1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/drop/props".
19 include "drop/fwd.ma".
21 include "lift/props.ma".
25 theorem drop_skip_bind:
26 \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
27 d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b)
28 (lift h d u)) (CHead e (Bind b) u))))))))
30 \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
31 (H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b)
32 d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e
33 (Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))).
35 theorem drop_skip_flat:
36 \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
37 (S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat
38 f) (lift h (S d) u)) (CHead e (Flat f) u))))))))
40 \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
41 (H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat
42 f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead
43 e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S
47 \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h:
48 nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
50 \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
51 C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to
52 (drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
53 T).(\lambda (h: nat).(\lambda (H: (drop h O (CSort n) (CHead e (Bind b)
54 u))).(and3_ind (eq C (CHead e (Bind b) u) (CSort n)) (eq nat h O) (eq nat O
55 O) (drop (S h) O (CSort n) e) (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort
56 n))).(\lambda (H1: (eq nat h O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O
57 (\lambda (n0: nat).(drop (S n0) O (CSort n) e)) (let H3 \def (eq_ind C (CHead
58 e (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
59 with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
60 (CSort n) H0) in (False_ind (drop (S O) O (CSort n) e) H3)) h H1))))
61 (drop_gen_sort n h O (CHead e (Bind b) u) H))))))) (\lambda (c0: C).(\lambda
62 (H: ((\forall (e: C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e
63 (Bind b) u)) \to (drop (S h) O c0 e))))))).(\lambda (k: K).(\lambda (t:
64 T).(\lambda (e: C).(\lambda (u: T).(\lambda (h: nat).(nat_ind (\lambda (n:
65 nat).((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead
66 c0 k t) e))) (\lambda (H0: (drop O O (CHead c0 k t) (CHead e (Bind b)
67 u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
68 (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _)
69 \Rightarrow c1])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead
70 c0 k t) (CHead e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0:
71 C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
72 (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k t) (CHead e (Bind b) u)
73 (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in ((let H3 \def
74 (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
75 [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k t)
76 (CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0))
77 in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0
78 (\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b)
79 (\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0
80 c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_:
81 (((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0
82 k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t) (CHead e (Bind b)
83 u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0:
84 nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0 (CHead e (Bind b)
85 u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)).
88 \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop
89 h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1)
92 \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d:
93 nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u:
94 T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda
95 (c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n)
96 c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h
97 O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0:
98 (eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d
99 O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n))
100 (CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u
101 (CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O
102 (CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2
103 H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2:
104 C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h:
105 nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k
106 u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3:
107 C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n
108 (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u
109 (CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n:
110 nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop
111 n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O
112 (CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t)
113 (\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c)))
114 (drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3
115 H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to
116 (\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail
117 k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0:
118 K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O
119 (r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n:
120 nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to
121 (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail
122 k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t)
123 c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda
124 (v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
125 (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e)))
126 (drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0:
127 C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2:
128 (eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4
129 \def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k
130 t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2
131 k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k
132 x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u
133 c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n
134 (CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop
135 h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4
136 (lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0:
137 T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1))))
138 (drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u)
139 x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d)))))))
143 \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h
144 d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2)))))))
146 \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d:
147 nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0
