update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / pr3 / props.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma
index 777f5d85d886636069e512ed5bd96c6b36196f26..c9cf9c301dc34b39fc8678c4463965e5da38b585 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr3/props.ma
@@ -14,13 +14,15 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr3/pr1.ma".
+include "basic_1/pr3/fwd.ma".
 
-include "Basic-1/pr2/props.ma".
+include "basic_1/pr3/pr1.ma".
 
-include "Basic-1/pr1/props.ma".
+include "basic_1/pr2/props.ma".
 
-theorem clear_pr3_trans:
+include "basic_1/pr1/props.ma".
+
+lemma clear_pr3_trans:
  \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to 
 (\forall (c1: C).((clear c1 c2) \to (pr3 c1 t1 t2))))))
 \def
@@ -30,19 +32,13 @@ T).(\lambda (t0: T).(pr3 c1 t t0))) (\lambda (t: T).(pr3_refl c1 t)) (\lambda
 (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c2 t4 t3)).(\lambda (t5: 
 T).(\lambda (_: (pr3 c2 t3 t5)).(\lambda (H3: (pr3 c1 t3 t5)).(pr3_sing c1 t3 
 t4 (clear_pr2_trans c2 t4 t3 H1 c1 H0) t5 H3))))))) t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 107
-END *)
 
-theorem pr3_pr2:
+lemma pr3_pr2:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (pr3 c 
 t1 t2))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 
 t2)).(pr3_sing c t2 t1 H t2 (pr3_refl c t2))))).
-(* COMMENTS
-Initial nodes: 31
-END *)
 
 theorem pr3_t:
  \forall (t2: T).(\forall (t1: T).(\forall (c: C).((pr3 c t1 t2) \to (\forall 
@@ -55,11 +51,8 @@ c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3
 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall 
 (t5: T).((pr3 c t4 t5) \to (pr3 c t0 t5))))).(\lambda (t5: T).(\lambda (H3: 
 (pr3 c t4 t5)).(pr3_sing c t0 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
-(* COMMENTS
-Initial nodes: 127
-END *)
 
-theorem pr3_thin_dx:
+lemma pr3_thin_dx:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
 (u: T).(\forall (f: F).(pr3 c (THead (Flat f) u t1) (THead (Flat f) u 
 t2)))))))
@@ -72,11 +65,8 @@ T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0
 t4)).(\lambda (H2: (pr3 c (THead (Flat f) u t0) (THead (Flat f) u 
 t4))).(pr3_sing c (THead (Flat f) u t0) (THead (Flat f) u t3) (pr2_thin_dx c 
 t3 t0 H0 u f) (THead (Flat f) u t4) H2))))))) t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 167
-END *)
 
-theorem pr3_head_1:
+lemma pr3_head_1:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
 (k: K).(\forall (t: T).(pr3 c (THead k u1 t) (THead k u2 t)))))))
 \def
@@ -89,11 +79,8 @@ T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda
 (THead k t2 t) (THead k t3 t)))))).(\lambda (k: K).(\lambda (t: T).(pr3_sing 
 c (THead k t2 t) (THead k t1 t) (pr2_head_1 c t1 t2 H0 k t) (THead k t3 t) 
 (H2 k t)))))))))) u1 u2 H)))).
-(* COMMENTS
-Initial nodes: 167
-END *)
 
-theorem pr3_head_2:
+lemma pr3_head_2:
  \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall 
 (k: K).((pr3 (CHead c k u) t1 t2) \to (pr3 c (THead k u t1) (THead k u 
 t2)))))))
@@ -106,9 +93,6 @@ T).(\lambda (H0: (pr2 (CHead c k u) t3 t0)).(\lambda (t4: T).(\lambda (_:
 (pr3 (CHead c k u) t0 t4)).(\lambda (H2: (pr3 c (THead k u t0) (THead k u 
 t4))).(pr3_sing c (THead k u t0) (THead k u t3) (pr2_head_2 c u t3 t0 k H0) 
 (THead k u t4) H2))))))) t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 175
-END *)
 
 theorem pr3_head_21:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
@@ -119,9 +103,6 @@ c (THead k u1 t1) (THead k u2 t2)))))))))
 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
 (CHead c k u1) t1 t2)).(pr3_t (THead k u1 t2) (THead k u1 t1) c (pr3_head_2 c 
 u1 t1 t2 k H0) (THead k u2 t2) (pr3_head_1 c u1 u2 H k t2))))))))).
-(* COMMENTS
-Initial nodes: 89
-END *)
 
 theorem pr3_head_12:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
@@ -132,11 +113,8 @@ c (THead k u1 t1) (THead k u2 t2)))))))))
 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 
 (CHead c k u2) t1 t2)).(pr3_t (THead k u2 t1) (THead k u1 t1) c (pr3_head_1 c 
 u1 u2 H k t1) (THead k u2 t2) (pr3_head_2 c u2 t1 t2 k H0))))))))).
-(* COMMENTS
-Initial nodes: 89
-END *)
 
