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
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Bool NZAxioms NZMulOrder.
(** Parity functions *)
Module Type NZParity (Import A : NZAxiomsSig').
Parameter Inline even odd : t -> bool.
Definition Even n := exists m, n == 2*m.
Definition Odd n := exists m, n == 2*m+1.
Axiom even_spec : forall n, even n = true <-> Even n.
Axiom odd_spec : forall n, odd n = true <-> Odd n.
End NZParity.
Module Type NZParityProp
(Import A : NZOrdAxiomsSig')
(Import B : NZParity A)
(Import C : NZMulOrderProp A).
(** Morphisms *)
Instance Even_wd : Proper (eq==>iff) Even.
Proof. unfold Even. solve_proper. Qed.
Instance Odd_wd : Proper (eq==>iff) Odd.
Proof. unfold Odd. solve_proper. Qed.
Instance even_wd : Proper (eq==>Logic.eq) even.
Proof.
intros x x' EQ. rewrite eq_iff_eq_true, 2 even_spec. now f_equiv.
Qed.
Instance odd_wd : Proper (eq==>Logic.eq) odd.
Proof.
intros x x' EQ. rewrite eq_iff_eq_true, 2 odd_spec. now f_equiv.
Qed.
(** Evenness and oddity are dual notions *)
Lemma Even_or_Odd : forall x, Even x \/ Odd x.
Proof.
nzinduct x.
left. exists 0. now nzsimpl.
intros x.
split; intros [(y,H)|(y,H)].
right. exists y. rewrite H. now nzsimpl.
left. exists (S y). rewrite H. now nzsimpl'.
right.
assert (LT : exists z, z<y).
destruct (lt_ge_cases 0 y) as [LT|GT]; [now exists 0 | exists x].
rewrite <- le_succ_l, H. nzsimpl'.
rewrite <- (add_0_r y) at 3. now apply add_le_mono_l.
destruct LT as (z,LT).
destruct (lt_exists_pred z y LT) as (y' & Hy' & _).
exists y'. rewrite <- succ_inj_wd, H, Hy'. now nzsimpl'.
left. exists y. rewrite <- succ_inj_wd. rewrite H. now nzsimpl.
Qed.
Lemma double_below : forall n m, n<=m -> 2*n < 2*m+1.
Proof.
intros. nzsimpl'. apply lt_succ_r. now apply add_le_mono.
Qed.
Lemma double_above : forall n m, n<m -> 2*n+1 < 2*m.
Proof.
intros. nzsimpl'.
rewrite <- le_succ_l, <- add_succ_l, <- add_succ_r.
apply add_le_mono; now apply le_succ_l.
Qed.
Lemma Even_Odd_False : forall x, Even x -> Odd x -> False.
Proof.
intros x (y,E) (z,O). rewrite O in E; clear O.
destruct (le_gt_cases y z) as [LE|GT].
generalize (double_below _ _ LE); order.
generalize (double_above _ _ GT); order.
Qed.
Lemma orb_even_odd : forall n, orb (even n) (odd n) = true.
Proof.
intros.
destruct (Even_or_Odd n) as [H|H].
rewrite <- even_spec in H. now rewrite H.
rewrite <- odd_spec in H. now rewrite H, orb_true_r.
Qed.
Lemma negb_odd : forall n, negb (odd n) = even n.
Proof.
intros.
generalize (Even_or_Odd n) (Even_Odd_False n).
rewrite <- even_spec, <- odd_spec.
destruct (odd n), (even n) ; simpl; intuition.
Qed.
Lemma negb_even : forall n, negb (even n) = odd n.
Proof.
intros. rewrite <- negb_odd. apply negb_involutive.
Qed.
(** Constants *)
Lemma even_0 : even 0 = true.
Proof.
rewrite even_spec. exists 0. now nzsimpl.
Qed.
Lemma odd_0 : odd 0 = false.
Proof.
now rewrite <- negb_even, even_0.
Qed.
Lemma odd_1 : odd 1 = true.
Proof.
rewrite odd_spec. exists 0. now nzsimpl'.
Qed.
Lemma even_1 : even 1 = false.
Proof.
now rewrite <- negb_odd, odd_1.
Qed.
Lemma even_2 : even 2 = true.
Proof.
rewrite even_spec. exists 1. now nzsimpl'.
Qed.
Lemma odd_2 : odd 2 = false.
Proof.
now rewrite <- negb_even, even_2.
Qed.
(** Parity and successor *)
Lemma Odd_succ : forall n, Odd (S n) <-> Even n.
Proof.
split; intros (m,H).
exists m. apply succ_inj. now rewrite add_1_r in H.
exists m. rewrite add_1_r. now f_equiv.
