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
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: PermutEq.v 10739 2008-04-01 14:45:20Z herbelin $ i*)
Require Import Omega Relations Setoid List Multiset Permutation.
Set Implicit Arguments.
(** This file is similar to [PermutSetoid], except that the equality used here
is Coq usual one instead of a setoid equality. In particular, we can then
prove the equivalence between [List.Permutation] and
[Permutation.permutation].
*)
Section Perm.
Variable A : Type.
Hypothesis eq_dec : forall x y:A, {x=y} + {~ x=y}.
Notation permutation := (permutation _ eq_dec).
Notation list_contents := (list_contents _ eq_dec).
(** we can use [multiplicity] to define [In] and [NoDup]. *)
Lemma multiplicity_In :
forall l a, In a l <-> 0 < multiplicity (list_contents l) a.
Proof.
induction l.
simpl.
split; inversion 1.
simpl.
split; intros.
inversion_clear H.
subst a0.
destruct (eq_dec a a) as [_|H]; auto with arith; destruct H; auto.
destruct (eq_dec a a0) as [H1|H1]; auto with arith; simpl.
rewrite <- IHl; auto.
destruct (eq_dec a a0); auto.
simpl in H.
right; rewrite IHl; auto.
Qed.
Lemma multiplicity_In_O :
forall l a, ~ In a l -> multiplicity (list_contents l) a = 0.
Proof.
intros l a; rewrite multiplicity_In;
destruct (multiplicity (list_contents l) a); auto.
destruct 1; auto with arith.
Qed.
Lemma multiplicity_In_S :
forall l a, In a l -> multiplicity (list_contents l) a >= 1.
Proof.
intros l a; rewrite multiplicity_In; auto.
Qed.
Lemma multiplicity_NoDup :
forall l, NoDup l <-> (forall a, multiplicity (list_contents l) a <= 1).
Proof.
induction l.
simpl.
split; auto with arith.
intros; apply NoDup_nil.
split; simpl.
inversion_clear 1.
rewrite IHl in H1.
intros; destruct (eq_dec a a0) as [H2|H2]; simpl; auto.
subst a0.
rewrite multiplicity_In_O; auto.
intros; constructor.
rewrite multiplicity_In.
generalize (H a).
destruct (eq_dec a a) as [H0|H0].
destruct (multiplicity (list_contents l) a); auto with arith.
simpl; inversion 1.
inversion H3.
destruct H0; auto.
rewrite IHl; intros.
generalize (H a0); auto with arith.
destruct (eq_dec a a0); simpl; auto with arith.
Qed.
Lemma NoDup_permut :
forall l l', NoDup l -> NoDup l' ->
(forall x, In x l <-> In x l') -> permutation l l'.
Proof.
intros.
red; unfold meq; intros.
rewrite multiplicity_NoDup in H, H0.
generalize (H a) (H0 a) (H1 a); clear H H0 H1.
do 2 rewrite multiplicity_In.
destruct 3; omega.
Qed.
(** Permutation is compatible with In. *)
Lemma permut_In_In :
forall l1 l2 e, permutation l1 l2 -> In e l1 -> In e l2.
Proof.
unfold Permutation.permutation, meq; intros l1 l2 e P IN.
generalize (P e); clear P.
destruct (In_dec eq_dec e l2) as [H|H]; auto.
rewrite (multiplicity_In_O _ _ H).
intros.
generalize (multiplicity_In_S _ _ IN).
rewrite H0.
inversion 1.
Qed.
Lemma permut_cons_In :
forall l1 l2 e, permutation (e :: l1) l2 -> In e l2.
Proof.
intros; eapply permut_In_In; eauto.
red; auto.
Qed.
(** Permutation of an empty list. *)
Lemma permut_nil :
forall l, permutation l nil -> l = nil.
Proof.
intro l; destruct l as [ | e l ]; trivial.
assert (In e (e::l)) by (red; auto).
intro Abs; generalize (permut_In_In _ Abs H).
inversion 1.
Qed.
(** When used with [eq], this permutation notion is equivalent to
the one defined in [List.v]. *)
Lemma permutation_Permutation :
forall l l', Permutation l l' <-> permutation l l'.
Proof.
split.
induction 1.
apply permut_refl.
apply permut_cons; auto.
change (permutation (y::x::l) ((x::nil)++y::l)).
apply permut_add_cons_inside; simpl; apply permut_refl.
apply permut_tran with l'; auto.
revert l'.
induction l.
intros.
rewrite (permut_nil (permut_sym H)).
apply Permutation_refl.
intros.
destruct (In_split _ _ (permut_cons_In H)) as (h2,(t2,H1)).
subst l'.
apply Permutation_cons_app.
apply IHl.
apply permut_remove_hd with a; auto.
Qed.
(** Permutation for short lists. *)
Lemma permut_length_1:
forall a b, permutation (a :: nil) (b :: nil) -> a=b.
Proof.
intros a b; unfold Permutation.permutation, meq; intro P;
generalize (P b); clear P; simpl.
destruct (eq_dec b b) as [H|H]; [ | destruct H; auto].
destruct (eq_dec a b); simpl; auto; intros; discriminate.
Qed.
Lemma permut_length_2 :
forall a1 b1 a2 b2, permutation (a1 :: b1 :: nil) (a2 :: b2 :: nil) ->
(a1=a2) /\ (b1=b2) \/ (a1=b2) /\ (a2=b1).
Proof.
intros a1 b1 a2 b2 P.
assert (H:=permut_cons_In P).
inversion_clear H.
left; split; auto.
apply permut_length_1.
red; red; intros.
generalize (P a); clear P; simpl.
destruct (eq_dec a1 a) as [H2|H2];
destruct (eq_dec a2 a) as [H3|H3]; auto.
destruct H3; transitivity a1; auto.
destruct H2; transitivity a2; auto.
right.
inversion_clear H0; [|inversion H].
split; auto.
apply permut_length_1.
red; red; intros.
generalize (P a); clear P; simpl.
destruct (eq_dec a1 a) as [H2|H2];
destruct (eq_dec b2 a) as [H3|H3]; auto.
simpl; rewrite <- plus_n_Sm; inversion 1; auto.
destruct H3; transitivity a1; auto.
destruct H2; transitivity b2; auto.
Qed.
(** Permutation is compatible with length. *)
Lemma permut_length :
forall l1 l2, permutation l1 l2 -> length l1 = length l2.
Proof.
induction l1; intros l2 H.
rewrite (permut_nil (permut_sym H)); auto.
destruct (In_split _ _ (permut_cons_In H)) as (h2,(t2,H1)).
subst l2.
rewrite app_length.
simpl; rewrite <- plus_n_Sm; f_equal.
rewrite <- app_length.
apply IHl1.
apply permut_remove_hd with a; auto.
Qed.
Variable B : Type.
Variable eqB_dec : forall x y:B, { x=y }+{ ~x=y }.
(** Permutation is compatible with map. *)
Lemma permutation_map :
forall f l1 l2, permutation l1 l2 ->
Permutation.permutation _ eqB_dec (map f l1) (map f l2).
Proof.
intros f; induction l1.
intros l2 P; rewrite (permut_nil (permut_sym P)); apply permut_refl.
intros l2 P.
simpl.
destruct (In_split _ _ (permut_cons_In P)) as (h2,(t2,H1)).
subst l2.
rewrite map_app.
simpl.
apply permut_add_cons_inside.
rewrite <- map_app.
apply IHl1; auto.
apply permut_remove_hd with a; auto.
Qed.
End Perm.
|
