update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / pr2 / fwd.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma
index 1a18f7dbcb5a99b0e6dd4810476d293e58e2f581..1c19f4a2f76ea2b083d50977562aa796fcf20b54 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr2/fwd.ma
@@ -14,15 +14,33 @@
 
 (* This file was automatically generated: do not edit *********************)
 
-include "Basic-1/pr2/defs.ma".
+include "basic_1/pr2/defs.ma".
 
-include "Basic-1/pr0/fwd.ma".
+include "basic_1/pr0/fwd.ma".
 
-include "Basic-1/getl/drop.ma".
+include "basic_1/getl/clear.ma".
 
-include "Basic-1/getl/clear.ma".
+include "basic_1/getl/drop.ma".
 
-theorem pr2_gen_sort:
+implied lemma pr2_ind:
+ \forall (P: ((C \to (T \to (T \to Prop))))).(((\forall (c: C).(\forall (t1: 
+T).(\forall (t2: T).((pr0 t1 t2) \to (P c t1 t2)))))) \to (((\forall (c: 
+C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d 
+(Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to 
+(\forall (t: T).((subst0 i u t2 t) \to (P c t1 t)))))))))))) \to (\forall (c: 
+C).(\forall (t: T).(\forall (t0: T).((pr2 c t t0) \to (P c t t0)))))))
+\def
+ \lambda (P: ((C \to (T \to (T \to Prop))))).(\lambda (f: ((\forall (c: 
+C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (P c t1 
+t2))))))).(\lambda (f0: ((\forall (c: C).(\forall (d: C).(\forall (u: 
+T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: 
+T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (t: T).((subst0 i u t2 t) \to 
+(P c t1 t))))))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: 
+T).(\lambda (p: (pr2 c t t0)).(match p with [(pr2_free x x0 x1 x2) 
+\Rightarrow (f x x0 x1 x2) | (pr2_delta x x0 x1 x2 x3 x4 x5 x6 x7 x8) 
+\Rightarrow (f0 x x0 x1 x2 x3 x4 x5 x6 x7 x8)]))))))).
+
+lemma pr2_gen_sort:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to 
 (eq T x (TSort n)))))
 \def
@@ -43,11 +61,8 @@ t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0))
 (let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n) 
 (pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n)))) 
 t1 H4))))))))))))) c y x H0))) H)))).
-(* COMMENTS
-Initial nodes: 347
-END *)
 
-theorem pr2_gen_lref:
+lemma pr2_gen_lref:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to 
 (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c 
 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S 
@@ -96,11 +111,8 @@ u))) H1 n H7) in (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T
 Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S 
 n) O u0)))) d u H9 (refl_equal T (lift (S n) O u))))) t H8))) 
 (subst0_gen_lref u t i n H6))) t1 H4))))))))))))) c y x H0))) H)))).
-(* COMMENTS
-Initial nodes: 1003
-END *)
 
-theorem pr2_gen_abst:
+lemma pr2_gen_abst:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
 (THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
 T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
@@ -221,11 +233,8 @@ H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
 (S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3 
 H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9)))))))) 
 (pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 2383
-END *)
 
-theorem pr2_gen_cast:
+lemma pr2_gen_cast:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c 
 (THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda 
 (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: 
@@ -356,11 +365,8 @@ t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t) 
 (pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2 
 H5)))))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 2659
-END *)
 
-theorem pr2_gen_csort:
+lemma pr2_gen_csort:
  \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) 
 \to (pr0 t1 t2))))
 \def
@@ -376,11 +382,8 @@ t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c
 (CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d 
 (Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr) 
 u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))).
-(* COMMENTS
-Initial nodes: 221
-END *)
 