Qed.
Lemma odd_succ : forall n, odd (S n) = even n.
Proof.
intros. apply eq_iff_eq_true. rewrite even_spec, odd_spec.
apply Odd_succ.
Qed.
Lemma even_succ : forall n, even (S n) = odd n.
Proof.
intros. now rewrite <- negb_odd, odd_succ, negb_even.
Qed.
Lemma Even_succ : forall n, Even (S n) <-> Odd n.
Proof.
intros. now rewrite <- even_spec, even_succ, odd_spec.
Qed.
(** Parity and successor of successor *)
Lemma Even_succ_succ : forall n, Even (S (S n)) <-> Even n.
Proof.
intros. now rewrite Even_succ, Odd_succ.
Qed.
Lemma Odd_succ_succ : forall n, Odd (S (S n)) <-> Odd n.
Proof.
intros. now rewrite Odd_succ, Even_succ.
Qed.
Lemma even_succ_succ : forall n, even (S (S n)) = even n.
Proof.
intros. now rewrite even_succ, odd_succ.
Qed.
Lemma odd_succ_succ : forall n, odd (S (S n)) = odd n.
Proof.
intros. now rewrite odd_succ, even_succ.
Qed.
(** Parity and addition *)
Lemma even_add : forall n m, even (n+m) = Bool.eqb (even n) (even m).
Proof.
intros.
case_eq (even n); case_eq (even m);
rewrite <- ?negb_true_iff, ?negb_even, ?odd_spec, ?even_spec;
intros (m',Hm) (n',Hn).
exists (n'+m'). now rewrite mul_add_distr_l, Hn, Hm.
exists (n'+m'). now rewrite mul_add_distr_l, Hn, Hm, add_assoc.
exists (n'+m'). now rewrite mul_add_distr_l, Hn, Hm, add_shuffle0.
exists (n'+m'+1). rewrite Hm,Hn. nzsimpl'. now rewrite add_shuffle1.
Qed.
Lemma odd_add : forall n m, odd (n+m) = xorb (odd n) (odd m).
Proof.
intros. rewrite <- !negb_even. rewrite even_add.
now destruct (even n), (even m).
Qed.
(** Parity and multiplication *)
Lemma even_mul : forall n m, even (mul n m) = even n || even m.
Proof.
intros.
case_eq (even n); simpl; rewrite ?even_spec.
intros (n',Hn). exists (n'*m). now rewrite Hn, mul_assoc.
case_eq (even m); simpl; rewrite ?even_spec.
intros (m',Hm). exists (n*m'). now rewrite Hm, !mul_assoc, (mul_comm 2).
(* odd / odd *)
rewrite <- !negb_true_iff, !negb_even, !odd_spec.
intros (m',Hm) (n',Hn). exists (n'*2*m' +n'+m').
rewrite Hn,Hm, !mul_add_distr_l, !mul_add_distr_r, !mul_1_l, !mul_1_r.
now rewrite add_shuffle1, add_assoc, !mul_assoc.
Qed.
Lemma odd_mul : forall n m, odd (mul n m) = odd n && odd m.
Proof.
intros. rewrite <- !negb_even. rewrite even_mul.
now destruct (even n), (even m).
Qed.
(** A particular case : adding by an even number *)
Lemma even_add_even : forall n m, Even m -> even (n+m) = even n.
Proof.
intros n m Hm. apply even_spec in Hm.
rewrite even_add, Hm. now destruct (even n).
Qed.
Lemma odd_add_even : forall n m, Even m -> odd (n+m) = odd n.
Proof.
intros n m Hm. apply even_spec in Hm.
rewrite odd_add, <- (negb_even m), Hm. now destruct (odd n).
Qed.
Lemma even_add_mul_even : forall n m p, Even m -> even (n+m*p) = even n.
Proof.
intros n m p Hm. apply even_spec in Hm.
apply even_add_even. apply even_spec. now rewrite even_mul, Hm.
Qed.
Lemma odd_add_mul_even : forall n m p, Even m -> odd (n+m*p) = odd n.
Proof.
intros n m p Hm. apply even_spec in Hm.
apply odd_add_even. apply even_spec. now rewrite even_mul, Hm.
Qed.
Lemma even_add_mul_2 : forall n m, even (n+2*m) = even n.
Proof.
intros. apply even_add_mul_even. apply even_spec, even_2.
Qed.
Lemma odd_add_mul_2 : forall n m, odd (n+2*m) = odd n.
Proof.
intros. apply odd_add_mul_even. apply even_spec, even_2.
Qed.
End NZParityProp.
|