-theorem pr3_cflat:
+lemma pr3_cflat:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall 
 (f: F).(\forall (v: T).(pr3 (CHead c (Flat f) v) t1 t2))))))
 \def
@@ -148,9 +126,6 @@ T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
 (_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (f: F).(\forall (v: T).(pr3 (CHead 
 c (Flat f) v) t3 t5))))).(\lambda (f: F).(\lambda (v: T).(pr3_sing (CHead c 
 (Flat f) v) t3 t4 (pr2_cflat c t4 t3 H0 f v) t5 (H2 f v)))))))))) t1 t2 H)))).
-(* COMMENTS
-Initial nodes: 151
-END *)
 
 theorem pr3_flat:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
@@ -161,11 +136,8 @@ theorem pr3_flat:
 u2)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda 
 (f: F).(pr3_head_12 c u1 u2 H (Flat f) t1 t2 (pr3_cflat c t1 t2 H0 f 
 u2))))))))).
-(* COMMENTS
-Initial nodes: 59
-END *)
 
-theorem pr3_pr0_pr2_t:
+lemma pr3_pr0_pr2_t:
  \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall 
 (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 
 (CHead c k u1) t1 t2))))))))
@@ -189,28 +161,26 @@ c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d
 (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d 
 (Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda 
 (H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \def 
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind 
-Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 
-(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 
-\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) 
-with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K 
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) 
+(f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead 
+c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) 
 (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) 
-u2) (CHead d (Bind Abbr) u) H9))) in ((let H12 \def (f_equal C T (\lambda (e: 
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | 
-(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) 
-u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind 
-b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda (H13: (eq B Abbr 
-b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0: 
-T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3 
-(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 
-t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t) 
-(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x 
-t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr) 
-u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind 
-Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) 
+u2) (CHead d (Bind Abbr) u) H9))) in ((let H11 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow 
+(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
+(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d 
+(Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) 
+H9))) in ((let H12 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) 
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 
+(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda 
+(H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u 
+(\lambda (t0: T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda 
+(b0: B).(pr3 (CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: 
+T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind 
+Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda 
+(H17: (pr0 x t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c 
+(Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 
+(CHead c (Bind Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17)))))) 
 (pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f: 
 F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) 
 u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O 
@@ -235,11 +205,8 @@ u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))
 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) 
 (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f 
 u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))).
-(* COMMENTS
-Initial nodes: 1557
-END *)
 
-theorem pr3_pr2_pr2_t:
+lemma pr3_pr2_pr2_t:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall 
 (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 
 (CHead c k u1) t1 t2))))))))
@@ -272,58 +239,54 @@ t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t)
 (\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 
 (CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0 
 (Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda 
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 
-| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind 
-b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 
-\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) 
-with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K 
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) 
-\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) 
-(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \def 
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with 
-[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind 
-Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) 
-u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let 
-H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in 
-(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind 
-T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i 
-O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda 
-(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let 
-H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O 
-i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n 
-u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7)) 
-(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda 
-(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0 
-x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind 
-Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead 
-c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i) 
-(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1) 
-(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x 
-O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14)) 
-H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0 
-(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 
-(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0 
-(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5 
-H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0) 
-H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0 
-(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4 
-t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr) 
-u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl 
-(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1) 
-t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t) 
-(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta 
-(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind 
-Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4 
-t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat 
-f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 
-(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f) 
-c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10))))) 
-i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))).
-(* COMMENTS
-Initial nodes: 1697
-END *)
+(e: C).(match e with [(CSort _) \Rightarrow d0 | (CHead c2 _ _) \Rightarrow 
+c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 
+(CHead d0 (Bind Abbr) u0) t H12)) in ((let H14 \def (f_equal C B (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow 
+(match k0 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) 
+(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead 
+d0 (Bind Abbr) u0) t H12)) in ((let H15 \def (f_equal C T (\lambda (e: 
+C).(match e with [(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) 
+(CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead 
+d0 (Bind Abbr) u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq 
+C d0 c0)).(let H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) 
+H11 t H15) in (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 
+t6)) (ex2_ind T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: 
+T).(subst0 (S (plus i O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) 
+(\lambda (x: T).(\lambda (H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S 
+(plus i O)) u x t6)).(let H21 \def (f_equal nat nat S (plus i O) i (sym_eq 
+nat i (plus i O) (plus_n_O i))) in (let H22 \def (eq_ind nat (S (plus i O)) 
+(\lambda (n: nat).(subst0 n u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda 
+(t7: T).(subst0 O t1 t5 t7)) (\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 
+(Bind Abbr) t1) t4 t6) (\lambda (x0: T).(\lambda (H23: (subst0 O t1 t5 
+x0)).(\lambda (H24: (pr0 x0 x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 
+(pr2_delta (CHead c0 (Bind Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 
+x0 H23) t6 (pr3_pr2 (CHead c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 
+(Bind Abbr) t1) d u (S i) (getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 
+t1 (clear_bind Abbr c0 t1) (CHead d (Bind Abbr) u) i H0) x0 x H24 t6 
+H22)))))) (pr0_subst0_back t2 t5 x O H19 t1 H1))))))) (subst0_subst0 t5 t6 t 
+O H18 t2 u i H2)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: 
+(clear (CHead c0 (Flat f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 
+(Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 O (getl_intro O c0 
+(CHead d0 (Bind Abbr) u0) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead d0 
+(Bind Abbr) u0) t H12)) t4 t5 H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k 
+t) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda (_: (((getl 
+i1 (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to 
+(pr3 (CHead c0 k t1) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) 
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind 
+(\lambda (k0: K).((getl (S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to 
+(pr3 (CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) 
+(CHead c0 (Bind b) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) 
+t1) t4 t6 (pr2_delta (CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) 
+i1 c0 (CHead d0 (Bind Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) 
+u0) t i1 H12) t1) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S 
+i1) (CHead c0 (Flat f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 
+(Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) 
+(getl_gen_S (Flat f) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) 
+f t1)))) k H10))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c 
+u1 u2 H)))).
 
