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
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
|
(************************************************************************)
(* 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 *)
(************************************************************************)
Require Import ZArith Nnat ZAxioms ZDivFloor ZSig.
(** * The interface [ZSig.ZType] implies the interface [ZAxiomsSig]
It also provides [sgn], [abs], [div], [mod]
*)
Module ZTypeIsZAxioms (Import Z : ZType').
Hint Rewrite
spec_0 spec_1 spec_2 spec_add spec_sub spec_pred spec_succ
spec_mul spec_opp spec_of_Z spec_div spec_modulo spec_sqrt
spec_compare spec_eq_bool spec_max spec_min spec_abs spec_sgn
spec_pow spec_log2 spec_even spec_odd
: zsimpl.
Ltac zsimpl := autorewrite with zsimpl.
Ltac zcongruence := repeat red; intros; zsimpl; congruence.
Ltac zify := unfold eq, lt, le in *; zsimpl.
Instance eq_equiv : Equivalence eq.
Proof. unfold eq. firstorder. Qed.
Local Obligation Tactic := zcongruence.
Program Instance succ_wd : Proper (eq ==> eq) succ.
Program Instance pred_wd : Proper (eq ==> eq) pred.
Program Instance add_wd : Proper (eq ==> eq ==> eq) add.
Program Instance sub_wd : Proper (eq ==> eq ==> eq) sub.
Program Instance mul_wd : Proper (eq ==> eq ==> eq) mul.
Theorem pred_succ : forall n, pred (succ n) == n.
Proof.
intros. zify. auto with zarith.
Qed.
Theorem one_succ : 1 == succ 0.
Proof.
now zify.
Qed.
Theorem two_succ : 2 == succ 1.
Proof.
now zify.
Qed.
Section Induction.
Variable A : Z.t -> Prop.
Hypothesis A_wd : Proper (eq==>iff) A.
Hypothesis A0 : A 0.
Hypothesis AS : forall n, A n <-> A (succ n).
Let B (z : Z) := A (of_Z z).
Lemma B0 : B 0.
Proof.
unfold B; simpl.
rewrite <- (A_wd 0); auto.
zify. auto.
Qed.
Lemma BS : forall z : Z, B z -> B (z + 1).
Proof.
intros z H.
unfold B in *. apply -> AS in H.
setoid_replace (of_Z (z + 1)) with (succ (of_Z z)); auto.
zify. auto.
Qed.
Lemma BP : forall z : Z, B z -> B (z - 1).
Proof.
intros z H.
unfold B in *. rewrite AS.
setoid_replace (succ (of_Z (z - 1))) with (of_Z z); auto.
zify. auto with zarith.
Qed.
Lemma B_holds : forall z : Z, B z.
Proof.
intros; destruct (Z_lt_le_dec 0 z).
apply natlike_ind; auto with zarith.
apply B0.
intros; apply BS; auto.
replace z with (-(-z))%Z in * by (auto with zarith).
remember (-z)%Z as z'.
pattern z'; apply natlike_ind.
apply B0.
intros; rewrite Zopp_succ; unfold Zpred; apply BP; auto.
subst z'; auto with zarith.
Qed.
Theorem bi_induction : forall n, A n.
Proof.
intro n. setoid_replace n with (of_Z (to_Z n)).
apply B_holds.
zify. auto.
Qed.
End Induction.
Theorem add_0_l : forall n, 0 + n == n.
Proof.
intros. zify. auto with zarith.
Qed.
Theorem add_succ_l : forall n m, (succ n) + m == succ (n + m).
Proof.
intros. zify. auto with zarith.
Qed.
Theorem sub_0_r : forall n, n - 0 == n.
Proof.
intros. zify. auto with zarith.
Qed.
Theorem sub_succ_r : forall n m, n - (succ m) == pred (n - m).
Proof.
intros. zify. auto with zarith.
Qed.
Theorem mul_0_l : forall n, 0 * n == 0.
