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/Level-1/LambdaDelta/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 c _ _) \Rightarrow
69 c])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead
70 e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: C).(match e0
71 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k _)
72 \Rightarrow k])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0
73 k t) (CHead e (Bind b) u) H0)) in ((let H3 \def (f_equal C T (\lambda (e0:
74 C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t |
75 (CHead _ _ t) \Rightarrow t])) (CHead c0 k t) (CHead e (Bind b) u)
76 (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in (\lambda (H4: (eq
77 K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0 (\lambda (c1: C).(drop
78 (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) (\lambda (k0: K).(drop (S O)
79 O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 c0 (drop_refl c0) t) k H4) e
80 H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c0 k t)
81 (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e)))).(\lambda (H1:
82 (drop (S n) O (CHead c0 k t) (CHead e (Bind b) u))).(drop_drop k (S n) c0 e
83 (eq_ind_r nat (S (r k n)) (\lambda (n0: nat).(drop n0 O c0 e)) (H e u (r k n)
84 (drop_gen_drop k c0 (CHead e (Bind b) u) t n H1)) (r k (S n)) (r_S k n))
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 (h: nat).((drop h n (CHead c2 k t)
130 c) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t))
131 (CTail k0 u c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k x1)
132 (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c)))
133 (let H5 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n (CHead
134 c2 k t) (CHead x0 k x1)) \to (\forall (k0: K).(\forall (u: T).(drop h n
135 (CTail k0 u (CHead c2 k t)) (CTail k0 u (CHead x0 k x1)))))))) H4 (lift h (r
136 k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: T).(drop h (S
137 n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) (drop_skip k h
138 n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) x1) t H2)) c3
139 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) c1).
142 \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h
143 d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2)))))))
145 \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d:
146 nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0
147 x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d:
148 nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2:
149 C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq
150 nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort
151 n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C
152 x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1
153 (CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r
154 C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h
155 (\lambda (n: nat).(eq nat n O)) H2 O H5) in (let H8 \def (eq_ind nat d
156 (\lambda (n: nat).(eq nat n O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda
157 (c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1))))
158 (drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda
159 (c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h:
160 nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1
161 x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d:
162 nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t)
163 x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2))))))
164 (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1)
165 \to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2)))))
166 (\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1:
167 (drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C
168 x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t)))
169 (refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2
170 (drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_:
171 (((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t)
172 x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t)
173 x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O
174 (r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n
175 H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
176 (CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq
177 C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t)
178 x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t)
179 x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v))))
180 (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
181 C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0:
182 C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4:
183 (eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0
184 x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v))))
185 (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
186 C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4:
187 C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7:
188 (eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r
189 C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1
190 (\lambda (c: C).(\forall (h: nat).((drop h n (CHead c0 k t) c) \to (\forall
191 (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C c x2)))))) H0 (CHead x4 k x5)
192 H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 (CHead x0 k
193 x3))) (let H10 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n
194 (CHead c0 k t) (CHead x4 k x5)) \to (\forall (x2: C).((drop h n (CHead c0 k
195 t) x2) \to (eq C (CHead x4 k x5) x2)))))) H9 (lift h (r k n) x5) H7) in (let
196 H11 \def (eq_ind T t (\lambda (t: T).(eq T t (lift h (r k n) x3))) H4 (lift h
197 (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda (t: T).(\forall (h0:
198 nat).((drop h0 n (CHead c0 k (lift h (r k n) t)) (CHead x4 k t)) \to (\forall
199 (x2: C).((drop h0 n (CHead c0 k (lift h (r k n) t)) x2) \to (eq C (CHead x4 k
200 t) x2)))))) H10 x3 (lift_inj x5 x3 h (r k n) H11)) in (eq_ind_r T x3 (\lambda
201 (t0: T).(eq C (CHead x4 k t0) (CHead x0 k x3))) (sym_equal C (CHead x0 k x3)
202 (CHead x4 k x3) (sym_equal C (CHead x4 k x3) (CHead x0 k x3) (sym_equal C
203 (CHead x0 k x3) (CHead x4 k x3) (f_equal3 C K T C CHead x0 x4 k k x3 x3 (H x0
204 (r k n) h H5 x4 H8) (refl_equal K k) (refl_equal T x3))))) x5 (lift_inj x5 x3
205 h (r k n) H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k
206 H1))))))) (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 (c: C).(drop h (S
225 (plus O d)) c 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 (c: C).(drop h d c e)) H0 a (drop_gen_refl c a H)) in (let H3 \def (match H1
352 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to (drop
353 (minus O h) O e a)))) with [le_n \Rightarrow (\lambda (H: (eq nat (plus d h)
354 O)).(let H3 \def (f_equal nat nat (\lambda (e0: nat).e0) (plus d h) O H) in
355 (eq_ind nat (plus d h) (\lambda (n: nat).(drop (minus n h) n e a)) (eq_ind_r
356 nat O (\lambda (n: nat).(drop (minus n h) n e a)) (and_ind (eq nat d O) (eq
357 nat h O) (drop O O e a) (\lambda (H0: (eq nat d O)).(\lambda (H1: (eq nat h
358 O)).(let H2 \def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H0) in
359 (let H4 \def (eq_ind nat h (\lambda (n: nat).(drop n O a e)) H2 O H1) in
360 (eq_ind C a (\lambda (c: C).(drop O O c a)) (drop_refl a) e (drop_gen_refl a
361 e H4)))))) (plus_O d h H3)) (plus d h) H3) O H3))) | (le_S m H) \Rightarrow
362 (\lambda (H2: (eq nat (S m) O)).((let H0 \def (eq_ind nat (S m) (\lambda (e0:
