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 include "Basic-1/drop/defs.ma".
19 theorem drop_gen_sort:
20 \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop
21 h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O))))))
23 \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (x:
24 C).(\lambda (H: (drop h d (CSort n) x)).(insert_eq C (CSort n) (\lambda (c:
25 C).(drop h d c x)) (\lambda (c: C).(and3 (eq C x c) (eq nat h O) (eq nat d
26 O))) (\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0:
27 nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n))
28 \to (and3 (eq C c0 c) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c:
29 C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e:
30 C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C
31 c0 c0) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort n)) (eq
32 nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O) (refl_equal
33 nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c: C).(\lambda
34 (e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c (CSort n))
35 \to (and3 (eq C e c) (eq nat (r k h0) O) (eq nat O O))))).(\lambda (u:
36 T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4 \def (eq_ind C
37 (CHead c k u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
38 with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
39 (CSort n) H3) in (False_ind (and3 (eq C e (CHead c k u)) (eq nat (S h0) O)
40 (eq nat O O)) H4)))))))))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (d0:
41 nat).(\lambda (c: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k d0) c
42 e)).(\lambda (_: (((eq C c (CSort n)) \to (and3 (eq C e c) (eq nat h0 O) (eq
43 nat (r k d0) O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k (lift h0
44 (r k d0) u)) (CSort n))).(let H4 \def (eq_ind C (CHead c k (lift h0 (r k d0)
45 u)) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort
46 _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H3) in
47 (False_ind (and3 (eq C (CHead e k u) (CHead c k (lift h0 (r k d0) u))) (eq
48 nat h0 O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) H))))).
53 theorem drop_gen_refl:
54 \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e)))
56 \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O
57 (\lambda (n: nat).(drop n O x e)) (\lambda (_: nat).(eq C x e)) (\lambda (y:
58 nat).(\lambda (H0: (drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y
59 n x e)) (\lambda (n: nat).((eq nat y n) \to (eq C x e))) (\lambda (y0:
60 nat).(\lambda (H1: (drop y y0 x e)).(drop_ind (\lambda (n: nat).(\lambda (n0:
61 nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to ((eq nat n n0) \to
62 (eq C c c0))))))) (\lambda (c: C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq
63 nat O O)).(refl_equal C c)))) (\lambda (k: K).(\lambda (h: nat).(\lambda (c:
64 C).(\lambda (e0: C).(\lambda (_: (drop (r k h) O c e0)).(\lambda (_: (((eq
65 nat O O) \to ((eq nat (r k h) O) \to (eq C c e0))))).(\lambda (u: T).(\lambda
66 (_: (eq nat O O)).(\lambda (H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S
67 h) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
68 \Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (eq C
69 (CHead c k u) e0) H6))))))))))) (\lambda (k: K).(\lambda (h: nat).(\lambda
70 (d: nat).(\lambda (c: C).(\lambda (e0: C).(\lambda (H2: (drop h (r k d) c
71 e0)).(\lambda (H3: (((eq nat (r k d) O) \to ((eq nat h (r k d)) \to (eq C c
72 e0))))).(\lambda (u: T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat
73 h (S d))).(let H6 \def (f_equal nat nat (\lambda (e1: nat).e1) h (S d) H5) in
74 (let H7 \def (eq_ind nat h (\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat
75 n (r k d)) \to (eq C c e0)))) H3 (S d) H6) in (let H8 \def (eq_ind nat h
76 (\lambda (n: nat).(drop n (r k d) c e0)) H2 (S d) H6) in (eq_ind_r nat (S d)
77 (\lambda (n: nat).(eq C (CHead c k (lift n (r k d) u)) (CHead e0 k u))) (let
78 H9 \def (eq_ind nat (S d) (\lambda (ee: nat).(match ee in nat return (\lambda
79 (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4)
