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 "LambdaDelta-1/csubc/fwd.ma".
19 include "LambdaDelta-1/sc3/props.ma".
21 theorem csubc_drop_conf_O:
22 \forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h
23 O c1 e1) \to (\forall (c2: C).((csubc g c1 c2) \to (ex2 C (\lambda (e2:
24 C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))))
26 \lambda (g: G).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (e1:
27 C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2)
28 \to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1
29 e2))))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (h: nat).(\lambda (H:
30 (drop h O (CSort n) e1)).(\lambda (c2: C).(\lambda (H0: (csubc g (CSort n)
31 c2)).(and3_ind (eq C e1 (CSort n)) (eq nat h O) (eq nat O O) (ex2 C (\lambda
32 (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (H1:
33 (eq C e1 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (_: (eq nat O
34 O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (e2: C).(drop n0 O c2
35 e2)) (\lambda (e2: C).(csubc g e1 e2)))) (eq_ind_r C (CSort n) (\lambda (c:
36 C).(ex2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2: C).(csubc g c
37 e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O c2 e2)) (\lambda (e2:
38 C).(csubc g (CSort n) e2)) c2 (drop_refl c2) H0) e1 H1) h H2))))
39 (drop_gen_sort n h O e1 H)))))))) (\lambda (c: C).(\lambda (H: ((\forall (e1:
40 C).(\forall (h: nat).((drop h O c e1) \to (\forall (c2: C).((csubc g c c2)
41 \to (ex2 C (\lambda (e2: C).(drop h O c2 e2)) (\lambda (e2: C).(csubc g e1
42 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (h:
43 nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e1) \to (\forall
44 (c2: C).((csubc g (CHead c k t) c2) \to (ex2 C (\lambda (e2: C).(drop n O c2
45 e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda (H0: (drop O O (CHead c
46 k t) e1)).(\lambda (c2: C).(\lambda (H1: (csubc g (CHead c k t) c2)).(eq_ind
47 C (CHead c k t) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop O O c2 e2))
48 (\lambda (e2: C).(csubc g c0 e2)))) (ex_intro2 C (\lambda (e2: C).(drop O O
49 c2 e2)) (\lambda (e2: C).(csubc g (CHead c k t) e2)) c2 (drop_refl c2) H1) e1
50 (drop_gen_refl (CHead c k t) e1 H0))))) (\lambda (n: nat).(\lambda (H0:
51 (((drop n O (CHead c k t) e1) \to (\forall (c2: C).((csubc g (CHead c k t)
52 c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g
53 e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2:
54 C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l
55 g c c2 t k H2) in (let H3 \def H_x in (or_ind (ex2 C (\lambda (c3: C).(eq C
56 c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda
57 (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
58 C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
59 (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda
60 (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda
61 (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex2 C (\lambda
62 (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda
63 (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) (\lambda (c3:
64 C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 k t)))
65 (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2: C).(drop (S n) O c2
66 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H5: (eq C
67 c2 (CHead x k t))).(\lambda (H6: (csubc g c x)).(eq_ind_r C (CHead x k t)
68 (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2:
69 C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k n) (drop_gen_drop k c e1 t n
70 H1) x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(drop (r k n) O
71 x e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n)
72 O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0:
73 C).(\lambda (H8: (drop (r k n) O x x0)).(\lambda (H9: (csubc g e1
74 x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda
75 (e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0 H8 t) H9)))) H7))) c2 H5))))
76 H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_:
77 A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
78 A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
79 T).(\lambda (_: A).(csubc g c c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
80 (a: A).(sc3 g (asucc g a) c t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda
81 (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_:
82 T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w:
83 T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
84 C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda (_: C).(\lambda
85 (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda (c3: C).(\lambda
86 (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(drop (S n)
87 O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (x1:
88 T).(\lambda (x2: A).(\lambda (H5: (eq K k (Bind Abst))).(\lambda (H6: (eq C
89 c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (csubc g c x0)).(\lambda (_:
90 (sc3 g (asucc g x2) c t)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C (CHead
91 x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop (S n) O c0
92 e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H10 \def (eq_ind K k (\lambda
93 (k0: K).(drop (r k0 n) O c e1)) (drop_gen_drop k c e1 t n H1) (Bind Abst) H5)
94 in (let H11 \def (eq_ind K k (\lambda (k0: K).((drop n O (CHead c k0 t) e1)
95 \to (\forall (c3: C).((csubc g (CHead c k0 t) c3) \to (ex2 C (\lambda (e2:
96 C).(drop n O c3 e2)) (\lambda (e2: C).(csubc g e1 e2))))))) H0 (Bind Abst)
97 H5) in (let H_x0 \def (H e1 (r (Bind Abst) n) H10 x0 H7) in (let H12 \def
98 H_x0 in (ex2_ind C (\lambda (e2: C).(drop (r (Bind Abst) n) O x0 e2))
99 (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) O
100 (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda
101 (x: C).(\lambda (H13: (drop (r (Bind Abst) n) O x0 x)).(\lambda (H14: (csubc
102 g e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr)