-theorem pr3_pr2_pr3_t:
+lemma pr3_pr2_pr3_t:
  \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall 
 (k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to 
 (pr3 (CHead c k u1) t1 t2))))))))
@@ -338,9 +301,6 @@ T).(\lambda (H0: (pr2 (CHead c k u2) t3 t0)).(\lambda (t4: T).(\lambda (_:
 \to (pr3 (CHead c k u1) t0 t4))))).(\lambda (u1: T).(\lambda (H3: (pr2 c u1 
 u2)).(pr3_t t0 t3 (CHead c k u1) (pr3_pr2_pr2_t c u1 u2 H3 t3 t0 k H0) t4 (H2 
 u1 H3)))))))))) t1 t2 H)))))).
-(* COMMENTS
-Initial nodes: 199
-END *)
 
 theorem pr3_pr3_pr3_t:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall 
@@ -357,11 +317,8 @@ t3)).(\lambda (H2: ((\forall (t4: T).(\forall (t5: T).(\forall (k: K).((pr3
 (CHead c k t3) t4 t5) \to (pr3 (CHead c k t2) t4 t5))))))).(\lambda (t0: 
 T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H3: (pr3 (CHead c k t3) t0 
 t4)).(pr3_pr2_pr3_t c t2 t0 t4 k (H2 t0 t4 k H3) t1 H0))))))))))) u1 u2 H)))).
-(* COMMENTS
-Initial nodes: 187
-END *)
 
-theorem pr3_lift:
+lemma pr3_lift:
  \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h 
 d c e) \to (\forall (t1: T).(\forall (t2: T).((pr3 e t1 t2) \to (pr3 c (lift 
 h d t1) (lift h d t2)))))))))
@@ -374,11 +331,8 @@ d t0)))) (\lambda (t: T).(pr3_refl c (lift h d t))) (\lambda (t0: T).(\lambda
 t4)).(\lambda (H3: (pr3 c (lift h d t0) (lift h d t4))).(pr3_sing c (lift h d 
 t0) (lift h d t3) (pr2_lift c e h d H t3 t0 H1) (lift h d t4) H3))))))) t1 t2 
 H0)))))))).
-(* COMMENTS
-Initial nodes: 167
-END *)
 
