we removed about 100 match-with costruction turning them into applications

[helm.git] / matita / contribs / LAMBDA-TYPES / Level-1 / LambdaDelta / csubst0 / drop.ma

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubst0/drop.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubst0/drop.ma
index 14e0bb2c8eb808542166a230782204bc1bb7f0f9..a1e7b22911fecb0b9ddca9b49e52d58faa8558d6 100644 (file)
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubst0/drop.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/csubst0/drop.ma
@@ -32,8 +32,8 @@ c1 e) \to (drop n O c2 e)))))))))
 \to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda 
 (i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
 (v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O 
-O c1 e)).(let H2 \def (match H in le return (\lambda (n: nat).(\lambda (_: 
-(le ? n)).((eq nat n O) \to (drop O O c2 e)))) with [le_n \Rightarrow 
+O c1 e)).(let H2 \def (match H in le return (\lambda (n0: nat).(\lambda (_: 
+(le ? n0)).((eq nat n0 O) \to (drop O O c2 e)))) with [le_n \Rightarrow 
 (\lambda (H2: (eq nat (S i) O)).(let H3 \def (eq_ind nat (S i) (\lambda (e0: 
 nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow 
 False | (S _) \Rightarrow True])) I O H2) in (False_ind (drop O O c2 e) H3))) 
@@ -78,21 +78,21 @@ c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S
 n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k 
 x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t 
 x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let 
-H8 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: 
-T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S 
-n0) O c2 e))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n: 
-nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k in K return (\lambda (k0: 
-K).((drop (r k0 n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 
-(s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O 
-c2 e))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0) 
-e))))) with [(Bind b) \Rightarrow (\lambda (H10: (drop (r (Bind b) n0) O c 
-e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) 
-v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 
-e)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c 
-e H10 x0)))) | (Flat f) \Rightarrow (\lambda (H10: (drop (r (Flat f) n0) O c 
-e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x1) 
-v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 
-e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) 
+H8 \def (eq_ind nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: 
+T).((csubst0 n1 v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop 
+(S n0) O c3 e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda 
+(n1: nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop 
+(r k0 n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) 
+v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0) 
+e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda 
+(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to 
+(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c 
+e H10 x0))))) (\lambda (f: F).(\lambda (H10: (drop (r (Flat f) n0) O c 
+e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x1) 
+v0 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) 
 (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) 
 (drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1 
 O)).(drop_drop (Flat f) n0 c e H10 x0)) (\lambda (H13: (ex2 nat (\lambda (m: 
@@ -100,68 +100,68 @@ nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
 (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O 
 (CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S 
 x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H10 x0)))) H13)) 
-(lt_gen_xS x1 n0 H12)))))]) (drop_gen_drop k c e t n0 H3) H8 H9))) c2 
+(lt_gen_xS x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 
 H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j: 
 nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
-c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c 
-c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
+c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
+c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
 j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
 (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda 
 (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: 
 (eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(eq_ind_r C 
 (CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind 
-nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) 
-\to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) H1 (s 
-k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 
-(s k x1) H5) in ((match k in K return (\lambda (k0: K).((drop (r k0 n0) O c 
-e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to 
-(\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt 
-(s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t) e))))) with [(Bind b) 
-\Rightarrow (\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: 
-((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to 
-(\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda 
-(H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 x0 e (H x1 (lt_S_n 
-x1 n0 H12) c x0 v H7 e H10) t)))) | (Flat f) \Rightarrow (\lambda (H10: (drop 
-(r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c2: C).(\forall (v: 
-T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) 
-\to (drop (S n0) O c2 e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S 
-n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) 
-(\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat f) t) e) 
-(\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) 
-t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: 
-nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda 
-(m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e) (\lambda (x: 
-nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat 
-f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS x1 n0 H12)))))]) 
-(drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4)) (\lambda (H4: (ex4_3 
-T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
-j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 
-k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
-u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c 
-c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
+nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c 
+c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n1: 
+nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).((drop (r k0 
+n0) O c e) \to (((\forall (c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c 
+c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t) 
+e))))) (\lambda (b: B).(\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda 
+(_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to 
+(\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 
+x0 e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H10) t))))) (\lambda (f: 
+F).(\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c3: 
+C).(\forall (v0: T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H12: (lt 
+(s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq 
+nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat 
+f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 
+e H10) t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) 
+(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S 
+m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e) 
+(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x 
+n0)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS 
+x1 n0 H12)))))) k (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4)) 
+(\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
 nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: 
 nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda 
 (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: 
-nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0: T).(\lambda 
-(x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: 
-(eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: 
-(csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(drop (S 
-n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: 
-C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c 
-e) \to (drop (S n0) O c2 e))))))) H1 (s k x2) H5) in (let H10 \def (eq_ind 
-nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x2) H5) in ((match k in K 
-return (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s k0 x2) v c c2) \to (\forall (e: C).((drop (S 
-n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt (s k0 x2) (S n0)) \to 
-(drop (S n0) O (CHead x1 k0 x0) e))))) with [(Bind b) \Rightarrow (\lambda 
-(H11: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall 
-(v: T).((csubst0 (s (Bind b) x2) v c c2) \to (\forall (e: C).((drop (S n0) O 
-c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S 
+nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: 
+C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0: 
+T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k 
+x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t 
+x0)).(\lambda (H8: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda 
+(c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n1: 
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
+(e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 e0))))))) H1 (s k x2) H5) 
+in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) 
+H5) in (K_ind (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c3: 
+C).(\forall (v0: T).((csubst0 (s k0 x2) v0 c c3) \to (\forall (e0: C).((drop 
+(S n0) O c e0) \to (drop (S n0) O c3 e0))))))) \to ((lt (s k0 x2) (S n0)) \to 
+(drop (S n0) O (CHead x1 k0 x0) e))))) (\lambda (b: B).(\lambda (H11: (drop 
+(r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c3: C).(\forall (v0: 
+T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c 
+e0) \to (drop (S n0) O c3 e0)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S 
 n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H11) 
-x0)))) | (Flat f) \Rightarrow (\lambda (H11: (drop (r (Flat f) n0) O c 
-e)).(\lambda (H12: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) 
-x2) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 
-e)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O) 
+x0))))) (\lambda (f: F).(\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda 
+(H12: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s (Flat f) x2) v0 c c3) 
+\to (\forall (e0: C).((drop (S n0) O c e0) \to (drop (S n0) O c3 
+e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O) 
 (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) 
 (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2 
 O)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)) (\lambda (H14: (ex2 
@@ -169,7 +169,7 @@ nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m
 n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: 
 nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x: 
 nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat 
-f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))]) 
+f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))) k 
 (drop_gen_drop k c e t n0 H3) H9 H10))) c2 H6)))))))) H4)) (csubst0_gen_head 
 k c c2 t v i H2))))))))))) c1)))))) n).
 
@@ -183,8 +183,8 @@ c2 e) \to (drop n O c1 e)))))))))
 \to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda 
 (i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda 
 (v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O 
-O c2 e)).(let H2 \def (match H in le return (\lambda (n: nat).(\lambda (_: 
-(le ? n)).((eq nat n O) \to (drop O O c1 e)))) with [le_n \Rightarrow 
+O c2 e)).(let H2 \def (match H in le return (\lambda (n0: nat).(\lambda (_: 
+(le ? n0)).((eq nat n0 O) \to (drop O O c1 e)))) with [le_n \Rightarrow 
 (\lambda (H2: (eq nat (S i) O)).(let H3 \def (eq_ind nat (S i) (\lambda (e0: 
 nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow 
 False | (S _) \Rightarrow True])) I O H2) in (False_ind (drop O O c1 e) H3))) 
@@ -222,54 +222,54 @@ T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_:
 nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
 v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: 
 nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k 
-x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda (c: 
-C).(drop (S n0) O c e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind nat i 
-(\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to 
-(\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) H1 (s k 
-x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s 
-k x1) H5) in ((match k in K return (\lambda (k0: K).(((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: C).((drop (S 
-n0) O c2 e) \to (drop (S n0) O c e))))))) \to ((lt (s k0 x1) (S n0)) \to 
-((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e))))) with [(Bind 
-b) \Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s 
-(Bind b) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S 
-n0) O c e)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda (H13: 
-(drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t)))) | (Flat f) 
-\Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat 
-f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c 
-e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda (H13: (drop (r 
-(Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq nat 
-x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead c (Flat f) t) 
-e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c e H13 t)) (\lambda 
-(H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m 
-n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: 
-nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (x: 
-nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat 
-f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))]) H9 H10 (drop_gen_drop k 
-c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: 
-C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: 
-nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j 
-v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k 
-j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
-(c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O (CHead c k t) 
-e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k 
-x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c 
-x0)).(let H8 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H3 (CHead 
-x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: 
-C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2 
-e) \to (drop (S n0) O c e))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat 
-i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k in K return 
-(\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c 
-c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) 
-\to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to (drop (S n0) O 
-(CHead c k0 t) e))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e: 
-C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H12: (lt (s 
-(Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0 
+x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda 
+(c0: C).(drop (S n0) O c0 e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind 
+nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c 
+c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c 
+e0))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n1: 
+nat).(lt n1 (S n0))) H0 (s k x1) H5) in (K_ind (\lambda (k0: K).(((\forall 
+(c3: C).(\forall (v0: T).((csubst0 (s k0 x1) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) \to ((lt (s k0 x1) 
+(S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e))))) 
+(\lambda (b: B).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s 
+(Bind b) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop 
+(S n0) O c e0)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda 
+(H13: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t))))) 
+(\lambda (f: F).(\lambda (_: ((\forall (c3: C).(\forall (v0: T).((csubst0 (s 
+(Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop 
+(S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda 
+(H13: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda 
+(m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O 
+(CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c 
+e H13 t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) 
+(\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S 
+m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) 
+(\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x 
+n0)).(drop_drop (Flat f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k 
+H9 H10 (drop_gen_drop k c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C 
+nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: 
+nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead 
+c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S 
+n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq 
+nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: 
+(csubst0 x1 v c x0)).(let H8 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) 
+O c0 e)) H3 (CHead x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n1: 
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
+(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x1) H5) 
+in (let H10 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x1) 
+H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 
+(s k0 x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S 
+n0) O c e0))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to 
+(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall 
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x1) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt 
+(s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0 
 e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H13) 
-t)))) | (Flat f) \Rightarrow (\lambda (H11: ((\forall (c2: C).(\forall (v: 
-T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 
-e) \to (drop (S n0) O c e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S 
+t))))) (\lambda (f: F).(\lambda (H11: ((\forall (c3: C).(\forall (v0: 
+T).((csubst0 (s (Flat f) x1) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 
+e0) \to (drop (S n0) O c e0)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S 
 n0))).(\lambda (H13: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O) 
 (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) 
 (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop 
@@ -278,12 +278,12 @@ nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
 (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O 
 (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S 
 x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13) 
-t)))) H14)) (lt_gen_xS x1 n0 H12)))))]) H9 H10 (drop_gen_drop k x0 e t n0 
+t)))) H14)) (lt_gen_xS x1 n0 H12)))))) k H9 H10 (drop_gen_drop k x0 e t n0 
 H8)))))))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: 
 C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: 
 C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
-(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: 
-C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: 
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: 
 T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: 
 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
@@ -291,22 +291,22 @@ T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O
 (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: 
 nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k 
 x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c 
-x1)).(let H9 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H3 (CHead 
-x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: 
-C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2 
-e) \to (drop (S n0) O c e))))))) H1 (s k x2) H5) in (let H11 \def (eq_ind nat 
-i (\lambda (n: nat).(lt n (S n0))) H0 (s k x2) H5) in ((match k in K return 
-(\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x2) v c 
-c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) 
-\to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to (drop (S n0) O 
-(CHead c k0 t) e))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s (Bind b) x2) v c c2) \to (\forall (e: 
-C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H13: (lt (s 
-(Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1 
+x1)).(let H9 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H3 
+(CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n1: 
+nat).(\forall (c3: C).(\forall (v0: T).((csubst0 n1 v0 c c3) \to (\forall 
+(e0: C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0))))))) H1 (s k x2) H5) 
+in (let H11 \def (eq_ind nat i (\lambda (n1: nat).(lt n1 (S n0))) H0 (s k x2) 
+H5) in (K_ind (\lambda (k0: K).(((\forall (c3: C).(\forall (v0: T).((csubst0 
+(s k0 x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 e0) \to (drop (S 
+n0) O c e0))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to 
+(drop (S n0) O (CHead c k0 t) e))))) (\lambda (b: B).(\lambda (_: ((\forall 
+(c3: C).(\forall (v0: T).((csubst0 (s (Bind b) x2) v0 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c3 e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt 
+(s (Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1 
 e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H14) 
-t)))) | (Flat f) \Rightarrow (\lambda (H12: ((\forall (c2: C).(\forall (v: 
-T).((csubst0 (s (Flat f) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c2 
-e) \to (drop (S n0) O c e)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S 
+t))))) (\lambda (f: F).(\lambda (H12: ((\forall (c3: C).(\forall (v0: 
+T).((csubst0 (s (Flat f) x2) v0 c c3) \to (\forall (e0: C).((drop (S n0) O c3 
+e0) \to (drop (S n0) O c e0)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S 
 n0))).(\lambda (H14: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O) 
 (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) 
 (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop 
@@ -315,7 +315,7 @@ nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda
 (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O 
 (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S 
 x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14) 
-t)))) H15)) (lt_gen_xS x2 n0 H13)))))]) H10 H11 (drop_gen_drop k x1 e x0 n0 
+t)))) H15)) (lt_gen_xS x2 n0 H13)))))) k H10 H11 (drop_gen_drop k x1 e x0 n0 
 H9)))))))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n).
 
 theorem csubst0_drop_lt:
@@ -392,7 +392,7 @@ C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w))))))
 (_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k: 
 K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: 
 T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r 
-nat i0 (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 (minus (s k i0) (s k O)) 
+nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 (minus (s k i0) (s k O)) 
 (s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1)) 
 (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
 T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
@@ -419,87 +419,88 @@ T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C
 (CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k: 
 K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: 
 T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 
-(s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k 
-O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2))))))))).(\lambda 
-(u: T).(let H4 \def (eq_ind_r nat i0 (\lambda (n: nat).(csubst0 n v0 c3 c4)) 
-H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H5 \def (eq_ind_r nat i0 
-(\lambda (n: nat).(or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus n (s k O)) v0 u w)))))) (ex3_4 K C C T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 
-(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k O)) v0 e1 e2)))))) 
-(ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C c3 (CHead e1 k u))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead 
-e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus n (s k O)) v0 u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus n (s k O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k 
-i0)) in (or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
-(CHead c3 k u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 
-w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k 
-u) (CHead e1 k0 u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s 
-k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) 
-(CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 
-w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda 
-(k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
-T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C 
-(CHead c3 k u) (CHead e1 k0 u0)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 k u) (CHead e2 k0 
-u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))) k c3 c4 u (refl_equal C 
-(CHead c3 k u)) (drop_refl (CHead c4 k u)) H4)))))))))))) (\lambda (k: 
-K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: 
-T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: 
-C).(\lambda (H3: (csubst0 i0 v0 c3 c4)).(\lambda (_: (or4 (drop O O c4 c3) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 
-(s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k 
-O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2))))))))).(let H5 
-\def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 (minus (s k 
-i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r nat i0 (\lambda (n: 
-nat).(csubst0 n v0 c3 c4)) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in 
-(or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1)) (ex3_4 K C T T 
-(\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C c3 (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 
+O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
+(CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 
+O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda 
+(n0: nat).(csubst0 n0 v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) 
+in (let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(or4 (drop O O c4 c3) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda 
+(_: T).(eq C c3 (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n0 (s 
+k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 k0 u0)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O c4 (CHead 
+e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 (minus n0 (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
+C c3 (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 (minus n0 (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0 
+(s k0 O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in 
+(or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda 
+(k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k 
+u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k 
+i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 
+u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) 
+v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0 
+u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda 
+(k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k 
+i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 
+u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) 
+v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k 
+u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: 
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 
+u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3 
+c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k0: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k0 
+u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop O O c4 (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k0 O)) v0 u w)))))) 
+(ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c3 (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k0 u)))))) (\lambda (k0: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k0 
+O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k0 
+u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop O O c4 (CHead e2 k0 w))))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i0 (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k0 O)) v0 e1 
+e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1 
+u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r 
+nat i0 (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 (minus (s k i0) (s k O)) 
+(s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1)) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: 
 C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0 
 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda 
@@ -635,1837 +636,1879 @@ K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0
 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda 
 (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda 
 (H2: (drop (S n0) O (CHead c k t) e)).(let H3 \def (match H1 in csubst0 
-return (\lambda (n: nat).(\lambda (t0: T).(\lambda (c0: C).(\lambda (c1: 
-C).(\lambda (_: (csubst0 n t0 c0 c1)).((eq nat n i) \to ((eq T t0 v) \to ((eq 
-C c0 (CHead c k t)) \to ((eq C c1 c2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K 
-C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S 
-n0))) v e1 e2))))))))))))))))) with [(csubst0_snd k0 i0 v0 u1 u2 H3 c0) 
+return (\lambda (n1: nat).(\lambda (t0: T).(\lambda (c0: C).(\lambda (c3: 
+C).(\lambda (_: (csubst0 n1 t0 c0 c3)).((eq nat n1 i) \to ((eq T t0 v) \to 
+((eq C c0 (CHead c k t)) \to ((eq C c3 c2) \to (or4 (drop (S n0) O c2 e) 
+(ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k0 w)))))) (\lambda (k0: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k0 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k0 
+u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k0 w))))))) 
+(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k0 
+(S n0))) v e1 e2))))))))))))))))) with [(csubst0_snd k0 i0 v0 u1 u2 H3 c0) 
 \Rightarrow (\lambda (H4: (eq nat (s k0 i0) i)).(\lambda (H5: (eq T v0 
 v)).(\lambda (H6: (eq C (CHead c0 k0 u1) (CHead c k t))).(\lambda (H7: (eq C 
-(CHead c0 k0 u2) c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) 
+(CHead c0 k0 u2) c2)).(eq_ind nat (s k0 i0) (\lambda (n1: nat).((eq T v0 v) 
 \to ((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) 
 \to ((subst0 i0 v0 u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k 
-(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 
-e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq 
-C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to 
-((subst0 i0 t0 u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))) (\lambda (H9: (eq C (CHead 
-c0 k0 u1) (CHead c k t))).(let H10 \def (f_equal C T (\lambda (e0: C).(match 
-e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ 
-t) \Rightarrow t])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H11 \def 
-(f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with 
-[(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u1) 
-(CHead c k t) H9) in ((let H12 \def (f_equal C C (\lambda (e0: C).(match e0 
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) 
