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_1A/clear/fwd.ma".
19 include "basic_1A/drop/fwd.ma".
22 \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to
23 (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead
24 e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e
27 \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i:
28 nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e:
29 C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda
30 (e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda
31 (c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind
32 (eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b:
33 B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v)))))
34 (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda
35 (_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat
36 O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee with [O
37 \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (ex2_3 B
38 C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e
39 (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e
40 c2))))) H3))))) (drop_gen_sort n (S i) O c2 H)))))) (\lambda (c: C).(\lambda
41 (H: ((\forall (c2: C).(\forall (i: nat).((drop (S i) O c c2) \to (ex2_3 B C T
42 (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b)
43 v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e
44 c2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (i:
45 nat).(\lambda (H0: (drop (S i) O (CHead c k t) c2)).(K_ind (\lambda (k0:
46 K).((drop (r k0 i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e:
47 C).(\lambda (v: T).(clear (CHead c k0 t) (CHead e (Bind b) v))))) (\lambda
48 (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))))) (\lambda (b:
49 B).(\lambda (H1: (drop (r (Bind b) i) O c c2)).(ex2_3_intro B C T (\lambda
50 (b0: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Bind b) t) (CHead e
51 (Bind b0) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e
52 c2)))) b c t (clear_bind b c t) H1))) (\lambda (f: F).(\lambda (H1: (drop (r
53 (Flat f) i) O c c2)).(let H2 \def (H c2 i H1) in (ex2_3_ind B C T (\lambda
54 (b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) v)))))
55 (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C
56 T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t)
57 (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_:
58 T).(drop i O e c2))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2:
59 T).(\lambda (H3: (clear c (CHead x1 (Bind x0) x2))).(\lambda (H4: (drop i O
60 x1 c2)).(ex2_3_intro B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v:
61 T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) (\lambda (_:
62 B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) x0 x1 x2 (clear_flat c
63 (CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))) k (drop_gen_drop k c c2 t i
67 \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c
68 (CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1
69 e2) \to (drop (S i) O c e2))))))))
71 \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1:
72 C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2:
73 C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2))))))))
74 (\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort
75 n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_:
76 (drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O
77 (CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1:
78 C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2:
79 C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0
80 e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u:
81 T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2:
82 C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).(K_ind (\lambda (k0:
83 K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S i) O (CHead c0
84 k0 t) e2))) (\lambda (b0: B).(\lambda (H2: (clear (CHead c0 (Bind b0) t)
85 (CHead e1 (Bind b) u))).(let H3 \def (f_equal C C (\lambda (e: C).(match e
86 with [(CSort _) \Rightarrow e1 | (CHead c1 _ _) \Rightarrow c1])) (CHead e1
87 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b)
88 u) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e with [(CSort
89 _) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match k0 with [(Bind b1)
90 \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) u) (CHead c0
91 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H5
92 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _) \Rightarrow u |
93 (CHead _ _ t0) \Rightarrow t0])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t)
94 (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b
95 b0)).(\lambda (H7: (eq C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c1:
96 C).(drop i O c1 e2)) H1 c0 H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O
97 (CHead c0 (Bind b1) t) e2)) (drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4))
98 H3)))) (\lambda (f: F).(\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1
99 (Bind b) u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead
100 e1 (Bind b) u) t H2) e2 i H1) t))) k H0))))))))))) c)).
103 \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop
104 h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear
105 x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1
106 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))))))))
108 \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h:
109 nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2:
110 C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1:
111 C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
112 c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda
113 (d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda
114 (c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b)
115 u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1:
116 C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
117 c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h:
118 nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2:
119 C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1:
120 C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
121 c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h:
122 nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k
123 t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear
124 (CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1
125 (CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C
126 (\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1:
127 C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift
128 h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k
129 (lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead
130 c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) (K_ind
131 (\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to ((drop h
132 (r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h (r k0 d)
133 t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))
134 (\lambda (b0: B).(\lambda (H4: (clear (CHead c (Bind b0) t) (CHead c2 (Bind
135 b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let H6 \def (f_equal C C
136 (\lambda (e: C).(match e with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
137 \Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind
138 b0 c (CHead c2 (Bind b) u) t H4)) in ((let H7 \def (f_equal C B (\lambda (e:
139 C).(match e with [(CSort _) \Rightarrow b | (CHead _ k0 _) \Rightarrow (match
140 k0 with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (CHead c2
141 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u)
142 t H4)) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _)
143 \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead
144 c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda
145 (H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0:
146 T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d)
147 t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2))))
148 (eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind
149 b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda
150 (c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda
151 (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind
152 b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda
153 (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind
154 b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x
155 (lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6))))) (\lambda (f:
156 F).(\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda
157 (H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u
158 (clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1:
159 C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1
160 c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d)
161 t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))
162 (\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d
163 u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear
164 (CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d
165 u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b)
166 (lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))) k H1 H3) x1
167 H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2).