148 x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d:
149 nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2:
150 C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq
151 nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort
152 n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C
153 x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1
154 (CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r
155 C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h
156 (\lambda (n0: nat).(eq nat n0 O)) H2 O H5) in (let H8 \def (eq_ind nat d
157 (\lambda (n0: nat).(eq nat n0 O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda
158 (c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1))))
159 (drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda
160 (c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h:
161 nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1
162 x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d:
163 nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t)
164 x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2))))))
165 (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1)
166 \to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2)))))
167 (\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1:
168 (drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C
169 x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t)))
170 (refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2
171 (drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_:
172 (((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t)
173 x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t)
174 x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O
175 (r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n
176 H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
177 (CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq
178 C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t)
179 x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t)
180 x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v))))
181 (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
182 C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0:
183 C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4:
184 (eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0
185 x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v))))
186 (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
187 C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4:
188 C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7:
189 (eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r
190 C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1
191 (\lambda (c1: C).(\forall (h0: nat).((drop h0 n (CHead c0 k t) c1) \to
192 (\forall (x6: C).((drop h0 n (CHead c0 k t) x6) \to (eq C c1 x6)))))) H0
193 (CHead x4 k x5) H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1
194 (CHead x0 k x3))) (let H10 \def (eq_ind T t (\lambda (t0: T).(\forall (h0:
195 nat).((drop h0 n (CHead c0 k t0) (CHead x4 k x5)) \to (\forall (x6: C).((drop
196 h0 n (CHead c0 k t0) x6) \to (eq C (CHead x4 k x5) x6)))))) H9 (lift h (r k
197 n) x5) H7) in (let H11 \def (eq_ind T t (\lambda (t0: T).(eq T t0 (lift h (r
198 k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda
199 (t0: T).(\forall (h0: nat).((drop h0 n (CHead c0 k (lift h (r k n) t0))
200 (CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n)
201 t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n)
202 H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k
203 x3))) (f_equal3 C K T C CHead x4 x0 k k x3 x3 (sym_eq C x0 x4 (H x0 (r k n) h
204 H5 x4 H8)) (refl_equal K k) (refl_equal T x3)) x5 (lift_inj x5 x3 h (r k n)
205 H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k H1)))))))
206 (drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c).
208 theorem drop_conf_lt:
209 \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall
210 (c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h:
211 nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
212 (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
213 (e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop
214 h (r k d) c0 e0)))))))))))))
216 \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u:
217 T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall
218 (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to
219 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v))))
220 (\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_:
221 T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u:
222 T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k
223 u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop
224 h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S
225 (plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in
226 (ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
227 (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v))))
228 (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C
229 (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
230 T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda
231 (e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
232 (H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d))
233 x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k
234 x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift
235 h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k
236 v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T
237 (lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v:
238 T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda
239 (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda
240 (e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda
241 (_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v:
242 T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_:
243 T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k
244 d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0
245 e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall
246 (u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to
247 (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d))
248 c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
249 v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v))))
250 (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda
251 (u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0)
252 O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
253 nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v:
254 T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
255 (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
256 C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S
257 i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda
258 (d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind
259 (eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C