-\Rightarrow c])) (CHead c0 k0 u1) (CHead c k t) H9) in (eq_ind C c (\lambda 
-(c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c k0 u2) c2) \to 
-((subst0 i0 v u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda (H13: (eq K k0 
-k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c k u2) c2) 
-\to ((subst0 i0 v u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T 
 (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
 (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
 T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s 
-k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k1 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k1 
+u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus n1 (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus n1 (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 
+(s k1 (S n0))) v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v 
+(\lambda (t0: T).((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 
+k0 u2) c2) \to ((subst0 i0 t0 u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C 
+T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) 
+(s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: 
 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead 
 e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
+T).(csubst0 (minus (s k0 i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
 (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
 (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
 e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) 
+T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k1 (S n0))) v u w)))))) 
 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda 
-(H14: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c k u2) c2) \to 
-((subst0 i0 v t u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
+(_: T).(csubst0 (minus (s k0 i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda 
+(H9: (eq C (CHead c0 k0 u1) (CHead c k t))).(let H10 \def (f_equal C T 
+(\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k0 u1) (CHead c k 
+t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
+\Rightarrow k1])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H12 \def 
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow c0 | (CHead c3 _ _) \Rightarrow c3])) (CHead c0 k0 u1) 
+(CHead c k t) H9) in (eq_ind C c (\lambda (c3: C).((eq K k0 k) \to ((eq T u1 
+t) \to ((eq C (CHead c3 k0 u2) c2) \to ((subst0 i0 v u1 u2) \to (or4 (drop (S 
+n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 
+w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s k0 i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 
+i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s k0 i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) 
+(s k1 (S n0))) v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k 
+(\lambda (k1: K).((eq T u1 t) \to ((eq C (CHead c k1 u2) c2) \to ((subst0 i0 
+v u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k2: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k2 
+u)))))) (\lambda (k2: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c2 (CHead e0 k2 w)))))) (\lambda (k2: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v 
+u w)))))) (ex3_4 K C C T (\lambda (k2: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 k2 u)))))) (\lambda (k2: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k2 u)))))) 
+(\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s k1 i0) (s k2 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k2: 
 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: 
+(CHead e1 k2 u))))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k2 w))))))) 
+(\lambda (k2: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v u w)))))) (\lambda (k2: 
 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) (\lambda (H15: (eq C (CHead 
-c k u2) c2)).(eq_ind C (CHead c k u2) (\lambda (c: C).((subst0 i0 v t u2) \to 
-(or4 (drop (S n0) O c e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c (CHead 
-e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+(minus (s k1 i0) (s k2 (S n0))) v e1 e2)))))))))))) (\lambda (H14: (eq T u1 
+t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c k u2) c2) \to ((subst0 i0 v 
+t0 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 
+K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) 
+(s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) 
+(s k1 (S n0))) v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c k u2) 
+c2)).(eq_ind C (CHead c k u2) (\lambda (c3: C).((subst0 i0 v t u2) \to (or4 
+(drop (S n0) O c3 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead 
+e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1: 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 
+u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O c3 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v 
-e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u 
-w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))) 
-(\lambda (_: (subst0 i0 v t u2)).(let H1 \def (eq_ind K k0 (\lambda (k: 
-K).(eq nat (s k i0) i)) H4 k H13) in (let H17 \def (eq_ind_r nat i (\lambda 
-(n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall 
-(e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k 
-(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 
-e2)))))))))))))) H0 (s k i0) H1) in (let H18 \def (eq_ind_r nat i (\lambda 
-(n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind (\lambda (k: K).((drop (r k 
-n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k i0) v c c2) 
-\to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 
-K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s 
-k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead 
-e2 k u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k0 (S n0))) v e1 e2)))))))))))))) \to ((lt (S n0) (s k 
-i0)) \to (or4 (drop (S n0) O (CHead c k u2) e) (ex3_4 K C T T (\lambda (k1: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 
-u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c k u2) (CHead e0 k1 w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s 
-k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c k 
-u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) 
-(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c k u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s 
 k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v 
-e1 e2)))))))))))) (\lambda (b: B).(\lambda (H2: (drop (r (Bind b) n0) O c 
-e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) i0) 
-v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 
-e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead 
-e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T 
-T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k 
-w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) i0))).(or4_intro0 (drop 
-(S n0) O (CHead c (Bind b) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-c (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) u2) 
-(CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S 
-n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c e H2 u2)))))) (\lambda (f: 
-F).(\lambda (H2: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s (Flat f) i0) v c c2) \to (\forall (e: 
-C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
+e1 e2)))))))))) (\lambda (_: (subst0 i0 v t u2)).(let H17 \def (eq_ind K k0 
+(\lambda (k1: K).(eq nat (s k1 i0) i)) H4 k H13) in (let H18 \def (eq_ind_r 
+nat i (\lambda (n1: nat).(\forall (c3: C).(\forall (v1: T).((csubst0 n1 v1 c 
+c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) 
+(ex3_4 K C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e0 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k1 (S 
+n0))) v1 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead 
+e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus n1 (s k1 (S n0))) v1 e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e0 (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus n1 (s k1 (S n0))) v1 u w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 
+(s k1 (S n0))) v1 e1 e2)))))))))))))) H0 (s k i0) H17) in (let H19 \def 
+(eq_ind_r nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k i0) H17) in (K_ind 
+(\lambda (k1: K).((drop (r k1 n0) O c e) \to (((\forall (c3: C).(\forall (v1: 
+T).((csubst0 (s k1 i0) v1 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) 
+\to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T (\lambda (k2: K).(\lambda (e1: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k2 u)))))) (\lambda 
+(k2: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 
+(CHead e1 k2 w)))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v1 u w)))))) 
+(ex3_4 K C C T (\lambda (k2: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e0 (CHead e1 k2 u)))))) (\lambda (k2: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k2 u)))))) (\lambda (k2: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k1 
+i0) (s k2 (S n0))) v1 e1 e2)))))) (ex4_5 K C C T T (\lambda (k2: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k2 
+u))))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k2 w))))))) (\lambda (k2: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s k1 i0) (s k2 (S n0))) v1 u w)))))) (\lambda (k2: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k1 i0) 
+(s k2 (S n0))) v1 e1 e2)))))))))))))) \to ((lt (S n0) (s k1 i0)) \to (or4 
+(drop (S n0) O (CHead c k1 u2) e) (ex3_4 K C T T (\lambda (k2: K).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k2 u)))))) (\lambda 
+(k2: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead c k1 u2) (CHead e0 k2 w)))))) (\lambda (k2: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v 
+u w)))))) (ex3_4 K C C T (\lambda (k2: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 k2 u)))))) (\lambda (k2: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c k1 u2) (CHead e2 
+k2 u)))))) (\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s k1 i0) (s k2 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
+(\lambda (k2: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e1 k2 u))))))) (\lambda (k2: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c 
+k1 u2) (CHead e2 k2 w))))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v 
+u w)))))) (\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus (s k1 i0) (s k2 (S n0))) v e1 
+e2)))))))))))) (\lambda (b: B).(\lambda (H20: (drop (r (Bind b) n0) O c 
+e)).(\lambda (_: ((\forall (c3: C).(\forall (v1: T).((csubst0 (s (Bind b) i0) 
+v1 c c3) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O 
+c3 e0) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 
+w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v1 u w)))))) (ex3_4 K C C T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) i0) (s k1 (S n0))) v1 e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v1 u w)))))) (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (_: (lt 
-(S n0) (s (Flat f) i0))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) u2) e) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: 
+(minus (s (Bind b) i0) (s k1 (S n0))) v1 e1 e2))))))))))))))).(\lambda (_: 
+(lt (S n0) (s (Bind b) i0))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) u2) 
+e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) u2) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) u2) (CHead e2 k1 
+u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+c (Bind b) u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 
+(S n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c e H20 u2)))))) (\lambda (f: 
+F).(\lambda (H20: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c3: 
+C).(\forall (v1: T).((csubst0 (s (Flat f) i0) v1 c c3) \to (\forall (e0: 
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c3 e0) (ex3_4 K C T T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e1 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
+i0) (s k1 (S n0))) v1 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 
+(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v1 e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
+i0) (s k1 (S n0))) v1 u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 
+(S n0))) v1 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) 
+i0))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) u2) e) (ex3_4 K C T T 
+(\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 
+u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e2 k1 u)))))) (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c 
-e H2 u2)))))) k (drop_gen_drop k c e t n0 H2) H17 H18))))) c2 H15)) u1 
-(sym_eq T u1 t H14))) k0 (sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11)) 
-H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 H3))))) | (csubst0_fst k0 i0 c1 
-c0 v0 H3 u) \Rightarrow (\lambda (H4: (eq nat (s k0 i0) i)).(\lambda (H5: (eq 
-T v0 v)).(\lambda (H6: (eq C (CHead c1 k0 u) (CHead c k t))).(\lambda (H7: 
-(eq C (CHead c0 k0 u) c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 
-v) \to ((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) 
-\to ((csubst0 i0 v0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n (s k (S 
-n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k 
-u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus n (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S 
-n0))) v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda 
-(t0: T).((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) 
-\to ((csubst0 i0 t0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) 
-(s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead 
-e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k 
-w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))) (\lambda 
-(H9: (eq C (CHead c1 k0 u) (CHead c k t))).(let H10 \def (f_equal C T 
+f) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) u2) 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2))))))) (drop_drop (Flat f) n0 c e H20 u2)))))) k (drop_gen_drop k c e t n0 
+H2) H18 H19))))) c2 H15)) u1 (sym_eq T u1 t H14))) k0 (sym_eq K k0 k H13))) 
+c0 (sym_eq C c0 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 
+H3))))) | (csubst0_fst k0 i0 c0 c3 v0 H3 u) \Rightarrow (\lambda (H4: (eq nat 
+(s k0 i0) i)).(\lambda (H5: (eq T v0 v)).(\lambda (H6: (eq C (CHead c0 k0 u) 
+(CHead c k t))).(\lambda (H7: (eq C (CHead c3 k0 u) c2)).(eq_ind nat (s k0 
+i0) (\lambda (n1: nat).((eq T v0 v) \to ((eq C (CHead c0 k0 u) (CHead c k t)) 
+\to ((eq C (CHead c3 k0 u) c2) \to ((csubst0 i0 v0 c0 c3) \to (or4 (drop (S 
+n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 
+w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 (minus n1 (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 
+u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop (S n0) O c2 (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n1 (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n1 (s k1 
+(S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 (s k1 (S n0))) v e1 
+e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq 
+C (CHead c0 k0 u) (CHead c k t)) \to ((eq C (CHead c3 k0 u) c2) \to ((csubst0 
+i0 t0 c0 c3) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k1 (S n0))) v 
+u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s k0 i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k1 (S n0))) v u0 w)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s k0 i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda 
+(H9: (eq C (CHead c0 k0 u) (CHead c k t))).(let H10 \def (f_equal C T 
 (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u) (CHead c k t) 
-H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 in C return 
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow 
-k])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H12 \def (f_equal C C 
-(\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u) (CHead c k t) 
-H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u t) \to ((eq C 
-(CHead c0 k0 u) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: 
-T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) 
-(s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead 
-e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k 
-w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda 
-(H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u t) \to ((eq C (CHead 
-c0 k u) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C 
-T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
-e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) 
-(s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda 
-(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 
-(CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) 
-(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k0 u) (CHead c k 
+t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
+\Rightarrow k1])) (CHead c0 k0 u) (CHead c k t) H9) in ((let H12 \def 
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0 k0 u) 
+(CHead c k t) H9) in (eq_ind C c (\lambda (c4: C).((eq K k0 k) \to ((eq T u 
+t) \to ((eq C (CHead c3 k0 u) c2) \to ((csubst0 i0 v c4 c3) \to (or4 (drop (S 
+n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 
+w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 (minus (s k0 i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
+(CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k1 u0)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 
+i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s k0 i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) 
+(s k1 (S n0))) v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k 
+(\lambda (k1: K).((eq T u t) \to ((eq C (CHead c3 k1 u) c2) \to ((csubst0 i0 
+v c c3) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k2: 
+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k2 
+u0)))))) (\lambda (k2: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c2 (CHead e0 k2 w)))))) (\lambda (k2: K).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v 
+u0 w)))))) (ex3_4 K C C T (\lambda (k2: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C e (CHead e1 k2 u0)))))) (\lambda (k2: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k2 
+u0)))))) (\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s k1 i0) (s k2 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
+(\lambda (k2: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(_: T).(eq C e (CHead e1 k2 u0))))))) (\lambda (k2: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e2 k2 w))))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v u0 w)))))) 
+(\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s k1 i0) (s k2 (S n0))) v e1 e2)))))))))))) (\lambda 
+(H14: (eq T u t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 k t0) c2) \to 
+((csubst0 i0 v c c3) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 
+k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
 C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v 
-u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 
-e2)))))))))))) (\lambda (H14: (eq T u t)).(eq_ind T t (\lambda (t: T).((eq C 
-(CHead c0 k t) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: 
-T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) 
-(s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead 
-e2 k u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k 
-w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
 T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) 
 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) (\lambda 
-(H15: (eq C (CHead c0 k t) c2)).(eq_ind C (CHead c0 k t) (\lambda (c2: 
-C).((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: 
+(H15: (eq C (CHead c3 k t) c2)).(eq_ind C (CHead c3 k t) (\lambda (c4: 
+C).((csubst0 i0 v c c3) \to (or4 (drop (S n0) O c4 e) (ex3_4 K C T T (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 
+k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c4 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
 C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v 
-u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))) (\lambda (H16: (csubst0 i0 v 
-c c0)).(let H1 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H4 k 
-H13) in (let H17 \def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: 
-C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c 
-e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
-e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
+u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c4 (CHead e2 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c4 (CHead 
+e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))) (\lambda 
+(H16: (csubst0 i0 v c c3)).(let H17 \def (eq_ind K k0 (\lambda (k1: K).(eq 
+nat (s k1 i0) i)) H4 k H13) in (let H18 \def (eq_ind_r nat i (\lambda (n1: 
+nat).(\forall (c4: C).(\forall (v1: T).((csubst0 n1 v1 c c4) \to (\forall 
+(e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c4 e0) (ex3_4 K C T T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (u0: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c4 (CHead e1 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n1 (s k1 
+(S n0))) v1 u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e0 (CHead e1 k1 u0)))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c4 
+(CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus n1 (s k1 (S n0))) v1 e1 e2)))))) (ex4_5 K 
+C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c4 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n1 (s k1 (S n0))) v1 u0 
+w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus n1 (s k1 (S n0))) v1 e1 e2)))))))))))))) 
+H0 (s k i0) H17) in (let H19 \def (eq_ind_r nat i (\lambda (n1: nat).(lt (S 
+n0) n1)) H (s k i0) H17) in (K_ind (\lambda (k1: K).((drop (r k1 n0) O c e) 
+\to (((\forall (c4: C).(\forall (v1: T).((csubst0 (s k1 i0) v1 c c4) \to 
+(\forall (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c4 e0) (ex3_4 
+K C T T (\lambda (k2: K).(\lambda (e1: C).(\lambda (u0: T).(\lambda (_: 
+T).(eq C e0 (CHead e1 k2 u0)))))) (\lambda (k2: K).(\lambda (e1: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O c4 (CHead e1 k2 w)))))) (\lambda (k2: 
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k1 i0) 
+(s k2 (S n0))) v1 u0 w)))))) (ex3_4 K C C T (\lambda (k2: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e0 (CHead e1 k2 u0)))))) (\lambda 
+(k2: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c4 
+(CHead e2 k2 u0)))))) (\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s k1 i0) (s k2 (S n0))) v1 e1 e2)))))) 
+(ex4_5 K C C T T (\lambda (k2: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(eq C e0 (CHead e1 k2 u0))))))) (\lambda (k2: 
 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) 
-in (let H18 \def (eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) 
-H1) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: 
-C).(\forall (v: T).((csubst0 (s k i0) v c c2) \to (\forall (e: C).((drop (S 
-n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k0: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) 
-(\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k0 (S n0))) v e1 e2)))))))))))))) \to ((lt (S n0) (s k 
-i0)) \to (or4 (drop (S n0) O (CHead c0 k t) e) (ex3_4 K C T T (\lambda (k1: 
+n0) O c4 (CHead e2 k2 w))))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) 
+v1 u0 w)))))) (\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus (s k1 i0) (s k2 (S n0))) v1 e1 
+e2)))))))))))))) \to ((lt (S n0) (s k1 i0)) \to (or4 (drop (S n0) O (CHead c3 
+k1 t) e) (ex3_4 K C T T (\lambda (k2: K).(\lambda (e0: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e (CHead e0 k2 u0)))))) (\lambda (k2: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 k1 t) (CHead 
+e0 k2 w)))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v u0 w)))))) (ex3_4 K C C T 
+(\lambda (k2: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
+(CHead e1 k2 u0)))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 k1 t) (CHead e2 k2 u0)))))) 
+(\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s k1 i0) (s k2 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k2: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
+(CHead e1 k2 u0))))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 k1 t) (CHead e2 
+k2 w))))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v u0 w)))))) 
+(\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s k1 i0) (s k2 (S n0))) v e1 e2)))))))))))) (\lambda 
+(b: B).(\lambda (H20: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall 
+(c4: C).(\forall (v1: T).((csubst0 (s (Bind b) i0) v1 c c4) \to (\forall (e0: 
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c4 e0) (ex3_4 K C T T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (u0: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c4 (CHead e1 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind 
+b) i0) (s k1 (S n0))) v1 u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e0 (CHead e1 k1 u0)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S 
+n0) O c4 (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v1 e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u0))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c4 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind 
+b) i0) (s k1 (S n0))) v1 u0 w)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) i0) (s k1 (S n0))) v1 e1 e2))))))))))))))).(\lambda (H22: (lt (S n0) (s 
+(Bind b) i0))).(let H23 \def (IHn i0 (le_S_n (S n0) i0 H22) c c3 v H16 e H20) 
+in (or4_ind (drop n0 O c3 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k1 u0)))))) (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c3 (CHead 
+e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 (minus i0 (s k1 n0)) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 
+u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop n0 O c3 (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k1 n0)) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k1 
+n0)) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus i0 (s k1 n0)) v e1 e2))))))) (or4 
+(drop (S n0) O (CHead c3 (Bind b) t) e) (ex3_4 K C T T (\lambda (k1: 
 K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k1 
 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 k t) (CHead e0 k1 w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) 
-(s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda 
-(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O 
-(CHead c0 k t) (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v 
-e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e0 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind 
+b) i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S 
+n0) O (CHead c3 (Bind b) t) (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s 
+k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))))) (\lambda 
+(H24: (drop n0 O c3 e)).(or4_intro0 (drop (S n0) O (CHead c3 (Bind b) t) e) 
+(ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda 
+(_: T).(eq C e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) t) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) t) 
+(CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda 
 (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 k t) (CHead e2 k1 w))))))) (\lambda (k1: 
+T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 w))))))) (\lambda (k1: 
 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) 
-(s k1 (S n0))) v e1 e2)))))))))))) (\lambda (b: B).(\lambda (H2: (drop (r 
-(Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 
-(s (Bind b) i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 
-(drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S 
-n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda 
 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (H: (lt (S n0) (s 
-(Bind b) i0))).(let H19 \def (IHn i0 (le_S_n (S n0) i0 H) c c0 v H16 e H2) in 
-(or4_ind (drop n0 O c0 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k 
-w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus i0 (s k n0)) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop n0 O c0 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: 
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-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop 
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-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u0 w)))))) 
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-(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda 
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-T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
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-T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
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-T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda 
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-T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w))))) 
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+t))) (\lambda (H24: (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda 
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+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c3 (CHead e0 
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+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k1 n0)) v u0 w))))) 
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+u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
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-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
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-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k 
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-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) 
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-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
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-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
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-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w))))))) 
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-T).(subst0 (minus i0 (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w))))))) (\lambda (k: 
+b) i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S 
+n0) O (CHead c3 (Bind b) t) (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda 
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+k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))))) (\lambda 
+(x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H25: (eq 
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+T).(\lambda (_: T).(eq C c4 (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda 
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+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) 
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+C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) t) 
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+C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e1 k1 u0))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 w))))))) (\lambda (k1: 
 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus i0 (s k n0)) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 
-e2)))))) (or4 (drop (S n0) O (CHead c0 (Bind b) t) e) (ex3_4 K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S 
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-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k 
