (S O) d (THead (Bind b) x0 x2)) (lift_bind b 
+x0 x2 (S O) d)) (eq_ind_r T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S 
+d) x3)) (\lambda (t0: T).(subst1 d u0 (THead (Bind b) u t4) t0)) (subst1_head 
+u0 u (lift (S O) d x0) d H8 (Bind b) t4 (lift (S O) (S d) x3) H13) (lift (S 
+O) d (THead (Bind b) x0 x3)) (lift_bind b x0 x3 (S O) d)) (ty3_bind g a x0 x1 
+H10 b x2 x3 H14))))))) H11))))))) H7)))))))))))))))))))) (\lambda (c0: 
+C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: 
+((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e 
+(Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to (\forall (a: 
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g 
+c0 v (THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: 
 T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: 
 C).((csubst1 d u0 c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 
-T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (v: 
-T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u 
-t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl 
-d c0 (CHead e (Bind Abbr) u0)) \to (\forall (a0: C).((csubst1 d u0 c0 a0) \to 
-(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda 
+T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u t) (lift 
+(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda 
+(H4: (getl d c0 (CHead e (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H5: 
+(csubst1 d u0 c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S O) d a0 a)).(let 
+H7 \def (H3 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
 (_: T).(subst1 d u0 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
 T).(subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda 
-(u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) 
-u0))).(\lambda (a0: C).(\lambda (H5: (csubst1 d u0 c0 a0)).(\lambda (a: 
-C).(\lambda (H6: (drop (S O) d a0 a)).(let H7 \def (H3 e u0 d H4 a0 H5 a H6) 
-in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 v (lift (S O) 
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Bind Abst) u 
-t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w 
-v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead 
-(Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H8: (subst1 d u0 v (lift (S O) d x0))).(\lambda (H9: (subst1 d 
-u0 (THead (Bind Abst) u t) (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 
-x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 d u0 w (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) (\lambda (y1: 
 T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
 (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: 
 T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u 
 t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
-(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w (lift (S O) d 
-x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d x3))).(\lambda (H14: (ty3 g 
-a x2 x3)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) 
-d x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d 
-u0 u u2))) (\lambda (_: T).(\lambda (t3: T).(subst1 (s (Bind Abst) d) u0 t 
-t3))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat 
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u0 v (lift (S O) d 
+x0))).(\lambda (H9: (subst1 d u0 (THead (Bind Abst) u t) (lift (S O) d 
+x1))).(\lambda (H10: (ty3 g a x0 x1)).(let H11 \def (H1 e u0 d H4 a0 H5 a H6) 
+in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 w (lift (S O) 
+d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 u (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead 
+(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (subst1 d u0 w 
+(lift (S O) d x2))).(\lambda (H13: (subst1 d u0 u (lift (S O) d 
+x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H_x \def (subst1_gen_head (Bind 
+Abst) u0 u t (lift (S O) d x1) d H9) in (let H15 \def H_x in (ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (lift (S O) d x1) (THead (Bind Abst) 
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst1 d u0 u u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst1 (S d) u0 t t3))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead 
+(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H16: (eq T (lift (S 
+O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H17: (subst1 d u0 u 
+x4)).(\lambda (H18: (subst1 (S d) u0 t x5)).(let H19 \def (sym_eq T (lift (S 
+O) d x1) (THead (Bind Abst) x4 x5) H16) in (ex3_2_ind T T (\lambda (y: 
+T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y: 
+T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T x5 (lift (S O) (S d) z)))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u0 (THead (Flat Appl) w v) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u0 (THead (Flat Appl) w (THead (Bind Abst) u 
+t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) 
+(\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x1 (THead (Bind Abst) 
+x6 x7))).(\lambda (H21: (eq T x4 (lift (S O) d x6))).(\lambda (H22: (eq T x5 