363 nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
364 False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (plus d h) m)
365 \to (drop (minus O h) O e a)) H0)) H))]) in (H3 (refl_equal nat O))))))))))))
366 (\lambda (i0: nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O
367 c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e)
368 \to ((le (plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a:
369 C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall
370 (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d
371 h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n:
372 nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h:
373 nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2:
374 (le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d
375 O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda
376 (H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n))
377 (eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6:
378 (eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O
379 O)).(let H9 \def (eq_ind nat d (\lambda (n: nat).(le (plus n h) (S i0))) H2 O
380 H5) in (let H10 \def (eq_ind nat h (\lambda (n: nat).(le (plus O n) (S i0)))
381 H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) O e
382 a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 a))
383 (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n)
384 c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat
385 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
386 True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n))
387 H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0)))))
388 (drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S
389 i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
390 d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e
391 a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S
392 i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
393 nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop
394 (minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1:
395 (drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h:
396 nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t)
397 e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h
398 n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S
399 i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda
400 (H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0
401 (Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e
402 a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le
403 (plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop
404 (minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b)
405 c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda
406 (h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O
407 h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0)
408 O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a
409 c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e
410 t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0:
411 nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h)
412 (S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0)
413 (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S
414 i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind
415 b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0)
416 v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop
417 (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C
418 e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0)
419 x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0
420 (Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S
421 (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a))
422 (drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t
423 i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h)
424 (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e
425 H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3)))))))))
426 (\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat
427 f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
428 (drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S
429 i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le
430 (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h
431 O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind
432 (\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S
433 i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0
434 (Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0
435 (Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat
436 f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead
437 c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead
438 c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O
439 e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7:
440 (le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S
441 h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0:
442 nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h)
443 (S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0)
444 (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S
445 i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat
446 f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0)
447 v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop
448 (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C
449 e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0)
450 x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0
451 (Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def
452 (eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0
453 (drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h))
454 (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in
455 (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f)
456 x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h)
457 (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e
458 H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3)))))))))
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)).((match k in K return (\lambda (k0:
496 K).((drop (r k0 j0) O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1
497 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 k0 t)))))) with [(Bind b)
498 \Rightarrow (\lambda (H1: (drop (r (Bind b) j0) O e2 e3)).(let H_x \def (IHj
499 e2 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop j0
500 O c1 c2)) (\lambda (c1: C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop
501 (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t))))
502 (\lambda (x: C).(\lambda (H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x
503 e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1:
504 C).(drop i (S j0) c1 (CHead e2 (Bind b) t))) (CHead x (Bind b) (lift i (r
505 (Bind b) j0) t)) (drop_drop (Bind b) j0 x c2 H3 (lift i (r (Bind b) j0) t))
506 (drop_skip (Bind b) i j0 x e2 H4 t))))) H2)))) | (Flat f) \Rightarrow
507 (\lambda (H1: (drop (r (Flat f) j0) O e2 e3)).(let H_x \def (IHe1 e3 H1 c2 i
508 H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (S j0) O c1 c2))
509 (\lambda (c1: C).(drop i (S j0) c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0)
510 O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat f) t))))
511 (\lambda (x: C).(\lambda (H3: (drop (S j0) O x c2)).(\lambda (H4: (drop i (S
512 j0) x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1:
513 C).(drop i (S j0) c1 (CHead e2 (Flat f) t))) (CHead x (Flat f) (lift i (r