260 (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
261 T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_:
262 T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort
263 n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d))
264 O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee
265 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
266 \Rightarrow True])) I O H4) in (False_ind (ex3_2 T C (\lambda (v: T).(\lambda
267 (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop
268 (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d)
269 c0 e0)))) H5))))) (drop_gen_sort n h (S (plus (S i0) d)) e H1))))))))
270 (\lambda (c1: C).(\lambda (H0: (((drop (S i0) O c1 (CHead c0 k u)) \to
271 (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0)
272 d)) c1 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k
273 d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v))))
274 (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))).(\lambda
275 (k0: K).(K_ind (\lambda (k1: K).(\forall (t: T).((drop (S i0) O (CHead c1 k1
276 t) (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
277 nat).((drop h (S (plus (S i0) d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda
278 (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
279 (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
280 C).(drop h (r k d) c0 e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda
281 (H1: (drop (S i0) O (CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e:
282 C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0)
283 d)) (CHead c1 (Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
284 T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
285 (lift h (r (Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_:
286 T).(drop h (r (Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v:
287 T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
288 (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
289 C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3:
290 (eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b)
291 (plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1
292 x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda
293 (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
294 (e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
295 C).(drop h (r k d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b)
296 c1 (CHead c0 k u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v:
297 T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
298 (e0: C).(drop i0 O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
299 C).(drop h (r k d) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T
300 u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O
301 (CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
302 C).(drop h (r k d) c0 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H7:
303 (eq T u (lift h (r k d) x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k
304 x2))).(\lambda (H9: (drop h (r k d) c0 x3)).(ex3_2_intro T C (\lambda (v:
305 T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
306 (e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_:
307 T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x2 x3 H7 (drop_drop (Bind b) i0
308 x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) e H3)))))) (drop_gen_skip_l c1 e t h
309 (plus (S i0) d) (Bind b) H2))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda
310 (H1: (drop (S i0) O (CHead c1 (Flat f) t) (CHead c0 k u))).(\lambda (e:
311 C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0)
312 d)) (CHead c1 (Flat f) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
313 T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
314 (lift h (r (Flat f) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_:
315 T).(drop h (r (Flat f) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v:
316 T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
317 (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
318 C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3:
319 (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f)
320 (plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1
321 x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda
322 (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
323 (e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
324 C).(drop h (r k d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
325 C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S
326 i0) O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d)
327 c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
328 v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1)
329 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))
330 (\lambda (x2: T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d)
331 x2))).(\lambda (H7: (drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h
332 (r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u
333 (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead
334 x0 (Flat f) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r
335 k d) c0 e0))) x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1)
336 H8)))))) (H0 (drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e
337 H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2)))))))))
340 theorem drop_conf_ge:
341 \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to
342 (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le
343 (plus d h) i) \to (drop (minus i h) O e a)))))))))
345 \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c:
346 C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
347 nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e
348 a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c
349 a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
350 d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda
351 (c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H3 \def (match
352 H1 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to
353 (drop (minus O h) O e a)))) with [le_n \Rightarrow (\lambda (H3: (eq nat
354 (plus d h) O)).(let H4 \def (f_equal nat nat (\lambda (e0: nat).e0) (plus d
355 h) O H3) in (eq_ind nat (plus d h) (\lambda (n: nat).(drop (minus n h) n e
356 a)) (eq_ind_r nat O (\lambda (n: nat).(drop (minus n h) n e a)) (and_ind (eq
357 nat d O) (eq nat h O) (drop O O e a) (\lambda (H5: (eq nat d O)).(\lambda
358 (H6: (eq nat h O)).(let H7 \def (eq_ind nat d (\lambda (n: nat).(drop h n a
359 e)) H2 O H5) in (let H8 \def (eq_ind nat h (\lambda (n: nat).(drop n O a e))
360 H7 O H6) in (eq_ind C a (\lambda (c0: C).(drop O O c0 a)) (drop_refl a) e
361 (drop_gen_refl a e H8)))))) (plus_O d h H4)) (plus d h) H4) O H4))) | (le_S m
362 H3) \Rightarrow (\lambda (H4: (eq nat (S m) O)).((let H5 \def (eq_ind nat (S
363 m) (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O