+(lift (S O) (S d) x7))).(let H23 \def (eq_ind T x5 (\lambda (t0: T).(subst1 
+(S d) u0 t t0)) H18 (lift (S O) (S d) x7) H22) in (let H24 \def (eq_ind T x4 
+(\lambda (t0: T).(subst1 d u0 u t0)) H17 (lift (S O) d x6) H21) in (let H25 
+\def (eq_ind T x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6 
+x7) H20) in (let H26 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead 
+(Bind Abst) t0 x7))) H25 x3 (subst1_confluence_lift u x6 u0 d H24 x3 H13)) in 
+(ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat 
 Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 
 (THead (Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H15: (eq T (lift (S O) d x1) (THead (Bind Abst) x4 
-x5))).(\lambda (H16: (subst1 d u0 u x4)).(\lambda (H17: (subst1 (s (Bind 
-Abst) d) u0 t x5)).(let H18 \def (sym_eq T (lift (S O) d x1) (THead (Bind 
-Abst) x4 x5) H15) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 
-(THead (Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T x4 (lift (S 
-O) d y)))) (\lambda (_: T).(\lambda (z: T).(eq T x5 (lift (S O) (S d) z)))) 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w 
-v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead 
-(Flat Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x6: T).(\lambda (x7: 
-T).(\lambda (H19: (eq T x1 (THead (Bind Abst) x6 x7))).(\lambda (H20: (eq T 
-x4 (lift (S O) d x6))).(\lambda (H21: (eq T x5 (lift (S O) (S d) x7))).(let 
-H22 \def (eq_ind T x5 (\lambda (t0: T).(subst1 (s (Bind Abst) d) u0 t t0)) 
-H17 (lift (S O) (S d) x7) H21) in (let H23 \def (eq_ind T x4 (\lambda (t0: 
-T).(subst1 d u0 u t0)) H16 (lift (S O) d x6) H20) in (let H24 \def (eq_ind T 
-x1 (\lambda (t0: T).(ty3 g a x0 t0)) H10 (THead (Bind Abst) x6 x7) H19) in 
-(let H25 \def (eq_ind T x6 (\lambda (t0: T).(ty3 g a x0 (THead (Bind Abst) t0 
-x7))) H24 x3 (subst1_confluence_lift u x6 u0 d H23 x3 H13)) in (ex3_2_intro T 
-T (\lambda (y1: T).(\lambda (_: T).(subst1 d u0 (THead (Flat Appl) w v) (lift 
-(S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (THead (Flat 
-Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead (Flat 
-Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl) (lift (S 
-O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead (Flat Appl) 
-w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v (lift (S O) 
-d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat Appl x2 x0 (S 
-O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift (S O) d (THead 
-(Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead (Flat Appl) w 
-(THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat 
-Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind Abst) x3 x7)) 
-(eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S d) x7)) 
-(\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t) t0)) 
-(subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t (lift 
-(S O) (S d) x7) H22) (lift (S O) d (THead (Bind Abst) x3 x7)) (lift_bind Abst 
-x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead (Bind Abst) x3 
-x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d)) (ty3_appl g a x2 
-x3 H14 x0 x7 H25))))))))))) (lift_gen_bind Abst x4 x5 x1 (S O) d H18)))))))) 
-(subst1_gen_head (Bind Abst) u0 u t (lift (S O) d x1) d H9))))))) H11))))))) 
-H7))))))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: 
-T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: ((\forall (e: C).(\forall (u: 
-T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: 
-C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T 
-T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (t0: 
-T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: 
-T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: 
-C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T 
-T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))))).(\lambda (e: 
-C).(\lambda (u: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind 
-Abbr) u))).(\lambda (a0: C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a: 
-C).(\lambda (H6: (drop (S O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6) 
-in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) 
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
-T).(\lambda (_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift 
-(S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d 
-x0))).(\lambda (H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g 
-a x0 x1)).(let H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda 
-(y1: T).(\lambda (_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: 
+(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Appl) x2 x0) (THead 
+(Flat Appl) x2 (THead (Bind Abst) x3 x7)) (eq_ind_r T (THead (Flat Appl) 
+(lift (S O) d x2) (lift (S O) d x0)) (\lambda (t0: T).(subst1 d u0 (THead 
+(Flat Appl) w v) t0)) (subst1_head u0 w (lift (S O) d x2) d H12 (Flat Appl) v 
+(lift (S O) d x0) H8) (lift (S O) d (THead (Flat Appl) x2 x0)) (lift_flat 
+Appl x2 x0 (S O) d)) (eq_ind_r T (THead (Flat Appl) (lift (S O) d x2) (lift 
