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/cimp/defs.ma".
19 include "LambdaDelta-1/getl/getl.ma".
22 \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v)
25 \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1:
26 C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f)
27 v) (CHead d1 (Bind b) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c (Flat
28 f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c (CHead d2
29 (Bind b) w)))))) (\lambda (H0: (getl O (CHead c (Flat f) v) (CHead d1 (Bind
30 b) w))).(ex_intro C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) w))) d1
31 (getl_intro O c (CHead d1 (Bind b) w) c (drop_refl c) (clear_gen_flat f c
32 (CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f) v) (CHead d1 (Bind b)
33 w) H0))))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c (Flat f) v)
34 (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h0 c (CHead d2 (Bind
35 b) w))))))).(\lambda (H0: (getl (S h0) (CHead c (Flat f) v) (CHead d1 (Bind
36 b) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) c (CHead d2 (Bind b) w)))
37 d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v h0 H0))))) h H)))))))).
40 \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f)
43 \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1:
44 C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h c (CHead d1 (Bind
45 b) w))).(ex_intro C (\lambda (d2: C).(getl h (CHead c (Flat f) v) (CHead d2
46 (Bind b) w))) d1 (getl_flat c (CHead d1 (Bind b) w) h H f v))))))))).
49 \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall
50 (v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v))))))
52 \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1:
53 C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to
54 (ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda
55 (b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w:
56 T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1
57 (Bind b0) w))).(nat_ind (\lambda (n: nat).((getl n (CHead c1 (Bind b) v)
58 (CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind b)
59 v) (CHead d2 (Bind b0) w)))))) (\lambda (H1: (getl O (CHead c1 (Bind b) v)
60 (CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C (\lambda (e: C).(match e
61 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _)
62 \Rightarrow c])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind
63 b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1
64 (Bind b0) w) H1))) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e in
65 C return (\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _)
66 \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
67 \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind b0) w) (CHead
68 c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O
69 (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H4 \def (f_equal
70 C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
71 \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) w) (CHead
72 c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O
73 (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in (\lambda (H5: (eq B b0
74 b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v (\lambda (t: T).(ex C (\lambda
75 (d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b0) t))))) (eq_ind_r B
76 b (\lambda (b1: B).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v)
77 (CHead d2 (Bind b1) v))))) (ex_intro C (\lambda (d2: C).(getl O (CHead c2
78 (Bind b) v) (CHead d2 (Bind b) v))) c2 (getl_refl b c2 v)) b0 H5) w H4))))
79 H3)) H2))) (\lambda (h0: nat).(\lambda (_: (((getl h0 (CHead c1 (Bind b) v)
80 (CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl h0 (CHead c2 (Bind
81 b) v) (CHead d2 (Bind b0) w))))))).(\lambda (H1: (getl (S h0) (CHead c1 (Bind
82 b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) h0)
83 (getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v h0 H1)) in (let H2 \def H_x
84 in (ex_ind C (\lambda (d2: C).(getl (r (Bind b) h0) c2 (CHead d2 (Bind b0)
85 w))) (ex C (\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b) v) (CHead d2
86 (Bind b0) w)))) (\lambda (x: C).(\lambda (H3: (getl (r (Bind b) h0) c2 (CHead
87 x (Bind b0) w))).(ex_intro C (\lambda (d2: C).(getl (S h0) (CHead c2 (Bind b)
88 v) (CHead d2 (Bind b0) w))) x (getl_head (Bind b) h0 c2 (CHead x (Bind b0) w)
89 H3 v)))) H2)))))) h H0)))))))))).
91 theorem cimp_getl_conf:
92 \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall
93 (d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w))
94 \to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead
95 d2 (Bind b) w)))))))))))
97 \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1:
98 C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to
99 (ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda
100 (b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (i: nat).(\lambda (H0: (getl
101 i c1 (CHead d1 (Bind b) w))).(let H_x \def (H b d1 w i H0) in (let H1 \def
102 H_x in (ex_ind C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) (ex2 C
103 (\lambda (d2: C).(\forall (b0: B).(\forall (d3: C).(\forall (w0: T).(\forall
104 (h: nat).((getl h d1 (CHead d3 (Bind b0) w0)) \to (ex C (\lambda (d4:
105 C).(getl h d2 (CHead d4 (Bind b0) w0)))))))))) (\lambda (d2: C).(getl i c2
106 (CHead d2 (Bind b) w)))) (\lambda (x: C).(\lambda (H2: (getl i c2 (CHead x
107 (Bind b) w))).(ex_intro2 C (\lambda (d2: C).(\forall (b0: B).(\forall (d3:
108 C).(\forall (w0: T).(\forall (h: nat).((getl h d1 (CHead d3 (Bind b0) w0))
109 \to (ex C (\lambda (d4: C).(getl h d2 (CHead d4 (Bind b0) w0))))))))))
110 (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w))) x (\lambda (b0:
111 B).(\lambda (d0: C).(\lambda (w0: T).(\lambda (h: nat).(\lambda (H3: (getl h
112 d1 (CHead d0 (Bind b0) w0))).(let H_y \def (getl_trans (S h) c1 (CHead d1
113 (Bind b) w) i H0) in (let H_x0 \def (H b0 d0 w0 (plus (S h) i) (H_y (CHead d0
114 (Bind b0) w0) (getl_head (Bind b) h d1 (CHead d0 (Bind b0) w0) H3 w))) in
115 (let H4 \def H_x0 in (ex_ind C (\lambda (d2: C).(getl (plus (S h) i) c2
116 (CHead d2 (Bind b0) w0))) (ex C (\lambda (d2: C).(getl h x (CHead d2 (Bind
117 b0) w0)))) (\lambda (x0: C).(\lambda (H5: (getl (plus (S h) i) c2 (CHead x0
118 (Bind b0) w0))).(let H_y0 \def (getl_conf_le (plus (S h) i) (CHead x0 (Bind
119 b0) w0) c2 H5 (CHead x (Bind b) w) i H2) in (let H6 \def (eq_ind nat (minus
120 (plus (S h) i) i) (\lambda (n: nat).(getl n (CHead x (Bind b) w) (CHead x0
121 (Bind b0) w0))) (H_y0 (le_plus_r (S h) i)) (S h) (minus_plus_r (S h) i)) in
122 (ex_intro C (\lambda (d2: C).(getl h x (CHead d2 (Bind b0) w0))) x0
123 (getl_gen_S (Bind b) x (CHead x0 (Bind b0) w0) w h H6)))))) H4))))))))) H2)))