364 \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind ((le
365 (plus d h) m) \to (drop (minus O h) O e a)) H5)) H3))]) in (H3 (refl_equal
366 nat O)))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall (a: C).(\forall
367 (c: C).((drop i0 O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
368 nat).((drop h d c e) \to ((le (plus d h) i0) \to (drop (minus i0 h) O e
369 a))))))))))).(\lambda (a: C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop
370 (S i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop
371 h d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e
372 a)))))))) (\lambda (n: nat).(\lambda (H0: (drop (S i0) O (CSort n)
373 a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop h
374 d (CSort n) e)).(\lambda (H2: (le (plus d h) (S i0))).(and3_ind (eq C e
375 (CSort n)) (eq nat h O) (eq nat d O) (drop (minus (S i0) h) O e a) (\lambda
376 (H3: (eq C e (CSort n))).(\lambda (H4: (eq nat h O)).(\lambda (H5: (eq nat d
377 O)).(and3_ind (eq C a (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (minus
378 (S i0) h) O e a) (\lambda (H6: (eq C a (CSort n))).(\lambda (H7: (eq nat (S
379 i0) O)).(\lambda (_: (eq nat O O)).(let H9 \def (eq_ind nat d (\lambda (n0:
380 nat).(le (plus n0 h) (S i0))) H2 O H5) in (let H10 \def (eq_ind nat h
381 (\lambda (n0: nat).(le (plus O n0) (S i0))) H9 O H4) in (eq_ind_r nat O
382 (\lambda (n0: nat).(drop (minus (S i0) n0) O e a)) (eq_ind_r C (CSort n)
383 (\lambda (c0: C).(drop (minus (S i0) O) O c0 a)) (eq_ind_r C (CSort n)
384 (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) c0)) (let H11 \def
385 (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_:
386 nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in
387 (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) H11)) a H6) e H3) h
388 H4)))))) (drop_gen_sort n (S i0) O a H0))))) (drop_gen_sort n h d e
389 H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S i0) O c0 a) \to
390 (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le
391 (plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))).(\lambda (k:
392 K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S i0) O (CHead c0 k0 t) a)
393 \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d (CHead c0
394 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e
395 a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O
396 (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d:
397 nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) e)).(\lambda (H3: (le
398 (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Bind b)
399 t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a))))
400 (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus O
401 h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Bind b) t) e)
402 \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda
403 (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le (plus O O) (S
404 i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop (minus (S i0) O)
405 O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) c0 a t i0 H1) t)
406 e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda (h0: nat).(\lambda
407 (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O h0) (S i0)) \to
408 (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0
409 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a c0
410 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e t
411 h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: nat).(\lambda
412 (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) (S i0)) \to
413 (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0
414 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T
415 (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind b) v)))) (\lambda
416 (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e0:
417 C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop (minus (S i0) h) O
418 e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C e (CHead x0 (Bind
419 b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda (H8:
420 (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda
421 (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S (minus i0 h)) (\lambda
422 (n: nat).(drop n O (CHead x0 (Bind b) x1) a)) (drop_drop (Bind b) (minus i0
423 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) x0 h d0 H8 (le_S_n
424 (plus d0 h) i0 H5)) x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0
425 h i0 (le_S_n (plus d0 h) i0 H5)))) e H6)))))) (drop_gen_skip_l c0 e t h d0
426 (Bind b) H4)))))) d H2 H3))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda
427 (H1: (drop (S i0) O (CHead c0 (Flat f) t) a)).(\lambda (e: C).(\lambda (h:
428 nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Flat f) t)
429 e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h
430 n (CHead c0 (Flat f) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S
431 i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Flat f) t) e)).(\lambda
432 (H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0
433 (Flat f) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e
434 a)))) (\lambda (H6: (drop O O (CHead c0 (Flat f) t) e)).(\lambda (_: (le
435 (plus O O) (S i0))).(eq_ind C (CHead c0 (Flat f) t) (\lambda (c1: C).(drop
436 (minus (S i0) O) O c1 a)) (drop_drop (Flat f) i0 c0 a (drop_gen_drop (Flat f)
437 c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Flat f) t) e H6)))) (\lambda
438 (h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Flat f) t) e) \to ((le (plus O
439 h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0)
440 O (CHead c0 (Flat f) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H0
441 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S h0) O (drop_gen_drop (Flat f) c0 e
442 t h0 H6) H7))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0
443 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0)
444 h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Flat f) t) e)).(\lambda
445 (H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda
446 (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T
447 t (lift h (r (Flat f) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r
448 (Flat f) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda
449 (x1: T).(\lambda (H6: (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t
450 (lift h (r (Flat f) d0) x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0
451 x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c1: C).(drop (minus (S i0)
452 h) O c1 a)) (let H9 \def (eq_ind_r nat (minus (S i0) h) (\lambda (n:
453 nat).(drop n O x0 a)) (H0 (drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0)
454 H8 H5) (S (minus i0 h)) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n
455 (plus d0 h) i0 H5)))) in (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop
456 n O (CHead x0 (Flat f) x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1)
