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4 (* ||A|| A project by Andrea Asperti *)
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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 *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr3/subst1".
19 include "pr3/defs.ma".
21 include "pr2/subst1.ma".
24 \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c
25 (CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2)
26 \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr3 c
27 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
29 \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
30 (H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2:
31 T).(\lambda (H0: (pr3 c t1 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0:
32 T).(\forall (w1: T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1
33 w2)) (\lambda (w2: T).(subst1 i v t0 w2))))))) (\lambda (t: T).(\lambda (w1:
34 T).(\lambda (H1: (subst1 i v t w1)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1
35 w2)) (\lambda (w2: T).(subst1 i v t w2)) w1 (pr3_refl c w1) H1)))) (\lambda
36 (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 c t4 t3)).(\lambda (t5:
37 T).(\lambda (_: (pr3 c t3 t5)).(\lambda (H3: ((\forall (w1: T).((subst1 i v
38 t3 w1) \to (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i
39 v t5 w2))))))).(\lambda (w1: T).(\lambda (H4: (subst1 i v t4 w1)).(ex2_ind T
40 (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T
41 (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5 w2)))
42 (\lambda (x: T).(\lambda (H5: (pr2 c w1 x)).(\lambda (H6: (subst1 i v t3
43 x)).(ex2_ind T (\lambda (w2: T).(pr3 c x w2)) (\lambda (w2: T).(subst1 i v t5
44 w2)) (ex2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1 i v t5
45 w2))) (\lambda (x0: T).(\lambda (H7: (pr3 c x x0)).(\lambda (H8: (subst1 i v
46 t5 x0)).(ex_intro2 T (\lambda (w2: T).(pr3 c w1 w2)) (\lambda (w2: T).(subst1
47 i v t5 w2)) x0 (pr3_sing c x w1 H5 x0 H7) H8)))) (H3 x H6))))) (pr2_subst1 c
48 e v i H t4 t3 H1 w1 H4)))))))))) t1 t2 H0)))))))).
50 theorem pr3_gen_cabbr:
51 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
52 (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
53 \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
54 a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T
55 (\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a
58 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
59 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (e: C).(\forall (u:
60 T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0:
61 C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall
62 (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1
63 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)))))))))))))))
64 (\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda
65 (_: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (_:
66 (csubst1 d u c a0)).(\lambda (a: C).(\lambda (_: (drop (S O) d a0
67 a)).(\lambda (x1: T).(\lambda (H3: (subst1 d u t (lift (S O) d
68 x1))).(ex_intro2 T (\lambda (x2: T).(subst1 d u t (lift (S O) d x2)))
69 (\lambda (x2: T).(pr3 a x1 x2)) x1 H3 (pr3_refl a x1))))))))))))) (\lambda
70 (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4:
71 T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: ((\forall (e: C).(\forall (u:
72 T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0:
73 C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall
74 (x1: T).((subst1 d u t0 (lift (S O) d x1)) \to (ex2 T (\lambda (x2:
75 T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1
76 x2))))))))))))))).(\lambda (e: C).(\lambda (u: T).(\lambda (d: nat).(\lambda
77 (H3: (getl d c (CHead e (Bind Abbr) u))).(\lambda (a0: C).(\lambda (H4:
78 (csubst1 d u c a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d a0
79 a)).(\lambda (x1: T).(\lambda (H6: (subst1 d u t3 (lift (S O) d
80 x1))).(ex2_ind T (\lambda (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda
81 (x2: T).(pr2 a x1 x2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d
82 x2))) (\lambda (x2: T).(pr3 a x1 x2))) (\lambda (x: T).(\lambda (H7: (subst1
83 d u t0 (lift (S O) d x))).(\lambda (H8: (pr2 a x1 x)).(ex2_ind T (\lambda
84 (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x x2))
85 (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2:
86 T).(pr3 a x1 x2))) (\lambda (x0: T).(\lambda (H9: (subst1 d u t4 (lift (S O)
87 d x0))).(\lambda (H10: (pr3 a x x0)).(ex_intro2 T (\lambda (x2: T).(subst1 d
88 u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr3 a x1 x2)) x0 H9 (pr3_sing a x
89 x1 H8 x0 H10))))) (H2 e u d H3 a0 H4 a H5 x H7))))) (pr2_gen_cabbr c t3 t0 H0
90 e u d H3 a0 H4 a H5 x1 H6)))))))))))))))))) t1 t2 H)))).