80 in (False_ind (eq C (CHead c k (lift (S d) (r k d) u)) (CHead e0 k u)) H9)) h
81 H6)))))))))))))) y y0 x e H1))) H0))) H))).
86 theorem drop_gen_drop:
87 \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h:
88 nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x))))))
90 \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h:
91 nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u)
92 (\lambda (c0: C).(drop (S h) O c0 x)) (\lambda (_: C).(drop (r k h) O c x))
93 (\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda
94 (n: nat).(drop (S h) n y x)) (\lambda (n: nat).((eq C y (CHead c k u)) \to
95 (drop (r k h) n c x))) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y
96 x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) (\lambda (_:
97 nat).((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) y0 c x))))
98 (\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n:
99 nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h))
100 \to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) n0 c
101 c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq
102 nat O O)).(\lambda (H5: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u)
103 (\lambda (c1: C).(drop (r k h) O c c1)) (let H6 \def (eq_ind nat O (\lambda
104 (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
105 True | (S _) \Rightarrow False])) I (S h) H3) in (False_ind (drop (r k h) O c
106 (CHead c k u)) H6)) c0 H5))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda
107 (c0: C).(\lambda (e: C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (H4:
108 (((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c0 (CHead c k u)) \to
109 (drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S
110 h))).(\lambda (_: (eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c
111 k u))).(let H8 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
112 (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _)
113 \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H7) in ((let H9 \def
114 (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with
115 [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0)
116 (CHead c k u) H7) in ((let H10 \def (f_equal C T (\lambda (e0: C).(match e0
117 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
118 \Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H7) in (\lambda (H11: (eq K
119 k0 k)).(\lambda (H12: (eq C c0 c)).(let H13 \def (eq_ind C c0 (\lambda (c1:
120 C).((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c1 (CHead c k u))
121 \to (drop (r k h) O c e))))) H4 c H12) in (let H14 \def (eq_ind C c0 (\lambda
122 (c1: C).(drop (r k0 h0) O c1 e)) H3 c H12) in (let H15 \def (eq_ind K k0
123 (\lambda (k1: K).((eq nat (r k1 h0) (S h)) \to ((eq nat O O) \to ((eq C c
124 (CHead c k u)) \to (drop (r k h) O c e))))) H13 k H11) in (let H16 \def
125 (eq_ind K k0 (\lambda (k1: K).(drop (r k1 h0) O c e)) H14 k H11) in (let H17
126 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_:
127 nat).nat) with [O \Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H5) in
128 (let H18 \def (eq_ind nat h0 (\lambda (n: nat).((eq nat (r k n) (S h)) \to
129 ((eq nat O O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) H15 h
130 H17) in (let H19 \def (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e))
131 H16 h H17) in H19)))))))))) H9)) H8)))))))))))) (\lambda (k0: K).(\lambda
132 (h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3:
133 (drop h0 (r k0 d) c0 e)).(\lambda (H4: (((eq nat h0 (S h)) \to ((eq nat (r k0
134 d) O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c
135 e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat h0 (S h))).(\lambda (H6: (eq
136 nat (S d) O)).(\lambda (H7: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead
137 c k u))).(let H8 \def (eq_ind nat h0 (\lambda (n: nat).(eq C (CHead c0 k0
138 (lift n (r k0 d) u0)) (CHead c k u))) H7 (S h) H5) in (let H9 \def (eq_ind
139 nat h0 (\lambda (n: nat).((eq nat n (S h)) \to ((eq nat (r k0 d) O) \to ((eq
140 C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H4 (S h) H5) in (let
141 H10 \def (eq_ind nat h0 (\lambda (n: nat).(drop n (r k0 d) c0 e)) H3 (S h)
142 H5) in (let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
143 (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _)
144 \Rightarrow c1])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in
145 ((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda
146 (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
147 (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in ((let H13 \def
148 (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
149 [(CSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t:
150 T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
151 \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | false
152 \Rightarrow (f i)])) | (THead k1 u1 t0) \Rightarrow (THead k1 (lref_map f d0
153 u1) (lref_map f (s k1 d0) t0))]) in lref_map) (\lambda (x0: nat).(plus x0 (S
154 h))) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 (lift (S h)
