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
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
|
(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
Require Import Equalities Bool SetoidList RelationPairs.
Set Implicit Arguments.
(** * Keys and datas used in MMap *)
Module KeyDecidableType(D:DecidableType).
Local Open Scope signature_scope.
Local Notation key := D.t.
Definition eqk {elt} : relation (key*elt) := D.eq @@1.
Definition eqke {elt} : relation (key*elt) := D.eq * Logic.eq.
Hint Unfold eqk eqke.
(** eqk, eqke are equalities *)
Instance eqk_equiv {elt} : Equivalence (@eqk elt) := _.
Instance eqke_equiv {elt} : Equivalence (@eqke elt) := _.
(** eqke is stricter than eqk *)
Instance eqke_eqk {elt} : subrelation (@eqke elt) (@eqk elt).
Proof. firstorder. Qed.
(** Alternative definitions of eqke and eqk *)
Lemma eqke_def {elt} k k' (e e':elt) :
eqke (k,e) (k',e') = (D.eq k k' /\ e = e').
Proof. reflexivity. Defined.
Lemma eqke_def' {elt} (p q:key*elt) :
eqke p q = (D.eq (fst p) (fst q) /\ snd p = snd q).
Proof. reflexivity. Defined.
Lemma eqke_1 {elt} k k' (e e':elt) : eqke (k,e) (k',e') -> D.eq k k'.
Proof. now destruct 1. Qed.
Lemma eqke_2 {elt} k k' (e e':elt) : eqke (k,e) (k',e') -> e=e'.
Proof. now destruct 1. Qed.
Lemma eqk_def {elt} k k' (e e':elt) : eqk (k,e) (k',e') = D.eq k k'.
Proof. reflexivity. Defined.
Lemma eqk_def' {elt} (p q:key*elt) : eqk p q = D.eq (fst p) (fst q).
Proof. reflexivity. Qed.
Lemma eqk_1 {elt} k k' (e e':elt) : eqk (k,e) (k',e') -> D.eq k k'.
Proof. trivial. Qed.
Hint Resolve eqke_1 eqke_2 eqk_1.
(* Additionnal facts *)
Lemma InA_eqke_eqk {elt} p (m:list (key*elt)) :
InA eqke p m -> InA eqk p m.
Proof.
induction 1; firstorder.
Qed.
Hint Resolve InA_eqke_eqk.
Lemma InA_eqk_eqke {elt} p (m:list (key*elt)) :
InA eqk p m -> exists q, eqk p q /\ InA eqke q m.
Proof.
induction 1; firstorder.
Qed.
Lemma InA_eqk {elt} p q (m:list (key*elt)) :
eqk p q -> InA eqk p m -> InA eqk q m.
Proof.
now intros <-.
Qed.
Definition MapsTo {elt} (k:key)(e:elt):= InA eqke (k,e).
Definition In {elt} k m := exists e:elt, MapsTo k e m.
Hint Unfold MapsTo In.
(* Alternative formulations for [In k l] *)
Lemma In_alt {elt} k (l:list (key*elt)) :
In k l <-> exists e, InA eqk (k,e) l.
Proof.
unfold In, MapsTo.
split; intros (e,H).
- exists e; auto.
- apply InA_eqk_eqke in H. destruct H as ((k',e'),(E,H)).
compute in E. exists e'. now rewrite E.
Qed.
Lemma In_alt' {elt} (l:list (key*elt)) k e :
In k l <-> InA eqk (k,e) l.
Proof.
rewrite In_alt. firstorder. eapply InA_eqk; eauto. now compute.
Qed.
Lemma In_alt2 {elt} k (l:list (key*elt)) :
In k l <-> Exists (fun p => D.eq k (fst p)) l.
Proof.
unfold In, MapsTo.
setoid_rewrite Exists_exists; setoid_rewrite InA_alt.
firstorder.
exists (snd x), x; auto.
Qed.
Lemma In_nil {elt} k : In k (@nil (key*elt)) <-> False.
Proof.
rewrite In_alt2; apply Exists_nil.
Qed.
Lemma In_cons {elt} k p (l:list (key*elt)) :
In k (p::l) <-> D.eq k (fst p) \/ In k l.
Proof.
rewrite !In_alt2, Exists_cons; intuition.
Qed.
Instance MapsTo_compat {elt} :
Proper (D.eq==>Logic.eq==>equivlistA eqke==>iff) (@MapsTo elt).
Proof.
intros x x' Hx e e' He l l' Hl. unfold MapsTo.
rewrite Hx, He, Hl; intuition.
Qed.
Instance In_compat {elt} : Proper (D.eq==>equivlistA eqk==>iff) (@In elt).
Proof.
intros x x' Hx l l' Hl. rewrite !In_alt.
setoid_rewrite Hl. setoid_rewrite Hx. intuition.
Qed.
Lemma MapsTo_eq {elt} (l:list (key*elt)) x y e :
D.eq x y -> MapsTo x e l -> MapsTo y e l.
Proof. now intros <-. Qed.
Lemma In_eq {elt} (l:list (key*elt)) x y :
D.eq x y -> In x l -> In y l.
Proof. now intros <-. Qed.
Lemma In_inv {elt} k k' e (l:list (key*elt)) :
In k ((k',e) :: l) -> D.eq k k' \/ In k l.
Proof.
intros (e',H). red in H. rewrite InA_cons, eqke_def in H.
intuition. right. now exists e'.
Qed.
Lemma In_inv_2 {elt} k k' e e' (l:list (key*elt)) :
InA eqk (k, e) ((k', e') :: l) -> ~ D.eq k k' -> InA eqk (k, e) l.
Proof.
rewrite InA_cons, eqk_def. intuition.
Qed.
Lemma In_inv_3 {elt} x x' (l:list (key*elt)) :
InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l.
Proof.
rewrite InA_cons. destruct 1 as [H|H]; trivial. destruct 1.
eauto with *.
Qed.
Hint Extern 2 (eqke ?a ?b) => split.
Hint Resolve InA_eqke_eqk.
Hint Resolve In_inv_2 In_inv_3.
End KeyDecidableType.
(** * PairDecidableType
From two decidable types, we can build a new DecidableType
over their cartesian product. *)
Module PairDecidableType(D1 D2:DecidableType) <: DecidableType.
Definition t := (D1.t * D2.t)%type.
Definition eq := (D1.eq * D2.eq)%signature.
Instance eq_equiv : Equivalence eq := _.
Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }.
Proof.
intros (x1,x2) (y1,y2); unfold eq; simpl.
destruct (D1.eq_dec x1 y1); destruct (D2.eq_dec x2 y2);
compute; intuition.
Defined.
End PairDecidableType.
(** Similarly for pairs of UsualDecidableType *)
Module PairUsualDecidableType(D1 D2:UsualDecidableType) <: UsualDecidableType.
Definition t := (D1.t * D2.t)%type.
Definition eq := @eq t.
Instance eq_equiv : Equivalence eq := _.
Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }.
Proof.
intros (x1,x2) (y1,y2);
destruct (D1.eq_dec x1 y1); destruct (D2.eq_dec x2 y2);
unfold eq, D1.eq, D2.eq in *; simpl;
(left; f_equal; auto; fail) ||
(right; intros [=]; auto).
Defined.
End PairUsualDecidableType.
|