-(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) 
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-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: 
-K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop n0 O c0 
-(CHead x2 x0 x4))).(\lambda (H23: (subst0 (minus i0 (s x0 n0)) v x3 
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-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k 
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-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) 
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-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
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-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S 
-n0) O (CHead c0 (Bind b) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Bind b) i0) (s k1 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O 
+(CHead c3 (Bind b) t) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) 
+(CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e0 k1 w)))))) 
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+(minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 
+x2) (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e1 k1 u0))))))) (\lambda 
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+T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Bind b) i0) (s k1 (S n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) 
+(CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u0 w))))) x0 x1 x2 x3 (refl_equal C 
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+nat (S (s x0 n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind b) i0) n1) v x2 
+x3)) H27 (s x0 (S n0)) (s_S x0 n0)))) e H25)))))))) H24)) (\lambda (H24: 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O c3 (CHead e2 k1 u0)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus i0 (s k1 n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 
+u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop n0 O c3 (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k1 n0)) v e1 
+e2))))) (or4 (drop (S n0) O (CHead c3 (Bind b) t) e) (ex3_4 K C T T (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 
+k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e0 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind 
+b) i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S 
+n0) O (CHead c3 (Bind b) t) (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s 
+k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
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+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))))) (\lambda 
+(x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H25: (eq 
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+T).(\lambda (_: T).(eq C c4 (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda 
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+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
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+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e1 k1 u0))))))) (\lambda 
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+T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 w))))))) (\lambda (k1: 
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+(Bind b) i0) (s k1 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O 
+(CHead c3 (Bind b) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k1: 
 K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) 
-(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) 
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+(CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e0 k1 w)))))) 
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+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 
+x3) (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 u0))))))) (\lambda 
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+T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda 
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+(Bind b) i0) (s k1 (S n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) 
-(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) 
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-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
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-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S 
-n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2))))))) (ex4_5_intro K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) 
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-(eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(subst0 (minus (s (Bind b) i0) 
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-(H2: (drop (r (Flat f) n0) O c e)).(\lambda (H0: ((\forall (c2: C).(\forall 
-(v: T).((csubst0 (s (Flat f) i0) v c c2) \to (\forall (e: C).((drop (S n0) O 
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-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 
-(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C 
-T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S 
-n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda 
+(CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2))))) x0 x1 x2 x3 
+(refl_equal C (CHead x1 x0 x3)) (drop_drop (Bind b) n0 c3 (CHead x2 x0 x3) 
+H26 t) (eq_ind_r nat (S (s x0 n0)) (\lambda (n1: nat).(csubst0 (minus (s 
+(Bind b) i0) n1) v x1 x2)) H27 (s x0 (S n0)) (s_S x0 n0)))) e H25)))))))) 
+H24)) (\lambda (H24: (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n0 O c3 (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus i0 (s k1 n0)) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k1 n0)) v e1 
+e2)))))))).(ex4_5_ind K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop n0 O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k1 
+n0)) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus i0 (s k1 n0)) v e1 e2)))))) (or4 (drop 
+(S n0) O (CHead c3 (Bind b) t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k1 u0)))))) 
+(\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead c3 (Bind b) t) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 
+(S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O 
+(CHead c3 (Bind b) t) (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S 
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))))) (\lambda 
+(x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
+T).(\lambda (H25: (eq C e (CHead x1 x0 x3))).(\lambda (H26: (drop n0 O c3 
+(CHead x2 x0 x4))).(\lambda (H27: (subst0 (minus i0 (s x0 n0)) v x3 
+x4)).(\lambda (H28: (csubst0 (minus i0 (s x0 n0)) v x1 x2)).(eq_ind_r C 
+(CHead x1 x0 x3) (\lambda (c4: C).(or4 (drop (S n0) O (CHead c3 (Bind b) t) 
+c4) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C c4 (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) t) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u0: T).(eq C c4 (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) t) 
+(CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e1 k1 u0))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda 
 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s 
-(Flat f) i0))).(let H19 \def (H0 c0 v H16 e H2) in (or4_ind (drop (S n0) O c0 
-e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k (S 
-n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c0 (CHead e2 k 
-u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus i0 (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-(S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K 
-C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
-e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: 
+(Bind b) i0) (s k1 (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O 
+(CHead c3 (Bind b) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) 
+(CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 
+x3) (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 u0))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Bind b) t) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Bind b) i0) (s k1 (S n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
+C (CHead x1 x0 x3) (CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 
+(Bind b) t) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 
+(S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S 
+n0))) v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop 
+(Bind b) n0 c3 (CHead x2 x0 x4) H26 t) (eq_ind_r nat (S (s x0 n0)) (\lambda 
+(n1: nat).(subst0 (minus (s (Bind b) i0) n1) v x3 x4)) H27 (s x0 (S n0)) (s_S 
+x0 n0)) (eq_ind_r nat (S (s x0 n0)) (\lambda (n1: nat).(csubst0 (minus (s 
+(Bind b) i0) n1) v x1 x2)) H28 (s x0 (S n0)) (s_S x0 n0)))) e H25)))))))))) 
+H24)) H23)))))) (\lambda (f: F).(\lambda (H20: (drop (r (Flat f) n0) O c 
+e)).(\lambda (H21: ((\forall (c4: C).(\forall (v1: T).((csubst0 (s (Flat f) 
+i0) v1 c c4) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) 
+O c4 e0) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c4 (CHead e1 k1 
+w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v1 u0 w)))))) (ex3_4 K C C T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e0 
+(CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (S n0) O c4 (CHead e2 k1 u0)))))) (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
+f) i0) (s k1 (S n0))) v1 e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c4 (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v1 u0 w)))))) (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v1 e1 
+e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) i0))).(let H23 \def 
+(H21 c3 v H16 e H20) in (or4_ind (drop (S n0) O c3 e) (ex3_4 K C T T (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 
+k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k1 (S n0))) v u0 
+w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead e2 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
+C e (CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 (minus i0 (s k1 (S n0))) v u0 w)))))) (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H20: (drop (S 
-n0) O c0 e)).(or4_intro0 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K C T 
-T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: 
+(minus i0 (s k1 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead c3 (Flat 
+f) t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) t) 
+(CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Flat f) i0) (s k1 (S n0))) v e1 e2)))))))) (\lambda (H24: (drop (S n0) O c3 
+e)).(or4_intro0 (drop (S n0) O (CHead c3 (Flat f) t) e) (ex3_4 K C T T 
+(\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
+(CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 
+k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 u0)))))) (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
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-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c0 
-e H20 t))) (\lambda (H20: (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead 
-e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (s k (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u0 
-w))))) (or4 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: 
+f) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
+(CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c3 e H24 t))) (\lambda (H24: 
+(ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda 
+(_: T).(eq C e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e0 k1 w)))))) 
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+(minus i0 (s k1 (S n0))) v u0 w))))))).(ex3_4_ind K C T T (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k1 (S n0))) v u0 
+w))))) (or4 (drop (S n0) O (CHead c3 (Flat f) t) e) (ex3_4 K C T T (\lambda 
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+k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e0 k1 w)))))) (\lambda (k1: 
 K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) 
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-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k 
-(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
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-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: 
-K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H21: (eq C e 
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-T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
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-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
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-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
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-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S 
-n0) O (CHead c0 (Flat f) t) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: 
+f) i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S 
+n0) O (CHead c3 (Flat f) t) (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda 
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+k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 w))))))) 
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+T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 e2)))))))) (\lambda 
+(x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H25: (eq 
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+x3))).(\lambda (H27: (subst0 (minus i0 (s x0 (S n0))) v x2 x3)).(eq_ind_r C 
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+T).(\lambda (_: T).(eq C c4 (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u0: T).(eq C c4 (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) t) 
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+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e1 k1 u0))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda 
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+(Flat f) i0) (s k1 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O 
+(CHead c3 (Flat f) t) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k1: 
 K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) 
-(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) 
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-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x2) 
-(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) 
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-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C (CHead x1 x0 x2) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S 
-n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2))))))) (ex3_4_intro K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u0)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w))))) 
-x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x2)) (drop_drop (Flat f) n0 c0 (CHead 
-x1 x0 x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: (ex3_4 K C C T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (S n0) O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 
-e2))))))).(ex3_4_ind K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c0 (CHead e2 k u0)))))) 
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-(minus i0 (s k (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead c0 (Flat f) 
-t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda 
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-T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
-(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) 
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-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 
-w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop (S n0) O c0 
-(CHead x2 x0 x3))).(\lambda (H23: (csubst0 (minus i0 (s x0 (S n0))) v x1 
-x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead 
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-(u0: T).(\lambda (_: T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C 
-T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c 
-(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) 
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-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C c (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 
-w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
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-e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 (Flat f) t) (CHead x1 x0 
-x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u0)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) 
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(eq C (CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
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-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C 
-(CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) 
-(CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))) 
-x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead 
-x2 x0 x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: (ex4_5 K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-(S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: 
+(CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 
+x2) (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e1 k1 u0))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Flat f) i0) (s k1 (S n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) 
+(CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u0 w))))) x0 x1 x2 x3 (refl_equal C 
+(CHead x1 x0 x2)) (drop_drop (Flat f) n0 c3 (CHead x1 x0 x3) H26 t) H27)) e 
+H25)))))))) H24)) (\lambda (H24: (ex3_4 K C C T (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S 
+n0) O c3 (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k1 (S n0))) v e1 
+e2))))))).(ex3_4_ind K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead e2 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (s k1 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead 
+c3 (Flat f) t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda 
+(u0: T).(\lambda (_: T).(eq C e (CHead e0 k1 u0)))))) (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 
+(Flat f) t) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u0 
+w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) t) 
+(CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 w))))))) (\lambda (k1: 
 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus i0 (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-(S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K 
-C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
-e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Flat f) i0) (s k1 (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
+C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H25: (eq C e (CHead x1 x0 
+x3))).(\lambda (H26: (drop (S n0) O c3 (CHead x2 x0 x3))).(\lambda (H27: 
+(csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) 
+(\lambda (c4: C).(or4 (drop (S n0) O (CHead c3 (Flat f) t) c4) (ex3_4 K C T T 
+(\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 
+(CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c4 (CHead e1 
+k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 u0)))))) (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: 
-K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop (S n0) O c0 
-(CHead x2 x0 x4))).(\lambda (H23: (subst0 (minus i0 (s x0 (S n0))) v x3 
-x4)).(\lambda (H24: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C 
-(CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) t) c) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: 
-T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k 
-u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: 
+f) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 
+(CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c3 (Flat f) t) 
+(CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda 
+(u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k1 u0)))))) (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead c3 (Flat f) t) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 u0)))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O 
+(CHead c3 (Flat f) t) (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) 
+(CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 
+u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 u0)))))) (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S 
-n0) O (CHead c0 (Flat f) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) 
-(CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) 
-(CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
-(k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S 
-n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2))))))) (ex4_5_intro K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k 
-u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal 
-C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x4) H22 t) H23 
-H24)) e H21)))))))))) H20)) H19)))))) k (drop_gen_drop k c e t n0 H2) H17 
-H18))))) c2 H15)) u (sym_eq T u t H14))) k0 (sym_eq K k0 k H13))) c1 (sym_eq 
-C c1 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 H3))))) | 
-(csubst0_both k0 i0 v0 u1 u2 H3 c1 c0 H4) \Rightarrow (\lambda (H5: (eq nat 
-(s k0 i0) i)).(\lambda (H6: (eq T v0 v)).(\lambda (H7: (eq C (CHead c1 k0 u1) 
-(CHead c k t))).(\lambda (H8: (eq C (CHead c0 k0 u2) c2)).(eq_ind nat (s k0 
-i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u1) (CHead c k t)) 
-\to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 v0 u1 u2) \to ((csubst0 i0 v0 
-c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 
-(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) (\lambda (H9: 
-(eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c1 k0 u1) (CHead c k 
-t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 t0 u1 u2) \to ((csubst0 
-i0 t0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda (H10: (eq C 
-(CHead c1 k0 u1) (CHead c k t))).(let H11 \def (f_equal C T (\lambda (e0: 
-C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | 
-(CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let 
-H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: 
-C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead 
-c1 k0 u1) (CHead c k t) H10) in ((let H13 \def (f_equal C C (\lambda (e0: 
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | 
-(CHead c _ _) \Rightarrow c])) (CHead c1 k0 u1) (CHead c k t) H10) in (eq_ind 
-C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c0 k0 u2) 
-c2) \to ((subst0 i0 v u1 u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O 
-c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s k0 i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v 
-e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u 
-w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 
-e2)))))))))))))) (\lambda (H14: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq 
-T u1 t) \to ((eq C (CHead c0 k u2) c2) \to ((subst0 i0 v u1 u2) \to ((csubst0 
-i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: 
+f) i0) (s k1 (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 
+x3)) (drop_drop (Flat f) n0 c3 (CHead x2 x0 x3) H26 t) H27)) e H25)))))))) 
+H24)) (\lambda (H24: (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus i0 (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k1 
+(S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus i0 (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k1 
+(S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead c3 (Flat f) t) e) (ex3_4 K 
+C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq 
+C e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 
+k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 u0)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
+f) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
+(CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: 
+C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H25: (eq C e (CHead x1 x0 
+x3))).(\lambda (H26: (drop (S n0) O c3 (CHead x2 x0 x4))).(\lambda (H27: 
+(subst0 (minus i0 (s x0 (S n0))) v x3 x4)).(\lambda (H28: (csubst0 (minus i0 
+(s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c4: C).(or4 
+(drop (S n0) O (CHead c3 (Flat f) t) c4) (ex3_4 K C T T (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e0 k1 
+u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e0 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat 
+f) i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c4 (CHead e1 k1 u0)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S 
+n0) O (CHead c3 (Flat f) t) (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s 
+k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e1 k1 
+u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 e2))))))))) 
+(or4_intro3 (drop (S n0) O (CHead c3 (Flat f) t) (CHead x1 x0 x3)) (ex3_4 K C 
+T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
+(CHead x1 x0 x3) (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u0: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 u0)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 
+(Flat f) t) (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 
+u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 e2))))))) (ex4_5_intro 
+K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 u0))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) t) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Flat f) i0) (s k1 (S n0))) v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead 
+x1 x0 x3)) (drop_drop (Flat f) n0 c3 (CHead x2 x0 x4) H26 t) H27 H28)) e 
+H25)))))))))) H24)) H23)))))) k (drop_gen_drop k c e t n0 H2) H18 H19))))) c2 
+H15)) u (sym_eq T u t H14))) k0 (sym_eq K k0 k H13))) c0 (sym_eq C c0 c 
+H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 H3))))) | 
+(csubst0_both k0 i0 v0 u1 u2 H3 c0 c3 H4) \Rightarrow (\lambda (H5: (eq nat 
+(s k0 i0) i)).(\lambda (H6: (eq T v0 v)).(\lambda (H7: (eq C (CHead c0 k0 u1) 
+(CHead c k t))).(\lambda (H8: (eq C (CHead c3 k0 u2) c2)).(eq_ind nat (s k0 
+i0) (\lambda (n1: nat).((eq T v0 v) \to ((eq C (CHead c0 k0 u1) (CHead c k 
+t)) \to ((eq C (CHead c3 k0 u2) c2) \to ((subst0 i0 v0 u1 u2) \to ((csubst0 
+i0 v0 c0 c3) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: 
 K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 
 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
 T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k1 (S n0))) v u 
 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: 
 C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k1 u)))))) 
 (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+(minus n1 (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
 (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
 C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) 
 (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))))) (\lambda (H15: (eq T u1 
-t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c0 k u2) c2) \to ((subst0 i0 v 
-t u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (H16: (eq C 
-(CHead c0 k u2) c2)).(eq_ind C (CHead c0 k u2) (\lambda (c2: C).((subst0 i0 v 
-t u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) 
-(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) (\lambda (_: (subst0 i0 v t 
-u2)).(\lambda (H18: (csubst0 i0 v c c0)).(let H1 \def (eq_ind K k0 (\lambda 
-(k: K).(eq nat (s k i0) i)) H5 k H14) in (let H19 \def (eq_ind_r nat i 
-(\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to 
-(\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C 
-T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) 
-(ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k 
-(S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: 
+T).(subst0 (minus n1 (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 
+(s k1 (S n0))) v e1 e2)))))))))))))) (\lambda (H9: (eq T v0 v)).(eq_ind T v 
+(\lambda (t0: T).((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c3 
+k0 u2) c2) \to ((subst0 i0 t0 u1 u2) \to ((csubst0 i0 t0 c0 c3) \to (or4 
+(drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s k0 i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 
+i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: 
 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 
-e2)))))))))))))) H0 (s k i0) H1) in (let H20 \def (eq_ind_r nat i (\lambda 
-(n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind (\lambda (k: K).((drop (r k 
-n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k i0) v c c2) 
-\to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 
-K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k0: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s 
-k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
+(minus (s k0 i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) 
+(s k1 (S n0))) v e1 e2))))))))))))) (\lambda (H10: (eq C (CHead c0 k0 u1) 
+(CHead c k t))).(let H11 \def (f_equal C T (\lambda (e0: C).(match e0 in C 
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0) 
+\Rightarrow t0])) (CHead c0 k0 u1) (CHead c k t) H10) in ((let H12 \def 
+(f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with 
+[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u1) 
+(CHead c k t) H10) in ((let H13 \def (f_equal C C (\lambda (e0: C).(match e0 
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c4 _ 
+_) \Rightarrow c4])) (CHead c0 k0 u1) (CHead c k t) H10) in (eq_ind C c 
+(\lambda (c4: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c3 k0 u2) 
+c2) \to ((subst0 i0 v u1 u2) \to ((csubst0 i0 v c4 c3) \to (or4 (drop (S n0) 
+O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 
+w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s k0 i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 
+i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s k0 i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) 
+(s k1 (S n0))) v e1 e2)))))))))))))) (\lambda (H14: (eq K k0 k)).(eq_ind K k 
+(\lambda (k1: K).((eq T u1 t) \to ((eq C (CHead c3 k1 u2) c2) \to ((subst0 i0 
+v u1 u2) \to ((csubst0 i0 v c c3) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T 
+T (\lambda (k2: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k2 u)))))) (\lambda (k2: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k2 w)))))) (\lambda (k2: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k1 i0) 
+(s k2 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k2: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k2 u)))))) (\lambda (k2: 
 K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead 
-e2 k u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) 
-(\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (\lambda (k0: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s k i0) (s k0 (S n0))) v e1 e2)))))))))))))) \to ((lt (S n0) (s k 
-i0)) \to (or4 (drop (S n0) O (CHead c0 k u2) e) (ex3_4 K C T T (\lambda (k1: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 
-u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 k u2) (CHead e0 k1 w)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s 
-k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-k u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) 
-(ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 k u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+e2 k2 u)))))) (\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s k1 i0) (s k2 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
+(\lambda (k2: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e1 k2 u))))))) (\lambda (k2: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead 
