+++ /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/drop/fwd.ma".
-
-lemma drop_skip_bind:
- \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
-d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b)
-(lift h d u)) (CHead e (Bind b) u))))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
-(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b)
-d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e
-(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))).
-
-lemma drop_skip_flat:
- \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
-(S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat
-f) (lift h (S d) u)) (CHead e (Flat f) u))))))))
-\def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
-(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat
-f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead
-e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S
-d))))))))).
-
-lemma drop_ctail:
- \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop
-h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1)
-(CTail k u c2))))))))
-\def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d:
-nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u:
-T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda
-(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n)
-c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h
-O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0:
-(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d
-O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n))
-(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u
-(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O
-(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2
-H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2:
-C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h:
-nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k
-u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3:
-C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n
-(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u
-(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n:
-nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop
-n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O
-(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t)
-(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c)))
-(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3
-H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to
-(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail
-k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0:
-K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O
-(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n:
-nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to
-(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail
-k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t)
-c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda
-(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
-(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e)))
-(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2:
-(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4
-\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k
-t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2
-k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k
-x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u
-c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n
-(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop
-h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4
-(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0:
-T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1))))
-(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u)
-x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d)))))))
-c1).
-
-theorem drop_conf_lt:
- \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall
-(c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h:
-nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
-(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop
-h (r k d) c0 e0)))))))))))))
-\def
- \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u:
-T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall
-(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v))))
-(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u:
-T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k
-u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop
-h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S
-(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in
-(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
-(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v))))
-(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
-T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda
-(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d))
-x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k
-x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift
-h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T
-(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v:
-T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda
-(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda
-(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v:
-T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k
-d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0
-e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall
-(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d))
-c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
-v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda
-(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0)
-O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
-nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v:
-T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S
-i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind
-(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
-T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_:
-T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort
-n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d))
-O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee
-with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v))))
-(\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda
-(_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) H5))))) (drop_gen_sort n h
-(S (plus (S i0) d)) e H1)))))))) (\lambda (c1: C).(\lambda (H0: (((drop (S
-i0) O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
-nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v:
-T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
-(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (r k d) c0 e0))))))))))).(\lambda (k0: K).(K_ind (\lambda (k1:
-K).(\forall (t: T).((drop (S i0) O (CHead c1 k1 t) (CHead c0 k u)) \to
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0)
-d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u
-(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e
-(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0
-e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O
-(CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e: C).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0) d)) (CHead c1
-(Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e
-(CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r
-(Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r
-(Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_:
-C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S
-i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0
-e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e (CHead x0
-(Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) (plus (S i0) d))
-x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1 x0)).(eq_ind_r C
-(CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda (v: T).(\lambda
-(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop
-(S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k
-d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b) c1 (CHead c0 k
-u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq
-T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O x0
-(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))
-(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v))))
-(\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead
-e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))
-(\lambda (x2: T).(\lambda (x3: C).(\lambda (H7: (eq T u (lift h (r k d)
-x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k x2))).(\lambda (H9: (drop h (r
-k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h
-(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Bind
-b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0
-e0))) x2 x3 H7 (drop_drop (Bind b) i0 x0 (CHead x3 k x2) H8 x1) H9)))))) H6))
-e H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Bind b) H2)))))))))
-(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c1 (Flat
-f) t) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H2: (drop h (S (plus (S i0) d)) (CHead c1 (Flat f) t)
-e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat
-f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) (plus (S
-i0) d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) (plus (S
-i0) d)) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h
-(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda
-(x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e (CHead x0 (Flat f)
-x1))).(\lambda (_: (eq T t (lift h (r (Flat f) (plus (S i0) d))
-x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1 x0)).(eq_ind_r C
-(CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda (v: T).(\lambda
-(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop
-(S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k
-d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h
-(r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O x0 (CHead e0 k
-v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))) (ex3_2 T C
-(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
-T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1) (CHead e0 k v))))
-(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (x2:
-T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d) x2))).(\lambda (H7:
-(drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h (r k d) c0
-x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
-v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1)
-(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))
-x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1) H8)))))) (H0
-(drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e H3))))))
-(drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2))))))))) k0)))) c))))))
-i)).