514 (Flat f) j0) t)) (drop_drop (Flat f) j0 x c2 H3 (lift i (r (Flat f) j0) t))
515 (drop_skip (Flat f) i j0 x e2 H4 t))))) H2))))]) (drop_gen_drop k e2 e3 t j0
516 H))))))))))) e1)))) j).
518 theorem drop_trans_le:
519 \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
520 (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O
521 c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h
522 (minus d i) e1 e2)))))))))))
524 \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to
525 (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to
526 (\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1
527 e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d:
528 nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h:
529 nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O
530 c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2
531 (drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda
532 (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C
533 (\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1
534 (drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0:
535 nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1:
536 C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
537 C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda
538 (e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind
539 (\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2:
540 C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O
541 c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
542 C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0)
543 O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h
544 O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(let H2 \def
545 (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
546 \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h
547 (minus O (S i0)) e1 e2)))))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S
548 i0) O)).(let H3 \def (eq_ind nat (S i0) (\lambda (e: nat).(match e in nat
549 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
550 True])) I O H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1))
551 (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))) H3))) | (le_S m H2)
552 \Rightarrow (\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m)
553 (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
554 \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S
555 i0) m) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
556 C).(drop h (minus O (S i0)) e1 e2)))) H4)) H2))]) in (H2 (refl_equal nat
557 O)))))))))) (\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall
558 (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall
559 (e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1
560 e1)) (\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda
561 (H: (le (S i0) (S d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
562 C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0)
563 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1:
564 C).(drop h (minus (S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda
565 (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CSort n)
566 c2)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2
567 (CSort n)) (eq nat h O) (eq nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S
568 i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))
569 (\lambda (H2: (eq C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_:
570 (eq nat (S d0) O)).(let H5 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c
571 e2)) H1 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq
572 nat O O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1:
573 C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort
574 n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C
575 (CSort n) (\lambda (c: C).(ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n)
576 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 c)))) (let H9 \def
577 (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_:
578 nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in
579 (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda
580 (e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6))))
581 (drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0))))))))
582 (\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h
583 (S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda
584 (e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0))
585 e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
586 T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to
587 (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S
588 i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
589 e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h:
590 nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2:
591 C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e:
592 C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda
593 (v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_:
594 T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O
595 (CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
596 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0
597 (Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda
598 (H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c:
599 C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h
600 (r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O
601 (CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
602 e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop
603 h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2
604 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S
605 d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda
606 (H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0)
607 O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop
608 h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r
609 (Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2
610 (drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3
611 t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3:
612 C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t)
613 c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T
614 (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_:
615 C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e:
616 C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1:
617 C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus
618 (S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C
619 c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0)
620 x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3
621 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in
622 (eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1:
623 C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus
624 (S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1))
625 (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1:
626 C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1))
627 (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x:
628 C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S
629 i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f)
630 (lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S
631 i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1))
632 H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3)))))))
633 (drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i).