+e2 k2 w))))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v u w)))))) 
+(\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s k1 i0) (s k2 (S n0))) v e1 e2))))))))))))) (\lambda 
+(H15: (eq T u1 t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 k u2) c2) 
+\to ((subst0 i0 v t0 u2) \to ((csubst0 i0 v c c3) \to (or4 (drop (S n0) O c2 
+e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 
+u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O c2 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v 
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s 
 k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v 
-e1 e2)))))))))))) (\lambda (b: B).(\lambda (H2: (drop (r (Bind b) n0) O c 
-e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) i0) 
-v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 
-e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead 
-e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T 
-T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k 
-w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2))))))))))))))).(\lambda (H: (lt (S n0) (s (Bind b) i0))).(let H21 \def 
-(IHn i0 (le_S_n (S n0) i0 H) c c0 v H18 e H2) in (or4_ind (drop n0 O c0 e) 
-(ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i0 (s k n0)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w))))))) (\lambda (k: 
+e1 e2)))))))))))) (\lambda (H16: (eq C (CHead c3 k u2) c2)).(eq_ind C (CHead 
+c3 k u2) (\lambda (c4: C).((subst0 i0 v t u2) \to ((csubst0 i0 v c c3) \to 
+(or4 (drop (S n0) O c4 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c4 (CHead 
+e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 
+u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O c4 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v 
+e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c4 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s 
+k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v 
+e1 e2))))))))))) (\lambda (_: (subst0 i0 v t u2)).(\lambda (H18: (csubst0 i0 
+v c c3)).(let H19 \def (eq_ind K k0 (\lambda (k1: K).(eq nat (s k1 i0) i)) H5 
+k H14) in (let H20 \def (eq_ind_r nat i (\lambda (n1: nat).(\forall (c4: 
+C).(\forall (v1: T).((csubst0 n1 v1 c c4) \to (\forall (e0: C).((drop (S n0) 
+O c e0) \to (or4 (drop (S n0) O c4 e0) (ex3_4 K C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 
+u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c4 (CHead e1 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n1 (s k1 (S n0))) v1 u 
+w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c4 (CHead e2 k1 
+u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus n1 (s k1 (S n0))) v1 e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e0 (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c4 (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus n1 (s k1 (S n0))) v1 u w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n1 
+(s k1 (S n0))) v1 e1 e2)))))))))))))) H0 (s k i0) H19) in (let H21 \def 
+(eq_ind_r nat i (\lambda (n1: nat).(lt (S n0) n1)) H (s k i0) H19) in (K_ind 
+(\lambda (k1: K).((drop (r k1 n0) O c e) \to (((\forall (c4: C).(\forall (v1: 
+T).((csubst0 (s k1 i0) v1 c c4) \to (\forall (e0: C).((drop (S n0) O c e0) 
+\to (or4 (drop (S n0) O c4 e0) (ex3_4 K C T T (\lambda (k2: K).(\lambda (e1: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k2 u)))))) (\lambda 
+(k2: K).(\lambda (e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c4 
+(CHead e1 k2 w)))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v1 u w)))))) 
+(ex3_4 K C C T (\lambda (k2: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e0 (CHead e1 k2 u)))))) (\lambda (k2: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O c4 (CHead e2 k2 u)))))) (\lambda (k2: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k1 
+i0) (s k2 (S n0))) v1 e1 e2)))))) (ex4_5 K C C T T (\lambda (k2: K).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k2 
+u))))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c4 (CHead e2 k2 w))))))) (\lambda (k2: 
 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i0 (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 
-e2))))))) (or4 (drop (S n0) O (CHead c0 (Bind b) u2) e) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H22: (drop n0 O c0 
-e)).(or4_intro0 (drop (S n0) O (CHead c0 (Bind b) u2) e) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H22 u2))) 
-(\lambda (H22: (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k 
-n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k 
-w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (s k n0)) v u w))))) (or4 (drop (S n0) O (CHead c0 (Bind 
-b) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) 
-(CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C 
-T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: 
+(minus (s k1 i0) (s k2 (S n0))) v1 u w)))))) (\lambda (k2: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k1 i0) 
+(s k2 (S n0))) v1 e1 e2)))))))))))))) \to ((lt (S n0) (s k1 i0)) \to (or4 
+(drop (S n0) O (CHead c3 k1 u2) e) (ex3_4 K C T T (\lambda (k2: K).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k2 u)))))) (\lambda 
+(k2: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead c3 k1 u2) (CHead e0 k2 w)))))) (\lambda (k2: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v 
+u w)))))) (ex3_4 K C C T (\lambda (k2: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 k2 u)))))) (\lambda (k2: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 k1 u2) (CHead e2 
+k2 u)))))) (\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s k1 i0) (s k2 (S n0))) v e1 e2)))))) (ex4_5 K C C T T 
+(\lambda (k2: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e1 k2 u))))))) (\lambda (k2: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 
+k1 u2) (CHead e2 k2 w))))))) (\lambda (k2: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k1 i0) (s k2 (S n0))) v 
+u w)))))) (\lambda (k2: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus (s k1 i0) (s k2 (S n0))) v e1 
+e2)))))))))))) (\lambda (b: B).(\lambda (H22: (drop (r (Bind b) n0) O c 
+e)).(\lambda (_: ((\forall (c4: C).(\forall (v1: T).((csubst0 (s (Bind b) i0) 
+v1 c c4) \to (\forall (e0: C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O 
+c4 e0) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c4 (CHead e1 k1 
+w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v1 u w)))))) (ex3_4 K C C T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e0 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c4 (CHead e2 k1 u)))))) (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: 
-K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H23: (eq C e 
-(CHead x1 x0 x2))).(\lambda (H24: (drop n0 O c0 (CHead x1 x0 x3))).(\lambda 
-(H25: (subst0 (minus i0 (s x0 n0)) v x2 x3)).(eq_ind_r C (CHead x1 x0 x2) 
-(\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Bind b) u2) c) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x2) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead 
-x1 x0 x2) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Bind b) i0) (s k (S n0))) v u w))))) x0 x1 x2 x3 (refl_equal C 
-(CHead x1 x0 x2)) (drop_drop (Bind b) n0 c0 (CHead x1 x0 x3) H24 u2) 
-(eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(subst0 (minus (s (Bind b) i0) 
-n) v x2 x3)) H25 (s x0 (S n0)) (s_S x0 n0)))) e H23)))))))) H22)) (\lambda 
-(H22: (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i0 (s k n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop n0 O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2))))) 
-(or4 (drop (S n0) O (CHead c0 (Bind b) u2) e) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
-C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H23: (eq C e (CHead x1 x0 
-x3))).(\lambda (H24: (drop n0 O c0 (CHead x2 x0 x3))).(\lambda (H25: (csubst0 
-(minus i0 (s x0 n0)) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: 
-C).(or4 (drop (S n0) O (CHead c0 (Bind b) u2) c) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
+b) i0) (s k1 (S n0))) v1 e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c4 (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v1 u w)))))) (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
-x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C 
-(CHead x1 x0 x3)) (drop_drop (Bind b) n0 c0 (CHead x2 x0 x3) H24 u2) 
-(eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(csubst0 (minus (s (Bind b) i0) 
-n) v x1 x2)) H25 (s x0 (S n0)) (s_S x0 n0)))) e H23)))))))) H22)) (\lambda 
-(H22: (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 
-O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 (minus i0 (s k n0)) v e1 e2)))))))).(ex4_5_ind K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus i0 (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead c0 (Bind b) u2) e) (ex3_4 K C T 
-T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
-C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H23: (eq C e 
-(CHead x1 x0 x3))).(\lambda (H24: (drop n0 O c0 (CHead x2 x0 x4))).(\lambda 
-(H25: (subst0 (minus i0 (s x0 n0)) v x3 x4)).(\lambda (H26: (csubst0 (minus 
-i0 (s x0 n0)) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 
-(drop (S n0) O (CHead c0 (Bind b) u2) c) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: 
+(minus (s (Bind b) i0) (s k1 (S n0))) v1 e1 e2))))))))))))))).(\lambda (H24: 
+(lt (S n0) (s (Bind b) i0))).(let H25 \def (IHn i0 (le_S_n (S n0) i0 H24) c 
+c3 v H18 e H22) in (or4_ind (drop n0 O c3 e) (ex3_4 K C T T (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 
+u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 O c3 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k1 n0)) v u w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop n0 O c3 (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k1 
+n0)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop n0 O c3 (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus i0 (s k1 n0)) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k1 n0)) v e1 
+e2))))))) (or4 (drop (S n0) O (CHead c3 (Bind b) u2) e) (ex3_4 K C T T 
+(\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 
+u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S 
+n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 
+e2)))))))) (\lambda (H26: (drop n0 O c3 e)).(or4_intro0 (drop (S n0) O (CHead 
+c3 (Bind b) u2) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 
+u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+c3 (Bind b) u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 
+(S n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c3 e H26 u2))) (\lambda (H26: 
+(ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop n0 O c3 (CHead e0 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k1 
+n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c3 (CHead e0 
+k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i0 (s k1 n0)) v u w))))) (or4 (drop (S n0) O (CHead c3 
+(Bind b) u2) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 
+u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+c3 (Bind b) u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 
+(S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: 
+T).(\lambda (x3: T).(\lambda (H27: (eq C e (CHead x1 x0 x2))).(\lambda (H28: 
+(drop n0 O c3 (CHead x1 x0 x3))).(\lambda (H29: (subst0 (minus i0 (s x0 n0)) 
+v x2 x3)).(eq_ind_r C (CHead x1 x0 x2) (\lambda (c4: C).(or4 (drop (S n0) O 
+(CHead c3 (Bind b) u2) c4) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c4 (CHead e0 k1 u)))))) (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead c3 (Bind b) u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C c4 (CHead e1 k1 u)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c4 (CHead e1 k1 u))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Bind b) i0) (s k1 (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O 
+(CHead c3 (Bind b) u2) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) 
+(CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 x0 
+x2) (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 u)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C (CHead x1 x0 x2) (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 
+(S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S 
+n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k1 
+u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e0 k1 w)))))) (\lambda (k1: 
 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind 
-b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
+i0) (s k1 (S n0))) v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x2)) 
+(drop_drop (Bind b) n0 c3 (CHead x1 x0 x3) H28 u2) (eq_ind_r nat (S (s x0 
+n0)) (\lambda (n1: nat).(subst0 (minus (s (Bind b) i0) n1) v x2 x3)) H29 (s 
+x0 (S n0)) (s_S x0 n0)))) e H27)))))))) H26)) (\lambda (H26: (ex3_4 K C C T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop n0 O c3 (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k1 
+n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c3 (CHead e2 
+k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (s k1 n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead c3 
+(Bind b) u2) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 
+u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+c3 (Bind b) u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 
+(S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: 
+C).(\lambda (x3: T).(\lambda (H27: (eq C e (CHead x1 x0 x3))).(\lambda (H28: 
+(drop n0 O c3 (CHead x2 x0 x3))).(\lambda (H29: (csubst0 (minus i0 (s x0 n0)) 
+v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c4: C).(or4 (drop (S n0) O 
+(CHead c3 (Bind b) u2) c4) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c4 (CHead e0 k1 u)))))) (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead c3 (Bind b) u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C c4 (CHead e1 k1 u)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c4 (CHead e1 k1 u))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Bind b) i0) (s k1 (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O 
+(CHead c3 (Bind b) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) 
+(CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 x0 
+x3) (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 u)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C (CHead x1 x0 x3) (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 
+(Bind b) u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 
+(S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S 
+n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 
+u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) i0) (s k1 (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 
+x3)) (drop_drop (Bind b) n0 c3 (CHead x2 x0 x3) H28 u2) (eq_ind_r nat (S (s 
+x0 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) i0) n1) v x1 x2)) H29 
+(s x0 (S n0)) (s_S x0 n0)))) e H27)))))))) H26)) (\lambda (H26: (ex4_5 K C C 
+T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c3 (CHead 
+e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus i0 (s k1 n0)) v u w)))))) (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (ex4_5_intro K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S 
-n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
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-e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Bind b) 
-n0 c0 (CHead x2 x0 x4) H24 u2) (eq_ind_r nat (S (s x0 n0)) (\lambda (n: 
-nat).(subst0 (minus (s (Bind b) i0) n) v x3 x4)) H25 (s x0 (S n0)) (s_S x0 
-n0)) (eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(csubst0 (minus (s (Bind 
-b) i0) n) v x1 x2)) H26 (s x0 (S n0)) (s_S x0 n0)))) e H23)))))))))) H22)) 
-H21)))))) (\lambda (f: F).(\lambda (H2: (drop (r (Flat f) n0) O c 
-e)).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) i0) 
-v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 
-e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead 
-e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T 
-T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k 
-w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) 
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-(_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) i0))).(let H21 \def (H0 
-c0 v H18 e H2) in (or4_ind (drop (S n0) O c0 e) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus i0 (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: 
+(minus i0 (s k1 n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k1: 
 K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) 
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-T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k 
-(S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 
-K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) 
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-(minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k 
-u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) 
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-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H22: (drop (S 
-n0) O c0 e)).(or4_intro0 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C 
-T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c3 (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i0 (s k1 n0)) v u w)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k1 
+n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead c3 (Bind b) u2) e) (ex3_4 K C T 
+T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 
+u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind 
+b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S 
+n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 
+e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: 
+T).(\lambda (x4: T).(\lambda (H27: (eq C e (CHead x1 x0 x3))).(\lambda (H28: 
+(drop n0 O c3 (CHead x2 x0 x4))).(\lambda (H29: (subst0 (minus i0 (s x0 n0)) 
+v x3 x4)).(\lambda (H30: (csubst0 (minus i0 (s x0 n0)) v x1 x2)).(eq_ind_r C 
+(CHead x1 x0 x3) (\lambda (c4: C).(or4 (drop (S n0) O (CHead c3 (Bind b) u2) 
+c4) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C c4 (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c4 (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 
+u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C c4 (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+c3 (Bind b) u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) 
+i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 
+(S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c3 (Bind b) u2) 
+(CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k1 u)))))) (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 
+(CHead c3 (Bind b) u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 u)))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
+(CHead c3 (Bind b) u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S 
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S 
+n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 
+e2))))))) (ex4_5_intro K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead e2 k1 
+w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k1 (S n0))) v u w)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Bind b) i0) (s k1 (S n0))) v e1 e2)))))) x0 x1 x2 
+x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Bind b) n0 c3 (CHead x2 x0 
+x4) H28 u2) (eq_ind_r nat (S (s x0 n0)) (\lambda (n1: nat).(subst0 (minus (s 
+(Bind b) i0) n1) v x3 x4)) H29 (s x0 (S n0)) (s_S x0 n0)) (eq_ind_r nat (S (s 
+x0 n0)) (\lambda (n1: nat).(csubst0 (minus (s (Bind b) i0) n1) v x1 x2)) H30 
+(s x0 (S n0)) (s_S x0 n0)))) e H27)))))))))) H26)) H25)))))) (\lambda (f: 
+F).(\lambda (H22: (drop (r (Flat f) n0) O c e)).(\lambda (H23: ((\forall (c4: 
+C).(\forall (v1: T).((csubst0 (s (Flat f) i0) v1 c c4) \to (\forall (e0: 
+C).((drop (S n0) O c e0) \to (or4 (drop (S n0) O c4 e0) (ex3_4 K C T T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (u: T).(\lambda (_: T).(eq C e0 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c4 (CHead e1 k1 w)))))) (\lambda (k1: 
 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H22 u2))) 
-(\lambda (H22: (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i0 (s k (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u w))))) 
-(or4 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C T T (\lambda (k: 
-K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
+i0) (s k1 (S n0))) v1 u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e0 (CHead e1 k1 u)))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c4 
+(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v1 e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e0 (CHead e1 k1 u))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c4 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
+i0) (s k1 (S n0))) v1 u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 
+(S n0))) v1 e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) 
+i0))).(let H25 \def (H23 c3 v H18 e H22) in (or4_ind (drop (S n0) O c3 e) 
+(ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e0 k1 w)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k1 (S 
+n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k1 
+u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 k1 w))))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i0 (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 
+(s k1 (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead c3 (Flat f) u2) e) 
+(ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e0 k1 
+w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 u)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2)))))))) (\lambda (H26: (drop (S n0) O c3 e)).(or4_intro0 (drop (S n0) O 
+(CHead c3 (Flat f) u2) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 
+(Flat f) u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u 
+w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Flat f) u2) 
+(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Flat f) i0) (s k1 (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c3 e H26 
+u2))) (\lambda (H26: (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead 
+e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 (minus i0 (s k1 (S n0))) v u w))))))).(ex3_4_ind K C T T (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 
+u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k1 (S n0))) v u 
+w))))) (or4 (drop (S n0) O (CHead c3 (Flat f) u2) e) (ex3_4 K C T T (\lambda 
+(k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 
+u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e0 k1 w)))))) (\lambda (k1: 
 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
-C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H23: (eq C e (CHead x1 x0 
-x2))).(\lambda (H24: (drop (S n0) O c0 (CHead x1 x0 x3))).(\lambda (H25: 
+i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 
+(Flat f) u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 w))))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Flat f) i0) (s k1 (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
+C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H27: (eq C e (CHead x1 x0 
+x2))).(\lambda (H28: (drop (S n0) O c3 (CHead x1 x0 x3))).(\lambda (H29: 
 (subst0 (minus i0 (s x0 (S n0))) v x2 x3)).(eq_ind_r C (CHead x1 x0 x2) 
-(\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) c) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x2) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead 
-x1 x0 x2) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus (s (Flat f) i0) (s k (S n0))) v u w))))) x0 x1 x2 x3 (refl_equal C 
-(CHead x1 x0 x2)) (drop_drop (Flat f) n0 c0 (CHead x1 x0 x3) H24 u2) H25)) e 
-H23)))))))) H22)) (\lambda (H22: (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead 
-e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 (minus i0 (s k (S n0))) v e1 e2))))))).(ex3_4_ind K C C T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (S n0) O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 
-e2))))) (or4 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C T T (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k 
-u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
-C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H23: (eq C e (CHead x1 x0 
-x3))).(\lambda (H24: (drop (S n0) O c0 (CHead x2 x0 x3))).(\lambda (H25: 
-(csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) 
-(\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) c) (ex3_4 K C T T 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
+(\lambda (c4: C).(or4 (drop (S n0) O (CHead c3 (Flat f) u2) c4) (ex3_4 K C T 
+T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c4 
+(CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c4 (CHead e1 
+k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
+f) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c4 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2))))))))) (or4_intro1 (drop (S n0) O (CHead c3 (Flat f) u2) (CHead x1 x0 
+x2)) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 
+(Flat f) u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u 
+w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead x1 x0 x2) (CHead e1 k1 u)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 
+(Flat f) u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e1 k1 
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 
+w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 e2))))))) 
+(ex3_4_intro K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 
+(Flat f) u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u 
+w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x2)) (drop_drop (Flat f) n0 c3 
+(CHead x1 x0 x3) H28 u2) H29)) e H27)))))))) H26)) (\lambda (H26: (ex3_4 K C 
+C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k1 
+(S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead 
+e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus i0 (s k1 (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead 
+c3 (Flat f) u2) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 
+u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C 
+T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
+c3 (Flat f) u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
+i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 
+(S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: 
+C).(\lambda (x3: T).(\lambda (H27: (eq C e (CHead x1 x0 x3))).(\lambda (H28: 
+(drop (S n0) O c3 (CHead x2 x0 x3))).(\lambda (H29: (csubst0 (minus i0 (s x0 
+(S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c4: C).(or4 (drop 
+(S n0) O (CHead c3 (Flat f) u2) c4) (ex3_4 K C T T (\lambda (k1: K).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c4 (CHead e0 k1 u)))))) 
+(\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O (CHead c3 (Flat f) u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 
+(S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c4 (CHead e1 k1 u)))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O 
+(CHead c3 (Flat f) u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c4 (CHead e1 k1 
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 
+w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 e2))))))))) 
+(or4_intro2 (drop (S n0) O (CHead c3 (Flat f) u2) (CHead x1 x0 x3)) (ex3_4 K 
+C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+(CHead x1 x0 x3) (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) 
+(CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) 
+(ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C (CHead x1 x0 x3) (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Flat f) u2) 
+(CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 
+w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 e2))))))) 
+(ex3_4_intro K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 u)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 
+(Flat f) u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c3 
+(CHead x2 x0 x3) H28 u2) H29)) e H27)))))))) H26)) (\lambda (H26: (ex4_5 K C 
+C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k1 (S n0))) v u 
+w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 (minus i0 (s k1 (S n0))) v e1 
+e2)))))))).(ex4_5_ind K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k1 (S 
+n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus i0 (s k1 (S n0))) v e1 e2)))))) (or4 
+(drop (S n0) O (CHead c3 (Flat f) u2) e) (ex3_4 K C T T (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 
+u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e0 k1 w)))))) (\lambda (k1: 
 K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
-x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C 
-(CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x3) H24 u2) H25)) e 
-H23)))))))) H22)) (\lambda (H22: (ex4_5 K C C T T (\lambda (k: K).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: 
+i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 
+(Flat f) u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda 
+(k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 w))))))) (\lambda (k1: 
 K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
-(minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 
-e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u 
-w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))) (or4 (drop 
-(S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda 
-(k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead 
-c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u 
-w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) 
-(CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
-C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H23: (eq C e 
-(CHead x1 x0 x3))).(\lambda (H24: (drop (S n0) O c0 (CHead x2 x0 
-x4))).(\lambda (H25: (subst0 (minus i0 (s x0 (S n0))) v x3 x4)).(\lambda 
-(H26: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 
-x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) c) (ex3_4 K C 
-T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
-(CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) 
-i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: 
-K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s 
-(Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat 
-f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) 
-(\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
-O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S 
-n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda 
-(k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead 
-c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k 
-u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) 
-(\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: 
-K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-(minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex4_5_intro K C C T T 
-(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 
-(Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S 
-n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 
-e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) 
-n0 c0 (CHead x2 x0 x4) H24 u2) H25 H26)) e H23)))))))))) H22)) H21)))))) k 
-(drop_gen_drop k c e t n0 H2) H19 H20)))))) c2 H16)) u1 (sym_eq T u1 t H15))) 
-k0 (sym_eq K k0 k H14))) c1 (sym_eq C c1 c H13))) H12)) H11))) v0 (sym_eq T 
-v0 v H9))) i H5 H6 H7 H8 H3 H4)))))]) in (H3 (refl_equal nat i) (refl_equal T 
-v) (refl_equal C (CHead c k t)) (refl_equal C c2)))))))))))) c1)))))) n).