103 x1) e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x
104 H13 x1) H14)))) H12))))) c2 H6))))))))) H4)) H3)))))))) h))))))) c1)).
106 theorem drop_csubc_trans:
107 \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall
108 (h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
109 (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))))
111 \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2:
112 C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1:
113 C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda
114 (c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda
115 (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda
116 (e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat
117 h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1:
118 C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2:
119 (eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0:
120 nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g
121 (CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1:
122 C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def
123 (eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C
124 (\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1))
125 e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H)))))))))
126 (\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h:
127 nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C
128 (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c
129 c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d:
130 nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t)
131 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h
132 n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h:
133 nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall
134 (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1))
135 (\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O
136 (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2
137 \def (eq_ind_r C e2 (\lambda (c0: C).(csubc g c0 e1)) H1 (CHead c k t)
138 (drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O
139 O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1)
140 H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to
141 (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1
142 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop
143 (S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2
144 e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in
145 (let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1))
146 (\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1
147 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda
148 (H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C
149 (\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k
150 t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t)))))
151 H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
152 (CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda
153 (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t)
154 c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t)
155 e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda
156 (e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
157 T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k
158 n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
159 C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
160 (H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n)
161 x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda
162 (c0: C).(csubc g c0 e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2
163 (\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to
164 (\forall (e3: C).((csubc g c0 e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1
165 e3)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1)
166 H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0
167 n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g (CHead x0 k
168 x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1:
169 C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r
170 T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n)
171 c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def
172 (csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or_ind (ex2 C
173 (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0
174 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
175 (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1
176 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
177 A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g
178 (asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
179 a c3 w))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
180 C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (H10: (ex2 C
181 (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0
182 c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3:
183 C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda
184 (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x:
185 C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x0
186 x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop
187 h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1))
188 c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in
189 (ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g
190 c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
191 (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2:
192 C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c
193 x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
194 (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r
195 k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g c x2 H15 k (lift h (r k
196 n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_:
197 C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
198 C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w)))))
199 (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda
200 (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda
201 (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A
202 (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
203 (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind
204 Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0
205 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0
206 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))
207 (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g
208 (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: C).(\lambda (x3:
209 T).(\lambda (x4: A).(\lambda (H11: (eq K k (Bind Abst))).(\lambda (H12: (eq C
210 e1 (CHead x2 (Bind Abbr) x3))).(\lambda (H13: (csubc g x0 x2)).(\lambda (H14:
211 (sc3 g (asucc g x4) x0 x1)).(\lambda (H15: (sc3 g x4 x2 x3)).(eq_ind_r C
212 (CHead x2 (Bind Abbr) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S
213 n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))))
214 (let H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n
215 (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3:
216 C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1
217 e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1))))))))
218 H8 (Bind Abst) H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r
219 k0 n) c x0)) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0:
220 K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3)))
221 (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0
222 \def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind
223 C (\lambda (c1: C).(drop h (r (Bind Abst) n) c1 x2)) (\lambda (c1: C).(csubc
224 g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3)))
225 (\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1))
226 c1))) (\lambda (x: C).(\lambda (H19: (drop h (r (Bind Abst) n) x
227 x2)).(\lambda (H20: (csubc g c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S
228 n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1: C).(csubc g (CHead c (Bind
229 Abst) (lift h (r (Bind Abst) n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3))
230 (drop_skip_bind h n x x2 H19 Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind
231 Abst) n) x1) x4 (sc3_lift g (asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17)
232 (lift h n x3) (sc3_lift g x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1