155 (r k0 d) u0)) (CHead c k u) H8) in (\lambda (H14: (eq K k0 k)).(\lambda (H15:
156 (eq C c0 c)).(let H16 \def (eq_ind C c0 (\lambda (c1: C).((eq nat (S h) (S
157 h)) \to ((eq nat (r k0 d) O) \to ((eq C c1 (CHead c k u)) \to (drop (r k h)
158 (r k0 d) c e))))) H9 c H15) in (let H17 \def (eq_ind C c0 (\lambda (c1:
159 C).(drop (S h) (r k0 d) c1 e)) H10 c H15) in (let H18 \def (eq_ind K k0
160 (\lambda (k1: K).(eq T (lift (S h) (r k1 d) u0) u)) H13 k H14) in (let H19
161 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (S h) (S h)) \to ((eq nat (r k1
162 d) O) \to ((eq C c (CHead c k u)) \to (drop (r k h) (r k1 d) c e))))) H16 k
163 H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(drop (S h) (r k1 d) c
164 e)) H17 k H14) in (eq_ind_r K k (\lambda (k1: K).(drop (r k h) (S d) c (CHead
165 e k1 u0))) (let H21 \def (eq_ind_r T u (\lambda (t: T).((eq nat (S h) (S h))
166 \to ((eq nat (r k d) O) \to ((eq C c (CHead c k t)) \to (drop (r k h) (r k d)
167 c e))))) H19 (lift (S h) (r k d) u0) H18) in (let H22 \def (eq_ind nat (S d)
168 (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
169 \Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind (drop (r
170 k h) (S d) c (CHead e k u0)) H22))) k0 H14))))))))) H12)) H11))))))))))))))))
171 y1 y0 y x H2))) H1))) H0))) H)))))).
176 theorem drop_gen_skip_r:
177 \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall
178 (d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda
179 (e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k
182 \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda
183 (d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k
184 u))).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) x c0))
185 (\lambda (_: C).(ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d)
186 u)))) (\lambda (e: C).(drop h (r k d) e c)))) (\lambda (y: C).(\lambda (H0:
187 (drop h (S d) x y)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n x y))
188 (\lambda (_: nat).((eq C y (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x
189 (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c)))))
190 (\lambda (y0: nat).(\lambda (H1: (drop h y0 x y)).(drop_ind (\lambda (n:
191 nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n0 (S d))
192 \to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C c0 (CHead e k
193 (lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))))) (\lambda
194 (c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C c0 (CHead c k
195 u))).(eq_ind_r C (CHead c k u) (\lambda (c1: C).(ex2 C (\lambda (e: C).(eq C
196 c1 (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))))
197 (let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
198 (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
199 I (S d) H2) in (False_ind (ex2 C (\lambda (e: C).(eq C (CHead c k u) (CHead e
200 k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))) H4)) c0 H3))))
201 (\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda
202 (H2: (drop (r k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C e
203 (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0
204 h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0
205 c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq C
206 e (CHead c k u))).(let H6 \def (eq_ind C e (\lambda (c1: C).((eq nat O (S d))
207 \to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k
208 (lift (r k0 h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0
209 c)))))) H3 (CHead c k u) H5) in (let H7 \def (eq_ind C e (\lambda (c1:
210 C).(drop (r k0 h0) O c0 c1)) H2 (CHead c k u) H5) in (let H8 \def (eq_ind nat
211 O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
212 \Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex2
213 C (\lambda (e0: C).(eq C (CHead c0 k0 u0) (CHead e0 k (lift (S h0) (r k d)
214 u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c))) H8))))))))))))) (\lambda
215 (k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e:
216 C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0)
217 (S d)) \to ((eq C e (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0
218 (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0
219 c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda (H5:
220 (eq C (CHead e k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0:
221 C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e |
222 (CHead c1 _ _) \Rightarrow c1])) (CHead e k0 u0) (CHead c k u) H5) in ((let
223 H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_:
224 C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
225 (CHead e k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0:
226 C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
227 (CHead _ _ t) \Rightarrow t])) (CHead e k0 u0) (CHead c k u) H5) in (\lambda
228 (H9: (eq K k0 k)).(\lambda (H10: (eq C e c)).(eq_ind_r T u (\lambda (t:
229 T).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k0 (lift h0 (r k0 d0) t)) (CHead
230 e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))) (let
231 H11 \def (eq_ind C e (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1
