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/pr3/defs.ma".
19 include "LambdaDelta-1/pr2/subst1.ma".
22 \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c
23 (CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2)
24 \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c
25 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
27 \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
28 (H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2:
29 T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0:
30 T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1
31 w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1:
32 T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1
33 w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda
34 (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5:
35 T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v
36 t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i
37 v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T
38 (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T
39 (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2)))
40 (\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3
41 x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5
42 w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5
43 w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v
44 t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1
45 i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c
46 e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))).
48 theorem pr3_gen_cabbr:
49 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
50 (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
51 \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
52 a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T
53 (\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a
56 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
57 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u:
58 T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0:
59 C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall
60 (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1
61 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)))))))))))))))
62 (\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda
63 (_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_:
64 (csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0
65 a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d
66 x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2)))
67 (\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda
68 (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4:
69 T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u:
70 T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0:
71 C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall
72 (x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2:
73 T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1
74 x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda
75 (H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4:
76 (csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0
77 a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d
78 x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda
79 (x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d
80 x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1
81 d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda
82 (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2))
83 (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2:
84 T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O)
85 d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d
86 u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x
87 x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0
88 e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))).