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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 *)
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15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/csubst1/defs.ma".
19 include "LambdaDelta-1/csubst0/fwd.ma".
21 include "LambdaDelta-1/subst1/props.ma".
23 theorem csubst1_gen_head:
24 \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
25 (v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T
26 C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2:
27 T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2:
28 C).(csubst1 i v c1 c2))))))))))
30 \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
31 (v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1)
32 x)).(let H0 \def (match H in csubst1 return (\lambda (c: C).(\lambda (_:
33 (csubst1 ? ? ? c)).((eq C c x) \to (ex3_2 T C (\lambda (u2: T).(\lambda (c2:
34 C).(eq C x (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1
35 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 c2))))))) with
36 [csubst1_refl \Rightarrow (\lambda (H0: (eq C (CHead c1 k u1) x)).(eq_ind C
37 (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c2:
38 C).(eq C c (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1
39 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 c2))))) (ex3_2_intro T
40 C (\lambda (u2: T).(\lambda (c2: C).(eq C (CHead c1 k u1) (CHead c2 k u2))))
41 (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
42 T).(\lambda (c2: C).(csubst1 i v c1 c2))) u1 c1 (refl_equal C (CHead c1 k
43 u1)) (subst1_refl i v u1) (csubst1_refl i v c1)) x H0)) | (csubst1_sing c2
44 H0) \Rightarrow (\lambda (H1: (eq C c2 x)).(eq_ind C x (\lambda (c:
45 C).((csubst0 (s k i) v (CHead c1 k u1) c) \to (ex3_2 T C (\lambda (u2:
46 T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_:
47 C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1
48 c3)))))) (\lambda (H2: (csubst0 (s k i) v (CHead c1 k u1) x)).(or3_ind (ex3_2
49 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda
50 (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2:
51 T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_:
52 C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_:
53 nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j
54 v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j:
55 nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda
56 (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_:
57 C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3:
58 C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C (\lambda (u2:
59 T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_:
60 C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1
61 c3)))) (\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat
62 (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k
63 u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))))).(ex3_2_ind T
64 nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda
65 (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2:
66 T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda (u2:
67 T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_:
68 C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1
69 c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq nat (s k i) (s k
70 x1))).(\lambda (H5: (eq C x (CHead c1 k x0))).(\lambda (H6: (subst0 x1 v u1
71 x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2 T C (\lambda (u2:
72 T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_:
73 C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1
74 c3))))) (let H7 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v u1 x0))
75 H6 i (s_inj k i x1 H4)) in (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3:
76 C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_:
77 C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1
78 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single i v u1 x0 H7)
79 (csubst1_refl i v c1))) x H5)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda
80 (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda
81 (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j:
82 nat).(csubst0 j v c1 c3))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j:
83 nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x
84 (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))
85 (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2))))
86 (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
87 T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1:
88 nat).(\lambda (H4: (eq nat (s k i) (s k x1))).(\lambda (H5: (eq C x (CHead x0
89 k u1))).(\lambda (H6: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1)
90 (\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead
91 c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda
92 (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H7 \def (eq_ind_r nat x1
93 (\lambda (n: nat).(csubst0 n v c1 x0)) H6 i (s_inj k i x1 H4)) in
94 (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x0 k u1)
95 (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
96 (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 (refl_equal C
97 (CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 H7))) x
98 H5)))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_:
99 C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda
100 (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2:
101 T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_:
102 T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))).(ex4_3_ind T C
103 nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k
104 j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3
105 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1
106 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1
107 c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k
108 u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_:
109 T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1:
110 C).(\lambda (x2: nat).(\lambda (H4: (eq nat (s k i) (s k x2))).(\lambda (H5:
111 (eq C x (CHead x1 k x0))).(\lambda (H6: (subst0 x2 v u1 x0)).(\lambda (H7:
112 (csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C
113 (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2:
114 T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3:
115 C).(csubst1 i v c1 c3))))) (let H8 \def (eq_ind_r nat x2 (\lambda (n:
116 nat).(csubst0 n v c1 x1)) H7 i (s_inj k i x2 H4)) in (let H9 \def (eq_ind_r
117 nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H6 i (s_inj k i x2 H4)) in
118 (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0)
119 (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2)))
120 (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C
121 (CHead x1 k x0)) (subst1_single i v u1 x0 H9) (csubst1_sing i v c1 x1 H8))))
122 x H5)))))))) H3)) (csubst0_gen_head k c1 x u1 v (s k i) H2))) c2 (sym_eq C c2
123 x H1) H0))]) in (H0 (refl_equal C x))))))))).