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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/arity/defs.ma".
19 include "LambdaDelta-1/cimp/props.ma".
21 theorem arity_cimp_conf:
22 \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
23 t a) \to (\forall (c2: C).((cimp c1 c2) \to (arity g c2 t a)))))))
25 \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
26 (arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
27 A).(\forall (c2: C).((cimp c c2) \to (arity g c2 t0 a0)))))) (\lambda (c:
28 C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (cimp c c2)).(arity_sort g
29 c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
30 nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr) u))).(\lambda (a0:
31 A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (c2: C).((cimp d
32 c2) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda (H3: (cimp c
33 c2)).(let H_x \def (H3 Abbr d u i H0) in (let H4 \def H_x in (ex_ind C
34 (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (arity g c2 (TLRef i)
35 a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abbr) u))).(let
36 H_x0 \def (cimp_getl_conf c c2 H3 Abbr d u i H0) in (let H6 \def H_x0 in
37 (ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2 (CHead
38 d2 (Bind Abbr) u))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (H7:
39 (cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(let H9 \def
40 (eq_ind C (CHead x (Bind Abbr) u) (\lambda (c0: C).(getl i c2 c0)) H5 (CHead
41 x0 (Bind Abbr) u) (getl_mono c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind
42 Abbr) u) H8)) in (let H10 \def (f_equal C C (\lambda (e: C).(match e in C
43 return (\lambda (_: C).C) with [(CSort _) \Rightarrow x | (CHead c0 _ _)
44 \Rightarrow c0])) (CHead x (Bind Abbr) u) (CHead x0 (Bind Abbr) u) (getl_mono
45 c2 (CHead x (Bind Abbr) u) i H5 (CHead x0 (Bind Abbr) u) H8)) in (let H11
46 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0 (Bind Abbr) u))) H9
47 x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0: C).(cimp d c0)) H7 x
48 H10) in (arity_abbr g c2 x u i H11 a0 (H2 x H12))))))))) H6)))))
49 H4))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
50 nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0:
51 A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (c2:
52 C).((cimp d c2) \to (arity g c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda
53 (H3: (cimp c c2)).(let H_x \def (H3 Abst d u i H0) in (let H4 \def H_x in
54 (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u))) (arity g c2
55 (TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x (Bind Abst)
56 u))).(let H_x0 \def (cimp_getl_conf c c2 H3 Abst d u i H0) in (let H6 \def
57 H_x0 in (ex2_ind C (\lambda (d2: C).(cimp d d2)) (\lambda (d2: C).(getl i c2
58 (CHead d2 (Bind Abst) u))) (arity g c2 (TLRef i) a0) (\lambda (x0:
59 C).(\lambda (H7: (cimp d x0)).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abst)
60 u))).(let H9 \def (eq_ind C (CHead x (Bind Abst) u) (\lambda (c0: C).(getl i
61 c2 c0)) H5 (CHead x0 (Bind Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i
62 H5 (CHead x0 (Bind Abst) u) H8)) in (let H10 \def (f_equal C C (\lambda (e:
63 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow x |
64 (CHead c0 _ _) \Rightarrow c0])) (CHead x (Bind Abst) u) (CHead x0 (Bind
65 Abst) u) (getl_mono c2 (CHead x (Bind Abst) u) i H5 (CHead x0 (Bind Abst) u)
66 H8)) in (let H11 \def (eq_ind_r C x0 (\lambda (c0: C).(getl i c2 (CHead c0
67 (Bind Abst) u))) H9 x H10) in (let H12 \def (eq_ind_r C x0 (\lambda (c0:
68 C).(cimp d c0)) H7 x H10) in (arity_abst g c2 x u i H11 a0 (H2 x H12)))))))))
69 H6))))) H4))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b
70 Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity
71 g c u a1)).(\lambda (H2: ((\forall (c2: C).((cimp c c2) \to (arity g c2 u
72 a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
73 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((cimp (CHead c (Bind b)
74 u) c2) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (cimp c
75 c2)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b) u)
76 (cimp_bind c c2 H5 b u)))))))))))))))) (\lambda (c: C).(\lambda (u:
77 T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda (H1:
78 ((\forall (c2: C).((cimp c c2) \to (arity g c2 u (asucc g a1)))))).(\lambda
79 (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c (Bind Abst) u) t0
80 a2)).(\lambda (H3: ((\forall (c2: C).((cimp (CHead c (Bind Abst) u) c2) \to
81 (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4: (cimp c
82 c2)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind Abst) u)
83 (cimp_bind c c2 H4 Abst u)))))))))))))) (\lambda (c: C).(\lambda (u:
84 T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H1: ((\forall
85 (c2: C).((cimp c c2) \to (arity g c2 u a1))))).(\lambda (t0: T).(\lambda (a2:
86 A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (c2:
87 C).((cimp c c2) \to (arity g c2 t0 (AHead a1 a2)))))).(\lambda (c2:
88 C).(\lambda (H4: (cimp c c2)).(arity_appl g c2 u a1 (H1 c2 H4) t0 a2 (H3 c2
89 H4))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_:
90 (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (c2: C).((cimp c c2) \to
91 (arity g c2 u (asucc g a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0
92 a0)).(\lambda (H3: ((\forall (c2: C).((cimp c c2) \to (arity g c2 t0
93 a0))))).(\lambda (c2: C).(\lambda (H4: (cimp c c2)).(arity_cast g c2 u a0 (H1
94 c2 H4) t0 (H3 c2 H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda
95 (a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (c2:
96 C).((cimp c c2) \to (arity g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2:
97 (leq g a1 a2)).(\lambda (c2: C).(\lambda (H3: (cimp c c2)).(arity_repl g c2
98 t0 a1 (H1 c2 H3) a2 H2)))))))))) c1 t a H))))).