-theorem pr3_eta:
+lemma pr3_eta:
  \forall (c: C).(\forall (w: T).(\forall (u: T).(let t \def (THead (Bind 
 Abst) w u) in (\forall (v: T).((pr3 c v w) \to (pr3 c (THead (Bind Abst) v 
 (THead (Flat Appl) (TLRef O) (lift (S O) O t))) t))))))
@@ -397,11 +351,1254 @@ O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u))) u (pr1_sing
 (THead (Bind Abbr) (TLRef O) (lift (S O) O u)) (THead (Bind Abbr) (TLRef O) 
 (lift (S O) (S O) u)) (pr0_delta1 (TLRef O) (TLRef O) (pr0_refl (TLRef O)) 
 (lift (S O) (S O) u) (lift (S O) (S O) u) (pr0_refl (lift (S O) (S O) u)) 
-(lift (S O) O u) (subst1_lift_S u O O (le_n O))) u (pr1_pr0 (THead (Bind 
+(lift (S O) O u) (subst1_lift_S u O O (le_O_n O))) u (pr1_pr0 (THead (Bind 
 Abbr) (TLRef O) (lift (S O) O u)) u (pr0_zeta Abbr not_abbr_abst u u 
 (pr0_refl u) (TLRef O))))) (CHead c (Bind Abst) w))) (lift (S O) O (THead 
 (Bind Abst) w u)) (lift_bind Abst w u (S O) O))))))).
-(* COMMENTS
-Initial nodes: 523
-END *)
+
+lemma pr3_gen_void:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c (Bind Void) u1) t1 
+(lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) t1 t2)))))) (pr3 (CHead c 
+(Bind Void) u1) t1 (lift (S O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y 
+x)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Bind Void) u1 t)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+t t2)))))) (pr3 (CHead c (Bind Void) u1) t (lift (S O) O x))))) (unintro T u1 
+(\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind Void) t x0)) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x0 t2)))))) (pr3 (CHead c (Bind Void) t) x0 (lift (S O) O x)))))) (pr3_ind c 
+(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t 
+(THead (Bind Void) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 
+(lift (S O) O t0)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: 
+T).(\lambda (H1: (eq T t (THead (Bind Void) x0 x1))).(eq_ind_r T (THead (Bind 
+Void) x0 x1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c (Bind Void) x0) x1 
+(lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t2)))))) (pr3 (CHead c 
+(Bind Void) x0) x1 (lift (S O) O (THead (Bind Void) x0 x1))) (ex3_2_intro T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) x0 x1) (THead 
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x1 t2))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) 
+(pr3_refl c x0) (\lambda (b: B).(\lambda (u: T).(pr3_refl (CHead c (Bind b) 
+u) x1))))) t H1))))) (\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c 
+t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall 
+(x0: T).(\forall (x1: T).((eq T t2 (THead (Bind Void) x0 x1)) \to (or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
+(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)))))))).(\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H4: (eq T t3 (THead (Bind Void) x0 x1))).(let 
+H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c t t2)) H1 (THead (Bind Void) x0 
+x1) H4) in (let H6 \def (pr2_gen_void c x0 x1 t2 H5) in (or_ind (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 t5)))))) 
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
+t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
+Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Void) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 t5))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
+(Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead 
+c (Bind b) u) x1 t5))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
+T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) 
+O t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Bind 
+Void) x2 x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H11 \def (eq_ind 
+T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
+Void) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x5 t5)))))) (pr3 (CHead c (Bind Void) x4) x5 (lift (S O) O 
+t4))))))) H3 (THead (Bind Void) x2 x3) H8) in (let H12 \def (H11 x2 x3 
+(refl_equal T (THead (Bind Void) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5)))))) (pr3 (CHead c 
+(Bind Void) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
+(Bind Void) x0) x1 (lift (S O) O t4))) (\lambda (H13: (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))))).(ex3_2_ind T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x3 t5))))) 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H14: (eq T t4 (THead (Bind Void) x4 
+x5))).(\lambda (H15: (pr3 c x2 x4)).(\lambda (H16: ((\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) x3 x5))))).(or_introl (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) (pr3 (CHead c 
+(Bind Void) x0) x1 (lift (S O) O t4)) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5))))) x4 x5 H14 
+(pr3_sing c x2 x0 H9 x4 H15) (\lambda (b: B).(\lambda (u: T).(pr3_sing (CHead 
+c (Bind b) u) x3 x1 (H10 b u) x5 (H16 b u))))))))))) H13)) (\lambda (H13: 
+(pr3 (CHead c (Bind Void) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Void) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x1 t5)))))) 
+(pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind 
+Void) x0) x3 x1 (H10 Void x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift 
+(S O) O t4) (Bind Void) H13 x0 H9)))) H12)))))))) H7)) (\lambda (H7: 
+((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 (lift (S O) O 
+t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Void) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b) u) x1 t5)))))) (pr3 (CHead c (Bind Void) x0) x1 (lift (S O) O t4)) 
+(pr3_sing (CHead c (Bind Void) x0) (lift (S O) O t2) x1 (H7 Void x0) (lift (S 
+O) O t4) (pr3_lift (CHead c (Bind Void) x0) c (S O) O (drop_drop (Bind Void) 
+O c c (drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+lemma pr3_gen_abbr:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
+O) O x)))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 (\lambda 
+(t: T).((eq T y (THead (Bind Abbr) u1 t)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) u1) t t2)))) (pr3 (CHead c (Bind Abbr) u1) t (lift (S O) 
+O x))))) (unintro T u1 (\lambda (t: T).(\forall (x0: T).((eq T y (THead (Bind 
+Abbr) t x0)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) t) x0 t2)))) (pr3 
+(CHead c (Bind Abbr) t) x0 (lift (S O) O x)))))) (pr3_ind c (\lambda (t: 
+T).(\lambda (t0: T).(\forall (x0: T).(\forall (x1: T).((eq T t (THead (Bind 
+Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t0)))))))) (\lambda (t: T).(\lambda 
+(x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t (THead (Bind Abbr) x0 
+x1))).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t0: T).(or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) 
+x1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t2)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x0 x1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t2)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t2))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 