+(S O) d (THead (Bind Abst) x3 x7))) (\lambda (t0: T).(subst1 d u0 (THead 
+(Flat Appl) w (THead (Bind Abst) u t)) t0)) (subst1_head u0 w (lift (S O) d 
+x2) d H12 (Flat Appl) (THead (Bind Abst) u t) (lift (S O) d (THead (Bind 
+Abst) x3 x7)) (eq_ind_r T (THead (Bind Abst) (lift (S O) d x3) (lift (S O) (S 
+d) x7)) (\lambda (t0: T).(subst1 (s (Flat Appl) d) u0 (THead (Bind Abst) u t) 
+t0)) (subst1_head u0 u (lift (S O) d x3) (s (Flat Appl) d) H13 (Bind Abst) t 
+(lift (S O) (S d) x7) H23) (lift (S O) d (THead (Bind Abst) x3 x7)) 
+(lift_bind Abst x3 x7 (S O) d))) (lift (S O) d (THead (Flat Appl) x2 (THead 
+(Bind Abst) x3 x7))) (lift_flat Appl x2 (THead (Bind Abst) x3 x7) (S O) d)) 
+(ty3_appl g a x2 x3 H14 x0 x7 H26))))))))))) (lift_gen_bind Abst x4 x5 x1 (S 
+O) d H19)))))))) H15)))))))) H11))))))) H7))))))))))))))))))) (\lambda (c0: 
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda 
+(H1: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e 
+(Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: 
+C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))))))))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 
+t0)).(\lambda (H3: ((\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl 
+d c0 (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c0 a0) \to 
+(\forall (a: C).((drop (S O) d a0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2)))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: 
+nat).(\lambda (H4: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: 
+C).(\lambda (H5: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (H6: (drop (S 
+O) d a0 a)).(let H7 \def (H3 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda 
+(y1: T).(\lambda (_: T).(subst1 d u t4 (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u t0 (lift (S O) d y2)))) (\lambda (y1: 
 T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda 
 (_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda 
 (_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d 
-y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: 
-T).(\lambda (x3: T).(\lambda (H12: (subst1 d u t3 (lift (S O) d 
-x2))).(\lambda (H13: (subst1 d u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g 
-a x2 x3)).(let H15 \def (eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 
-(subst1_confluence_lift t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda 
-(y1: T).(\lambda (_: T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) 
-(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
-(THead (Flat Cast) x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat 
-Cast) (lift (S O) d x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead 
-(Flat Cast) t4 t3) t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) 
-t3 (lift (S O) d x2) H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat 
-Cast x0 x2 (S O) d)) (eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift 
-(S O) d x0)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) 
-(subst1_head u t0 (lift (S O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) 
-(lift (S O) d (THead (Flat Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) 
-(ty3_cast g a x2 x0 H15 x1 H10)))))))) H11))))))) H7)))))))))))))))))) c t1 
-t2 H))))).
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H8: (subst1 d u t4 (lift (S O) d x0))).(\lambda 
+(H9: (subst1 d u t0 (lift (S O) d x1))).(\lambda (H10: (ty3 g a x0 x1)).(let 
+H11 \def (H1 e u d H4 a0 H5 a H6) in (ex3_2_ind T T (\lambda (y1: T).(\lambda 
+(_: T).(subst1 d u t3 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u t4 (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
+g a y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(subst1 d u (THead 
+(Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
+T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H12: (subst1 d u t3 (lift (S O) d x2))).(\lambda (H13: (subst1 d 
+u t4 (lift (S O) d x3))).(\lambda (H14: (ty3 g a x2 x3)).(let H15 \def 
+(eq_ind T x3 (\lambda (t: T).(ty3 g a x2 t)) H14 x0 (subst1_confluence_lift 
+t4 x3 u d H13 x0 H8)) in (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: 
+T).(subst1 d u (THead (Flat Cast) t4 t3) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(subst1 d u (THead (Flat Cast) t0 t4) (lift (S O) d 
+y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (THead (Flat Cast) 
+x0 x2) (THead (Flat Cast) x1 x0) (eq_ind_r T (THead (Flat Cast) (lift (S O) d 
+x0) (lift (S O) d x2)) (\lambda (t: T).(subst1 d u (THead (Flat Cast) t4 t3) 
+t)) (subst1_head u t4 (lift (S O) d x0) d H8 (Flat Cast) t3 (lift (S O) d x2) 
+H12) (lift (S O) d (THead (Flat Cast) x0 x2)) (lift_flat Cast x0 x2 (S O) d)) 
+(eq_ind_r T (THead (Flat Cast) (lift (S O) d x1) (lift (S O) d x0)) (\lambda 
+(t: T).(subst1 d u (THead (Flat Cast) t0 t4) t)) (subst1_head u t0 (lift (S 
+O) d x1) d H9 (Flat Cast) t4 (lift (S O) d x0) H8) (lift (S O) d (THead (Flat 
+Cast) x1 x0)) (lift_flat Cast x1 x0 (S O) d)) (ty3_cast g a x2 x0 H15 x1 
+H10)))))))) H11))))))) H7)))))))))))))))))) c t1 t2 H))))).