-
-theorem drop_conf_ge:
- \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le
-(plus d h) i) \to (drop (minus i h) O e a)))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c:
-C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
-nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e
-a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c
-a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
-d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda
-(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H_y \def
-(le_n_O_eq (plus d h) H1) in (land_ind (eq nat d O) (eq nat h O) (drop (minus
-O h) O e a) (\lambda (H3: (eq nat d O)).(\lambda (H4: (eq nat h O)).(let H5
-\def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H3) in (let H6 \def
-(eq_ind nat h (\lambda (n: nat).(drop n O a e)) H5 O H4) in (eq_ind_r nat O
-(\lambda (n: nat).(drop (minus O n) O e a)) (eq_ind C a (\lambda (c0:
-C).(drop (minus O O) O c0 a)) (drop_refl a) e (drop_gen_refl a e H6)) h
-H4))))) (plus_O d h (sym_eq nat O (plus d h) H_y))))))))))))) (\lambda (i0:
-nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le
-(plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a:
-C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall
-(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d
-h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n:
-nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h:
-nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2:
-(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d
-O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda
-(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n))
-(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6:
-(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O
-O)).(let H9 \def (eq_ind nat d (\lambda (n0: nat).(le (plus n0 h) (S i0))) H2
-O H5) in (let H10 \def (eq_ind nat h (\lambda (n0: nat).(le (plus O n0) (S
-i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0)
-O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0
-a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n)
-c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H7) in (False_ind (drop
-(minus (S i0) O) O (CSort n) (CSort n)) H11)) a H6) e H3) h H4))))))
-(drop_gen_sort n (S i0) O a H0))))) (drop_gen_sort n h d e H1)))))))))
-(\lambda (c0: C).(\lambda (H0: (((drop (S i0) O c0 a) \to (\forall (e:
-C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d h)
-(S i0)) \to (drop (minus (S i0) h) O e a))))))))).(\lambda (k: K).(K_ind
-(\lambda (k0: K).(\forall (t: T).((drop (S i0) O (CHead c0 k0 t) a) \to
-(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d (CHead c0 k0
-t) e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))))
-(\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Bind
-b) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
-(drop h d (CHead c0 (Bind b) t) e)).(\lambda (H3: (le (plus d h) (S
-i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Bind b) t) e) \to ((le
-(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h
-O (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind
-(\lambda (n: nat).((drop n O (CHead c0 (Bind b) t) e) \to ((le (plus O n) (S
-i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0
-(Bind b) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0
-(Bind b) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Bind
-b) i0 c0 a (drop_gen_drop (Bind b) c0 a t i0 H1) t) e (drop_gen_refl (CHead
-c0 (Bind b) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead
-c0 (Bind b) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O
-e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Bind b) t) e)).(\lambda (H7:
-(le (plus O (S h0)) (S i0))).(H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e
-h0 O (drop_gen_drop (Bind b) c0 e t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h
-H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t)
-e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e
-a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Bind b) t) e)).(\lambda (H5:
-(le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
-T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
-(lift h (r (Bind b) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r
-(Bind b) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H6: (eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t
-(lift h (r (Bind b) d0) x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0
-x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c1: C).(drop (minus (S i0)
-h) O c1 a)) (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead
-x0 (Bind b) x1) a)) (drop_drop (Bind b) (minus i0 h) x0 a (H a c0
-(drop_gen_drop (Bind b) c0 a t i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5))
-x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus
-d0 h) i0 H5)))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d
-H2 H3))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O
-(CHead c0 (Flat f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d:
-nat).(\lambda (H2: (drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le
-(plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f)
-t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a))))
-(\lambda (H4: (drop h O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O
-h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e)
-\to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda
-(H6: (drop O O (CHead c0 (Flat f) t) e)).(\lambda (_: (le (plus O O) (S
-i0))).(eq_ind C (CHead c0 (Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O)
-O c1 a)) (drop_drop (Flat f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t)
-e (drop_gen_refl (CHead c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda
-(_: (((drop h0 O (CHead c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to
-(drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0
-(Flat f) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop
-(Flat f) c0 a t i0 H1) e (S h0) O (drop_gen_drop (Flat f) c0 e t h0 H6)
-H7))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0
-(Flat f) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e
-a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Flat f) t) e)).(\lambda (H5:
-(le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
-T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
-(lift h (r (Flat f) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r
-(Flat f) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H6: (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t
-(lift h (r (Flat f) d0) x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0
-x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c1: C).(drop (minus (S i0)
-h) O c1 a)) (let H9 \def (eq_ind_r nat (minus (S i0) h) (\lambda (n:
-nat).(drop n O x0 a)) (H0 (drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0)
-H8 H5) (S (minus i0 h)) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n
-(plus d0 h) i0 H5)))) in (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop
-n O (CHead x0 (Flat f) x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1)
-(minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0
-h) i0 H5))))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2
-H3))))))))) k)))) c))))) i).