+x1)) (pr3_refl c x0) (pr3_refl (CHead c (Bind Abbr) x0) x1))) t H1))))) 
+(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: 
+T).(\lambda (H2: (pr3 c t2 t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: 
+T).((eq T t2 (THead (Bind Abbr) x0 x1)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4)))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
+(THead (Bind Abbr) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
+t t2)) H1 (THead (Bind Abbr) x0 x1) H4) in (let H6 \def (pr2_gen_abbr c x0 x1 
+t2 H5) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
+(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
+T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
+(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z t5)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 
+(lift (S O) O t2)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H7: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind 
+b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead 
+c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z 
+t5))))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 
+(CHead c (Bind b) u) x1 t5))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: 
+T).(pr2 (CHead c (Bind Abbr) u) x1 t5))) (ex3_2 T T (\lambda (y0: T).(\lambda 
+(_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z t5))))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr2 
+c x0 x2)).(\lambda (H10: (or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c 
+(Bind b) u) x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 
+(CHead c (Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: 
+T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: 
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) 
+z x3)))))).(or3_ind (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 x3))) (ex2 T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
+(Bind Abbr) u) x1 x3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
+(CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) x0) z x3)))) 
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H11: ((\forall (b: 
+B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x1 x3))))).(let H12 \def (eq_ind 
+T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t (THead (Bind 
+Abbr) x4 x5)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x4 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x4) x5 t5)))) (pr3 
+(CHead c (Bind Abbr) x4) x5 (lift (S O) O t4))))))) H3 (THead (Bind Abbr) x2 
+x3) H8) in (let H13 \def (H12 x2 x3 (refl_equal T (THead (Bind Abbr) x2 x3))) 
+in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind 
+Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c 
+(Bind Abbr) x2) x3 (lift (S O) O t4)) (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4))) (\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 
+t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))) (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 
+t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c 
+(Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H15: (eq T t4 (THead (Bind Abbr) x4 x5))).(\lambda (H16: (pr3 c 
+x2 x4)).(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 x5)).(eq_ind_r T 
+(THead (Bind Abbr) x4 x5) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+(THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x4 x5))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x4 x5 (refl_equal T (THead (Bind Abbr) x4 
+x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing (CHead c (Bind Abbr) x0) x3 x1 
+(H11 Abbr x0) x5 (pr3_pr2_pr3_t c x2 x3 x5 (Bind Abbr) H17 x0 H9)))) t4 
+H15)))))) H14)) (\lambda (H14: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
+x0) x3 x1 (H11 Abbr x0) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) 
+O t4) (Bind Abbr) H14 x0 H9)))) H13)))) (\lambda (H11: (ex2 T (\lambda (u: 
+T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) x1 
+x3)))).(ex2_ind T (\lambda (u: T).(pr0 x0 u)) (\lambda (u: T).(pr2 (CHead c 
+(Bind Abbr) u) x1 x3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
+T).(\lambda (H12: (pr0 x0 x4)).(\lambda (H13: (pr2 (CHead c (Bind Abbr) x4) 
+x1 x3)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(\forall (x5: T).(\forall 
+(x6: T).((eq T t (THead (Bind Abbr) x5 x6)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x5 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x5) x6 t5)))) (pr3 (CHead c (Bind Abbr) x5) x6 (lift (S 
+O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H15 \def (H14 x2 x3 
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H16: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x2) x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5))) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda 
+(x5: T).(\lambda (x6: T).(\lambda (H17: (eq T t4 (THead (Bind Abbr) x5 
+x6))).(\lambda (H18: (pr3 c x2 x5)).(\lambda (H19: (pr3 (CHead c (Bind Abbr) 
+x2) x3 x6)).(eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(or (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) 
+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x5 x6))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x5 x6) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x5 x6 (refl_equal T (THead (Bind Abbr) x5 
+x6)) (pr3_sing c x2 x0 H9 x5 H18) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) 
+(pr3_pr0_pr2_t x0 x4 H12 c x1 x3 (Bind Abbr) H13) x6 (pr3_pr2_pr3_t c x2 x3 
+x6 (Bind Abbr) H19 x0 H9)))) t4 H17)))))) H16)) (\lambda (H16: (pr3 (CHead c 
+(Bind Abbr) x2) x3 (lift (S O) O t4))).(or_intror (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O t4)) (pr3_t x3 x1 (CHead c (Bind Abbr) x0) (pr3_pr0_pr2_t x0 x4 H12 c x1 
+x3 (Bind Abbr) H13) (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 (lift (S O) O 
+t4) (Bind Abbr) H16 x0 H9)))) H15)))))) H11)) (\lambda (H11: (ex3_2 T T 
+(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) 
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
+T).(pr2 (CHead c (Bind Abbr) x0) z x3))))).(ex3_2_ind T T (\lambda (y0: 
+T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) x0) x1 y0))) (\lambda (y0: 
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c 
+(Bind Abbr) x0) z x3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq 
+T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 
+t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (x4: 
+T).(\lambda (x5: T).(\lambda (H12: (pr2 (CHead c (Bind Abbr) x0) x1 
+x4)).(\lambda (H13: (pr0 x4 x5)).(\lambda (H14: (pr2 (CHead c (Bind Abbr) x0) 
+x5 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall 
+(x7: T).((eq T t (THead (Bind Abbr) x6 x7)) \to (or (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x6) x7 t5)))) (pr3 (CHead c (Bind Abbr) x6) x7 (lift (S 
+O) O t4))))))) H3 (THead (Bind Abbr) x2 x3) H8) in (let H16 \def (H15 x2 x3 