+(minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s 
+(Flat f) i0) (s k1 (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: 
+C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H27: (eq C e 
+(CHead x1 x0 x3))).(\lambda (H28: (drop (S n0) O c3 (CHead x2 x0 
+x4))).(\lambda (H29: (subst0 (minus i0 (s x0 (S n0))) v x3 x4)).(\lambda 
+(H30: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 
+x3) (\lambda (c4: C).(or4 (drop (S n0) O (CHead c3 (Flat f) u2) c4) (ex3_4 K 
+C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+c4 (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e0 k1 w)))))) 
+(\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 
+(minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda 
+(k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c4 (CHead e1 
+k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat 
+f) i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: 
+K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c4 
+(CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) 
+(CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S 
+n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2))))))))) (or4_intro3 (drop (S n0) O (CHead c3 (Flat f) u2) (CHead x1 x0 
+x3)) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k1 u)))))) (\lambda (k1: 
+K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c3 
+(Flat f) u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u 
+w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 u)))))) (\lambda (k1: 
+K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 
+(Flat f) u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 
+e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 
+w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 e2))))))) 
+(ex4_5_intro K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k1 
+u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O (CHead c3 (Flat f) u2) (CHead e2 k1 
+w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k1 (S n0))) v u w)))))) 
+(\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 (minus (s (Flat f) i0) (s k1 (S n0))) v e1 e2)))))) x0 x1 x2 
+x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c3 (CHead x2 x0 
+x4) H28 u2) H29 H30)) e H27)))))))))) H26)) H25)))))) k (drop_gen_drop k c e 
+t n0 H2) H20 H21)))))) c2 H16)) u1 (sym_eq T u1 t H15))) k0 (sym_eq K k0 k 
+H14))) c0 (sym_eq C c0 c H13))) H12)) H11))) v0 (sym_eq T v0 v H9))) i H5 H6 
+H7 H8 H3 H4)))))]) in (H3 (refl_equal nat i) (refl_equal T v) (refl_equal C 
+(CHead c k t)) (refl_equal C c2)))))))))))) c1)))))) n).
 
 theorem csubst0_drop_eq:
  \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 
@@ -2596,8 +2639,8 @@ C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))))
 (_: T).(csubst0 O v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: 
 nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 
 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind 
-nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H3) in (or4_intro1 (drop O 
-O (CHead c (Flat f) u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: 
+nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H3) in (or4_intro1 (drop 
+O O (CHead c (Flat f) u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: 
 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) 
 u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
 (_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) 
@@ -2694,39 +2737,39 @@ C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w))))))
 (_: T).(csubst0 O v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: 
 nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: 
 (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop O O c4 c3) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C c3 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f0) w)))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 
-u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: 
+u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
+(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda 
 (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: 
+(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda 
-(H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) 
-\to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 
-(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+(H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 
+O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f0) u0)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O 
+c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c3 (CHead e1 (Flat 
+f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
+T).(drop O O c4 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T 
-T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
+T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(_: T).(eq C c3 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: 
+(Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 
-e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n: 
-nat).(csubst0 n v0 c3 c4)) H2 O H4) in (or4_intro2 (drop O O (CHead c4 (Flat 
+e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n0: 
+nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in (or4_intro2 (drop O O (CHead c4 (Flat 
 f) u) (CHead c3 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
 C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 
 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
@@ -2827,39 +2870,39 @@ e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0:
 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 
 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 
 c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
-(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 
+(CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f0) w)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f0) u)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f0) u))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop O O c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 
-\def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c4 c3) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) 
+\def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 (drop O O c4 
+c3) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C c3 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f0) w)))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 
-u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: 
+u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
+(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda 
 (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: 
+(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f0) w))))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
 v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H4 O H5) in (let H7 \def 
-(eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 O H5) in (let H8 
-\def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H5) in 
+(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 
+\def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in 
 (or4_intro3 (drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F 
 C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
 (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
@@ -2994,11 +3037,11 @@ v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
 T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda 
 (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v 
 (CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CHead c k t) 
-e)).(let H2 \def (match H0 in csubst0 return (\lambda (n: nat).(\lambda (t0: 
-T).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubst0 n t0 c0 c1)).((eq 
-nat n (S n0)) \to ((eq T t0 v) \to ((eq C c0 (CHead c k t)) \to ((eq C c1 c2) 
-\to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+e)).(let H2 \def (match H0 in csubst0 return (\lambda (n1: nat).(\lambda (t0: 
+T).(\lambda (c0: C).(\lambda (c3: C).(\lambda (_: (csubst0 n1 t0 c0 c3)).((eq 
+nat n1 (S n0)) \to ((eq T t0 v) \to ((eq C c0 (CHead c k t)) \to ((eq C c3 
+c2) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) 
 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
@@ -3016,20 +3059,20 @@ e2))))))))))))))))) with [(csubst0_snd k0 i v0 u1 u2 H2 c0) \Rightarrow
 (\lambda (H3: (eq nat (s k0 i) (S n0))).(\lambda (H4: (eq T v0 v)).(\lambda 
 (H5: (eq C (CHead c0 k0 u1) (CHead c k t))).(\lambda (H6: (eq C (CHead c0 k0 
 u2) c2)).((let H7 \def (f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S 
-n0) H3) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C 
+n0) H3) in (eq_ind nat (s k0 i) (\lambda (n1: nat).((eq T v0 v) \to ((eq C 
 (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 
-i v0 u1 u2) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+i v0 u1 u2) \to (or4 (drop n1 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 
-(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n1 O 
+c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop n O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+T).(drop n1 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
 T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n1 O c2 (CHead e2 
 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
@@ -3053,117 +3096,118 @@ T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda 
 (H9: (eq C (CHead c0 k0 u1) (CHead c k t))).(let H10 \def (f_equal C T 
 (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) 
-\Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u1) (CHead c k 
+\Rightarrow u1 | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k0 u1) (CHead c k 
 t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 in C return 
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow 
-k])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H12 \def (f_equal C C 
-(\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) 
-\Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u1) (CHead c k 
-t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq 
-C (CHead c k0 u2) c2) \to ((subst0 i v u1 u2) \to (or4 (drop (s k0 i) O c2 e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 
-i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq 
-T u1 t) \to ((eq C (CHead c k u2) c2) \to ((subst0 i v u1 u2) \to (or4 (drop 
-(s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
+\Rightarrow k1])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H12 \def 
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with 
+[(CSort _) \Rightarrow c0 | (CHead c3 _ _) \Rightarrow c3])) (CHead c0 k0 u1) 
+(CHead c k t) H9) in (eq_ind C c (\lambda (c3: C).((eq K k0 k) \to ((eq T u1 
+t) \to ((eq C (CHead c3 k0 u2) c2) \to ((subst0 i v u1 u2) \to (or4 (drop (s 
+k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead 
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k 
-i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))))))) (\lambda (H14: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C 
-(CHead c k u2) c2) \to ((subst0 i v t u2) \to (or4 (drop (s k i) O c2 e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k 
-i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))) (\lambda (H15: (eq C (CHead c k u2) c2)).(eq_ind C (CHead c k 
-u2) (\lambda (c: C).((subst0 i v t u2) \to (or4 (drop (s k i) O c e) (ex3_4 F 
-C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s k i) O c (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c (CHead e2 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
-u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s k i) O c (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))) (\lambda (H16: (subst0 i v t 
-u2)).(let H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i) (S n0))) H7 k 
-H13) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to ((eq nat (s k i) (S 
-n0)) \to (or4 (drop (s k i) O (CHead c k u2) e) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 
+(CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K 
+k (\lambda (k1: K).((eq T u1 t) \to ((eq C (CHead c k1 u2) c2) \to ((subst0 i 
+v u1 u2) \to (or4 (drop (s k1 i) O c2 e) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (s k1 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(drop (s k1 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (s k1 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H14: (eq T u1 t)).(eq_ind T t 
+(\lambda (t0: T).((eq C (CHead c k u2) c2) \to ((subst0 i v t0 u2) \to (or4 
+(drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k 
+i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c k u2) 
+c2)).(eq_ind C (CHead c k u2) (\lambda (c3: C).((subst0 i v t u2) \to (or4 
+(drop (s k i) O c3 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k 
+i) O c3 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (s k i) O c3 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+k i) O c3 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))))) (\lambda (H16: (subst0 i v t u2)).(let 
+H17 \def (eq_ind K k0 (\lambda (k1: K).(eq nat (s k1 i) (S n0))) H7 k H13) in 
+(K_ind (\lambda (k1: K).((drop (r k1 n0) O c e) \to ((eq nat (s k1 i) (S n0)) 
+\to (or4 (drop (s k1 i) O (CHead c k1 u2) e) (ex3_4 F C T T (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
 f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s k i) O (CHead c k u2) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+T).(drop (s k1 i) O (CHead c k1 u2) (CHead e0 (Flat f) w)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O (CHead c k u2) (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s k i) O (CHead c k u2) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s k1 i) O (CHead c k1 u2) (CHead 
+e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k1 i) O (CHead c k1 u2) (CHead e2 
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) n0) O c 
-e)).(\lambda (H17: (eq nat (s (Bind b) i) (S n0))).(let H18 \def (f_equal nat 
+e2))))))))))) (\lambda (b: B).(\lambda (H18: (drop (r (Bind b) n0) O c 
+e)).(\lambda (H19: (eq nat (s (Bind b) i) (S n0))).(let H20 \def (f_equal nat 
 nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O 
-\Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H17) in (let H19 \def 
-(eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H16 n0 H18) in (eq_ind_r 
-nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c (Bind b) u2) e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c (Bind b) 
+\Rightarrow i | (S n1) \Rightarrow n1])) (S i) (S n0) H19) in (let H21 \def 
+(eq_ind nat i (\lambda (n1: nat).(subst0 n1 v t u2)) H16 n0 H20) in (eq_ind_r 
+nat n0 (\lambda (n1: nat).(or4 (drop (s (Bind b) n1) O (CHead c (Bind b) u2) 
+e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n1) O (CHead c (Bind b) 
 u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (s (Bind b) n) O (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) 
+T).(drop (s (Bind b) n1) O (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) 
 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Bind b) n) O (CHead c (Bind b) u2) (CHead e2 (Flat f) w))))))) 
+T).(drop (s (Bind b) n1) O (CHead c (Bind b) u2) (CHead e2 (Flat f) w))))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro0 
@@ -3183,97 +3227,98 @@ F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
 (Bind b) n0) O (CHead c (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H1 
-u2)) i H18)))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) n0) O c 
-e)).(\lambda (H17: (eq nat (s (Flat f) i) (S n0))).(let H18 \def (f_equal nat 
-nat (\lambda (e0: nat).e0) i (S n0) H17) in (let H19 \def (eq_ind nat i 
-(\lambda (n: nat).(subst0 n v t u2)) H16 (S n0) H18) in (eq_ind_r nat (S n0) 
-(\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead c (Flat f) u2) e) (ex3_4 
-F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c (Flat f) u2) (CHead e0 
-(Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s 
-(Flat f) n) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) n) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro0 (drop (s (Flat f) 
-(S n0)) O (CHead c (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c 
-(Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) 
-O (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H1 u2)) i 
-H18)))))) k (drop_gen_drop k c e t n0 H1) H0))) c2 H15)) u1 (sym_eq T u1 t 
-H14))) k0 (sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11)) H10))) v0 
-(sym_eq T v0 v H8))) (S n0) H7)) H4 H5 H6 H2))))) | (csubst0_fst k0 i c1 c0 
-v0 H2 u) \Rightarrow (\lambda (H3: (eq nat (s k0 i) (S n0))).(\lambda (H4: 
-(eq T v0 v)).(\lambda (H5: (eq C (CHead c1 k0 u) (CHead c k t))).(\lambda 
-(H6: (eq C (CHead c0 k0 u) c2)).((let H7 \def (f_equal nat nat (\lambda (e0: 
-nat).e0) (s k0 i) (S n0) H3) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T 
-v0 v) \to ((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) 
-c2) \to ((csubst0 i v0 c1 c0) \to (or4 (drop n O c2 e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop n O c2 (CHead e2 (Flat f) u0)))))) 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H18 
+u2)) i H20)))))) (\lambda (f: F).(\lambda (H18: (drop (r (Flat f) n0) O c 
+e)).(\lambda (H19: (eq nat (s (Flat f) i) (S n0))).(let H20 \def (f_equal nat 
+nat (\lambda (e0: nat).e0) i (S n0) H19) in (let H21 \def (eq_ind nat i 
+(\lambda (n1: nat).(subst0 n1 v t u2)) H16 (S n0) H20) in (eq_ind_r nat (S 
+n0) (\lambda (n1: nat).(or4 (drop (s (Flat f) n1) O (CHead c (Flat f) u2) e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n1) O (CHead c (Flat f) 
+u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (s (Flat f) n1) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v 
-(\lambda (t0: T).((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 
-k0 u) c2) \to ((csubst0 i t0 c1 c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F 
-C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
-e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 i) O c2 
-(CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (s (Flat f) n1) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))))))) (\lambda (H9: (eq C (CHead c1 k0 u) (CHead c k t))).(let H10 
-\def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) 
-with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 
-u) (CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match 
-e0 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k 
-_) \Rightarrow k])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H12 \def 
+e2))))))))) (or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c (Flat f) u2) e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c (Flat 
+f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (s (Flat f) (S n0)) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat 
+f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c (Flat f) u2) (CHead 
+e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))) (drop_drop (Flat f) n0 c e H18 u2)) i H20)))))) k 
+(drop_gen_drop k c e t n0 H1) H17))) c2 H15)) u1 (sym_eq T u1 t H14))) k0 
+(sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11)) H10))) v0 (sym_eq T v0 v 
+H8))) (S n0) H7)) H4 H5 H6 H2))))) | (csubst0_fst k0 i c0 c3 v0 H2 u) 
+\Rightarrow (\lambda (H3: (eq nat (s k0 i) (S n0))).(\lambda (H4: (eq T v0 
+v)).(\lambda (H5: (eq C (CHead c0 k0 u) (CHead c k t))).(\lambda (H6: (eq C 
+(CHead c3 k0 u) c2)).((let H7 \def (f_equal nat nat (\lambda (e0: nat).e0) (s 
+k0 i) (S n0) H3) in (eq_ind nat (s k0 i) (\lambda (n1: nat).((eq T v0 v) \to 
+((eq C (CHead c0 k0 u) (CHead c k t)) \to ((eq C (CHead c3 k0 u) c2) \to 
+((csubst0 i v0 c0 c3) \to (or4 (drop n1 O c2 e) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n1 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
+(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop n1 O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n1 
+O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: 
+T).((eq C (CHead c0 k0 u) (CHead c k t)) \to ((eq C (CHead c3 k0 u) c2) \to 
+((csubst0 i t0 c0 c3) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 
+(Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) 
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat 
+f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda 
+(H9: (eq C (CHead c0 k0 u) (CHead c k t))).(let H10 \def (f_equal C T 
+(\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) 
+\Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k0 u) (CHead c k 
+t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 in C return 
+(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) 
+\Rightarrow k1])) (CHead c0 k0 u) (CHead c k t) H9) in ((let H12 \def 
 (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with 
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u) 
-(CHead c k t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u t) 
-\to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i v c c0) \to (or4 (drop (s k0 
-i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: 
+[(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0 k0 u) 
+(CHead c k t) H9) in (eq_ind C c (\lambda (c4: C).((eq K k0 k) \to ((eq T u 
+t) \to ((eq C (CHead c3 k0 u) c2) \to ((csubst0 i v c4 c3) \to (or4 (drop (s 
+k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: 
 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 
 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
@@ -3289,8 +3334,25 @@ k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_:
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K 
-k (\lambda (k: K).((eq T u t) \to ((eq C (CHead c0 k u) c2) \to ((csubst0 i v 
-c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+k (\lambda (k1: K).((eq T u t) \to ((eq C (CHead c3 k1 u) c2) \to ((csubst0 i 
+v c c3) \to (or4 (drop (s k1 i) O c2 e) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (s k1 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
+(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (s k1 i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (s k1 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 
+w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H14: (eq T u 
+t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 k t0) c2) \to ((csubst0 i v 
+c c3) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
 (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k 
 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
@@ -3305,335 +3367,319 @@ F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s
 k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H14: (eq T u t)).(eq_ind T t 
-(\lambda (t: T).((eq C (CHead c0 k t) c2) \to ((csubst0 i v c c0) \to (or4 
-(drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k 
-i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat 
-f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (s k i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c0 k t) 
-c2)).(eq_ind C (CHead c0 k t) (\lambda (c2: C).((csubst0 i v c c0) \to (or4 
-(drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+(_: T).(csubst0 O v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c3 k t) 
+c2)).(eq_ind C (CHead c3 k t) (\lambda (c4: C).((csubst0 i v c c3) \to (or4 
+(drop (s k i) O c4 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
 C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k 
-i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+i) O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
 (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat 
 f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (s k i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda 
+T).(drop (s k i) O c4 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda 
 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
 T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+k i) O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))))) (\lambda (H16: (csubst0 i v c c0)).(let 
-H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i) (S n0))) H7 k H13) in 
-(K_ind (\lambda (k: K).((drop (r k n0) O c e) \to ((eq nat (s k i) (S n0)) 
-\to (or4 (drop (s k i) O (CHead c0 k t) e) (ex3_4 F C T T (\lambda (f: 
+(_: T).(csubst0 O v e1 e2)))))))))) (\lambda (H16: (csubst0 i v c c3)).(let 
+H17 \def (eq_ind K k0 (\lambda (k1: K).(eq nat (s k1 i) (S n0))) H7 k H13) in 
+(K_ind (\lambda (k1: K).((drop (r k1 n0) O c e) \to ((eq nat (s k1 i) (S n0)) 
+\to (or4 (drop (s k1 i) O (CHead c3 k1 t) e) (ex3_4 F C T T (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
 f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s k i) O (CHead c0 k t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+T).(drop (s k1 i) O (CHead c3 k1 t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
 T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k i) O (CHead c0 k t) (CHead e2 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k1 i) O (CHead c3 k1 t) (CHead 
+e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k1 i) O (CHead c3 k1 t) (CHead e2 
+(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))))) (\lambda (b: B).(\lambda (H18: (drop (r (Bind b) n0) O c 
+e)).(\lambda (H19: (eq nat (s (Bind b) i) (S n0))).(let H20 \def (f_equal nat 
+nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O 
+\Rightarrow i | (S n1) \Rightarrow n1])) (S i) (S n0) H19) in (let H21 \def 
+(eq_ind nat i (\lambda (n1: nat).(csubst0 n1 v c c3)) H16 n0 H20) in 
+(eq_ind_r nat n0 (\lambda (n1: nat).(or4 (drop (s (Bind b) n1) O (CHead c3 
+(Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n1) O 
+(CHead c3 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
+(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (s (Bind b) n1) O (CHead c3 (Bind b) t) (CHead e2 
 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
 (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
 (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 
 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 k t) (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
+(_: T).(\lambda (w: T).(drop (s (Bind b) n1) O (CHead c3 (Bind b) t) (CHead 
+e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
+(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) n0) O c 
-e)).(\lambda (H17: (eq nat (s (Bind b) i) (S n0))).(let H18 \def (f_equal nat 
-nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O 
-\Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H17) in (let H19 \def 
-(eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H16 n0 H18) in (eq_ind_r 
-nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c0 (Bind b) t) e) 
+e2))))))))) (let H22 \def (IHn c c3 v H21 e H18) in (or4_ind (drop n0 O c3 e) 
 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: 
 T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) 
-t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat 
-f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H \def 
-(IHn c c0 v H19 e H1) in (or4_ind (drop n0 O c0 e) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 
-(Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
-(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop n0 O c0 (CHead e2 (Flat f) u0)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 
-O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) e) (ex3_4 
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda 
-(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c3 (CHead e0 (Flat f) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v 
+u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O c3 (CHead e2 
+(Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 
+(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop n0 O c3 (CHead e2 (Flat f) w))))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
 v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H20: (drop n0 O c0 
-e)).(or4_intro0 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) e) (ex3_4 F C T 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Bind b) n0) O 
+(CHead c3 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+(Bind b) n0) O (CHead c3 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c3 (Bind 
+b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
+C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 
+(Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (H23: (drop n0 O c3 
+e)).(or4_intro0 (drop (s (Bind b) n0) O (CHead c3 (Bind b) t) e) (ex3_4 F C T 
 T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
 (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead e0 
 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda 
 (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) 
 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
+(Bind b) n0) O (CHead c3 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
 (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
 (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: 
+(Bind b) n0) O (CHead c3 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
 v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H20 
-t))) (\lambda (H20: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c3 e H23 
+t))) (\lambda (H23: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c3 (CHead e0 
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 O v u0 w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O 
-c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
+c3 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
 T).(\lambda (w: T).(subst0 O v u0 w))))) (or4 (drop (s (Bind b) n0) O (CHead 
-c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
+c3 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
 (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+(CHead c3 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
 (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
+C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead e2 
 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
 (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
 (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 
 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
+(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead 
 e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
 (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: 
-T).(\lambda (H21: (eq C e (CHead x1 (Flat x0) x2))).(\lambda (H22: (drop n0 O 
-c0 (CHead x1 (Flat x0) x3))).(\lambda (H23: (subst0 O v x2 x3)).(eq_ind_r C 
-(CHead x1 (Flat x0) x2) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead 
-c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0)))))) (\lambda (f: 
+T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x2))).(\lambda (H25: (drop n0 O 
+c3 (CHead x1 (Flat x0) x3))).(\lambda (H26: (subst0 O v x2 x3)).(eq_ind_r C 