-theorem pr2_gen_appl:
+lemma pr2_gen_appl:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 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 (_: 
@@ -1747,17 +1750,16 @@ T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7 
-(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u 
-x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
-x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) (S O) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead 
+(Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) z1 t3)))))))) (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 
@@ -1771,131 +1773,179 @@ T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H20: (eq T x7 (lift (S O) O 
-x8))).(\lambda (H21: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H22 
-\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u 
-x3 x8)) H21 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3: 
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
-x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind 
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+y2) z1 z2))))))))) (\lambda (H20: (lt (s (Bind x0) i) (S 
+O))).(subst0_gen_lift_false x3 u x7 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
+x0) i)) H20 H19 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t3)))))))) (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 (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (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 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) 
+(S O) O H19 Hle) in (let H20 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x7 
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead 
+(Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind 
 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
-(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+(Flat Appl) x7 x5)) (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 c0 u1 u2))))))) (\lambda (_: 
 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) 
-(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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) 
-O x8) x5)) (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 c0 u1 u2))))))) (\lambda (_: 
+y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H21: (eq T x7 (lift (S O) O 
+x8))).(\lambda (H22: (subst0 (minus i O) u x3 x8)).(eq_ind_r T (lift (S O) O 
+x8) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) 
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t4)))))))) (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 (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (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 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H23 \def (eq_ind_r 
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x8)) H22 i (minus_n_O i)) in 
+(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
+x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift 
+(S O) O x8) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (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 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) 
+(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 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (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).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat 
+Appl) (lift (S O) O x8) x5)) (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 c0 u1 u2))))))) (\lambda (_: 
 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (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).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
-(Flat Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) 
+y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) 
+(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5))) 
+(pr2_delta c0 d u i H1 u1 x3 H10 x8 H23) (pr2_free c0 x1 x4 H11) (pr2_free 
+(CHead c0 (Bind x0) x4) x2 x5 H12)))) x7 H21)))) H20))))) x6 H18)))) H17)) 
+(\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S 
+O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O 
+x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)) 
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 
+x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
+x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (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 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x4 x6) (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 c0 u1 u2))))))) 
 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead 
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift 
-(S O) O x8) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x8 H22) (pr2_free c0 x1 x4 
-H11) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 H12))) x7 H20))))) 
-(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H19 (le_n_S O i (le_O_n 
-i)))) x6 H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead 
-(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) 
-(s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead 
-(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) 
-(s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T 
+x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19: (subst0 (s (Flat 
+Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O 
+x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: 
+T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda 
+(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda 
 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (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 c0 
 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda 
-(H18: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19: 
-(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat 
-Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2 
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
-x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (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 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T (THead (Bind x0) x4 t3) (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 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) 
-O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) 
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead 
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat 
+Appl) (lift (S O) O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
@@ -1981,9 +2031,32 @@ T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: 
-T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) 
-i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) 
+(S O) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
+x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
+(Flat Appl) x7 x8)) (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 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (H21: (lt (s (Bind x0) i) (S 
+O))).(subst0_gen_lift_false x3 u x7 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
+x0) i)) H21 H19 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
 (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) 
 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
 (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
@@ -2005,115 +2078,139 @@ T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda 
-(H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) 
-i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) 
-(\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T 
-(lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat 
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
-(_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
-x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) 
-(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 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
-z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
-(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) 
-(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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) 
-O x9) x8)) (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 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (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 (_: 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) 
+(S O) O H19 Hle) in (let H21 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x7 
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead 
+(Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind 
 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead 
-(Flat Appl) (lift (S O) O x9) x8)) (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 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead 
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift 
-(S O) O x9) x8))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_free c0 x1 x4 
-H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead 
-c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2 
-x5 H12 x8 H20))) x7 H21))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S 
-O) O H19 (le_n_S O i (le_O_n i)))) x6 H18)))))) H17)) (subst0_gen_head (Flat 
-Appl) u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14)) 
-(\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
-(Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) 
-(lift (S O) O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) 
-i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (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 
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
-(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+(Flat Appl) x7 x8)) (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 c0 u1 u2))))))) (\lambda (_: 
 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t 
-(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17: 
-(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
-x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
+y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H22: (eq T x7 (lift (S O) O 
+x9))).(\lambda (H23: (subst0 (minus i O) u x3 x9)).(eq_ind_r T (lift (S O) O 
+x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat Appl) u2 t4)))) 
 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
 (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) 
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t4)))))))) (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 (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (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 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H24 \def (eq_ind_r 
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x9)) H23 i (minus_n_O i)) in 
+(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
+x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift 
+(S O) O x9) x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (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 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) 