+(refl_equal T (THead (Bind Abbr) x2 x3))) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x2) x3 t5)))) (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S 
+O) O t4)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4))) (\lambda (H17: (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda 
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+x7))).(\lambda (H19: (pr3 c x2 x6)).(\lambda (H20: (pr3 (CHead c (Bind Abbr) 
+x2) x3 x7)).(eq_ind_r T (THead (Bind Abbr) x6 x7) (\lambda (t: T).(or (ex3_2 
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+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
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+x1 (lift (S O) O t)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
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+(CHead c (Bind Abbr) x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S 
+O) O (THead (Bind Abbr) x6 x7))) (ex3_2_intro T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T (THead (Bind Abbr) x6 x7) (THead (Bind Abbr) u2 t5)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 
+(CHead c (Bind Abbr) x0) x1 t5))) x6 x7 (refl_equal T (THead (Bind Abbr) x6 
+x7)) (pr3_sing c x2 x0 H9 x6 H19) (pr3_sing (CHead c (Bind Abbr) x0) x4 x1 
+H12 x7 (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 (pr2_free (CHead c (Bind 
+Abbr) x0) x4 x5 H13) x7 (pr3_sing (CHead c (Bind Abbr) x0) x3 x5 H14 x7 
+(pr3_pr2_pr3_t c x2 x3 x7 (Bind Abbr) H20 x0 H9)))))) t4 H18)))))) H17)) 
+(\lambda (H17: (pr3 (CHead c (Bind Abbr) x2) x3 (lift (S O) O 
+t4))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) x0) x1 t5)))) (pr3 
+(CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing (CHead c (Bind Abbr) 
+x0) x4 x1 H12 (lift (S O) O t4) (pr3_sing (CHead c (Bind Abbr) x0) x5 x4 
+(pr2_free (CHead c (Bind Abbr) x0) x4 x5 H13) (lift (S O) O t4) (pr3_sing 
+(CHead c (Bind Abbr) x0) x3 x5 H14 (lift (S O) O t4) (pr3_pr2_pr3_t c x2 x3 
+(lift (S O) O t4) (Bind Abbr) H17 x0 H9)))))) H16)))))))) H11)) H10)))))) 
+H7)) (\lambda (H7: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+x1 (lift (S O) O t2)))))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Bind Abbr) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 (CHead c (Bind Abbr) 
+x0) x1 t5)))) (pr3 (CHead c (Bind Abbr) x0) x1 (lift (S O) O t4)) (pr3_sing 
+(CHead c (Bind Abbr) x0) (lift (S O) O t2) x1 (H7 Abbr x0) (lift (S O) O t4) 
+(pr3_lift (CHead c (Bind Abbr) x0) c (S O) O (drop_drop (Bind Abbr) O c c 
+(drop_refl c) x0) t2 t4 H2)))) H6)))))))))))) y x H0))))) H))))).
+
+lemma pr3_gen_appl:
+ \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c 
+(THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c u1 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))
+\def
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
+(H: (pr3 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 
+t1) (\lambda (t: T).(pr3 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c t1 
+t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr3 c y x)).(unintro T t1 
+(\lambda (t: T).((eq T y (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c t t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2)) x))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))) (unintro T u1 (\lambda 
+(t: T).(\forall (x0: T).((eq T y (THead (Flat Appl) t x0)) \to (or3 (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) (\lambda (_: T).(\lambda (t2: 
+T).(pr3 c x0 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) x))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x0 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+x))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))))))) (pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall 
+(x0: T).(\forall (x1: T).((eq T t (THead (Flat Appl) x0 x1)) \to (or3 (ex3_2 
+T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind Abbr) u2 t2) t0))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))))))) 
+(\lambda (t: T).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T t 
+(THead (Flat Appl) x0 x1))).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda 
+(t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead 
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind 
+Abbr) u2 t2) t0))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t0))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) 
+(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat 
+Appl) x0 x1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 c x1 t2)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 
+c (THead (Bind Abbr) u2 t2) (THead (Flat Appl) x0 x1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+(THead (Flat Appl) x0 x1)))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 c x1 t2))) x0 x1 (refl_equal T (THead (Flat Appl) x0 
+x1)) (pr3_refl c x0) (pr3_refl c x1))) t H1))))) (\lambda (t2: T).(\lambda 
+(t3: T).(\lambda (H1: (pr2 c t3 t2)).(\lambda (t4: T).(\lambda (H2: (pr3 c t2 
+t4)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: T).((eq T t2 (THead (Flat 
+Appl) x0 x1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
+z2)))))))))))))).(\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t3 
+(THead (Flat Appl) x0 x1))).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr2 c 
+t t2)) H1 (THead (Flat Appl) x0 x1) H4) in (let H6 \def (pr2_gen_appl c x0 x1 
+t2 H5) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr2 c x1 t5)))) (ex4_4 T T T T (\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(eq T t2 (THead (Bind Abbr) u2 t5)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq 
+T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead 
+(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (or3 (ex3_2 
+T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr2 c x1 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t2 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr2 c x1 
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t2 (THead (Flat Appl) x2 
+x3))).(\lambda (H9: (pr2 c x0 x2)).(\lambda (H10: (pr2 c x1 x3)).(let H11 
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: T).((eq T t 
+(THead (Flat Appl) x4 x5)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x4 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x5 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x4 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x5 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x5 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x4 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Flat Appl) x2 x3) H8) in (let H12 \def (eq_ind T t2 (\lambda (t: 
+T).(pr3 c t t4)) H2 (THead (Flat Appl) x2 x3) H8) in (let H13 \def (H11 x2 x3 
+(refl_equal T (THead (Flat Appl) x2 x3))) in (or3_ind (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x3 t5))))).(ex3_2_ind T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x3 