232 (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k
233 d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H3 c H10) in (let H12
234 \def (eq_ind C e (\lambda (c1: C).(drop h0 (r k0 d0) c0 c1)) H2 c H10) in
235 (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to
236 ((eq C c (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k
237 (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H11 k H9)
238 in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c0 c)) H12
239 k H9) in (eq_ind_r K k (\lambda (k1: K).(ex2 C (\lambda (e0: C).(eq C (CHead
240 c0 k1 (lift h0 (r k1 d0) u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0:
241 C).(drop h0 (r k d) e0 c)))) (let H15 \def (f_equal nat nat (\lambda (e0:
242 nat).(match e0 in nat return (\lambda (_: nat).nat) with [O \Rightarrow d0 |
243 (S n) \Rightarrow n])) (S d0) (S d) H4) in (let H16 \def (eq_ind nat d0
244 (\lambda (n: nat).((eq nat (r k n) (S d)) \to ((eq C c (CHead c k u)) \to
245 (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) (\lambda
246 (e0: C).(drop h0 (r k d) e0 c)))))) H13 d H15) in (let H17 \def (eq_ind nat
247 d0 (\lambda (n: nat).(drop h0 (r k n) c0 c)) H14 d H15) in (eq_ind_r nat d
248 (\lambda (n: nat).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k n)
249 u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0
250 c)))) (ex_intro2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k d) u))
251 (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c))
252 c0 (refl_equal C (CHead c0 k (lift h0 (r k d) u))) H17) d0 H15)))) k0 H9)))))
253 u0 H8)))) H7)) H6)))))))))))) h y0 x y H1))) H0))) H))))))).
258 theorem drop_gen_skip_l:
259 \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall
260 (d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T
261 (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_:
262 C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_:
263 T).(drop h (r k d) c e))))))))))
265 \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda
266 (d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u)
267 x)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) c0 x)) (\lambda
268 (_: C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v))))
269 (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e:
270 C).(\lambda (_: T).(drop h (r k d) c e))))) (\lambda (y: C).(\lambda (H0:
271 (drop h (S d) y x)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n y x))
272 (\lambda (_: nat).((eq C y (CHead c k u)) \to (ex3_2 C T (\lambda (e:
273 C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v:
274 T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k
275 d) c e)))))) (\lambda (y0: nat).(\lambda (H1: (drop h y0 y x)).(drop_ind
276 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq
277 nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e:
278 C).(\lambda (v: T).(eq C c1 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
279 T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop n (r k
280 d) c e)))))))))) (\lambda (c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda
281 (H3: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) (\lambda (c1:
282 C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C c1 (CHead e k v))))
283 (\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e:
284 C).(\lambda (_: T).(drop O (r k d) c e))))) (let H4 \def (eq_ind nat O
285 (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
286 \Rightarrow True | (S _) \Rightarrow False])) I (S d) H2) in (False_ind
287 (ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C (CHead c k u) (CHead e k
288 v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda
289 (e: C).(\lambda (_: T).(drop O (r k d) c e)))) H4)) c0 H3)))) (\lambda (k0:
290 K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop (r
291 k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C c0 (CHead c k u))
292 \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
293 (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v))))
294 (\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c
295 e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq
296 C (CHead c0 k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0:
297 C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
298 (CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H5) in ((let
299 H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_:
300 C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
301 (CHead c0 k0 u0) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda
302 (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow
303 u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H5) in
304 (\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind
305 C c0 (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k u)) \to
306 (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
307 (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v))))
308 (\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c e0))))))) H3 c
309 H10) in (let H12 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e))
