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
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \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 ZArith.
Require Import Coq.Arith.Max.
Require Import List.
Set Implicit Arguments.
(* I have addded a Leaf constructor to the varmap data structure (/plugins/ring/Quote.v)
-- this is harmless and spares a lot of Empty.
This means smaller proof-terms.
BTW, by dropping the polymorphism, I get small (yet noticeable) speed-up.
*)
Section S.
Variable D :Type.
Definition Env := positive -> D.
Definition jump (j:positive) (e:Env) := fun x => e (Pplus x j).
Definition nth (n:positive) (e : Env ) := e n.
Definition hd (x:D) (e: Env) := nth xH e.
Definition tail (e: Env) := jump xH e.
Lemma psucc : forall p, (match p with
| xI y' => xO (Psucc y')
| xO y' => xI y'
| 1%positive => 2%positive
end) = (p+1)%positive.
Proof.
destruct p.
auto with zarith.
rewrite xI_succ_xO.
auto with zarith.
reflexivity.
Qed.
Lemma jump_Pplus : forall i j l,
forall x, jump (i + j) l x = jump i (jump j l) x.
Proof.
unfold jump.
intros.
rewrite Pplus_assoc.
reflexivity.
Qed.
Lemma jump_simpl : forall p l,
forall 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 ; intros ; repeat rewrite <- jump_Pplus.
(* xI p = p + p + 1 *)
rewrite xI_succ_xO.
rewrite Pplus_diag.
rewrite <- Pplus_one_succ_r.
reflexivity.
(* xO p = p + p *)
rewrite Pplus_diag.
reflexivity.
reflexivity.
Qed.
Ltac jump_s :=
repeat
match goal with
| |- context [jump xH ?e] => rewrite (jump_simpl xH)
| |- context [jump (xO ?p) ?e] => rewrite (jump_simpl (xO p))
| |- context [jump (xI ?p) ?e] => rewrite (jump_simpl (xI p))
end.
Lemma jump_tl : forall j l, forall x, tail (jump j l) x = jump j (tail l) x.
Proof.
unfold tail.
intros.
repeat rewrite <- jump_Pplus.
rewrite Pplus_comm.
reflexivity.
Qed.
Lemma jump_Psucc : forall j l,
forall x, (jump (Psucc j) l x) = (jump 1 (jump j l) x).
Proof.
intros.
rewrite <- jump_Pplus.
rewrite Pplus_one_succ_r.
rewrite Pplus_comm.
reflexivity.
Qed.
Lemma jump_Pdouble_minus_one : forall i l,
forall x, (jump (Pdouble_minus_one i) (tail l)) x = (jump i (jump i l)) x.
Proof.
unfold tail.
intros.
repeat rewrite <- jump_Pplus.
rewrite <- Pplus_one_succ_r.
rewrite Psucc_o_double_minus_one_eq_xO.
rewrite Pplus_diag.
reflexivity.
Qed.
Lemma jump_x0_tail : forall p l, forall x, jump (xO p) (tail l) x = jump (xI p) l x.
Proof.
intros.
unfold jump.
unfold tail.
unfold jump.
rewrite <- Pplus_assoc.
simpl.
reflexivity.
Qed.
Lemma nth_spec : forall p l x,
nth p l =
match p with
| xH => hd x l
| xO p => nth p (jump p l)
| xI p => nth p (jump p (tail l))
end.
Proof.
unfold nth.
destruct p.
intros.
unfold jump, tail.
unfold jump.
rewrite Pplus_diag.
rewrite xI_succ_xO.
simpl.
reflexivity.
unfold jump.
rewrite Pplus_diag.
reflexivity.
unfold hd.
unfold nth.
reflexivity.
Qed.
Lemma nth_jump : forall p l x, nth p (tail l) = hd x (jump p l).
Proof.
unfold tail.
unfold hd.
unfold jump.
unfold nth.
intros.
rewrite Pplus_comm.
reflexivity.
Qed.
Lemma nth_Pdouble_minus_one :
forall p l, nth (Pdouble_minus_one p) (tail l) = nth p (jump p l).
Proof.
intros.
unfold tail.
unfold nth, jump.
rewrite Pplus_diag.
rewrite <- Psucc_o_double_minus_one_eq_xO.
rewrite Pplus_one_succ_r.
reflexivity.
Qed.
End S.
|
