+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "Basic-1/csubst0/props.ma".
-
-include "Basic-1/csubst0/fwd.ma".
-
-include "Basic-1/clear/fwd.ma".
-
-theorem csubst0_clear_O:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to
-(\forall (c: C).((clear c1 c) \to (clear c2 c))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
-T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2
-c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H:
-(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n)
-c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda
-(H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0:
-C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t:
-T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t)
-c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2
-T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (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 O (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 O (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))))) (clear c2 c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j:
-nat).(eq nat O (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 O (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))) (clear c2 c0) (\lambda (x0:
-T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C
-c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k
-x0) (\lambda (c3: C).(clear c3 c0)) (K_ind (\lambda (k0: K).((clear (CHead c
-k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead c k0 x0) c0))))
-(\lambda (b: B).(\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H7:
-(eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O (\lambda (ee:
-nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
-| (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind
-b) x0) c0) H8))))) (\lambda (f: F).(\lambda (H6: (clear (CHead c (Flat f) t)
-c0)).(\lambda (H7: (eq nat O (s (Flat f) x1))).(let H8 \def (eq_ind_r nat x1
-(\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0
-(clear_gen_flat f c c0 t H6) f x0))))) k H1 H3) c2 H4)))))) H2)) (\lambda
-(H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (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 O (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))) (clear c2 c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq
-nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5:
-(csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(clear c3
-c0)) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0
-x1)) \to (clear (CHead x0 k0 t) c0)))) (\lambda (b: B).(\lambda (_: (clear
-(CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8
-\def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7)
-in (False_ind (clear (CHead x0 (Bind b) t) c0) H8))))) (\lambda (f:
-F).(\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s
-(Flat f) x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c
-x0)) H5 O H7) in (clear_flat x0 c0 (H x0 v H8 c0 (clear_gen_flat f c c0 t
-H6)) f t))))) k H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda
-(_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (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 O (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)))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2:
-nat).(\lambda (H3: (eq nat O (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)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(clear c3 c0)) (K_ind
-(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x2)) \to
-(clear (CHead x1 k0 x0) c0)))) (\lambda (b: B).(\lambda (_: (clear (CHead c
-(Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x2))).(let H9 \def
-(eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H8)
-in (False_ind (clear (CHead x1 (Bind b) x0) c0) H9))))) (\lambda (f:
-F).(\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat O (s
-(Flat f) x2))).(let H9 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c
-x1)) H6 O H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v
-t x0)) H5 O H8) in (clear_flat x1 c0 (H x1 v H9 c0 (clear_gen_flat f c c0 t
-H7)) f x0)))))) k H1 H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v O
-H0))))))))))) c1).
-(* COMMENTS
-Initial nodes: 1582
-END *)
-
-theorem csubst0_clear_O_back:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to
-(\forall (c: C).((clear c2 c) \to (clear c1 c))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
-T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c
-c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H:
-(csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2
-c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c:
-C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to
-(\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O
-v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(or3_ind
-(ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (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 O (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 O (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))))) (clear (CHead c k t) c0) (\lambda (H2: (ex3_2 T
-nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (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 O (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))) (clear
-(CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O
-(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 (c3: C).(clear c3 c0)) H1 (CHead c
-k x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead
-c k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7:
-(eq nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead c (Bind b) x0)
-c0)).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
-I (S x1) H7) in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda
-(f: F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead
-c (Flat f) x0) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n
-v t x0)) H5 O H7) in (clear_flat c c0 (clear_gen_flat f c c0 x0 H8) f t)))))
-k H3 H6))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j:
-nat).(eq nat O (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 O (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))) (clear (CHead c k t) c0)
-(\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (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 (c3: C).(clear c3 c0)) H1 (CHead x0 k
-t) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead x0
-k0 t) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H7: (eq
-nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let
-H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return (\lambda (_:
-nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7)
-in (False_ind (clear (CHead c (Bind b) t) c0) H9))))) (\lambda (f:
-F).(\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead x0
-(Flat f) t) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v
-c x0)) H5 O H7) in (clear_flat c c0 (H x0 v H9 c0 (clear_gen_flat f x0 c0 t
-H8)) f t))))) k H3 H6))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (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 O (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)))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda
-(x2: nat).(\lambda (H3: (eq nat O (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 (c3: C).(clear c3 c0)) H1 (CHead x1 k
-x0) H4) in (K_ind (\lambda (k0: K).((eq nat O (s k0 x2)) \to ((clear (CHead
-x1 k0 x0) c0) \to (clear (CHead c k0 t) c0)))) (\lambda (b: B).(\lambda (H8:
-(eq nat O (s (Bind b) x2))).(\lambda (_: (clear (CHead x1 (Bind b) x0)
-c0)).(let H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
-(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
-I (S x2) H8) in (False_ind (clear (CHead c (Bind b) t) c0) H10))))) (\lambda
-(f: F).(\lambda (H8: (eq nat O (s (Flat f) x2))).(\lambda (H9: (clear (CHead
-x1 (Flat f) x0) c0)).(let H10 \def (eq_ind_r nat x2 (\lambda (n:
-nat).(csubst0 n v c x1)) H6 O H8) in (let H11 \def (eq_ind_r nat x2 (\lambda
-(n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat c c0 (H x1 v H10 c0
-(clear_gen_flat f x1 c0 x0 H9)) f t)))))) k H3 H7))))))))) H2))
-(csubst0_gen_head k c c2 t v O H0))))))))))) c1).