+(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 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (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).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat 
+Appl) (lift (S O) O x9) x8)) (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 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) 
+(refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8))) 
+(pr2_delta c0 d u i H1 u1 x3 H10 x9 H24) (pr2_free c0 x1 x4 H11) (pr2_delta 
+(CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) 
+c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 
+H20)))) x7 H22)))) H21))))) x6 H18)))))) H17)) (subst0_gen_head (Flat Appl) u 
+(lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14)) (\lambda (H14: 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind x0) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) 
+O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
+(THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat 
+Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) 
+x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (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 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T t (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 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t 
+(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17: 
+(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) 
+x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T 
+(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
 (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (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: 
@@ -2204,17 +2301,16 @@ T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8 
-(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u 
-x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
-x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind 
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) (S O) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
 T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
 u0) z1 t3)))))))) (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 
@@ -2228,131 +2324,180 @@ T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H21: (eq T x8 (lift (S O) O 
-x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H23 
-\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u 
-x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3: 
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
-x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind 
-Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda 
+y2) z1 z2))))))))) (\lambda (H21: (lt (s (Bind x0) i) (S 
+O))).(subst0_gen_lift_false x3 u x8 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
+x0) i)) H21 H20 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) 
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t3)))))))) (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 (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (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 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) 
+(S O) O H20 Hle) in (let H21 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x8 
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind 
 b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
-(Flat Appl) t3 x5)) (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 c0 u1 u2))))))) (\lambda (_: 
-B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
-y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead 
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
-x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3)))))) 
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: 
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) 
-(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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) 
-O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) 
+(Flat Appl) x8 x5)) (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 c0 u1 u2))))))) (\lambda (_: 
 B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
 (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (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).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
-(Flat Appl) (lift (S O) O x9) x5)) (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 c0 u1 u2))))))) 
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
-T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
-(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead 
-(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift 
-(S O) O x9) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_delta c0 d u 
-i H1 x1 x4 H11 x6 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) x8 
-H21))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i 
-(le_O_n i)))) x7 H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T 
-x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s 
-(Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 
-(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat 
-Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda 
-(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H22: (eq T x8 (lift (S O) O 
+x9))).(\lambda (H23: (subst0 (minus i O) u x3 x9)).(eq_ind_r T (lift (S O) O 
+x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
-(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind 
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) 
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) 
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t4)))))))) (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 (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (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 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H24 \def (eq_ind_r 
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x9)) H23 i (minus_n_O i)) in 
+(or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
+x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Flat Appl) u2 t3)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift 
+(S O) O x9) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (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 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) 
 (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 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
-z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) (lift 
-(S O) O x3) x8))).(\lambda (H20: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 
-x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3: 
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
-x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) 
-t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: 
-T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (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).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 
+z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: 
+T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat 
+Appl) (lift (S O) O x9) x5)) (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 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) 
+(refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5))) 
+(pr2_delta c0 d u i H1 u1 x3 H10 x9 H24) (pr2_delta c0 d u i H1 x1 x4 H11 x6 
+H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12)))) x8 H22)))) H21))))) x7 
+H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat 
+Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s 
+(Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7 (THead (Flat 
+Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s 
+(Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
+T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall 
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
 (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead 
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (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 c0 
 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat 
-Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
-(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
-T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
-(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) 
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda 
+(H19: (eq T x7 (THead (Flat Appl) (lift (S O) O x3) x8))).(\lambda (H20: 
+(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat 
+Appl) (lift (S O) O x3) x8) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Flat Appl) u2 
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T 
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
+(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
+x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall 
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (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 
+(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda 
+(y2: T).(eq T (THead (Bind x0) x6 t3) (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 c0 u1 u2))))))) 
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) 
+O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) 
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead 
+(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
 T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
 T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: 
@@ -2438,9 +2583,32 @@ T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: 
-T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) 
-i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (lt_le_e (s (Bind x0) i) 
+(S O) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
+x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind 
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind 
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x9)) (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 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (H22: (lt (s (Bind x0) i) (S 
+O))).(subst0_gen_lift_false x3 u x8 (S O) O (s (Bind x0) i) (le_O_n (s (Bind 
+x0) i)) H22 H20 (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
 (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) 
 (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
 (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: 
@@ -2462,37 +2630,62 @@ T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
 T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
 B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
-(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda 
-(H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind 
-x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S 
-O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in 
-(eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) 
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) 
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: 
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) 
