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
|
(***********************************************************************)
(* 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 *)
(***********************************************************************)
(*i $Id$ i*)
(* G. Huet 1-9-95 *)
Require Import Permut.
Set Implicit Arguments.
Section multiset_defs.
Variable A : Set.
Variable eqA : A -> A -> Prop.
Hypothesis Aeq_dec : forall x y:A, {eqA x y} + {~ eqA x y}.
Inductive multiset : Set :=
Bag : (A -> nat) -> multiset.
Definition EmptyBag := Bag (fun a:A => 0).
Definition SingletonBag (a:A) :=
Bag (fun a':A => match Aeq_dec a a' with
| left _ => 1
| right _ => 0
end).
Definition multiplicity (m:multiset) (a:A) : nat := let (f) := m in f a.
(** multiset equality *)
Definition meq (m1 m2:multiset) :=
forall a:A, multiplicity m1 a = multiplicity m2 a.
Hint Unfold meq multiplicity.
Lemma meq_refl : forall x:multiset, meq x x.
Proof.
destruct x; auto.
Qed.
Hint Resolve meq_refl.
Lemma meq_trans : forall x y z:multiset, meq x y -> meq y z -> meq x z.
Proof.
unfold meq in |- *.
destruct x; destruct y; destruct z.
intros; rewrite H; auto.
Qed.
Lemma meq_sym : forall x y:multiset, meq x y -> meq y x.
Proof.
unfold meq in |- *.
destruct x; destruct y; auto.
Qed.
Hint Immediate meq_sym.
(** multiset union *)
Definition munion (m1 m2:multiset) :=
Bag (fun a:A => multiplicity m1 a + multiplicity m2 a).
Lemma munion_empty_left : forall x:multiset, meq x (munion EmptyBag x).
Proof.
unfold meq in |- *; unfold munion in |- *; simpl in |- *; auto.
Qed.
Hint Resolve munion_empty_left.
Lemma munion_empty_right : forall x:multiset, meq x (munion x EmptyBag).
Proof.
unfold meq in |- *; unfold munion in |- *; simpl in |- *; auto.
Qed.
Require Import Plus. (* comm. and ass. of plus *)
Lemma munion_comm : forall x y:multiset, meq (munion x y) (munion y x).
Proof.
unfold meq in |- *; unfold multiplicity in |- *; unfold munion in |- *.
destruct x; destruct y; auto with arith.
Qed.
Hint Resolve munion_comm.
Lemma munion_ass :
forall x y z:multiset, meq (munion (munion x y) z) (munion x (munion y z)).
Proof.
unfold meq in |- *; unfold munion in |- *; unfold multiplicity in |- *.
destruct x; destruct y; destruct z; auto with arith.
Qed.
Hint Resolve munion_ass.
Lemma meq_left :
forall x y z:multiset, meq x y -> meq (munion x z) (munion y z).
Proof.
unfold meq in |- *; unfold munion in |- *; unfold multiplicity in |- *.
destruct x; destruct y; destruct z.
intros; elim H; auto with arith.
Qed.
Hint Resolve meq_left.
Lemma meq_right :
forall x y z:multiset, meq x y -> meq (munion z x) (munion z y).
Proof.
unfold meq in |- *; unfold munion in |- *; unfold multiplicity in |- *.
destruct x; destruct y; destruct z.
intros; elim H; auto.
Qed.
Hint Resolve meq_right.
(** Here we should make multiset an abstract datatype, by hiding [Bag],
[munion], [multiplicity]; all further properties are proved abstractly *)
Lemma munion_rotate :
forall x y z:multiset, meq (munion x (munion y z)) (munion z (munion x y)).
Proof.
intros; apply (op_rotate multiset munion meq); auto.
exact meq_trans.
Qed.
Lemma meq_congr :
forall x y z t:multiset, meq x y -> meq z t -> meq (munion x z) (munion y t).
Proof.
intros; apply (cong_congr multiset munion meq); auto.
exact meq_trans.
Qed.
Lemma munion_perm_left :
forall x y z:multiset, meq (munion x (munion y z)) (munion y (munion x z)).
Proof.
intros; apply (perm_left multiset munion meq); auto.
exact meq_trans.
Qed.
Lemma multiset_twist1 :
forall x y z t:multiset,
meq (munion x (munion (munion y z) t)) (munion (munion y (munion x t)) z).
Proof.
intros; apply (twist multiset munion meq); auto.
exact meq_trans.
Qed.
Lemma multiset_twist2 :
forall x y z t:multiset,
meq (munion x (munion (munion y z) t)) (munion (munion y (munion x z)) t).
Proof.
intros; apply meq_trans with (munion (munion x (munion y z)) t).
apply meq_sym; apply munion_ass.
apply meq_left; apply munion_perm_left.
Qed.
(** specific for treesort *)
Lemma treesort_twist1 :
forall x y z t u:multiset,
meq u (munion y z) ->
meq (munion x (munion u t)) (munion (munion y (munion x t)) z).
Proof.
intros; apply meq_trans with (munion x (munion (munion y z) t)).
apply meq_right; apply meq_left; trivial.
apply multiset_twist1.
Qed.
Lemma treesort_twist2 :
forall x y z t u:multiset,
meq u (munion y z) ->
meq (munion x (munion u t)) (munion (munion y (munion x z)) t).
Proof.
intros; apply meq_trans with (munion x (munion (munion y z) t)).
apply meq_right; apply meq_left; trivial.
apply multiset_twist2.
Qed.
(*i theory of minter to do similarly
Require Min.
(* multiset intersection *)
Definition minter := [m1,m2:multiset]
(Bag [a:A](min (multiplicity m1 a)(multiplicity m2 a))).
i*)
End multiset_defs.
Unset Implicit Arguments.
Hint Unfold meq multiplicity: v62 datatypes.
Hint Resolve munion_empty_right munion_comm munion_ass meq_left meq_right
munion_empty_left: v62 datatypes.
Hint Immediate meq_sym: v62 datatypes.
|