-(* COMMENTS
-Initial nodes: 1606
-END *)
-
-theorem csubst0_clear_S:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0
-(S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4
-B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq
-C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v e1 e2))))))))))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v:
-T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c
-c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda
-(i: nat).(\lambda (H: (csubst0 (S i) v (CSort n) c2)).(\lambda (c:
-C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v (S i) n H (or4
-(clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))))))))
-(\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).(\forall (i:
-nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4
-(clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda
-(v: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c k t)
-c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2
-T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S 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)))) (ex3_2 C nat (\lambda (_:
-C).(\lambda (j: nat).(eq nat (S 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)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
-nat).(eq nat (S 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))))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
-T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-i v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j:
-nat).(eq nat (S 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))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S 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))) (or4 (clear c2 c0) (ex3_4 B C T
-T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1:
-nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k
-x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda
-(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat
-(S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T (\lambda
-(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead c k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c k0 x0) (CHead e2 (Bind b)
-u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))
-(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7:
-(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e:
-nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S
-n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1
-(\lambda (n: nat).(subst0 n v t x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind
-b) t) (\lambda (c3: C).(or4 (clear (CHead c (Bind b) x0) c3) (ex3_4 B C T T
-(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3
-(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b)
-x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda
-(b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq
-C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead
-e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0
-i v e1 e2))))))))) (or4_intro1 (clear (CHead c (Bind b) x0) (CHead c (Bind b)
-t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b)
-x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b)
-t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1
-(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2
-(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1))))))
-(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) b c t x0
-(refl_equal C (CHead c (Bind b) t)) (clear_bind b c x0) H9)) c0
-(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear
-(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let
-H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in
-(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i)
-H8) in (or4_intro0 (clear (CHead c (Flat f) x0) c0) (ex3_4 B C T T (\lambda
-(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead c (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Flat f) x0) (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))
-(clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))))) k H1 H3) c2
-H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j:
-nat).(eq nat (S 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 (S 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))) (or4 (clear c2 c0) (ex3_4 B C
-T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1:
-nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0
-k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda
-(c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))))) (K_ind (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat
-(S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4 B C T T (\lambda
-(b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e
-(Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b)
-u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))
-(\lambda (b: B).(\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7:
-(eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e:
-nat).(match e in nat return (\lambda (_: nat).nat) with [O \Rightarrow i | (S
-n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1
-(\lambda (n: nat).(csubst0 n v c x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind
-b) t) (\lambda (c3: C).(or4 (clear (CHead x0 (Bind b) t) c3) (ex3_4 B C T T
-(\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3
-(CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b)
-t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3
-(CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2
-(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))))) (or4_intro2 (clear (CHead x0 (Bind b) t) (CHead c (Bind b) t))
-(ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_:
-T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b)
-t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b)
-t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1
-(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2
-(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u))))))
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear
-(CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) b c x0 t
-(refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t) H9)) c0
-(clear_gen_bind b c c0 t H6))))))) (\lambda (f: F).(\lambda (H6: (clear
-(CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let
-H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in
-(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 (S i)
-H8) in (let H10 \def (H x0 v i H9 c0 (clear_gen_flat f c c0 t H6)) in
-(or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
-T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (H11: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat
-f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f)
-t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x0 c0 H11 f t)))
-(\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x0 (Flat f) t)
-c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e
-(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x2: B).(\lambda (x3:
-C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind
-x2) x4))).(\lambda (H13: (clear x0 (CHead x3 (Bind x2) x5))).(\lambda (H14:
-(subst0 i v x4 x5)).(or4_intro1 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T
-T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f)
-t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f)
-t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) x2 x3 x4 x5 H12 (clear_flat x0
-(CHead x3 (Bind x2) x5) H13 f t) H14))))))))) H11)) (\lambda (H11: (ex3_4 B C
-C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1
-e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T
-T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f)
-t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
-T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H13: (clear x0
-(CHead x4 (Bind x2) x5))).(\lambda (H14: (csubst0 i v x3 x4)).(or4_intro2
-(clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C
-C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x4 (Bind x2) x5) H13 f t)
-H14))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
-T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5:
-T).(\lambda (x6: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2)
-x5))).(\lambda (H13: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H14:
-(subst0 i v x5 x6)).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro3 (clear
