1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* *)
(* Micromega: A reflexive tactic using the Positivstellensatz *)
(* *)
(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
(* *)
(************************************************************************)
Require Import BinInt List.
Set Implicit Arguments.
Local Open Scope positive_scope.
Section S.
Variable D :Type.
Definition Env := positive -> D.
Definition jump (j:positive) (e:Env) := fun x => e (x+j).
Definition nth (n:positive) (e:Env) := e n.
Definition hd (e:Env) := nth 1 e.
Definition tail (e:Env) := jump 1 e.
Lemma jump_add i j l x : jump (i + j) l x = jump i (jump j l) x.
Proof.
unfold jump. f_equal. apply Pos.add_assoc.
Qed.
Lemma jump_simpl p l x :
jump p l x =
match p with
| xH => tail l x
| xO p => jump p (jump p l) x
| xI p => jump p (jump p (tail l)) x
end.
Proof.
destruct p; unfold tail; rewrite <- ?jump_add; f_equal;
now rewrite Pos.add_diag.
Qed.
Lemma jump_tl j l x : tail (jump j l) x = jump j (tail l) x.
Proof.
unfold tail. rewrite <- !jump_add. f_equal. apply Pos.add_comm.
Qed.
Lemma jump_succ j l x : jump (Pos.succ j) l x = jump 1 (jump j l) x.
Proof.
rewrite <- jump_add. f_equal. symmetry. apply Pos.add_1_l.
Qed.
Lemma jump_pred_double i l x :
jump (Pos.pred_double i) (tail l) x = jump i (jump i l) x.
Proof.
unfold tail. rewrite <- !jump_add. f_equal.
now rewrite Pos.add_1_r, Pos.succ_pred_double, Pos.add_diag.
Qed.
Lemma nth_spec p l :
nth p l =
match p with
| xH => hd l
| xO p => nth p (jump p l)
| xI p => nth p (jump p (tail l))
end.
Proof.
unfold hd, nth, tail, jump.
destruct p; f_equal; now rewrite Pos.add_diag.
Qed.
Lemma nth_jump p l : nth p (tail l) = hd (jump p l).
Proof.
unfold hd, nth, tail, jump. f_equal. apply Pos.add_comm.
Qed.
Lemma nth_pred_double p l :
nth (Pos.pred_double p) (tail l) = nth p (jump p l).
Proof.
unfold nth, tail, jump. f_equal.
now rewrite Pos.add_1_r, Pos.succ_pred_double, Pos.add_diag.
Qed.
End S.
Ltac jump_simpl :=
repeat
match goal with
| |- appcontext [jump xH] => rewrite (jump_simpl xH)
| |- appcontext [jump (xO ?p)] => rewrite (jump_simpl (xO p))
| |- appcontext [jump (xI ?p)] => rewrite (jump_simpl (xI p))
end.
|