310 H2 c H10) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat O (S d))
311 \to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v:
312 T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r
313 k1 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (r k1 h0) (r k d)
314 c e0))))))) H11 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop (r
315 k1 h0) O c e)) H12 k H9) in (let H15 \def (eq_ind nat O (\lambda (ee:
316 nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
317 | (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex3_2 C T (\lambda
318 (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda
319 (v: T).(eq T u (lift (S h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_:
320 T).(drop (S h0) (r k d) c e0)))) H15))))))))) H7)) H6))))))))))) (\lambda
321 (k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e:
322 C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0)
323 (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda
324 (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u
325 (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c
326 e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda
327 (H5: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(let H6 \def
328 (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
329 [(CSort _) \Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0
330 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in ((let H7 \def (f_equal C K
331 (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with [(CSort _)
332 \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 (lift h0 (r k0
333 d0) u0)) (CHead c k u) H5) in ((let H8 \def (f_equal C T (\lambda (e0:
334 C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow ((let
335 rec lref_map (f: ((nat \to nat))) (d1: nat) (t: T) on t: T \def (match t with
336 [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
337 d1) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u1 t0)
338 \Rightarrow (THead k1 (lref_map f d1 u1) (lref_map f (s k1 d1) t0))]) in
339 lref_map) (\lambda (x0: nat).(plus x0 h0)) (r k0 d0) u0) | (CHead _ _ t)
340 \Rightarrow t])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in
341 (\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind
342 C c0 (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead c k u))
343 \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
344 (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0:
345 C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H3 c H10) in (let H12 \def
346 (eq_ind C c0 (\lambda (c1: C).(drop h0 (r k0 d0) c1 e)) H2 c H10) in (let H13
347 \def (eq_ind K k0 (\lambda (k1: K).(eq T (lift h0 (r k1 d0) u0) u)) H8 k H9)
348 in (let H14 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to
349 ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C
350 e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d)
351 v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H11 k H9)
352 in (let H15 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c e)) H12 k
353 H9) in (eq_ind_r K k (\lambda (k1: K).(ex3_2 C T (\lambda (e0: C).(\lambda
354 (v: T).(eq C (CHead e k1 u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
355 T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0
356 (r k d) c e0))))) (let H16 \def (eq_ind_r T u (\lambda (t: T).((eq nat (r k
357 d0) (S d)) \to ((eq C c (CHead c k t)) \to (ex3_2 C T (\lambda (e0:
358 C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
359 T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0
360 (r k d) c e0))))))) H14 (lift h0 (r k d0) u0) H13) in (eq_ind T (lift h0 (r k
361 d0) u0) (\lambda (t: T).(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C
362 (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
363 (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c
364 e0))))) (let H17 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat
365 return (\lambda (_: nat).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n]))
366 (S d0) (S d) H4) in (let H18 \def (eq_ind nat d0 (\lambda (n: nat).((eq nat
367 (r k n) (S d)) \to ((eq C c (CHead c k (lift h0 (r k n) u0))) \to (ex3_2 C T
368 (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_:
369 C).(\lambda (v: T).(eq T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda
370 (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H16 d H17) in (let H19
371 \def (eq_ind nat d0 (\lambda (n: nat).(drop h0 (r k n) c e)) H15 d H17) in
372 (eq_ind_r nat d (\lambda (n: nat).(ex3_2 C T (\lambda (e0: C).(\lambda (v:
373 T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq
374 T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_:
375 T).(drop h0 (r k d) c e0))))) (ex3_2_intro C T (\lambda (e0: C).(\lambda (v:
376 T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq
377 T (lift h0 (r k d) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_:
378 T).(drop h0 (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal T
379 (lift h0 (r k d) u0)) H19) d0 H17)))) u H13)) k0 H9))))))))) H7))
380 H6)))))))))))) h y0 y x H1))) H0))) H))))))).