+t5))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat 
+Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (H15: (eq T t4 (THead (Flat Appl) x4 
+x5))).(\lambda (H16: (pr3 c x2 x4)).(\lambda (H17: (pr3 c x3 x5)).(eq_ind_r T 
+(THead (Flat Appl) x4 x5) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t5: T).(eq T t (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T (THead (Flat Appl) x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 
+t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) (THead (Flat Appl) x4 
+x5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda 
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) (THead (Flat Appl) x4 x5)))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) 
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t5: T).(eq T (THead (Flat Appl) 
+x4 x5) (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c 
+x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5))) x4 x5 (refl_equal T 
+(THead (Flat Appl) x4 x5)) (pr3_sing c x2 x0 H9 x4 H16) (pr3_sing c x3 x1 H10 
+x5 H17))) t4 H15)))))) H14)) (\lambda (H14: (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))))).(ex4_4_ind T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x2 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5))))))) (or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: 
+(pr3 c (THead (Bind Abbr) x6 x7) t4)).(\lambda (H16: (pr3 c x2 x6)).(\lambda 
+(H17: (pr3 c x3 (THead (Bind Abst) x4 x5))).(\lambda (H18: ((\forall (b: 
+B).(\forall (u: T).(pr3 (CHead c (Bind b) u) x5 x7))))).(or3_intro1 (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro 
+T T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: 
+T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 
+t5))))))) x4 x5 x6 x7 H15 (pr3_sing c x2 x0 H9 x6 H16) (pr3_sing c x3 x1 H10 
+(THead (Bind Abst) x4 x5) H17) H18)))))))))) H14)) (\lambda (H14: (ex6_6 B T 
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x2 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 
+z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b 
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x3 (THead (Bind b) y1 z1)))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda 
+(u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift 
+(S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x2 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 
+(CHead c (Bind b) y2) z1 z2))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x4: B).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: 
+T).(\lambda (x9: T).(\lambda (H15: (not (eq B x4 Abst))).(\lambda (H16: (pr3 
+c x3 (THead (Bind x4) x5 x6))).(\lambda (H17: (pr3 c (THead (Bind x4) x9 
+(THead (Flat Appl) (lift (S O) O x8) x7)) t4)).(\lambda (H18: (pr3 c x2 
+x8)).(\lambda (H19: (pr3 c x5 x9)).(\lambda (H20: (pr3 (CHead c (Bind x4) x9) 
+x6 x7)).(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro 
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(pr3 c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c 
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))) 
+x4 x5 x6 x7 x8 x9 H15 (pr3_sing c x3 x1 H10 (THead (Bind x4) x5 x6) H16) H17 
+(pr3_sing c x2 x0 H9 x8 H18) H19 H20)))))))))))))) H14)) H13))))))))) H7)) 
+(\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
+Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t5))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(eq T t2 (THead (Bind 
+Abbr) u2 t5)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) 
+z1 t5))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 
+(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H8: (eq 
+T x1 (THead (Bind Abst) x2 x3))).(\lambda (H9: (eq T t2 (THead (Bind Abbr) x4 
+x5))).(\lambda (H10: (pr2 c x0 x4)).(\lambda (H11: ((\forall (b: B).(\forall 
+(u: T).(pr2 (CHead c (Bind b) u) x3 x5))))).(eq_ind_r T (THead (Bind Abst) x2 
+x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T 
+t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c t t5)))) (ex4_4 T T T T 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H12 
+\def (eq_ind T t2 (\lambda (t: T).(\forall (x6: T).(\forall (x7: T).((eq T t 
+(THead (Flat Appl) x6 x7)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: 
+T).(eq T t4 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c x6 u2))) (\lambda (_: T).(\lambda (t5: T).(pr3 c x7 t5)))) (ex4_4 T 
+T T T (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 
+c (THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x6 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x7 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x7 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x6 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Bind Abbr) x4 x5) H9) in (let H13 \def (eq_ind T t2 (\lambda (t: 
+T).(pr3 c t t4)) H2 (THead (Bind Abbr) x4 x5) H9) in (or3_intro1 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c (THead (Bind Abst) x2 x3) t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c 
+(THead (Bind Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind Abst) x2 x3) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5))))))) x2 x3 x4 x5 H13 (pr3_pr2 c x0 x4 H10) (pr3_refl c (THead (Bind 
+Abst) x2 x3)) (\lambda (b: B).(\lambda (u: T).(pr3_pr2 (CHead c (Bind b) u) 
+x3 x5 (H11 b u)))))))) x1 H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x1 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind 
+B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: 
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(eq T x1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) 
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 
+y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) 
+u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: 
+T).(\lambda (t5: T).(pr3 c x1 t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda 
+(_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) 
+t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 
+c x0 u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c x1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c x1 (THead 
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) 
+y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda 
+(x2: B).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: 
+T).(\lambda (x7: T).(\lambda (H8: (not (eq B x2 Abst))).(\lambda (H9: (eq T 
+x1 (THead (Bind x2) x3 x4))).(\lambda (H10: (eq T t2 (THead (Bind x2) x7 
+(THead (Flat Appl) (lift (S O) O x6) x5)))).(\lambda (H11: (pr2 c x0 
+x6)).(\lambda (H12: (pr2 c x3 x7)).(\lambda (H13: (pr2 (CHead c (Bind x2) x7) 
+x4 x5)).(eq_ind_r T (THead (Bind x2) x3 x4) (\lambda (t: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c t t5)))) (ex4_4 T T T T (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind Abbr) u2 t5) t4))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(pr3 c t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 