 
 theorem ty3_gen_cvoid:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c 
@@ -769,12 +739,12 @@ n) (le_S (S d0) (plus O n) H5)) (le_O_n d0))) (eq_ind_r nat (S (minus n (S
 O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n (S O))) (lift n0 O t))) 
 (ty3_abbr g (minus n (S O)) a d u (getl_drop_conf_ge n (CHead d (Bind Abbr) 
 u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) (\lambda (n0: nat).(le 
-n0 n)) H5 (plus d0 (S O)) (plus_comm d0 (S O)))) t H1) n (minus_x_SO n 
-(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) 
-(plus_comm (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal 
-nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n 
-(le_O_n d0) H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda 
-(d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) 
+n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) n (minus_x_SO n 
+(le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n (S O))) (plus_sym 
+(S O) (minus n (S O)))) (S (plus O (minus n (S O)))) (refl_equal nat (S (plus 
+O (minus n (S O)))))) n (lt_plus_minus O n (le_lt_trans O d0 n (le_O_n d0) 
+H5))))))))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: 
+C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) 
 u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2: ((\forall (e: 
 C).(\forall (u0: T).(\forall (d0: nat).((getl d0 d (CHead e (Bind Void) u0)) 
 \to (\forall (a: C).((drop (S O) d0 d a) \to (ex3_2 T T (\lambda (y1: 
@@ -886,9 +856,9 @@ O))) (lift n O u) (eq_ind_r T (TLRef (plus (minus n (S O)) (S O))) (\lambda
 (eq_ind_r nat (S (minus n (S O))) (\lambda (n0: nat).(ty3 g a (TLRef (minus n 
 (S O))) (lift n0 O u))) (ty3_abst g (minus n (S O)) a d u (getl_drop_conf_ge 
 n (CHead d (Bind Abst) u) c0 H0 a (S O) d0 H4 (eq_ind_r nat (plus (S O) d0) 
-(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_comm d0 (S O)))) t H1) 
+(\lambda (n0: nat).(le n0 n)) H5 (plus d0 (S O)) (plus_sym d0 (S O)))) t H1) 
 n (minus_x_SO n (le_lt_trans O d0 n (le_O_n d0) H5)))) (plus (S O) (minus n 
-(S O))) (plus_comm (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) 
+(S O))) (plus_sym (S O) (minus n (S O)))) (S (plus O (minus n (S O)))) 
 (refl_equal nat (S (plus O (minus n (S O)))))) n (lt_plus_minus O n 
 (le_lt_trans O d0 n (le_O_n d0) H5))))))))))))))))))) (\lambda (c0: 
 C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u t)).(\lambda 
@@ -902,130 +872,89 @@ T).(\lambda (y2: T).(eq T t (lift (S O) d y2)))) (\lambda (y1: T).(\lambda
 b) u) (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 
 (Bind b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift 
 (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (t0: 
-T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t4 t0)).(\lambda (H5: ((\forall 
-(e: C).(\forall (u0: T).(\forall (d: nat).((getl d (CHead c0 (Bind b) u) 
-(CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d (CHead c0 (Bind 
-b) u) a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t4 (lift (S O) 
-d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 (lift (S O) d y2)))) 
 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: 
-C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H6: (getl d c0 (CHead e (Bind 
-Void) u0))).(\lambda (a: C).(\lambda (H7: (drop (S O) d c0 a)).(let H8 \def 
-(H1 e u0 d H6 a H7) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind 
+Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def 
+(H1 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
 u (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t (lift (S O) d 
 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T 
 (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) u t3) (lift (S O) d 
 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) u t4) (lift (S 
 O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda 
-(x0: T).(\lambda (x1: T).(\lambda (H9: (eq T u (lift (S O) d x0))).(\lambda 
-(H10: (eq T t (lift (S O) d x1))).(\lambda (H11: (ty3 g a x0 x1)).(let H12 
-\def (eq_ind T t (\lambda (t5: T).(ty3 g c0 u t5)) H0 (lift (S O) d x1) H10) 
-in (let H13 \def (eq_ind T u (\lambda (t5: T).(ty3 g c0 t5 (lift (S O) d 