-x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: 
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) 
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: 
-T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) 
-(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 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: 
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 
-y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: 
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 
-z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat 
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))) (\lambda (Hle: (le (S O) 
+(s (Bind x0) i))).(let H_x \def (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) 
+(S O) O H20 Hle) in (let H22 \def H_x in (ex2_ind T (\lambda (t3: T).(eq T x8 
+(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus i O) u x3 t3)) (or3 
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda 
+(_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind 
+x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: 
+T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind Abbr) u2 
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
+u0) z1 t3)))))))) (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 (THead (Bind x0) x1 x2) (THead (Bind 
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead 
+(Flat Appl) x8 x9)) (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 c0 u1 u2))))))) (\lambda (_: 
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: 
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) 
+y2) z1 z2))))))))) (\lambda (x10: T).(\lambda (H23: (eq T x8 (lift (S O) O 
+x10))).(\lambda (H24: (subst0 (minus i O) u x3 x10)).(eq_ind_r T (lift (S O) 
+O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq 
+T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) (THead (Flat Appl) u2 t4)))) 
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda 
+(u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) 
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: 
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: 
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 
+(CHead c0 (Bind b) u0) z1 t4)))))))) (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 (THead (Bind 
+x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T 
+(THead (Bind x0) x6 (THead (Flat Appl) t3 x9)) (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 c0 
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda 
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (let H25 \def (eq_ind_r 
+nat (minus i O) (\lambda (n: nat).(subst0 n u x3 x10)) H24 i (minus_n_O i)) 
+in (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead 
+(Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat Appl) u2 
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
+T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T 
 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 
 (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: 
 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) 
@@ -2526,748 +2719,16 @@ T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 
 (CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T 
 (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) 
-(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta 
+(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H25) (pr2_delta 
 c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i) 
 (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6) 
-(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22))))) 
-(subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n 
-i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 
-x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 
-(THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) 
+(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21)))) x8 H23)))) H22))))) x7 
+H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s 
+(Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4 (THead 
+(Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6)) 
 (pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 38859
-END *)
-
-theorem pr2_gen_abbr:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: 
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T 
-T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) 
-(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 
-t1) (\lambda (t: T).(pr2 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T 
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 
-t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) 
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) 
-(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda 
-(t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \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).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind 
-Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 
-(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) 
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t2)))))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O 
-t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: 
-(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(let H3 \def 
-(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abbr) u1 t1) H2) in 
-(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)) (or 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind 
-b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead 
-c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
-(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z 
-t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-(lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T 
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
-t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead 
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
-(u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t3)))))) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T 
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 
-t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) 
-u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) 
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Bind Abbr) 
-x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T 
-(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 
-x1))))).(or_ind (pr0 t1 x1) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda 
-(y0: T).(subst0 O x0 y0 x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: 
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 
-T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) 
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (H7: (pr0 t1 
-x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind 
-Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 
-(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) 
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O 
-t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead 
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: 
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 
-T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) 
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1))))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) 
-x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) 
-(\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda 
-(y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1)) 
-(pr2_free c0 u1 x0 H6) (or3_intro0 (\forall (b: B).(\forall (u: T).(pr2 
-(CHead c0 (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda 
-(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z x1)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead 
-c0 (Bind b) u) t1 x1 H7)))))) t2 H5)) (\lambda (H_x0: (ex2 T (\lambda (y0: 
-T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)))).(ex2_ind T (\lambda 
-(y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O x0 y0 x1)) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind 
-Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 
-(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) 
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O 
-t2))))) (\lambda (x2: T).(\lambda (H7: (pr0 t1 x2)).(\lambda (H8: (subst0 O 
-x0 x2 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind 
-Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 
-(Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) 
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O 
-t)))))) (ex2_ind T (\lambda (t: T).(subst0 O u1 x2 t)) (\lambda (t: T).(pr0 t 
-x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind 
-Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: 
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 
-T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) 
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1)))))) 
-(\lambda (x3: T).(\lambda (_: (subst0 O u1 x2 x3)).(\lambda (_: (pr0 x3 
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead 
-(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: 
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 
-T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) 
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1))))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) 
-x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) 
-(\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda 
-(y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1)) 
-(pr2_free c0 u1 x0 H6) (or3_intro1 (\forall (b: B).(\forall (u: T).(pr2 
-(CHead c0 (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda 
-(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z x1)))) (ex_intro2 T (\lambda (u: T).(pr0 u1 u)) (\lambda 
-(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1)) x0 H6 (pr2_delta (CHead c0 (Bind 
-Abbr) x0) c0 x0 O (getl_refl Abbr c0 x0) t1 x2 H7 x1 H8)))))))) 
-(pr0_subst0_back x0 x2 x1 O H8 u1 H6)) t2 H5)))) H_x0)) H_x)))))) H4)) 
-(\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T 
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 
-t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) 
-u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2)))) 
-(\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 (lift (S 
-O) O t2) H4))))) (pr0_gen_abbr u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda 
-(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d 
-(Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 
-t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 
-(THead (Bind Abbr) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 
-t3 t2)) H2 (THead (Bind Abbr) u1 t1) H4) in (or_ind (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 
-t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 
-t3))))))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: 
-T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) 
-(ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 
-y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda 