-
-theorem drop_conf_rev:
- \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to
-(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1:
-C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1)))))))))
-\def
- \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2:
-C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2)
-\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1
-e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1
-e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let
-H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl
-e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1:
-C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda
-(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2:
-C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O
-c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind
-(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2:
-C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S
-j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n:
-nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda
-(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2
-(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S
-j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1:
-(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O
-O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n)
-H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee with [O
-\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (ex2 C
-(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1
-(CSort n)))) H5)))))) (drop_gen_sort n (S j0) O e2 H)))))))) (\lambda (e2:
-C).(\lambda (IHe1: ((\forall (e3: C).((drop (S j0) O e2 e3) \to (\forall (c2:
-C).(\forall (i: nat).((drop i O c2 e3) \to (ex2 C (\lambda (c1: C).(drop (S
-j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)))))))))).(\lambda (k:
-K).(\lambda (t: T).(\lambda (e3: C).(\lambda (H: (drop (S j0) O (CHead e2 k
-t) e3)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2
-e3)).(K_ind (\lambda (k0: K).((drop (r k0 j0) O e2 e3) \to (ex2 C (\lambda
-(c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2
-k0 t)))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) j0) O e2 e3)).(let
-H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda
-(c1: C).(drop j0 O c1 c2)) (\lambda (c1: C).(drop i j0 c1 e2)) (ex2 C
-(\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1
-(CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda (H3: (drop j0 O x
-c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S
-j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t)))
-(CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind b) j0 x c2 H3
-(lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4 t))))) H2)))))
-(\lambda (f: F).(\lambda (H1: (drop (r (Flat f) j0) O e2 e3)).(let H_x \def
-(IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop
-(S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)) (ex2 C (\lambda (c1:
-C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat
-f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0) O x c2)).(\lambda (H4:
-(drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2))
-(\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat f) t))) (CHead x (Flat f)
-(lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0 x c2 H3 (lift i (r (Flat
-f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t))))) H2))))) k (drop_gen_drop k
-e2 e3 t j0 H))))))))))) e1)))) j).
-
-theorem drop_trans_le:
- \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
-(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O
-c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h
-(minus d i) e1 e2)))))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to
-(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to
-(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1
-e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d:
-nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h:
-nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O
-c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2
-(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda
-(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C
-(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1
-(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0:
-nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1:
-C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
-C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda
-(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind
-(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2:
-C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O
-c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
-C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0)
-O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h
-O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(ex2_ind nat
-(\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le i0 n)) (ex2 C
-(\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S
-i0)) e1 e2))) (\lambda (x: nat).(\lambda (H2: (eq nat O (S x))).(\lambda (_:
-(le i0 x)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee with [O
-\Rightarrow True | (S _) \Rightarrow False])) I (S x) H2) in (False_ind (ex2
-C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O
-(S i0)) e1 e2))) H4))))) (le_gen_S i0 O H))))))))) (\lambda (d0:
-nat).(\lambda (_: (((le (S i0) d0) \to (\forall (c1: C).(\forall (c2:
-C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall (e2: C).((drop (S i0) O
-c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
-C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda (H: (le (S i0) (S
-d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h:
-nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to
-(ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1: C).(drop h (minus
-(S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (h:
-nat).(\lambda (H0: (drop h (S d0) (CSort n) c2)).(\lambda (e2: C).(\lambda
-(H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2 (CSort n)) (eq nat h O) (eq
-nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda
-(e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H2: (eq C c2 (CSort
-n))).(\lambda (_: (eq nat h O)).(\lambda (_: (eq nat (S d0) O)).(let H5 \def
-(eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CSort n) H2) in
-(and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (ex2 C (\lambda
-(e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0)
-(S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort n))).(\lambda (H7: (eq nat (S
-i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda (c: C).(ex2
-C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h
-(minus (S d0) (S i0)) e1 c)))) (let H9 \def (eq_ind nat (S i0) (\lambda (ee:
-nat).(match ee with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7)
-in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda
-(e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6))))
-(drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0))))))))
-(\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h
-(S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda
-(e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0))
-e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
-T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to
-(\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S
-i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
-e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h:
-nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2:
-C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e:
-C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda
-(v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_:
-T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O
-(CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
-e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0
-(Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda
-(H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c:
-C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h
-(r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O
-(CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
-e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop
-h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2
-(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S
-d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda
-(H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0)
-O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop
-h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r
-(Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2
-(drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3
-t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3:
-C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t)
-c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T
-(\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_:
-C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e:
-C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1:
-C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus
-(S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C
-c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0)
-x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3
-(\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in
-(eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1:
-C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus
-(S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1))
-(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1:
-C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1))
-(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x:
-C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S
-i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f)
-(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S
-i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1))
-H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3)))))))
-(drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i).