457 (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0
458 h) i0 H5))))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2
459 H3))))))))) k)))) c))))) i).
461 theorem drop_conf_rev:
462 \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to
463 (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1:
464 C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1)))))))))
466 \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2:
467 C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2)
468 \to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1
469 e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1
470 e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let
471 H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl
472 e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1:
473 C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda
474 (IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2:
475 C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O
476 c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind
477 (\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2:
478 C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S
479 j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n:
480 nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda
481 (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2
482 (CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S
483 j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1:
484 (eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O
485 O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n)
486 H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee in nat
487 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
488 True])) I O H2) in (False_ind (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2))
489 (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) H5)))))) (drop_gen_sort n (S
490 j0) O e2 H)))))))) (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop
491 (S j0) O e2 e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to
492 (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S
493 j0) c1 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda
494 (H: (drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i:
495 nat).(\lambda (H0: (drop i O c2 e3)).(K_ind (\lambda (k0: K).((drop (r k0 j0)
496 O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1:
497 C).(drop i (S j0) c1 (CHead e2 k0 t)))))) (\lambda (b: B).(\lambda (H1: (drop
498 (r (Bind b) j0) O e2 e3)).(let H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2
499 \def H_x in (ex2_ind C (\lambda (c1: C).(drop j0 O c1 c2)) (\lambda (c1:
500 C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda
501 (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda
502 (H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda
503 (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2
504 (Bind b) t))) (CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind
505 b) j0 x c2 H3 (lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4
506 t))))) H2))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) j0) O e2
507 e3)).(let H_x \def (IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C
508 (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1
509 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i
510 (S j0) c1 (CHead e2 (Flat f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0)
511 O x c2)).(\lambda (H4: (drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1:
512 C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat
513 f) t))) (CHead x (Flat f) (lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0
514 x c2 H3 (lift i (r (Flat f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t)))))
515 H2))))) k (drop_gen_drop k e2 e3 t j0 H))))))))))) e1)))) j).
517 theorem drop_trans_le:
518 \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
519 (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O
520 c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h
521 (minus d i) e1 e2)))))))))))
523 \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to
524 (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to
525 (\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1
526 e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d:
527 nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h:
528 nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O
529 c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2
530 (drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda
531 (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C
532 (\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1
533 (drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0:
534 nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1:
535 C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
536 C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda
537 (e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind
538 (\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2:
539 C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O
540 c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
541 C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0)
542 O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h
543 O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(let H2 \def
544 (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
545 \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h
546 (minus O (S i0)) e1 e2)))))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S
547 i0) O)).(let H3 \def (eq_ind nat (S i0) (\lambda (e: nat).(match e in nat
548 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
549 True])) I O H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1))
550 (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))) H3))) | (le_S m H2)
551 \Rightarrow (\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m)
552 (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
553 \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S
554 i0) m) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
555 C).(drop h (minus O (S i0)) e1 e2)))) H4)) H2))]) in (H2 (refl_equal nat
556 O)))))))))) (\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall
557 (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall
558 (e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1
559 e1)) (\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda
560 (H: (le (S i0) (S d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
561 C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0)
562 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1:
563 C).(drop h (minus (S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda
564 (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CSort n)
565 c2)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2
566 (CSort n)) (eq nat h O) (eq nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S
567 i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))
568 (\lambda (H2: (eq C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_:
569 (eq nat (S d0) O)).(let H5 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c
570 e2)) H1 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq
571 nat O O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1:
572 C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort
573 n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C
574 (CSort n) (\lambda (c: C).(ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n)
575 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 c)))) (let H9 \def
576 (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_:
577 nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in
578 (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda
579 (e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6))))
580 (drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0))))))))
581 (\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h
582 (S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda
583 (e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0))
584 e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
585 T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to
586 (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S
587 i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
588 e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h:
589 nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2:
590 C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e:
591 C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda
592 (v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_:
593 T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O
594 (CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
595 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0
596 (Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda
597 (H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c:
598 C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h
599 (r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O
600 (CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
601 e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop
602 h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2
603 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S
604 d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda
605 (H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0)
606 O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop
607 h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r
608 (Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2
609 (drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3
610 t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3:
611 C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t)
612 c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T
613 (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_:
614 C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e:
615 C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1:
616 C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus
617 (S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C
618 c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0)
619 x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3
620 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in
621 (eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1:
622 C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus
623 (S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1))
624 (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1:
625 C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1))
626 (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x:
627 C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S
628 i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f)
629 (lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S
630 i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1))
631 H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3)))))))
632 (drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i).