-(z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (H6: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: 
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) 
-(\lambda (y0: T).(subst0 O u2 y0 t3)))))))).(ex3_2_ind T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 
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-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 
-(Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: 
-T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z t3))))))) x3 x1 H13 (pr2_delta c0 d u i H1 u1 x0 H8 x3 
-H14) (or3_intro1 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) 
-t1 x1))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 
-(Bind Abbr) u0) t1 x1))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
-(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z x1)))) 
-(ex_intro2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 
-(Bind Abbr) u0) t1 x1)) x0 H8 (pr2_delta (CHead c0 (Bind Abbr) x0) c0 x0 O 
-(getl_refl Abbr c0 x0) t1 x2 H9 x1 H10)))))))) (pr0_subst0_back x0 x2 x1 O 
-H10 u1 H8))))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t (THead 
-(Bind Abbr) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x0 t3))) 
-(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3)) (or (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
-(\lambda (x3: T).(\lambda (H13: (eq T t (THead (Bind Abbr) x0 x3))).(\lambda 
-(H14: (subst0 (s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t3: T).(subst0 
-O u1 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
-(\lambda (x4: T).(\lambda (H15: (subst0 O u1 x2 x4)).(\lambda (H16: (pr0 x4 
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
-(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13 
-(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
-(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_: 
-T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: 
-T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) 
-u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl 
-Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S 
-i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl 
-x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda 
-(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 
-(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 
-(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 
-z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z 
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t 
-(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15: 
-(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1 
-x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
-(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5 
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead 
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) 
-(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda 
-(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead 
-c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13 
-(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall 
-(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 
-u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T 
-(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) 
-(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: 
-T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0: 
-T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: 
-T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead 
-c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O 
-(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr) 
-u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 
-x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))))) 
-H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x)))))) 
-H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T 
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) 
-t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind 
-Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: 
-T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
-(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) 
-(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) 
-H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) 
-(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 11671
-END *)
-
-theorem pr2_gen_void:
- \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall 
-(u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: 
-T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
-\def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda 
-(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 
-t1) (\lambda (t: T).(pr2 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).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall 
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: 
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))) 
-(\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda 
-(t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \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).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift 
-(S O) O t0))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 
-t1))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind 
-Void) u1 t1) H2) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq 
-T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O 
-t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind 
-Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda 
-(_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind 
-b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) 
-t1 (lift (S O) O t2))))) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda 
-(t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
-(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c0 (Bind b) u) t1 (lift (S O) O t2))))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H5: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H6: (pr0 u1 
-x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) 
-(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 
-(CHead c0 (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead 
-c0 (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
-u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-(lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
-u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c0 u1 
-x0 H6) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 
-H7))))) t2 H5)))))) H4)) (\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) 
-u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 
-(lift (S O) O t2)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 
-(Bind b) u) t1 (lift (S O) O t2) H4))))) (pr0_gen_void u1 t1 t2 H3)))))))) 
-(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda 
-(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: 
-T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 
-t)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(let H5 \def (eq_ind T 
-t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Bind Void) u1 t1) H4) in (or_ind 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-(lift (S O) O t))))) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: 
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind 
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t 
-(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1: 
-T).(\lambda (H7: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 
-x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3: 
-T).(subst0 i u t3 t)) H3 (THead (Bind Void) x0 x1) H7) in (or3_ind (ex2 T 
-(\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 
-i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) 
-(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: 
-T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
-(\lambda (H11: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Void) u2 x1))) 
-(\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t 
-(THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T 
-T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-(lift (S O) O t))))) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind 
-Void) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-(lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T 
-t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 
-(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8 
-x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1 
-x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead 
-(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3))) 
-(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda 
-(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) 
-(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda 
-(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0: 
-T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0 
-(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11: 
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T 
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) 
-(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda 
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 
-(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t 
-(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: 
-(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: 
-T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: 
-T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall 
-(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: 
-B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) 
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) 
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: 
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) 
-u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b: 
-B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head 
-(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11)) 
-(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0 
-t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: 
-T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 
-c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: 
-T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b: 
-B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head 
-(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S 
-O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1 
-t2 H5)))))))))))))) c y x H0))) H))))).
-(* COMMENTS
-Initial nodes: 3467
-END *)
 
-theorem pr2_gen_lift:
+lemma pr2_gen_lift:
  \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall 
 (d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to 
 (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 
@@ -3337,7 +2798,4 @@ t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h)
 (getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1 
 H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1 
 t2 h d H6)))))))))))))))) c y x H0))) H)))))).
-(* COMMENTS
-Initial nodes: 1579
-END *)