-(CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H12 (clear_flat x0
-(CHead x4 (Bind x2) x6) H13 f t) H14 H15))))))))))) H11)) H10))))))) k H1 H3)
-c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
-C).(\lambda (j: nat).(eq nat (S 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)))))).(ex4_3_ind T C nat (\lambda (_:
-T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S 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)))) (or4 (clear c2
-c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
-C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind
-b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda
-(x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (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)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4
-(clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3
-(CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2)))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda
-(e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (K_ind
-(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x2)) \to
-(or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) (\lambda (b: B).(\lambda
-(H7: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat (S i) (s (Bind b)
-x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return
-(\lambda (_: nat).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i)
-(S x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c
-x1)) H6 i H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v
-t x0)) H5 i H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4
-(clear (CHead x1 (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda
-(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1))))))
-(\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
-T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3
-(CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1)))))))
-(\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro3
-(clear (CHead x1 (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda
-(b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind
-b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0)
-u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind
-b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0)
-u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))
-(ex4_5_intro B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1
-(Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2
-(Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))) b c x1 t x0 (refl_equal C (CHead c (Bind b) t)) (clear_bind b x1 x0)
-H11 H10)) c0 (clear_gen_bind b c c0 t H7)))))))) (\lambda (f: F).(\lambda
-(H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f)
-x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f)
-x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c
-x1)) H6 (S i) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0
-n v t x0)) H5 (S i) H9) in (let H12 \def (H x1 v i H10 c0 (clear_gen_flat f c
-c0 t H7)) in (or4_ind (clear x1 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda
-(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
-T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (H13: (clear x1 c0)).(or4_intro0 (clear (CHead x1 (Flat
-f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f)
-x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind
-b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u:
-T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))
-(ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x1 c0 H13 f x0)))
-(\lambda (H13: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind
-b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1
-(CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x1 (Flat f) x0)
-c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e
-(Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1:
-C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4:
-C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind
-x3) x5))).(\lambda (H15: (clear x1 (CHead x4 (Bind x3) x6))).(\lambda (H16:
-(subst0 i v x5 x6)).(or4_intro1 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C
-T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f)
-x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b:
-B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f)
-x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2))))) x3 x4 x5 x6 H14 (clear_flat x1
-(CHead x4 (Bind x3) x6) H15 f x0) H16))))))))) H13)) (\lambda (H13: (ex3_4 B
-C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C
-c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_:
-B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1
-e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind
-b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C
-T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_:
-T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2))))))
-(\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v
-u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f)
-x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b:
-B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0
-(CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
-T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H15: (clear x1
-(CHead x5 (Bind x3) x6))).(\lambda (H16: (csubst0 i v x4 x5)).(or4_intro2
-(clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C
-C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2))))) x3 x4 x5 x6 H14 (clear_flat x1 (CHead x5 (Bind x3) x6) H15 f x0)
-H16))))))))) H13)) (\lambda (H13: (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0
-i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T
-(\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda
-(_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2
-(Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda
-(u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda
-(e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b:
-B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind
-b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2:
-T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_:
-B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))
-(ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u:
-T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_:
-C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2
-(Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda
-(_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda
-(e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1
-(Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda
-(_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b)
-u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1:
-C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1
-e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6:
-T).(\lambda (x7: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3)
-x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H16:
-(subst0 i v x6 x7)).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro3 (clear
-(CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1))))))
-(\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C
-T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0
-(CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2:
-C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u))))))
-(\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i
-v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_:
-C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1)))))))
-(\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda
-(u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda
-(_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2:
-T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2:
-C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C
-C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1:
-T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b:
-B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear
-(CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda
-(_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1
-u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_:
-T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H14 (clear_flat x1
-(CHead x5 (Bind x3) x7) H15 f x0) H16 H17))))))))))) H13)) H12)))))))) k H1
-H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v (S i) H0)))))))))))) c1).