+(CHead x1 (Flat x0) x2) (\lambda (c4: C).(or4 (drop (s (Bind b) n0) O (CHead 
+c3 (Bind b) t) c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u0: T).(\lambda (_: T).(eq C c4 (CHead e0 (Flat f) u0)))))) (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+(CHead c3 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c4 
 (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
+C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead e2 
 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
 (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 
+(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e1 
 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
+(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead 
 e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
 (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro1 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
+e2))))))))) (or4_intro1 (drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead 
 x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
 (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) 
 u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) 
+T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead e0 (Flat f) w)))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v 
 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f) u0)))))) 
 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
+(Bind b) n0) O (CHead c3 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
 (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
 (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f) 
 u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead e2 
 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
 (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
 (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: 
 T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) u0)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+(Bind b) n0) O (CHead c3 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))) x0 
-x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Bind b) n0 c0 
-(CHead x1 (Flat x0) x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O c0 (CHead e2 
-(Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind 
-b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
-(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
-(Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 
-(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
-e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (H21: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H22: (drop n0 O 
-c0 (CHead x2 (Flat x0) x3))).(\lambda (H23: (csubst0 O v x1 x2)).(eq_ind_r C 
-(CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead 
-c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c 
-(CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
-(Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 
-(Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
-e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro2 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
-x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f) 
-u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v 
-u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) 
-u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(ex3_4_intro F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
-x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0 
-(CHead x2 (Flat x0) x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 
-O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat 
-f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
-v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Bind b) n0 c3 
+(CHead x1 (Flat x0) x3) H25 t) H26)) e H24)))))))) H23)) (\lambda (H23: 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O c3 (CHead e2 (Flat f) 
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O 
+c3 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Bind b) n0) O 
+(CHead c3 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
 C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+(Bind b) n0) O (CHead c3 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
 T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c3 (Bind 
 b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
 C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 
 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda 
-(x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H21: (eq C e (CHead x1 
-(Flat x0) x3))).(\lambda (H22: (drop n0 O c0 (CHead x2 (Flat x0) 
-x4))).(\lambda (H23: (subst0 O v x3 x4)).(\lambda (H24: (csubst0 O v x1 
-x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind 
-b) n0) O (CHead c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0)))))) 
+(x2: C).(\lambda (x3: T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) 
+x3))).(\lambda (H25: (drop n0 O c3 (CHead x2 (Flat x0) x3))).(\lambda (H26: 
+(csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c4: 
+C).(or4 (drop (s (Bind b) n0) O (CHead c3 (Bind b) t) c4) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 
+(CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead e0 
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C c4 (CHead e1 (Flat f) u0)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
+(Bind b) n0) O (CHead c3 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(eq C c4 (CHead e1 (Flat f) u0))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+(Bind b) n0) O (CHead c3 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
+v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (s (Bind b) 
+n0) O (CHead c3 (Bind b) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 
+(Flat x0) x3) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) 
+t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat 
+x0) x3) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead 
+e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
+(CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) 
+O (CHead c3 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 
+w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 
+(Flat x0) x3) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c3 (Bind 
+b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C 
+(CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c3 (CHead x2 (Flat x0) x3) 
+H25 t) H26)) e H24)))))))) H23)) (\lambda (H23: (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c3 (CHead e2 (Flat f) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C 
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 
+O c3 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))) (or4 (drop (s (Bind b) n0) O (CHead c3 (Bind b) t) e) (ex3_4 
+F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq 
+C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead e0 
+(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda 
+(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
+(Bind b) n0) O (CHead c3 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+(Bind b) n0) O (CHead c3 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
+v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: 
+C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H24: (eq C e 
+(CHead x1 (Flat x0) x3))).(\lambda (H25: (drop n0 O c3 (CHead x2 (Flat x0) 
+x4))).(\lambda (H26: (subst0 O v x3 x4)).(\lambda (H27: (csubst0 O v x1 
+x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c4: C).(or4 (drop (s (Bind 
+b) n0) O (CHead c3 (Bind b) t) c4) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e0 (Flat f) u0)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+(Bind b) n0) O (CHead c3 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(eq C c (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
+T).(eq C c4 (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c3 (Bind 
 b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C c (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 
+C c4 (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 
 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Bind b) n0) O (CHead 
-c0 (Bind b) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: 
+c3 (Bind b) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat 
 x0) x3) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead 
+(_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead 
 e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda 
 (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 
 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) 
+(u0: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) t) (CHead e2 (Flat f) 
 u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
 x3) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 
 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
@@ -3641,383 +3687,401 @@ C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))))
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
 (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda 
 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) 
-O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
+O (CHead c3 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
 (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 
 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C 
-(CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x4) 
-H22 t) H23 H24)) e H21)))))))))) H20)) H)) i H18)))))) (\lambda (f: 
-F).(\lambda (H1: (drop (r (Flat f) n0) O c e)).(\lambda (H17: (eq nat (s 
-(Flat f) i) (S n0))).(let H18 \def (f_equal nat nat (\lambda (e0: nat).e0) i 
-(S n0) H17) in (let H19 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c 
-c0)) H16 (S n0) H18) in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s 
-(Flat f) n) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
-f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Flat f) n) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v 
-u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) n) O 
-(CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) n) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
-v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H20 \def (H c0 v H19 e 
-H1) in (or4_ind (drop (S n0) O c0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda 
-(e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat 
-f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O c0 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
-n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) 
-O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
-(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
-e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (H21: (drop (S n0) O c0 
-e)).(or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F 
-C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq 
-C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) 
-(CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat 
-f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) 
-u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat 
-f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
-e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H21 t))) (\lambda (H21: (ex3_4 F 
-C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
-w))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead 
-e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
-T).(\lambda (w: T).(subst0 O v u0 w))))) (or4 (drop (s (Flat f) (S n0)) O 
-(CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C 
+(CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c3 (CHead x2 (Flat x0) x4) 
+H25 t) H26 H27)) e H24)))))))))) H23)) H22)) i H20)))))) (\lambda (f: 
+F).(\lambda (H18: (drop (r (Flat f) n0) O c e)).(\lambda (H19: (eq nat (s 
+(Flat f) i) (S n0))).(let H20 \def (f_equal nat nat (\lambda (e0: nat).e0) i 
+(S n0) H19) in (let H21 \def (eq_ind nat i (\lambda (n1: nat).(csubst0 n1 v c 
+c3)) H16 (S n0) H20) in (eq_ind_r nat (S n0) (\lambda (n1: nat).(or4 (drop (s 
+(Flat f) n1) O (CHead c3 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (s (Flat f) n1) O (CHead c3 (Flat f) t) (CHead e0 (Flat f0) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v 
+u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) n1) O 
+(CHead c3 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+(Flat f) n1) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H22 \def 
+(H c3 v H21 e H18) in (or4_ind (drop (S n0) O c3 e) (ex3_4 F C T T (\lambda 
+(f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 
+(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S n0) O c3 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
+(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (S n0) O c3 (CHead e2 (Flat f0) u0)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (S n0) O c3 (CHead e2 (Flat f0) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
+v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Flat f) (S n0)) O 
+(CHead c3 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
 C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) 
 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda 
+(Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda 
 (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 
 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S 
-n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: 
+n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
 (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) 
+(Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) w))))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
 T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: 
-F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H22: (eq C e 
-(CHead x1 (Flat x0) x2))).(\lambda (H23: (drop (S n0) O c0 (CHead x1 (Flat 
-x0) x3))).(\lambda (H24: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) 
-x2) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) c) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C c (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H23: 
+(drop (S n0) O c3 e)).(or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat 
+f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) 
+O (CHead c3 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
 (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
+C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead 
 e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (s (Flat f) (S n0)) O 
-(CHead c0 (Flat f) t) (CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat 
-x0) x2) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C 
-(CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S 
-n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) 
-u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
-e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 
-(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
-T).(subst0 O v u0 w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) 
-(drop_drop (Flat f) n0 c0 (CHead x1 (Flat x0) x3) H23 t) H24)) e H22)))))))) 
-H21)) (\lambda (H21: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead 
-e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat 
-f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: 
-T).(drop (S n0) O c0 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop 
-(s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: 
+(_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c3 e H23 t))) 
+(\lambda (H23: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead 
+e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 O v u0 w))))))).(ex3_4_ind F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (S n0) O c3 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))) (or4 (drop (s (Flat 
+f) (S n0)) O (CHead c3 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: 
 F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
 f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) 
+T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e0 (Flat f0) 
 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
 T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) 
 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) 
+(Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) u0)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) 
 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) 
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) 
 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
 T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (H22: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H23: (drop (S 
-n0) O c0 (CHead x2 (Flat x0) x3))).(\lambda (H24: (csubst0 O v x1 
-x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat 
-f) (S n0)) O (CHead c0 (Flat f) t) c) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat 
+e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x2))).(\lambda (H25: (drop (S 
+n0) O c3 (CHead x1 (Flat x0) x3))).(\lambda (H26: (subst0 O v x2 
+x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) (\lambda (c4: C).(or4 (drop (s (Flat 
+f) (S n0)) O (CHead c3 (Flat f) t) c4) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e0 (Flat 
 f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) 
+T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e0 (Flat f0) 
 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
 T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 (Flat f0) u0)))))) 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C c4 (CHead e1 (Flat f0) u0)))))) 
 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) 
+(Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) u0)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u0))))))) 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e1 (Flat f0) u0))))))) 
 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) 
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) 
 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
 T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro2 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) 
-(CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 
+e2))))))))) (or4_intro1 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) 
+(CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 
 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) 
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e0 (Flat f0) 
 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
 T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 
 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) 
+(u0: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) 
 u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
-x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
+x2) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat 
-x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 
-(refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 
-(Flat x0) x3) H23 t) H24)) e H22)))))))) H21)) (\lambda (H21: (ex4_5 F C C T 
-T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead 
-e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O 
-v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Flat f) (S n0)) O 
-(CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat 
+x0) x2) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
+(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))) x0 x1 x2 x3 
+(refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Flat f) n0 c3 (CHead x1 
+(Flat x0) x3) H25 t) H26)) e H24)))))))) H23)) (\lambda (H23: (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
+(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (S n0) O c3 (CHead e2 (Flat f0) u0)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2))))))).(ex3_4_ind F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead 
+e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Flat f) (S n0)) O 
+(CHead c3 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
 C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) 
 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda 
+(Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda 
 (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 
 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S 
-n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: 
+n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
 (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) 
+(Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) w))))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
 T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: 
-F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H22: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H23: (drop (S 
-n0) O c0 (CHead x2 (Flat x0) x4))).(\lambda (H24: (subst0 O v x3 
-x4)).(\lambda (H25: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) 
-(\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) c) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda 
-(_: T).(eq C c (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c 
+F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H24: (eq C e 
+(CHead x1 (Flat x0) x3))).(\lambda (H25: (drop (S n0) O c3 (CHead x2 (Flat 
+x0) x3))).(\lambda (H26: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) 
+x3) (\lambda (c4: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) 
+c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C c4 (CHead e0 (Flat f0) u0)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) 
+O (CHead c3 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c4 
 (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
+C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead 
 e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 
 (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Flat f) (S n0)) O 
-(CHead c0 (Flat f) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: 
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (s (Flat f) (S n0)) O 
+(CHead c3 (Flat f) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: 
 F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat 
 x0) x3) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C 
 (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S 
-n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: 
+n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
 (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
 (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) 
 u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead 
+T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead 
 e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
 C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: 
+O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 
+(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u0: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) 
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) 
+(drop_drop (Flat f) n0 c3 (CHead x2 (Flat x0) x3) H25 t) H26)) e H24)))))))) 
+H23)) (\lambda (H23: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat 
+f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 (Flat f0) w))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C 
+C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O c3 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c3 
+(Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) 
+O (CHead c3 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e 
+(CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead 
+e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
+(Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H24: (eq C e (CHead x1 
+(Flat x0) x3))).(\lambda (H25: (drop (S n0) O c3 (CHead x2 (Flat x0) 
+x4))).(\lambda (H26: (subst0 O v x3 x4)).(\lambda (H27: (csubst0 O v x1 
+x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c4: C).(or4 (drop (s (Flat 
+f) (S n0)) O (CHead c3 (Flat f) t) c4) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e0 (Flat 
+f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e0 (Flat f0) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C c4 (CHead e1 (Flat f0) u0)))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s 
+(Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) u0)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C c4 (CHead e1 (Flat f0) u0))))))) 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) 
+w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: 
+T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))) (or4_intro3 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) 
+(CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 
+(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e0 (Flat f0) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 
+(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u0: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) 
+u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
 x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat 
-x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x4) H23 t) H24 H25)) e 
-H22)))))))))) H21)) H20)) i H18)))))) k (drop_gen_drop k c e t n0 H1) H0))) 
-c2 H15)) u (sym_eq T u t H14))) k0 (sym_eq K k0 k H13))) c1 (sym_eq C c1 c 
-H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) (S n0) H7)) H4 H5 H6 H2))))) | 
-(csubst0_both k0 i v0 u1 u2 H2 c1 c0 H3) \Rightarrow (\lambda (H4: (eq nat (s 
-k0 i) (S n0))).(\lambda (H5: (eq T v0 v)).(\lambda (H6: (eq C (CHead c1 k0 
-u1) (CHead c k t))).(\lambda (H7: (eq C (CHead c0 k0 u2) c2)).((let H8 \def 
-(f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S n0) H4) in (eq_ind nat (s 
-k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u1) (CHead c k 
-t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v0 u1 u2) \to ((csubst0 i 
-v0 c1 c0) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C 
+(CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+(Flat f) (S n0)) O (CHead c3 (Flat f) t) (CHead e2 (Flat f0) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: 
+T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 
+(refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c3 (CHead x2 
+(Flat x0) x4) H25 t) H26 H27)) e H24)))))))))) H23)) H22)) i H20)))))) k 
+(drop_gen_drop k c e t n0 H1) H17))) c2 H15)) u (sym_eq T u t H14))) k0 
+(sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11)) H10))) v0 (sym_eq T v0 v 
+H8))) (S n0) H7)) H4 H5 H6 H2))))) | (csubst0_both k0 i v0 u1 u2 H2 c0 c3 H3) 
+\Rightarrow (\lambda (H4: (eq nat (s k0 i) (S n0))).(\lambda (H5: (eq T v0 
+v)).(\lambda (H6: (eq C (CHead c0 k0 u1) (CHead c k t))).(\lambda (H7: (eq C 
+(CHead c3 k0 u2) c2)).((let H8 \def (f_equal nat nat (\lambda (e0: nat).e0) 
+(s k0 i) (S n0) H4) in (eq_ind nat (s k0 i) (\lambda (n1: nat).((eq T v0 v) 
+\to ((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c3 k0 u2) c2) 
+\to ((subst0 i v0 u1 u2) \to ((csubst0 i v0 c0 c3) \to (or4 (drop n1 O c2 e) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n1 O c2 (CHead e0 (Flat f) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n1 O c2 (CHead e2 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
+(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 
+(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (w: T).(drop n1 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
+v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (H9: (eq T v0 
+v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c0 k0 u1) (CHead c k t)) \to 
+((eq C (CHead c3 k0 u2) c2) \to ((subst0 i t0 u1 u2) \to ((csubst0 i t0 c0 
+c3) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
 (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 
+i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H10: (eq C (CHead c0 k0 u1) 
+(CHead c k t))).(let H11 \def (f_equal C T (\lambda (e0: C).(match e0 in C 
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t0) 
+\Rightarrow t0])) (CHead c0 k0 u1) (CHead c k t) H10) in ((let H12 \def 
+(f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with 
+[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u1) 
+(CHead c k t) H10) in ((let H13 \def (f_equal C C (\lambda (e0: C).(match e0 
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c4 _ 
+_) \Rightarrow c4])) (CHead c0 k0 u1) (CHead c k t) H10) in (eq_ind C c 
+(\lambda (c4: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c3 k0 u2) 
+c2) \to ((subst0 i v u1 u2) \to ((csubst0 i v c4 c3) \to (or4 (drop (s k0 i) 
+O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 
 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop n O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 
-(Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))))))))) (\lambda (H9: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq 
-C (CHead c1 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to 
-((subst0 i t0 u1 u2) \to ((csubst0 i t0 c1 c0) \to (or4 (drop (s k0 i) O c2 
-e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 
-i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))))) (\lambda (H10: (eq C (CHead c1 k0 u1) (CHead c k t))).(let 
-H11 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: 
-C).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead 
-c1 k0 u1) (CHead c k t) H10) in ((let H12 \def (f_equal C K (\lambda (e0: 
-C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | 
-(CHead _ k _) \Rightarrow k])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let 
-H13 \def (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: 
-C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead 
-c1 k0 u1) (CHead c k t) H10) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to 
-((eq T u1 t) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v u1 u2) \to 
-((csubst0 i v c c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (H14: (eq K k0 k)).(eq_ind K 
+k (\lambda (k1: K).((eq T u1 t) \to ((eq C (CHead c3 k1 u2) c2) \to ((subst0 
+i v u1 u2) \to ((csubst0 i v c c3) \to (or4 (drop (s k1 i) O c2 e) (ex3_4 F C 
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (w: T).(drop (s k1 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s k1 i) O c2 (CHead e2 (Flat f) 
 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda 
+T).(\lambda (w: T).(drop (s k1 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda 
 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda 
-(H14: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead 
-c0 k u2) c2) \to ((subst0 i v u1 u2) \to ((csubst0 i v c c0) \to (or4 (drop 
-(s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead 
-e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda 
+(H15: (eq T u1 t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 k u2) c2) 
+\to ((subst0 i v t0 u2) \to ((csubst0 i v c c3) \to (or4 (drop (s k i) O c2 