635 theorem drop_trans_ge:
636 \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
637 nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2
638 e2) \to ((le d i) \to (drop (plus i h) O c1 e2)))))))))
640 \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2:
641 C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
642 C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2))))))))))
643 (\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
644 nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O
645 c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h)
646 O c1 c)) (let H2 \def (match H1 in le return (\lambda (n: nat).(\lambda (_:
647 (le ? n)).((eq nat n O) \to (drop (plus O h) O c1 c2)))) with [le_n
648 \Rightarrow (\lambda (H0: (eq nat d O)).(eq_ind nat O (\lambda (_: nat).(drop
649 (plus O h) O c1 c2)) (let H2 \def (eq_ind nat d (\lambda (n: nat).(le n O))
650 H1 O H0) in (let H3 \def (eq_ind nat d (\lambda (n: nat).(drop h n c1 c2)) H
651 O H0) in H3)) d (sym_eq nat d O H0))) | (le_S m H0) \Rightarrow (\lambda (H2:
652 (eq nat (S m) O)).((let H1 \def (eq_ind nat (S m) (\lambda (e: nat).(match e
653 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
654 \Rightarrow True])) I O H2) in (False_ind ((le d m) \to (drop (plus O h) O c1
655 c2)) H1)) H0))]) in (H2 (refl_equal nat O))) e2 (drop_gen_refl c2 e2
656 H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall
657 (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall
658 (e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1
659 e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
660 C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2:
661 C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c
662 e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
663 nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0:
664 (drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort
665 n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda
666 (H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d
667 O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n)
668 e2)) (let H5 \def (eq_ind nat d (\lambda (n: nat).(le n (S i0))) H1 O H4) in
669 (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort n)
670 H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (S
671 (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda (H8:
672 (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda
673 (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind nat (S i0)
674 (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
675 \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind (drop (S
676 (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n (S i0) O
677 e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2:
678 C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h:
679 nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d
680 (S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda
681 (t: T).(\lambda (c3: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall
682 (h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O
683 c3 e2) \to ((le n (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t)
684 e2))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c2 k
685 t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to
686 (drop (S (plus i0 n)) O (CHead c2 k t) e2)))))) (\lambda (H: (drop O O (CHead
687 c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda
688 (_: (le O (S i0))).(let H2 \def (eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O
689 c e2)) H0 (CHead c2 k t) (drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat
690 i0 (\lambda (n: nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2
691 (drop_gen_drop k c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda
692 (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2:
693 C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O
694 (CHead c2 k t) e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t)
695 c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le
696 O (S i0))).(eq_ind nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O
697 (CHead c2 k t) e2)) (drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k
698 (plus i0 n))) (\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r
699 k n)) (\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n)
700 (drop_gen_drop k c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0
701 n)) (r k (S (plus i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm
702 i0 n)))))))) h)) (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop
703 h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le
704 d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h:
705 nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2:
706 C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S
707 i0))).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e k
708 v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda
709 (e: C).(\lambda (_: T).(drop h (r k d0) c2 e))) (drop (S (plus i0 h)) O
710 (CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3
711 (CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4:
712 (drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall
713 (h: nat).((drop h d0 (CHead c2 k t) c) \to (\forall (e2: C).((drop (S i0) O c
714 e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))
715 IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: C).(drop (S
716 i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t (\lambda (t:
717 T).(\forall (h: nat).((drop h d0 (CHead c2 k t) (CHead x0 k x1)) \to (\forall
718 (e2: C).((drop (S i0) O (CHead x0 k x1) e2) \to ((le d0 (S i0)) \to (drop (S
719 (plus i0 h)) O (CHead c2 k t) e2))))))) H5 (lift h (r k d0) x1) H3) in
720 (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S (plus i0 h)) O
721 (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind (\lambda (k0:
722 K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to (drop (r k0
723 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r (Bind b)
724 d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 (r (Bind
725 b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: F).(\lambda
726 (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0
727 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 (drop_gen_drop k x0 e2 x1
728 i0 H6)) (lift h (r k d0) x1)) t H3))))))))) (drop_gen_skip_l c2 c3 t h d0 k
729 H))))))))) d))))))) c1)))) i).
projects / helm.git / blob