-
-theorem drop_trans_ge:
- \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
-nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2
-e2) \to ((le d i) \to (drop (plus i h) O c1 e2)))))))))
-\def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2:
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
-C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2))))))))))
-(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
-nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O
-c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h)
-O c1 c)) (let H_y \def (le_n_O_eq d H1) in (let H2 \def (eq_ind_r nat d
-(\lambda (n: nat).(drop h n c1 c2)) H O H_y) in H2)) e2 (drop_gen_refl c2 e2
-H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall
-(c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall
-(e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1
-e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
-C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2:
-C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c
-e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
-nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0:
-(drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort
-n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda
-(H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d
-O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n)
-e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4)
-in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort
-n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop
-(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda
-(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n)
-(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind
-nat (S i0) (\lambda (ee: nat).(match ee with [O \Rightarrow False | (S _)
-\Rightarrow True])) I O H8) in (False_ind (drop (S (plus i0 O)) O (CSort n)
-(CSort n)) H10)) e2 H7)))) (drop_gen_sort n (S i0) O e2 H6)))) h H3))))
-(drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: C).(\lambda (IHc: ((\forall
-(c3: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 c3) \to (\forall
-(e2: C).((drop (S i0) O c3 e2) \to ((le d (S i0)) \to (drop (S (plus i0 h)) O
-c2 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: C).(\lambda (d:
-nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c2 k t)
-c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le n (S i0)) \to (drop
-(S (plus i0 h)) O (CHead c2 k t) e2))))))) (\lambda (h: nat).(nat_ind
-(\lambda (n: nat).((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop
-(S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t)
-e2)))))) (\lambda (H: (drop O O (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda
-(H0: (drop (S i0) O c3 e2)).(\lambda (_: (le O (S i0))).(let H2 \def
-(eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead c2 k t)
-(drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat i0 (\lambda (n:
-nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2 (drop_gen_drop k
-c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda (n: nat).(\lambda
-(_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3
-e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t)
-e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (e2:
-C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le O (S i0))).(eq_ind
-nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O (CHead c2 k t) e2))
-(drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k (plus i0 n)))
-(\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r k n))
-(\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n) (drop_gen_drop k
-c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0 n)) (r k (S (plus
-i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm i0 n)))))))) h))
-(\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop h d0 (CHead c2 k
-t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d0 (S i0)) \to
-(drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h: nat).(\lambda
-(H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop
-(S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S i0))).(ex3_2_ind C T (\lambda
-(e: C).(\lambda (v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
-T).(eq T t (lift h (r k d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r
-k d0) c2 e))) (drop (S (plus i0 h)) O (CHead c2 k t) e2) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 k x1))).(\lambda (H3:
-(eq T t (lift h (r k d0) x1))).(\lambda (H4: (drop h (r k d0) c2 x0)).(let H5
-\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 d0 (CHead c2 k
-t) c) \to (\forall (e3: C).((drop (S i0) O c e3) \to ((le d0 (S i0)) \to
-(drop (S (plus i0 h0)) O (CHead c2 k t) e3))))))) IHd (CHead x0 k x1) H2) in
-(let H6 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead x0
-k x1) H2) in (let H7 \def (eq_ind T t (\lambda (t0: T).(\forall (h0:
-nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k x1)) \to (\forall (e3:
-C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S i0)) \to (drop (S (plus
-i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k d0) x1) H3) in (eq_ind_r
-T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S (plus i0 h)) O (CHead c2 k
-t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind (\lambda (k0: K).((drop h (r
-k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to (drop (r k0 (plus i0 h)) O c2
-e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r (Bind b) d0) c2 x0)).(\lambda
-(H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 (r (Bind b) d0) h H8 e2 H9
-(le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: F).(\lambda (H8: (drop h (r
-(Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0 e2)).(IHc x0 (r
-(Flat f) d0) h H8 e2 H9 H1)))) k H4 (drop_gen_drop k x0 e2 x1 i0 H6)) (lift h
-(r k d0) x1)) t H3))))))))) (drop_gen_skip_l c2 c3 t h d0 k H)))))))))
-d))))))) c1)))) i).
-
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