-x1))) H12 (lift (S O) d x0) H9) in (let H14 \def (eq_ind T u (\lambda (t5: 
+(x0: T).(\lambda (x1: T).(\lambda (H7: (eq T u (lift (S O) d x0))).(\lambda 
+(H8: (eq T t (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def 
+(eq_ind T t (\lambda (t0: T).(ty3 g c0 u t0)) H0 (lift (S O) d x1) H8) in 
+(let H11 \def (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 (lift (S O) d x1))) 
+H10 (lift (S O) d x0) H7) in (let H12 \def (eq_ind T u (\lambda (t0: 
 T).(\forall (e0: C).(\forall (u1: T).(\forall (d0: nat).((getl d0 (CHead c0 
-(Bind b) t5) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 
-(CHead c0 (Bind b) t5) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T t4 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 
+(Bind b) t0) (CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 
+(CHead c0 (Bind b) t0) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T t3 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 
 (lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 
-y2))))))))))) H5 (lift (S O) d x0) H9) in (let H15 \def (eq_ind T u (\lambda 
-(t5: T).(ty3 g (CHead c0 (Bind b) t5) t4 t0)) H4 (lift (S O) d x0) H9) in 
-(let H16 \def (eq_ind T u (\lambda (t5: T).(\forall (e0: C).(\forall (u1: 
-T).(\forall (d0: nat).((getl d0 (CHead c0 (Bind b) t5) (CHead e0 (Bind Void) 
-u1)) \to (\forall (a0: C).((drop (S O) d0 (CHead c0 (Bind b) t5) a0) \to 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) d0 y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S O) d0 y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a0 y1 y2))))))))))) H3 (lift (S O) d x0) H9) in 
-(let H17 \def (eq_ind T u (\lambda (t5: T).(ty3 g (CHead c0 (Bind b) t5) t3 
-t4)) H2 (lift (S O) d x0) H9) in (eq_ind_r T (lift (S O) d x0) (\lambda (t5: 
-T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind b) t5 t3) 
-(lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) 
-t5 t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2))))) (let H18 \def (H16 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind 
-Void) u0) H6 (lift (S O) d x0)) (CHead a (Bind b) x0) (drop_skip_bind (S O) d 
-c0 a H7 b x0)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T t3 
-(lift (S O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t4 (lift (S 
-O) (S d) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) 
-x0) y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind 
-b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: 
-T).(eq T (THead (Bind b) (lift (S O) d x0) t4) (lift (S O) d y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H19: (eq T t3 (lift (S O) (S d) x2))).(\lambda (H20: (eq T t4 
-(lift (S O) (S d) x3))).(\lambda (H21: (ty3 g (CHead a (Bind b) x0) x2 
-x3)).(let H22 \def (eq_ind T t4 (\lambda (t5: T).(\forall (e0: C).(\forall 
-(u1: T).(\forall (d0: nat).((getl d0 (CHead c0 (Bind b) (lift (S O) d x0)) 
-(CHead e0 (Bind Void) u1)) \to (\forall (a0: C).((drop (S O) d0 (CHead c0 
-(Bind b) (lift (S O) d x0)) a0) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T t5 (lift (S O) d0 y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t0 
-(lift (S O) d0 y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a0 y1 
-y2))))))))))) H14 (lift (S O) (S d) x3) H20) in (eq_ind_r T (lift (S O) (S d) 
-x3) (\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead 
+y2))))))))))) H3 (lift (S O) d x0) H7) in (let H13 \def (eq_ind T u (\lambda 
+(t0: T).(ty3 g (CHead c0 (Bind b) t0) t3 t4)) H2 (lift (S O) d x0) H7) in 
+(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Bind b) t0 t3) (lift (S O) d y1)))) (\lambda 
+(_: T).(\lambda (y2: T).(eq T (THead (Bind b) t0 t4) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (let H14 \def (H12 e u0 
+(S d) (getl_head (Bind b) d c0 (CHead e (Bind Void) u0) H4 (lift (S O) d x0)) 
+(CHead a (Bind b) x0) (drop_skip_bind (S O) d c0 a H5 b x0)) in (ex3_2_ind T 
+T (\lambda (y1: T).(\lambda (_: T).(eq T t3 (lift (S O) (S d) y1)))) (\lambda 
+(_: T).(\lambda (y2: T).(eq T t4 (lift (S O) (S d) y2)))) (\lambda (y1: 