634 theorem drop_trans_ge:
635 \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
636 nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2
637 e2) \to ((le d i) \to (drop (plus i h) O c1 e2)))))))))
639 \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2:
640 C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
641 C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2))))))))))
642 (\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
643 nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O
644 c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h)
645 O c1 c)) (let H2 \def (match H1 in le return (\lambda (n: nat).(\lambda (_:
646 (le ? n)).((eq nat n O) \to (drop (plus O h) O c1 c2)))) with [le_n
647 \Rightarrow (\lambda (H2: (eq nat d O)).(eq_ind nat O (\lambda (_: nat).(drop
648 (plus O h) O c1 c2)) (let H3 \def (eq_ind nat d (\lambda (n: nat).(le n O))
649 H1 O H2) in (let H4 \def (eq_ind nat d (\lambda (n: nat).(drop h n c1 c2)) H
650 O H2) in H4)) d (sym_eq nat d O H2))) | (le_S m H2) \Rightarrow (\lambda (H3:
651 (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e: nat).(match e
652 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
653 \Rightarrow True])) I O H3) in (False_ind ((le d m) \to (drop (plus O h) O c1
654 c2)) H4)) H2))]) in (H2 (refl_equal nat O))) e2 (drop_gen_refl c2 e2
655 H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall
656 (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall
657 (e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1
658 e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
659 C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2:
660 C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c
661 e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
662 nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0:
663 (drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort
664 n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda
665 (H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d
666 O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n)
667 e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4)
668 in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort
669 n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop
670 (S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda
671 (H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n)
672 (\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind
673 nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop)
674 with [O \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind
675 (drop (S (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n
676 (S i0) O e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2:
677 C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h:
678 nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d
679 (S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda
680 (t: T).(\lambda (c3: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall
681 (h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O
682 c3 e2) \to ((le n (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t)
683 e2))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c2 k
684 t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to
685 (drop (S (plus i0 n)) O (CHead c2 k t) e2)))))) (\lambda (H: (drop O O (CHead
686 c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda
687 (_: (le O (S i0))).(let H2 \def (eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O
688 c e2)) H0 (CHead c2 k t) (drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat
689 i0 (\lambda (n: nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2
690 (drop_gen_drop k c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda
691 (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2:
692 C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O
693 (CHead c2 k t) e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t)
694 c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le
695 O (S i0))).(eq_ind nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O
696 (CHead c2 k t) e2)) (drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k
697 (plus i0 n))) (\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r
698 k n)) (\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n)
699 (drop_gen_drop k c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0
700 n)) (r k (S (plus i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm
701 i0 n)))))))) h)) (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop
702 h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le
703 d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h:
704 nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2:
705 C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S
706 i0))).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e k
707 v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda
708 (e: C).(\lambda (_: T).(drop h (r k d0) c2 e))) (drop (S (plus i0 h)) O
709 (CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3
710 (CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4:
711 (drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall
712 (h0: nat).((drop h0 d0 (CHead c2 k t) c) \to (\forall (e3: C).((drop (S i0) O
713 c e3) \to ((le d0 (S i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t)
714 e3))))))) IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c:
715 C).(drop (S i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t
716 (\lambda (t0: T).(\forall (h0: nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k
717 x1)) \to (\forall (e3: C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S
718 i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k
719 d0) x1) H3) in (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S
720 (plus i0 h)) O (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind
721 (\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to
722 (drop (r k0 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r
723 (Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0
724 (r (Bind b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f:
725 F).(\lambda (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat
726 f) i0) O x0 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4
727 (drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0) x1)) t H3)))))))))
728 (drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1)))) i).