+e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda 
+(_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k 
 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
@@ -4028,647 +4092,631 @@ C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f)
 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))))))) (\lambda (H15: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq 
-C (CHead c0 k u2) c2) \to ((subst0 i v t u2) \to ((csubst0 i v c c0) \to (or4 
-(drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+e2)))))))))))) (\lambda (H16: (eq C (CHead c3 k u2) c2)).(eq_ind C (CHead c3 
+k u2) (\lambda (c4: C).((subst0 i v t u2) \to ((csubst0 i v c c3) \to (or4 
+(drop (s k i) O c4 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k 
-i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+i) O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
 (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+T).(drop (s k i) O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H16: (eq C (CHead c0 k u2) 
-c2)).(eq_ind C (CHead c0 k u2) (\lambda (c2: C).((subst0 i v t u2) \to 
-((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 
-(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
+k i) O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (H17: (subst0 i v t 
-u2)).(\lambda (H18: (csubst0 i v c c0)).(let H0 \def (eq_ind K k0 (\lambda 
-(k: K).(eq nat (s k i) (S n0))) H8 k H14) in (K_ind (\lambda (k: K).((drop (r 
-k n0) O c e) \to ((eq nat (s k i) (S n0)) \to (or4 (drop (s k i) O (CHead c0 
-k u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
-T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 
-k u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (s k i) O (CHead c0 k u2) (CHead e2 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-k i) O (CHead c0 k u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r 
-(Bind b) n0) O c e)).(\lambda (H19: (eq nat (s (Bind b) i) (S n0))).(let H20 
-\def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: 
-nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H19) in 
-(let H21 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H18 n0 H20) 
-in (let H22 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H17 n0 
-H20) in (eq_ind_r nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead 
-c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
+u2)).(\lambda (H18: (csubst0 i v c c3)).(let H19 \def (eq_ind K k0 (\lambda 
+(k1: K).(eq nat (s k1 i) (S n0))) H8 k H14) in (K_ind (\lambda (k1: K).((drop 
+(r k1 n0) O c e) \to ((eq nat (s k1 i) (S n0)) \to (or4 (drop (s k1 i) O 
+(CHead c3 k1 u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
 (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O 
-(CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) u2) (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) u2) (CHead 
-e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (let H \def (IHn c c0 v H21 e H1) in (or4_ind (drop n0 O c0 e) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
-w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 
-(Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
-v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k1 i) O (CHead 
+c3 k1 u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (s k1 i) O (CHead c3 k1 u2) (CHead e2 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+k1 i) O (CHead c3 k1 u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H20: (drop (r 
+(Bind b) n0) O c e)).(\lambda (H21: (eq nat (s (Bind b) i) (S n0))).(let H22 
+\def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: 
+nat).nat) with [O \Rightarrow i | (S n1) \Rightarrow n1])) (S i) (S n0) H21) 
+in (let H23 \def (eq_ind nat i (\lambda (n1: nat).(csubst0 n1 v c c3)) H18 n0 
+H22) in (let H24 \def (eq_ind nat i (\lambda (n1: nat).(subst0 n1 v t u2)) 
+H17 n0 H22) in (eq_ind_r nat n0 (\lambda (n1: nat).(or4 (drop (s (Bind b) n1) 
+O (CHead c3 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
-b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) 
-u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (H23: (drop n0 O c0 e)).(or4_intro0 (drop (s (Bind 
-b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+(Bind b) n1) O (CHead c3 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n1) O (CHead c3 (Bind 
 b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n1) O (CHead c3 (Bind b) 
 u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H23 u2))) (\lambda (H23: (ex3_4 
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
-w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
+O v e1 e2))))))))) (let H25 \def (IHn c c3 v H23 e H20) in (or4_ind (drop n0 
+O c3 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: 
 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c3 (CHead e0 
 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O 
+c3 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c3 (CHead e2 (Flat f) w))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s 
+(Bind b) n0) O (CHead c3 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e0 (Flat f) w)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
+w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O 
+(CHead c3 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+(Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H26: 
+(drop n0 O c3 e)).(or4_intro0 (drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) 
 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda 
 (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) 
 u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
 f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) 
+T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e2 (Flat f) u)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) 
 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) 
+T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e2 (Flat f) w))))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: 
-F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H24: (eq C e 
-(CHead x1 (Flat x0) x2))).(\lambda (H25: (drop n0 O c0 (CHead x1 (Flat x0) 
-x3))).(\lambda (H26: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) 
-(\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4 
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 
-(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead x1 (Flat x0) x2)) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
-x1 (Flat x0) x2) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind 
+b) n0 c3 e H26 u2))) (\lambda (H26: (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O 
+c3 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
+T).(drop n0 O c3 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (s (Bind 
+b) n0) O (CHead c3 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+(Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c3 (Bind 
 b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-(CHead x1 (Flat x0) x2) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) 
-O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
-x2) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 
-(Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) 
+u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H27: (eq C e (CHead x1 (Flat x0) x2))).(\lambda (H28: (drop 
+n0 O c3 (CHead x1 (Flat x0) x3))).(\lambda (H29: (subst0 O v x2 
+x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) (\lambda (c4: C).(or4 (drop (s (Bind 
+b) n0) O (CHead c3 (Bind b) u2) c4) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c4 (CHead e0 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+(Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C c4 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c3 (Bind 
+b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
+c4 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) 
+u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))))) (or4_intro1 (drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) 
+(CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
+T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(u: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e2 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
+x2) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) 
+u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e0 (Flat f) 
+w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) 
-(drop_drop (Bind b) n0 c0 (CHead x1 (Flat x0) x3) H25 u2) H26)) e H24)))))))) 
-H23)) (\lambda (H23: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda 
+(drop_drop (Bind b) n0 c3 (CHead x1 (Flat x0) x3) H28 u2) H29)) e H27)))))))) 
+H26)) (\lambda (H26: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda 
 (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c3 (CHead e2 
 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
 (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda 
 (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O 
-c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+c3 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+(CHead c3 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+(Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
 T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c3 (Bind 
 b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) 
 u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda 
-(x3: T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H25: (drop 
-n0 O c0 (CHead x2 (Flat x0) x3))).(\lambda (H26: (csubst0 O v x1 
-x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind 
-b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) 
+(x3: T).(\lambda (H27: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H28: (drop 
+n0 O c3 (CHead x2 (Flat x0) x3))).(\lambda (H29: (csubst0 O v x1 
+x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c4: C).(or4 (drop (s (Bind 
+b) n0) O (CHead c3 (Bind b) u2) c4) (ex3_4 F C T T (\lambda (f: F).(\lambda 
+(e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c4 (CHead e0 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: 
+(Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
+T).(eq C c4 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c3 (Bind 
 b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
-c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) 
+c4 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) 
 u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))) (or4_intro2 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) 
+O v e1 e2))))))))) (or4_intro2 (drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) 
 (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 
 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) 
+(w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e0 (Flat f) 
 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 
 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) 
+(u: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e2 (Flat f) 
 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
 x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) 
 u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 
 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) 
+(u: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e2 (Flat f) 
 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) 
-(drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x3) H25 u2) H26)) e H24)))))))) 
-H23)) (\lambda (H23: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
+(drop_drop (Bind b) n0 c3 (CHead x2 (Flat x0) x3) H28 u2) H29)) e H27)))))))) 
+H26)) (\lambda (H26: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) 
 u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: 
+T).(\lambda (w: T).(drop n0 O c3 (CHead e2 (Flat f) w))))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda 
 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
 e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) 
+C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c3 (CHead e2 (Flat f) w))))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: 
+(Bind b) n0) O (CHead c3 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat 
 f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) 
+T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e0 (Flat f) w)))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(CHead c3 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda 
 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
 C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda 
+(Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda 
 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: 
 F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H25: (drop n0 O 
-c0 (CHead x2 (Flat x0) x4))).(\lambda (H26: (subst0 O v x3 x4)).(\lambda 
-(H27: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: 
-C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
+T).(\lambda (H27: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H28: (drop n0 O 
+c3 (CHead x2 (Flat x0) x4))).(\lambda (H29: (subst0 O v x3 x4)).(\lambda 
+(H30: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c4: 
+C).(or4 (drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) c4) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c4 
 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 
+T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e0 
 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c4 (CHead e1 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: 
+(Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
 (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: 
+(u: T).(\lambda (_: T).(eq C c4 (CHead e1 (Flat f) u))))))) (\lambda (f: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda 
+(Bind b) n0) O (CHead c3 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda 
 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 
-(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead x1 (Flat x0) x3)) 
+(drop (s (Bind b) n0) O (CHead c3 (Bind b) u2) (CHead x1 (Flat x0) x3)) 
 (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
 T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f) u)))))) (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O 
-(CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
+(CHead c3 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda 
 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead 
 x1 (Flat x0) x3) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c3 (Bind 
 b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
 (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
 (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
 (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) 
-O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
+O (CHead c3 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
 (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
 (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda 
 (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) 
-O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
+O (CHead c3 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda 
 (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat 
-x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x4) H25 u2) H26 H27)) 
-e H24)))))))))) H23)) H)) i H20))))))) (\lambda (f: F).(\lambda (H1: (drop (r 
-(Flat f) n0) O c e)).(\lambda (H19: (eq nat (s (Flat f) i) (S n0))).(let H20 
-\def (f_equal nat nat (\lambda (e0: nat).e0) i (S n0) H19) in (let H21 \def 
-(eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H18 (S n0) H20) in (let 
-H22 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H17 (S n0) H20) 
-in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead 
-c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O 
-(CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
-(CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) u2) (CHead e2 
-(Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 
-(Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) u2) (CHead 
-e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))) (let H23 \def (H c0 v H21 e H1) in (or4_ind (drop (S n0) O 
-c0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
+x0) x3)) (drop_drop (Bind b) n0 c3 (CHead x2 (Flat x0) x4) H28 u2) H29 H30)) 
+e H27)))))))))) H26)) H25)) i H22))))))) (\lambda (f: F).(\lambda (H20: (drop 
+(r (Flat f) n0) O c e)).(\lambda (H21: (eq nat (s (Flat f) i) (S n0))).(let 
+H22 \def (f_equal nat nat (\lambda (e0: nat).e0) i (S n0) H21) in (let H23 
+\def (eq_ind nat i (\lambda (n1: nat).(csubst0 n1 v c c3)) H18 (S n0) H22) in 
+(let H24 \def (eq_ind nat i (\lambda (n1: nat).(subst0 n1 v t u2)) H17 (S n0) 
+H22) in (eq_ind_r nat (S n0) (\lambda (n1: nat).(or4 (drop (s (Flat f) n1) O 
+(CHead c3 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
+(Flat f) n1) O (CHead c3 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) n1) O (CHead c3 (Flat 
+f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n1) O (CHead c3 
+(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))))) (let H25 \def (H c3 v H23 e H20) in 
+(or4_ind (drop (S n0) O c3 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O c3 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O c3 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S 
+n0) O c3 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c3 
+(Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead 
-e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S 
-n0) O c0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead 
-e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
-(Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) 
+O (CHead c3 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead 
+C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead 
 e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (H24: (drop (S n0) O c0 
-e)).(or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (H26: (drop (S n0) O c3 
+e)).(or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) e) (ex3_4 
 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
 C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
+(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) 
 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
 f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) 
+T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e2 (Flat f0) 
 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat 
 f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead 
+T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead 
 e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H24 u2))) (\lambda (H24: (ex3_4 
-F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
-C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
+O v e1 e2))))))) (drop_drop (Flat f) n0 c3 e H26 u2))) (\lambda (H26: (ex3_4 
+F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq 
+C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e0 (Flat f0) w)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
 w))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead 
 e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
-(w: T).(subst0 O v u w))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat 
+(w: T).(subst0 O v u w))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat 
 f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: 
 T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: 
 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) 
-O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
+O (CHead c3 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda 
 (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
 (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead 
+C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead 
 e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda 
-(x2: T).(\lambda (x3: T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) 
-x2))).(\lambda (H26: (drop (S n0) O c0 (CHead x1 (Flat x0) x3))).(\lambda 
-(H27: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) (\lambda (c: 
-C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
+(x2: T).(\lambda (x3: T).(\lambda (H27: (eq C e (CHead x1 (Flat x0) 
+x2))).(\lambda (H28: (drop (S n0) O c3 (CHead x1 (Flat x0) x3))).(\lambda 
+(H29: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) (\lambda (c4: 
+C).(or4 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) c4) (ex3_4 F C T T 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c4 
 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead 
+T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead 
 e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f0) u)))))) 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c4 (CHead e1 (Flat f0) u)))))) 
 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) 
+(Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e2 (Flat f0) u)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u))))))) 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c4 (CHead e1 (Flat f0) u))))))) 
 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) 
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e2 (Flat f0) 
 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro1 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
+e2))))))))) (or4_intro1 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) 
 (CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 
 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) 
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e0 (Flat f0) 
 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 
 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) 
+(u: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e2 (Flat f0) 
 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
 x2) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: 
 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
 x2) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
+(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) 
 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 
-(Flat x0) x2)) (drop_drop (Flat f) n0 c0 (CHead x1 (Flat x0) x3) H26 u2) 
-H27)) e H25)))))))) H24)) (\lambda (H24: (ex3_4 F C C T (\lambda (f: 
+(Flat x0) x2)) (drop_drop (Flat f) n0 c3 (CHead x1 (Flat x0) x3) H28 u2) 
+H29)) e H27)))))))) H26)) (\lambda (H26: (ex3_4 F C C T (\lambda (f0: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(drop (S n0) O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C 
-C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
+f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(drop (S n0) O c3 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind 
+F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq 
+C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead e2 (Flat f0) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
+T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
+(Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e 
 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
-(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead 
+e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
+(Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda 
+(x2: C).(\lambda (x3: T).(\lambda (H27: (eq C e (CHead x1 (Flat x0) 
+x3))).(\lambda (H28: (drop (S n0) O c3 (CHead x2 (Flat x0) x3))).(\lambda 
+(H29: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c4: 
+C).(or4 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) c4) (ex3_4 F C T T 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c4 
 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead 
+T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead 
 e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda 
 (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) 
-w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
-T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: 
-T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H26: (drop (S 
-n0) O c0 (CHead x2 (Flat x0) x3))).(\lambda (H27: (csubst0 O v x1 
-x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat 
-f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat 
-f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: 
-T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) 
-w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
-T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f0) u)))))) 
+C).(\lambda (_: C).(\lambda (u: T).(eq C c4 (CHead e1 (Flat f0) u)))))) 
 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) 
+(Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e2 (Flat f0) u)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
 v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u))))))) 
+C).(\lambda (u: T).(\lambda (_: T).(eq C c4 (CHead e1 (Flat f0) u))))))) 
 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) 
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e2 (Flat f0) 
 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro2 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
+e2))))))))) (or4_intro2 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) 
 (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
 C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 
 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) 
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e0 (Flat f0) 
 w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 
 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) 
+(u: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e2 (Flat f0) 
 u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
 x3) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
+(e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) 
 x3) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
+(e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) 
 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead 
-x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x3) H26 u2) 
-H27)) e H25)))))))) H24)) (\lambda (H24: (ex4_5 F C C T T (\lambda (f: 
+x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c3 (CHead x2 (Flat x0) x3) H28 u2) 
+H29)) e H27)))))))) H26)) (\lambda (H26: (ex4_5 F C C T T (\lambda (f0: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e 
-(CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) 
+(CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c3 (CHead e2 (Flat f0) 
 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda 
 (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) 
 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: 
+(w: T).(drop (S n0) O c3 (CHead e2 (Flat f0) w))))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O 
 v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
 T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Flat f) (S n0)) O 
-(CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+(CHead c3 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
 C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) 
 (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) 
+(Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e0 (Flat f0) w)))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u 
 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) 
-O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda 
+O (CHead c3 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda 
 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
 C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) 
+(Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e2 (Flat f0) w))))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: 
 F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: 
-T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H26: (drop (S 
-n0) O c0 (CHead x2 (Flat x0) x4))).(\lambda (H27: (subst0 O v x3 
-x4)).(\lambda (H28: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) 
-(\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c) 
+T).(\lambda (H27: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H28: (drop (S 
+n0) O c3 (CHead x2 (Flat x0) x4))).(\lambda (H29: (subst0 O v x3 
+x4)).(\lambda (H30: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) 
+(\lambda (c4: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) c4) 
 (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: 
-T).(eq C c (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
+T).(eq C c4 (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c4 
 (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead 
+C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead 
 e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c4 
 (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 
+C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 
 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Flat f) (S n0)) O 
-(CHead c0 (Flat f) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: 
+(CHead c3 (Flat f) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: 
 F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) 
 x3) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
+(_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) 
 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) 
 x3) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) 
+(e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) 
 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C 
 (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u))))))) (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s 
-(Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) 
+(Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e2 (Flat f0) w))))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: 
 T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C 
 C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u))))))) 
 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) 
+(w: T).(drop (s (Flat f) (S n0)) O (CHead c3 (Flat f) u2) (CHead e2 (Flat f0) 
 w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: 
 T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
 x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 
-c0 (CHead x2 (Flat x0) x4) H26 u2) H27 H28)) e H25)))))))))) H24)) H23)) i 
-H20))))))) k (drop_gen_drop k c e t n0 H1) H0)))) c2 H16)) u1 (sym_eq T u1 t 
-H15))) k0 (sym_eq K k0 k H14))) c1 (sym_eq C c1 c H13))) H12)) H11))) v0 
+c3 (CHead x2 (Flat x0) x4) H28 u2) H29 H30)) e H27)))))))))) H26)) H25)) i 
+H22))))))) k (drop_gen_drop k c e t n0 H1) H19)))) c2 H16)) u1 (sym_eq T u1 t 
+H15))) k0 (sym_eq K k0 k H14))) c0 (sym_eq C c0 c H13))) H12)) H11))) v0 
 (sym_eq T v0 v H9))) (S n0) H8)) H5 H6 H7 H2 H3)))))]) in (H2 (refl_equal nat 
 (S n0)) (refl_equal T v) (refl_equal C (CHead c k t)) (refl_equal C 
 c2)))))))))))) c1)))) n).
