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