-(* COMMENTS
-Initial nodes: 14968
-END *)
-
-theorem csubst0_clear_trans:
- \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0
-i v c1 c2) \to (\forall (e2: C).((clear c2 e2) \to (or (clear c1 e2) (ex2 C
-(\lambda (e1: C).(csubst0 i v e1 e2)) (\lambda (e1: C).(clear c1 e1))))))))))
-\def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H: (csubst0 i v c1 c2)).(csubst0_ind (\lambda (n: nat).(\lambda (t:
-T).(\lambda (c: C).(\lambda (c0: C).(\forall (e2: C).((clear c0 e2) \to (or
-(clear c e2) (ex2 C (\lambda (e1: C).(csubst0 n t e1 e2)) (\lambda (e1:
-C).(clear c e1)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0:
-T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0 i0 v0 u1
-u2)).(\lambda (c: C).(\lambda (e2: C).(\lambda (H1: (clear (CHead c k u2)
-e2)).(K_ind (\lambda (k0: K).((clear (CHead c k0 u2) e2) \to (or (clear
-(CHead c k0 u1) e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2))
-(\lambda (e1: C).(clear (CHead c k0 u1) e1)))))) (\lambda (b: B).(\lambda
-(H2: (clear (CHead c (Bind b) u2) e2)).(eq_ind_r C (CHead c (Bind b) u2)
-(\lambda (c0: C).(or (clear (CHead c (Bind b) u1) c0) (ex2 C (\lambda (e1:
-C).(csubst0 (s (Bind b) i0) v0 e1 c0)) (\lambda (e1: C).(clear (CHead c (Bind
-b) u1) e1))))) (or_intror (clear (CHead c (Bind b) u1) (CHead c (Bind b) u2))
-(ex2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2)))
-(\lambda (e1: C).(clear (CHead c (Bind b) u1) e1))) (ex_intro2 C (\lambda
-(e1: C).(csubst0 (S i0) v0 e1 (CHead c (Bind b) u2))) (\lambda (e1: C).(clear
-(CHead c (Bind b) u1) e1)) (CHead c (Bind b) u1) (csubst0_snd_bind b i0 v0 u1
-u2 H0 c) (clear_bind b c u1))) e2 (clear_gen_bind b c e2 u2 H2)))) (\lambda
-(f: F).(\lambda (H2: (clear (CHead c (Flat f) u2) e2)).(or_introl (clear
-(CHead c (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
-(\lambda (e1: C).(clear (CHead c (Flat f) u1) e1))) (clear_flat c e2
-(clear_gen_flat f c e2 u2 H2) f u1)))) k H1)))))))))) (\lambda (k:
-K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0:
-T).(\lambda (H0: (csubst0 i0 v0 c3 c4)).(\lambda (H1: ((\forall (e2:
-C).((clear c4 e2) \to (or (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0
-v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))))))).(\lambda (u: T).(\lambda
-(e2: C).(\lambda (H2: (clear (CHead c4 k u) e2)).(K_ind (\lambda (k0:
-K).((clear (CHead c4 k0 u) e2) \to (or (clear (CHead c3 k0 u) e2) (ex2 C
-(\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1: C).(clear (CHead
-c3 k0 u) e1)))))) (\lambda (b: B).(\lambda (H3: (clear (CHead c4 (Bind b) u)
-e2)).(eq_ind_r C (CHead c4 (Bind b) u) (\lambda (c: C).(or (clear (CHead c3
-(Bind b) u) c) (ex2 C (\lambda (e1: C).(csubst0 (s (Bind b) i0) v0 e1 c))
-(\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1))))) (or_intror (clear
-(CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex2 C (\lambda (e1: C).(csubst0
-(S i0) v0 e1 (CHead c4 (Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind
-b) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4
-(Bind b) u))) (\lambda (e1: C).(clear (CHead c3 (Bind b) u) e1)) (CHead c3
-(Bind b) u) (csubst0_fst_bind b i0 c3 c4 v0 H0 u) (clear_bind b c3 u))) e2
-(clear_gen_bind b c4 e2 u H3)))) (\lambda (f: F).(\lambda (H3: (clear (CHead
-c4 (Flat f) u) e2)).(let H_x \def (H1 e2 (clear_gen_flat f c4 e2 u H3)) in
-(let H4 \def H_x in (or_ind (clear c3 e2) (ex2 C (\lambda (e1: C).(csubst0 i0
-v0 e1 e2)) (\lambda (e1: C).(clear c3 e1))) (or (clear (CHead c3 (Flat f) u)
-e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear
-(CHead c3 (Flat f) u) e1)))) (\lambda (H5: (clear c3 e2)).(or_introl (clear
-(CHead c3 (Flat f) u) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
-(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1))) (clear_flat c3 e2 H5 f
-u))) (\lambda (H5: (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda
-(e1: C).(clear c3 e1)))).(ex2_ind C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
-(\lambda (e1: C).(clear c3 e1)) (or (clear (CHead c3 (Flat f) u) e2) (ex2 C
-(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3
-(Flat f) u) e1)))) (\lambda (x: C).(\lambda (H6: (csubst0 i0 v0 x
-e2)).(\lambda (H7: (clear c3 x)).(or_intror (clear (CHead c3 (Flat f) u) e2)
-(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead
-c3 (Flat f) u) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2))
-(\lambda (e1: C).(clear (CHead c3 (Flat f) u) e1)) x H6 (clear_flat c3 x H7 f
-u)))))) H5)) H4))))) k H2))))))))))) (\lambda (k: K).(\lambda (i0:
-nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (subst0
-i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H1: (csubst0 i0 v0
-c3 c4)).(\lambda (H2: ((\forall (e2: C).((clear c4 e2) \to (or (clear c3 e2)
-(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3
-e1)))))))).(\lambda (e2: C).(\lambda (H3: (clear (CHead c4 k u2) e2)).(K_ind
-(\lambda (k0: K).((clear (CHead c4 k0 u2) e2) \to (or (clear (CHead c3 k0 u1)
-e2) (ex2 C (\lambda (e1: C).(csubst0 (s k0 i0) v0 e1 e2)) (\lambda (e1:
-C).(clear (CHead c3 k0 u1) e1)))))) (\lambda (b: B).(\lambda (H4: (clear
-(CHead c4 (Bind b) u2) e2)).(eq_ind_r C (CHead c4 (Bind b) u2) (\lambda (c:
-C).(or (clear (CHead c3 (Bind b) u1) c) (ex2 C (\lambda (e1: C).(csubst0 (s
-(Bind b) i0) v0 e1 c)) (\lambda (e1: C).(clear (CHead c3 (Bind b) u1) e1)))))
-(or_intror (clear (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex2 C
-(\lambda (e1: C).(csubst0 (S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1:
-C).(clear (CHead c3 (Bind b) u1) e1))) (ex_intro2 C (\lambda (e1: C).(csubst0
-(S i0) v0 e1 (CHead c4 (Bind b) u2))) (\lambda (e1: C).(clear (CHead c3 (Bind
-b) u1) e1)) (CHead c3 (Bind b) u1) (csubst0_both_bind b i0 v0 u1 u2 H0 c3 c4
-H1) (clear_bind b c3 u1))) e2 (clear_gen_bind b c4 e2 u2 H4)))) (\lambda (f:
-F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) e2)).(let H_x \def (H2 e2
-(clear_gen_flat f c4 e2 u2 H4)) in (let H5 \def H_x in (or_ind (clear c3 e2)
-(ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3
-e1))) (or (clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0
-i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda
-(H6: (clear c3 e2)).(or_introl (clear (CHead c3 (Flat f) u1) e2) (ex2 C
-(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3
-(Flat f) u1) e1))) (clear_flat c3 e2 H6 f u1))) (\lambda (H6: (ex2 C (\lambda
-(e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)))).(ex2_ind C
-(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear c3 e1)) (or
-(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1
-e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1)))) (\lambda (x:
-C).(\lambda (H7: (csubst0 i0 v0 x e2)).(\lambda (H8: (clear c3 x)).(or_intror
-(clear (CHead c3 (Flat f) u1) e2) (ex2 C (\lambda (e1: C).(csubst0 i0 v0 e1
-e2)) (\lambda (e1: C).(clear (CHead c3 (Flat f) u1) e1))) (ex_intro2 C
-(\lambda (e1: C).(csubst0 i0 v0 e1 e2)) (\lambda (e1: C).(clear (CHead c3
-(Flat f) u1) e1)) x H7 (clear_flat c3 x H8 f u1)))))) H6)) H5))))) k
-H3))))))))))))) i v c1 c2 H))))).
-(* COMMENTS
-Initial nodes: 2085
-END *)
-
projects / helm.git / blobdiff