+T).(\lambda (y2: T).(ty3 g (CHead a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda 
+(y1: T).(\lambda (_: T).(eq T (THead (Bind b) (lift (S O) d x0) t3) (lift (S 
+O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S 
+O) d x0) t4) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H15: (eq T t3 (lift (S 
+O) (S d) x2))).(\lambda (H16: (eq T t4 (lift (S O) (S d) x3))).(\lambda (H17: 
+(ty3 g (CHead a (Bind b) x0) x2 x3)).(eq_ind_r T (lift (S O) (S d) x3) 
+(\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead 
 (Bind b) (lift (S O) d x0) t3) (lift (S O) d y1)))) (\lambda (_: T).(\lambda 
-(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t5) (lift (S O) d y2)))) 
+(y2: T).(eq T (THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y2)))) 
 (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind_r T (lift (S O) 
-(S d) x2) (\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-(THead (Bind b) (lift (S O) d x0) t5) (lift (S O) d y1)))) (\lambda (_: 
+(S d) x2) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+(THead (Bind b) (lift (S O) d x0) t0) (lift (S O) d y1)))) (\lambda (_: 
 T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) 
 x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2))))) (let H23 \def (H22 e u0 (S d) (getl_head (Bind b) d c0 (CHead e (Bind 
-Void) u0) H6 (lift (S O) d x0)) (CHead a (Bind b) x0) (drop_skip_bind (S O) d 
-c0 a H7 b x0)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift 
-(S O) (S d) x3) (lift (S O) (S d) y1)))) (\lambda (_: T).(\lambda (y2: T).(eq 
-T t0 (lift (S O) (S d) y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g (CHead 
-a (Bind b) x0) y1 y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) 
-(lift (S O) (S d) x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: 
-T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H24: (eq T 
-(lift (S O) (S d) x3) (lift (S O) (S d) x4))).(\lambda (_: (eq T t0 (lift (S 
-O) (S d) x5))).(\lambda (H26: (ty3 g (CHead a (Bind b) x0) x4 x5)).(let H27 
-\def (eq_ind_r T x4 (\lambda (t5: T).(ty3 g (CHead a (Bind b) x0) t5 x5)) H26 
-x3 (lift_inj x3 x4 (S O) (S d) H24)) in (eq_ind T (lift (S O) d (THead (Bind 
-b) x0 x2)) (\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
-t5 (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind 
-b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift (S O) d y2)))) (\lambda 
-(y1: T).(\lambda (y2: T).(ty3 g a y1 y2))))) (eq_ind T (lift (S O) d (THead 
-(Bind b) x0 x3)) (\lambda (t5: T).(ex3_2 T T (\lambda (y1: T).(\lambda (_: 
-T).(eq T (lift (S O) d (THead (Bind b) x0 x2)) (lift (S O) d y1)))) (\lambda 
-(_: T).(\lambda (y2: T).(eq T t5 (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2))))) (ex3_2_intro T T (\lambda (y1: 
-T).(\lambda (_: T).(eq T (lift (S O) d (THead (Bind b) x0 x2)) (lift (S O) d 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (lift (S O) d (THead (Bind b) 
-x0 x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2))) (THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (refl_equal T (lift (S O) 
-d (THead (Bind b) x0 x2))) (refl_equal T (lift (S O) d (THead (Bind b) x0 
-x3))) (ty3_bind g a x0 x1 H11 b x2 x3 H21 x5 H27)) (THead (Bind b) (lift (S 
-O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3 (S O) d)) (THead (Bind b) 
-(lift (S O) d x0) (lift (S O) (S d) x2)) (lift_bind b x0 x2 (S O) d)))))))) 
-H23)) t3 H19) t4 H20))))))) H18)) u H9)))))))))))) H8))))))))))))))))))))) 
-(\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w 
-u)).(\lambda (H1: ((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl 
-d c0 (CHead e (Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to 
-(ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T w (lift (S O) d y1)))) 
-(\lambda (_: T).(\lambda (y2: T).(eq T u (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (v: T).(\lambda (t: 
-T).(\lambda (H2: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (H3: 
-((\forall (e: C).(\forall (u0: T).(\forall (d: nat).((getl d c0 (CHead e 
-(Bind Void) u0)) \to (\forall (a: C).((drop (S O) d c0 a) \to (ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_: 