+(CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c t (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead 
+(Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))))) (let H14 \def (eq_ind T t2 
+(\lambda (t: T).(\forall (x8: T).(\forall (x9: T).((eq T t (THead (Flat Appl) 
+x8 x9)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead 
+(Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c x8 u2))) 
+(\lambda (_: T).(\lambda (t5: T).(pr3 c x9 t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x8 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c x9 (THead (Bind Abst) y1 z1)))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t5: T).(\forall 
+(b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) z1 t5)))))))) (ex6_6 B T T T 
+T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda 
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 
+c x9 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind 
+b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr3 c x8 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) 
+(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 
+(_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) y2) z1 z2))))))))))))) H3 
+(THead (Bind x2) x7 (THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (let 
+H15 \def (eq_ind T t2 (\lambda (t: T).(pr3 c t t4)) H2 (THead (Bind x2) x7 
+(THead (Flat Appl) (lift (S O) O x6) x5)) H10) in (or3_intro2 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead (Flat Appl) u2 t5)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda 
+(t5: T).(pr3 c (THead (Bind x2) x3 x4) t5)))) (ex4_4 T T T T (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t5: T).(pr3 c (THead (Bind 
+Abbr) u2 t5) t4))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr3 c x0 u2))))) (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead 
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t5: T).(\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind b) u) 
+z1 t5)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) 
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda 
+(_: T).(\lambda (_: T).(pr3 c (THead (Bind x2) x3 x4) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat 
+Appl) (lift (S O) O u2) z2)) t4))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T 
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(pr3 c 
+(THead (Bind x2) x3 x4) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)) 
+t4))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr3 c x0 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3 (CHead c (Bind b) 
+y2) z1 z2))))))) x2 x3 x4 x5 x6 x7 H8 (pr3_refl c (THead (Bind x2) x3 x4)) 
+H15 (pr3_pr2 c x0 x6 H11) (pr3_pr2 c x3 x7 H12) (pr3_pr2 (CHead c (Bind x2) 
+x7) x4 x5 H13))))) x1 H9))))))))))))) H7)) H6)))))))))))) y x H0))))) H))))).
+
+lemma pr3_gen_bind:
+ \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u1: 
+T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind b) u1 t1) x) \to (or 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind b) u2 
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind b) u1) t1 t2)))) (pr3 (CHead c (Bind 
+b) u1) t1 (lift (S O) O x)))))))))
+\def
+ \lambda (b: B).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to (\forall 
+(c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr3 c (THead (Bind 
+b0) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind b0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind b0) u1) t1 t2)))) (pr3 
+(CHead c (Bind b0) u1) t1 (lift (S O) O x)))))))))) (\lambda (_: (not (eq B 
+Abbr Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: 
+T).(\lambda (H0: (pr3 c (THead (Bind Abbr) u1 t1) x)).(let H1 \def 
+(pr3_gen_abbr c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) (or (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
+t1 (lift (S O) O x))) (\lambda (H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 
+t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind 
+Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) (or (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda 
+(t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) 
+t1 (lift (S O) O x))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T x 
+(THead (Bind Abbr) x0 x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: (pr3 
+(CHead c (Bind Abbr) u1) t1 x1)).(or_introl (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Abbr) u1) t1 t2)))) (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
+O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2))) x0 x1 
+H3 H4 H5))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Abbr) u1) t1 (lift (S 
+O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Abbr) u1) t1 t2)))) (pr3 
+(CHead c (Bind Abbr) u1) t1 (lift (S O) O x)) H2)) H1)))))))) (\lambda (H: 
+(not (eq B Abst Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
+T).(\lambda (x: T).(\lambda (_: (pr3 c (THead (Bind Abst) u1 t1) x)).(let H1 
+\def (match (H (refl_equal B Abst)) in False with []) in H1))))))) (\lambda 
+(_: (not (eq B Void Abst))).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: 
+T).(\lambda (x: T).(\lambda (H0: (pr3 c (THead (Bind Void) u1 t1) x)).(let H1 
+\def (pr3_gen_void c u1 t1 x H0) in (or_ind (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
+(b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) u) t1 t2)))))) (pr3 (CHead c 
+(Bind Void) u1) t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) 
+u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda 
+(H2: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) 
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: 
+T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead c (Bind b0) 
+u) t1 t2))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t2: T).(eq T x 
+(THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(\forall (b0: B).(\forall (u: T).(pr3 (CHead 
+c (Bind b0) u) t1 t2))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x))) (\lambda (x0: 
+T).(\lambda (x1: T).(\lambda (H3: (eq T x (THead (Bind Void) x0 
+x1))).(\lambda (H4: (pr3 c u1 x0)).(\lambda (H5: ((\forall (b0: B).(\forall 
+(u: T).(pr3 (CHead c (Bind b0) u) t1 x1))))).(or_introl (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr3 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 
+(CHead c (Bind Void) u1) t1 t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S 
+O) O x)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead 
+(Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c u1 u2))) 
+(\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 t2))) x0 x1 
+H3 H4 (H5 Void u1)))))))) H2)) (\lambda (H2: (pr3 (CHead c (Bind Void) u1) t1 
+(lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 
+c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr3 (CHead c (Bind Void) u1) t1 
+t2)))) (pr3 (CHead c (Bind Void) u1) t1 (lift (S O) O x)) H2)) H1)))))))) b).