233 H12))))))))) H10)) H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k
234 H1)))))))) d))))))) c2)).
236 theorem csubc_drop_conf_rev:
237 \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall
238 (h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
239 (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))))
241 \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2:
242 C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1:
243 C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda
244 (c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda
245 (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda
246 (e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat
247 h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1:
248 C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2:
249 (eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0:
250 nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1
251 (CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1:
252 C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def
253 (eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C
254 (\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n)))
255 e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H)))))))))
256 (\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h:
257 nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C
258 (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1
259 c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d:
260 nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t)
261 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h
262 n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h:
263 nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall
264 (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1))
265 (\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O
266 (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2
267 \def (eq_ind_r C e2 (\lambda (c0: C).(csubc g e1 c0)) H1 (CHead c k t)
268 (drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O
269 O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1)
270 H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to
271 (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1
272 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop
273 (S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1
274 e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in
275 (let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1))
276 (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1
277 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda
278 (H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C
279 (\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c
280 k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t)))))
281 H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
282 (CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda
283 (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k
284 t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t)
285 e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda
286 (e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v:
287 T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k
288 n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
289 C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
290 (H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n)
291 x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda
292 (c0: C).(csubc g e1 c0)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2
293 (\lambda (c0: C).(\forall (h0: nat).((drop h0 n (CHead c k t) c0) \to
294 (\forall (e3: C).((csubc g e3 c0) \to (ex2 C (\lambda (c1: C).(drop h0 n c1
295 e3)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1)
296 H3) in (let H8 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0
297 n (CHead c k t0) (CHead x0 k x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0
298 k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc
299 g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h
300 (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1))
301 (\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def
302 (csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or_ind (ex2 C
303 (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1
304 x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
305 (Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1
306 (CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_:
307 A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g
308 (asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a
309 x0 x1))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
310 C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C
311 (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1
312 x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1:
313 C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda
314 (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x:
315 C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x
316 x0)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1:
317 C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k
318 n) x1)))))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in
319 (ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g
320 c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
321 (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x2:
322 C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g x2
323 c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
324 (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))) (CHead x2 k (lift h (r
325 k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g x2 c H15 k (lift h (r k
326 n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_:
327 C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1:
328 C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v)))))
329 (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda
330 (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
331 (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))))).(ex5_3_ind C T A
332 (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
333 (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind
334 Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
335 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1
336 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))) (ex2
337 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead
338 c k (lift h (r k n) x1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
339 A).(\lambda (H11: (eq K k (Bind Abbr))).(\lambda (H12: (eq C e1 (CHead x2
340 (Bind Abst) x3))).(\lambda (H13: (csubc g x2 x0)).(\lambda (H14: (sc3 g
341 (asucc g x4) x2 x3)).(\lambda (H15: (sc3 g x4 x0 x1)).(eq_ind_r C (CHead x2
342 (Bind Abst) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1
343 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let
344 H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c
345 k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3
346 (CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda
347 (c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr)
348 H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5
349 (Bind Abbr) H11) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda
350 (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc
351 g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind
352 Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1:
353 C).(drop h (r (Bind Abbr) n) c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C
354 (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1:
355 C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))))
356 (\lambda (x: C).(\lambda (H19: (drop h (r (Bind Abbr) n) x x2)).(\lambda
357 (H20: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2
358 (Bind Abst) x3))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind Abbr) (lift h
359 (r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n x3)) (drop_skip_bind
360 h n x x2 H19 Abst x3) (csubc_abst g x c H20 (lift h n x3) x4 (sc3_lift g
361 (asucc g x4) x2 x3 H14 x h n H19) (lift h (r (Bind Abbr) n) x1) (sc3_lift g
362 x4 x0 x1 H15 c h (r (Bind Abbr) n) H17)))))) H18))) k H11))) e1 H12)))))))))
363 H10)) H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d)))))))