-T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2)))) 
-(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))))))))))).(\lambda (e: 
-C).(\lambda (u0: T).(\lambda (d: nat).(\lambda (H4: (getl d c0 (CHead e (Bind 
-Void) u0))).(\lambda (a: C).(\lambda (H5: (drop (S O) d c0 a)).(let H6 \def 
-(H3 e u0 d H4 a H5) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T 
+y2))))) (ex3_2_intro T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Bind 
+b) (lift (S O) d x0) (lift (S O) (S d) x2)) (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) 
+x3)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) 
+(THead (Bind b) x0 x2) (THead (Bind b) x0 x3) (sym_eq T (lift (S O) d (THead 
+(Bind b) x0 x2)) (THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x2)) 
+(lift_bind b x0 x2 (S O) d)) (sym_eq T (lift (S O) d (THead (Bind b) x0 x3)) 
+(THead (Bind b) (lift (S O) d x0) (lift (S O) (S d) x3)) (lift_bind b x0 x3 
+(S O) d)) (ty3_bind g a x0 x1 H9 b x2 x3 H17)) t3 H15) t4 H16)))))) H14)) u 
+H7)))))))))) H6)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda 
+(u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: ((\forall (e: C).(\forall 
+(u0: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall 
+(a: C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: 
+T).(eq T w (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T u 
+(lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
+y2)))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v 
+(THead (Bind Abst) u t))).(\lambda (H3: ((\forall (e: C).(\forall (u0: 
+T).(\forall (d: nat).((getl d c0 (CHead e (Bind Void) u0)) \to (\forall (a: 
+C).((drop (S O) d c0 a) \to (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T 
 v (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Bind 
 Abst) u t) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 
-y2))) (ex3_2 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w 
-v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat 
-Appl) w (THead (Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: 
-T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H7: (eq T v (lift (S O) d x0))).(\lambda (H8: (eq T (THead (Bind 
-Abst) u t) (lift (S O) d x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def 
-(eq_ind T v (\lambda (t0: T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift 
-(S O) d x0) H7) in (eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T 
-(\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d 
-y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead 
-(Bind Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 
-g a y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead 
+y2)))))))))))).(\lambda (e: C).(\lambda (u0: T).(\lambda (d: nat).(\lambda 
+(H4: (getl d c0 (CHead e (Bind Void) u0))).(\lambda (a: C).(\lambda (H5: 
+(drop (S O) d c0 a)).(let H6 \def (H3 e u0 d H4 a H5) in (ex3_2_ind T T 
+(\lambda (y1: T).(\lambda (_: T).(eq T v (lift (S O) d y1)))) (\lambda (_: 
+T).(\lambda (y2: T).(eq T (THead (Bind Abst) u t) (lift (S O) d y2)))) 
+(\lambda (y1: T).(\lambda (y2: T).(ty3 g a y1 y2))) (ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w v) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind 
+Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T v (lift (S O) 
+d x0))).(\lambda (H8: (eq T (THead (Bind Abst) u t) (lift (S O) d 
+x1))).(\lambda (H9: (ty3 g a x0 x1)).(let H10 \def (eq_ind T v (\lambda (t0: 
+T).(ty3 g c0 t0 (THead (Bind Abst) u t))) H2 (lift (S O) d x0) H7) in 
+(eq_ind_r T (lift (S O) d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y1: 
+T).(\lambda (_: T).(eq T (THead (Flat Appl) w t0) (lift (S O) d y1)))) 
+(\lambda (_: T).(\lambda (y2: T).(eq T (THead (Flat Appl) w (THead (Bind 
+Abst) u t)) (lift (S O) d y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g a 
+y1 y2))))) (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead 
 (Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift (S O) d 
 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift (S O) (S d) z)))) (ex3_2 
 T T (\lambda (y1: T).(\lambda (_: T).(eq T (THead (Flat Appl) w (lift (S O) d