@@ -4802,7 +4850,7 @@ T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2:
 C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H4)))))))))) 
 (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda 
 (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq 
-nat i O)).(let H4 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) 
+nat i O)).(let H4 \def (eq_ind nat i (\lambda (n0: nat).(subst0 n0 v0 u1 u2)) 
 H2 O H3) in (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c (Flat f) 
 u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
 T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) 
@@ -4901,78 +4949,79 @@ C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Bind b) u) (CHead e1
 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3: 
 C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3 
 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
+(CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f0) u)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i 
-O)).(let H5 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop 
-O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 
-(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 
-(CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H3 O 
-H4) in (let H6 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H2 
-O H4) in (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda 
-(f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 
-(Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f0) 
+u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f0) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: 
+T).(\lambda (H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n0: 
+nat).((eq nat n0 O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat 
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop O O c3 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C 
-(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) 
-(CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
-(CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 
-(Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C c4 
+(CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u0: T).(drop O O c3 (CHead e1 (Flat f0) u0)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f0) u2))))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop O O c3 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 
+u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H3 O H4) in (let H6 \def 
+(eq_ind nat i (\lambda (n0: nat).(csubst0 n0 v0 c3 c4)) H2 O H4) in 
+(or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C 
+T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
+(CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda 
+(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u) 
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) 
 u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
 (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) 
 u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u)) 
-(drop_refl (CHead c3 (Flat f) u)) H6))))))))))))) k)) (\lambda (k: K).(K_ind 
-(\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall 
-(u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: C).((csubst0 
-i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) 
+T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) 
+u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u) 
+(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) u) (CHead e2 
+(Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 
+e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u)) (drop_refl (CHead c3 
+(Flat f) u)) H6))))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: 
+K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: 
+T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: C).((csubst0 i v0 
+c3 c4) \to ((((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e0 
+(Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda 
+(_: T).(drop O O c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda 
+(_: T).(drop O O c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop 
 O O (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda (f: 
@@ -4995,19 +5044,19 @@ O v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0:
 T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 
 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 
 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 
+(CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: 
+T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u3)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
 T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
 v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) 
+C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f) u4))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda 
+(_: T).(drop O O c3 (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 
 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee in nat return (\lambda 
@@ -5023,283 +5072,199 @@ c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1:
 C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c3 (Bind b) u1) (CHead e1 
 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
 (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 
-(Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Bind 
-b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda 
-(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
-T).(csubst0 O v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: 
-nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 
-i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 
-c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
-v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 
-\def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c3 c4) 
-(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
-T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O 
-v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda 
-(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 
-(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H4 O H5) in 
-(let H7 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 O H5) 
-in (let H8 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H5) 
-in (or4_intro3 (drop O O (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) 
-(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
-(u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda 
-(f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 
-(Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C 
-(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 
-(e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c3 (Flat f) u1) 
-(CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C 
-(CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O 
-O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: 
+(_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 
+(Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Bind 
+b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+(_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: 
+T).(csubst0 O v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: 
+nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 
+i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 
+c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 
+(CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: 
+T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f0) u3)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) 
+(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C c4 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f0) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 (CHead e2 (Flat f0) 
+u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: 
+T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f0) u3))))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 
 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex4_5_intro F C C T T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda 
-(f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: 
-T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: 
-T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) f c3 c4 u1 u2 
-(refl_equal C (CHead c4 (Flat f) u2)) (drop_refl (CHead c3 (Flat f) u1)) H8 
-H7)))))))))))))))) k)) y v c1 c2 H1))) H) e (drop_gen_refl c2 e H0)))))))) 
-(\lambda (n0: nat).(\lambda (IHn: ((\forall (c1: C).(\forall (c2: C).(\forall 
-(v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (or4 
-(drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e0 
-(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O 
-c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 (Flat f) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: 
-C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) 
-O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead 
-e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+(_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat i 
+O)).(let H6 \def (eq_ind nat i (\lambda (n0: nat).((eq nat n0 O) \to (or4 
+(drop O O c3 c4) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda 
+(_: T).(\lambda (u4: T).(eq C c4 (CHead e0 (Flat f0) u4)))))) (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O c3 (CHead e0 
+(Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda 
+(u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f0) u)))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O 
+c3 (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C c4 
+(CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f0) 
+u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: 
+T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 
+e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n0: 
+nat).(csubst0 n0 v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda 
+(n0: nat).(subst0 n0 v0 u1 u2)) H2 O H5) in (or4_intro3 (drop O O (CHead c3 
+(Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda 
+(e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead 
+e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: 
+T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) 
+u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: 
+T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 
+(Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead 
+e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: 
-T).(\lambda (H: (csubst0 (S n0) v (CSort n1) c2)).(\lambda (e: C).(\lambda 
-(_: (drop (S n0) O c2 e)).(csubst0_gen_sort c2 v (S n0) n1 H (or4 (drop (S 
-n0) O (CSort n1) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CSort 
-n1) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CSort n1) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))))))))) (\lambda (c: C).(\lambda (H: 
-((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e: 
-C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C 
-T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) 
+u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) 
+(CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: 
-C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) 
-c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T 
-nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: 
-T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
-nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq 
-nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k 
-t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat 
-(\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) 
-(\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k 
-u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t 
-u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c 
-c3))))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2)) 
+(drop_refl (CHead c3 (Flat f) u1)) H8 H7)))))))))))))))) k)) y v c1 c2 H1))) 
+H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: 
+((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to 
+(\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda 
+(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
+u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind 
+(\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to 
+(\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C 
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) 
+u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
-(H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) 
-(\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: 
-T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: 
-T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: 
-nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
-v t u2))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: 
+nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort 
+n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort 
+c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda 
+(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
+(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
 T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 
-(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat 
+f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
+T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
 f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) 
+T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1))))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
-(x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda 
-(H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def 
-(eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H1 (CHead c k x0) H4) in 
-((match k in K return (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop 
-(r k0 n0) O c e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
-(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
-(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq 
-C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead 
-e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))))) with [(Bind b) \Rightarrow (\lambda (H7: (eq nat (S n0) 
-(s (Bind b) x1))).(\lambda (H8: (drop (r (Bind b) n0) O c e)).(let H9 \def 
-(f_equal nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: 
-nat).nat) with [O \Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x1) H7) 
-in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 n0 
-H9) in (or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) 
+(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S 
+n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O 
+c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) 
 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
 T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
 C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
-O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
-C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
+e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda 
+(v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: 
+C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: 
+T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: 
+nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j 
+v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k 
+j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda 
+(c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: 
+T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: 
+T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
+(u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
+T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S 
+n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e 
-H8 t)))))) | (Flat f) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Flat f) 
-x1))).(\lambda (H8: (drop (r (Flat f) n0) O c e)).(let H9 \def (f_equal nat 
-nat (\lambda (e0: nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 
-(\lambda (n: nat).(subst0 n v t x0)) H5 (S n0) H9) in (or4_intro0 (drop (S 
-n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead 
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
-(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))) (drop_drop (Flat f) n0 c e H8 t))))))]) H3 (drop_gen_drop k 
-c e x0 n0 H6)))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda 
-(j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C 
-c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c 
-c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s 
-k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) 
-(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O 
-(CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: 
-nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 
-k t))).(\lambda (H5: (csubst0 x1 v c x0)).(let H6 \def (eq_ind C c2 (\lambda 
-(c: C).(drop (S n0) O c e)) H1 (CHead x0 k t) H4) in ((match k in K return 
-(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O x0 e) \to 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat 
+(\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: 
+T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: 
+nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: 
+nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 
+(CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 
+(drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
+(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
+(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda 
+(x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2 
+(CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2 
+(\lambda (c0: C).(drop (S n0) O c0 e)) H1 (CHead c k x0) H4) in (K_ind 
+(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to 
 (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
 (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
 (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
@@ -5315,139 +5280,154 @@ C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2)))))))
 (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda 
 (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
 T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) with [(Bind 
-b) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: 
-(drop (r (Bind b) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0: 
-nat).(match e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow n0 | 
-(S n) \Rightarrow n])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1 
-(\lambda (n: nat).(csubst0 n v c x0)) H5 n0 H9) in (let H11 \def (IHn c x0 v 
-H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O 
-c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: 
+B).(\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r 
+(Bind b) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match 
+e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow n0 | (S n1) 
+\Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1 
+(\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H9) in (or4_intro0 (drop (S n0) 
+O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead 
+e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+(CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))) (drop_drop (Bind b) n0 c e H8 t))))))) (\lambda (f: 
+F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r 
+(Flat f) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S 
+n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: nat).(subst0 n1 v 
+t x0)) H5 (S n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop 
+(Flat f) n0 c e H8 t))))))) k H3 (drop_gen_drop k c e x0 n0 H6)))))))) H2)) 
+(\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) 
+(s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) 
+(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))).(ex3_2_ind C nat 
+(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: 
+nat).(csubst0 j v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k 
+t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 
 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
 (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda 
 (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) 
+e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) 
+(s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 
+v c x0)).(let H6 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 
+(CHead x0 k t) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to 
+((drop (r k0 n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C 
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop n0 O c e)).(or4_intro0 
-(drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H12 t))) (\lambda 
-(H12: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 
-(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O 
-c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Bind 
-b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: 
-T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H14: (drop n0 O 
-c (CHead x3 (Flat x2) x4))).(\lambda (H15: (subst0 O v x4 x5)).(eq_ind_r C 
-(CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind 
-b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind b) t) (CHead x3 (Flat 
-x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq 
+C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead 
+e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H7: (eq nat (S n0) (s (Bind b) 
+x1))).(\lambda (H8: (drop (r (Bind b) n0) O x0 e)).(let H9 \def (f_equal nat 
+nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O 
+\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x1) H7) in (let H10 \def 
+(eq_ind_r nat x1 (\lambda (n1: nat).(csubst0 n1 v c x0)) H5 n0 H9) in (let 
+H11 \def (IHn c x0 v H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u2))))))) (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
-(CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead 
-x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 (Flat x2) x4) H14 t) 
-H15)) e H13)))))))) H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C 
-C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
+(H12: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop 
+(Bind b) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C 
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) 
 (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
@@ -5465,10 +5445,10 @@ C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda 
-(x4: C).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) 
-x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) x5))).(\lambda (H15: 
-(csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: 
-C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: 
+(x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) 
+x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) x4))).(\lambda (H15: 
+(subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 
+(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat 
 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda 
@@ -5484,24 +5464,93 @@ C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
 (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Bind 
-b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind 
+b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 
 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) 
+C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) 
+T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) 
 u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) 
+x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c 
+(CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
+(x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H13: (eq 
+C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) 
+x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) 
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O 
+(CHead c (Bind b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat 
+x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat 
+f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
 (ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
 C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
@@ -5584,11 +5633,11 @@ f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
 x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0 
-c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11)))))) | 
-(Flat f) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda 
-(H8: (drop (r (Flat f) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda 
-(e0: nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: 
-nat).(csubst0 n v c x0)) H5 (S n0) H9) in (let H11 \def (H x0 v H10 e H8) in 
+c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11))))))) 
+(\lambda (f: F).(\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: 
+(drop (r (Flat f) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0: 
+nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n1: 
+nat).(csubst0 n1 v c x0)) H5 (S n0) H9) in (let H11 \def (H x0 v H10 e H8) in 
 (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
 C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
@@ -5638,10 +5687,10 @@ n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_:
 F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
 (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e 
-H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
+H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
+(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
 (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda 
 (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
@@ -5708,10 +5757,10 @@ T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0)
 u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
 T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) 
 (drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) 
-H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda 
-(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
+H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead 
-e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
 f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
@@ -5778,58 +5827,58 @@ n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_:
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
 x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Flat f) n0 c 
 (CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) 
-O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) 
+n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) 
+u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
+u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
+(Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
+(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda 
+(x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4 
+(Flat x2) x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) 
+x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3 
+x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O 
+(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) 
 (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
 n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
 (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
+T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead 
 e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
 C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda 
-(x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) 
-x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x5))).(\lambda 
-(H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C 
-(CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat 
-f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 
-(CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat 
-f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 (Flat 
-x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
+O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 
+(Flat x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
 T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0) 
 u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
@@ -5855,12 +5904,12 @@ C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_:
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
 O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) 
 (drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e 
-H13)))))))))) H12)) H11))))))]) H3 (drop_gen_drop k x0 e t n0 H6)))))))) H2)) 
-(\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: 
-nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda 
-(_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: 
-C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: 
-C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: 
+H13)))))))))) H12)) H11))))))) k H3 (drop_gen_drop k x0 e t n0 H6)))))))) 
+H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda 
+(j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: 
+C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda 
+(_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: 
+C).(\lambda (j: nat).(csubst0 j v c c3)))))).(ex4_3_ind T C nat (\lambda (_: 
 T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: 
 T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda 
 (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: 
@@ -5873,168 +5922,98 @@ k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
 T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda 
-(x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2 
-(CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 
-v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H1 
-(CHead x1 k x0) H4) in ((match k in K return (\lambda (k0: K).((eq nat (S n0) 
-(s k0 x2)) \to ((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 
-t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) with [(Bind b) \Rightarrow 
-(\lambda (H8: (eq nat (S n0) (s (Bind b) x2))).(\lambda (H9: (drop (r (Bind 
-b) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0: nat).(match e0 
-in nat return (\lambda (_: nat).nat) with [O \Rightarrow n0 | (S n) 
-\Rightarrow n])) (S n0) (S x2) H8) in (let H11 \def (eq_ind_r nat x2 (\lambda 
-(n: nat).(csubst0 n v c x1)) H6 n0 H10) in (let H12 \def (eq_ind_r nat x2 
-(\lambda (n: nat).(subst0 n v t x0)) H5 n0 H10) in (let H13 \def (IHn c x1 v 
-H11 e H9) in (or4_ind (drop n0 O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O 
-c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 
-(Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda 
-(u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda 
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda 
-(f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 
-(Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) 
-(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
-u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
+(ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (H14: (drop n0 O c e)).(or4_intro0 
-(drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda 
+(x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2 
+(CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 
+v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(drop (S n0) O c0 e)) H1 
+(CHead x1 k x0) H4) in (K_ind (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to 
+((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C 
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
+u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
 C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H14 t))) (\lambda 
-(H14: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 
-(Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O 
-c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Bind 
-b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: 
-T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H16: (drop n0 O 
-c (CHead x4 (Flat x3) x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C 
-(CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind 
-b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: 
-F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat 
-f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 
-e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 (Flat 
-x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) 
-(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda 
+(f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq 
+C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda 
+(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead 
+e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
+F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
+O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H8: (eq nat (S n0) (s (Bind b) 
+x2))).(\lambda (H9: (drop (r (Bind b) n0) O x1 e)).(let H10 \def (f_equal nat 
+nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O 
+\Rightarrow n0 | (S n1) \Rightarrow n1])) (S n0) (S x2) H8) in (let H11 \def 
+(eq_ind_r nat x2 (\lambda (n1: nat).(csubst0 n1 v c x1)) H6 n0 H10) in (let 
+H12 \def (eq_ind_r nat x2 (\lambda (n1: nat).(subst0 n1 v t x0)) H5 n0 H10) 
+in (let H13 \def (IHn c x1 v H11 e H9) in (or4_ind (drop n0 O c e) (ex3_4 F C 
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: 
 F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
 (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u)))))) (\lambda (f: 
-F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
-(Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
-(\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u2))))))) (\lambda 
-(f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
-T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C 
-T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
-(CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda 
-(e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead 
-x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 (Flat x3) x5) H16 t) 
-H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
 f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C 
-C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
-(CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
+(H14: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop 
+(Bind b) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
+f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C 
+T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) 
 (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
@@ -6052,10 +6031,10 @@ C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
 (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda 
-(x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) 
-x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x6))).(\lambda (H17: 
-(csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0: 
-C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: 
+(x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) 
+x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H17: 
+(subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 
+(drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: 
 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat 
 f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
 T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda 
@@ -6071,24 +6050,93 @@ C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c
 (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Bind 
-b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0 
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind 
+b) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 
 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda 
 (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) 
 (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v 
 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u)))))) 
+C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
 O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
 (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
 C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) 
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) 
 u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
 T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) 
 u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
+(ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) 
+x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c 
+(CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: 
+(ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
+T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 
+(Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) 
+(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: 
+T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 
+(CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
+(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq 
+C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) 
+x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) 
+(\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T 
+(\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
+(CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) 
+(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
+O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda 
+(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C 
+C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: 
+F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 
+O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda 
+(_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O 
+(CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat 
+x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat 
+f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
+(Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 
+(Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat 
+f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
 (ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
 C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u)))))) 
 (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) 
@@ -6171,13 +6219,13 @@ f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
 T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
 x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0 
-c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13))))))) | 
-(Flat f) \Rightarrow (\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda 
-(H9: (drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda 
-(e0: nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: 
-nat).(csubst0 n v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2 
-(\lambda (n: nat).(subst0 n v t x0)) H5 (S n0) H10) in (let H13 \def (H x1 v 
-H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: 
+c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13)))))))) 
+(\lambda (f: F).(\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H9: 
+(drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0: 
+nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n1: 
+nat).(csubst0 n1 v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2 
+(\lambda (n1: nat).(subst0 n1 v t x0)) H5 (S n0) H10) in (let H13 \def (H x1 
+v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: 
 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
 T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
@@ -6226,10 +6274,10 @@ C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2)))))))
 (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
 (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
 (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop 
-(Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: 
+(Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f0: 
 F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
-f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 
+f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
+T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C 
 T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C 
 e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda 
@@ -6296,10 +6344,10 @@ C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t)
 (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
 T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead 
 x4 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H16 t) 
-H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f: 
+H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f0: 
 F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
-f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
-T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: 
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C 
 C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e 
 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
@@ -6366,83 +6414,83 @@ C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
 v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop 
 (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) 
-(\lambda (H14: (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) 
-(\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
-(_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda 
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: 
-T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda 
-(f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq 
-C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda 
-(_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: 
-F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat 
+(\lambda (H14: (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) 
+u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda 
+(_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C 
+C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 
+T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: 
+F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
+n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) 
+(ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda 
+(u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
+(CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: 
+T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: 
+F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat 
+f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: 
+T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda 
+(_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 
+e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) 
+(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) 
+(\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda 
+(u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda 
+(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda 
+(x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: 
+T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop (S 
+n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda 
+(H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: 
+C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: 
+F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat 
 f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
 T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
 (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
+C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: 
 F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c 
 (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: 
 C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T 
 (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: 
+(u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: 
 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
 (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
 C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
 (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
-(_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda 
-(x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x5 
-(Flat x3) x7))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) 
-x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18: (csubst0 O v x4 
-x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: C).(or4 (drop (S n0) O 
-(CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
-C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) 
-(\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: 
-F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
-(ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: 
-T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: 
-C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead 
-e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: 
-C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: 
-F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 
-(CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
-(CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x5 
-(Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: 
-T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 (Flat f0) 
-u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: 
-T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda 
-(_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 
-u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
-C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u)))))) 
-(\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S 
-n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: 
-F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) 
-(ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda 
-(_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) 
-u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) 
-u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: 
-T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: 
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) 
-(ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
+(_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat 
+f) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: 
+C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 
+(Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: 
+T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) 
+u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: 
+T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 
+(Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda 
+(u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O 
+v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: 
 C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 
 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: 
 C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) 
 (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: 
 C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: 
 F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 
-O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) 
-(drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e 
-H15)))))))))) H14)) H13)))))))]) H3 (drop_gen_drop k x1 e x0 n0 H7)))))))))) 
+O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: 
+C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat 
+x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c 
+(Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: 
+C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) 
+(\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 
+(_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat 
+x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e 
+H15)))))))))) H14)) H13)))))))) k H3 (drop_gen_drop k x1 e x0 n0 H7)))))))))) 
 H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n).