Proof.
intros. zify. auto with zarith.
Qed.
Theorem mul_succ_l : forall n m, (succ n) * m == n * m + m.
Proof.
intros. zify. ring.
Qed.
(** Order *)
Lemma compare_spec : forall x y, CompSpec eq lt x y (compare x y).
Proof.
intros. zify. destruct (Zcompare_spec [x] [y]); auto.
Qed.
Definition eqb := eq_bool.
Lemma eqb_eq : forall x y, eq_bool x y = true <-> x == y.
Proof.
intros. zify. symmetry. apply Zeq_is_eq_bool.
Qed.
Instance compare_wd : Proper (eq ==> eq ==> Logic.eq) compare.
Proof.
intros x x' Hx y y' Hy. rewrite 2 spec_compare, Hx, Hy; intuition.
Qed.
Instance lt_wd : Proper (eq ==> eq ==> iff) lt.
Proof.
intros x x' Hx y y' Hy; unfold lt; rewrite Hx, Hy; intuition.
Qed.
Theorem lt_eq_cases : forall n m, n <= m <-> n < m \/ n == m.
Proof.
intros. zify. omega.
Qed.
Theorem lt_irrefl : forall n, ~ n < n.
Proof.
intros. zify. omega.
Qed.
Theorem lt_succ_r : forall n m, n < (succ m) <-> n <= m.
Proof.
intros. zify. omega.
Qed.
Theorem min_l : forall n m, n <= m -> min n m == n.
Proof.
intros n m. zify. omega with *.
Qed.
Theorem min_r : forall n m, m <= n -> min n m == m.
Proof.
intros n m. zify. omega with *.
Qed.
Theorem max_l : forall n m, m <= n -> max n m == n.
Proof.
intros n m. zify. omega with *.
Qed.
Theorem max_r : forall n m, n <= m -> max n m == m.
Proof.
intros n m. zify. omega with *.
Qed.
(** Part specific to integers, not natural numbers *)
Theorem succ_pred : forall n, succ (pred n) == n.
Proof.
intros. zify. auto with zarith.
Qed.
(** Opp *)
Program Instance opp_wd : Proper (eq ==> eq) opp.
Theorem opp_0 : - 0 == 0.
Proof.
intros. zify. auto with zarith.
Qed.
Theorem opp_succ : forall n, - (succ n) == pred (- n).
Proof.
intros. zify. auto with zarith.
Qed.
(** Abs / Sgn *)
Theorem abs_eq : forall n, 0 <= n -> abs n == n.
Proof.
intros n. zify. omega with *.
Qed.
Theorem abs_neq : forall n, n <= 0 -> abs n == -n.
Proof.
intros n. zify. omega with *.
Qed.
Theorem sgn_null : forall n, n==0 -> sgn n == 0.
Proof.
intros n. zify. omega with *.
Qed.
Theorem sgn_pos : forall n, 0<n -> sgn n == 1.
Proof.
intros n. zify. omega with *.
Qed.
Theorem sgn_neg : forall n, n<0 -> sgn n == opp 1.
Proof.
intros n. zify. omega with *.
Qed.
(** Power *)
Program Instance pow_wd : Proper (eq==>eq==>eq) pow.
Lemma pow_0_r : forall a, a^0 == 1.
Proof.
intros. now zify.
Qed.
Lemma pow_succ_r : forall a b, 0<=b -> a^(succ b) == a * a^b.
Proof.
intros a b. zify. intros Hb.
rewrite Zpower_exp; auto with zarith.
simpl. unfold Zpower_pos; simpl. ring.
Qed.
Lemma pow_neg_r : forall a b, b<0 -> a^b == 0.
Proof.
intros a b. zify. intros Hb.
destruct [b]; reflexivity || discriminate.
Qed.
Lemma pow_pow_N : forall a b, 0<=b -> a^b == pow_N a (Zabs_N (to_Z b)).
Proof.
intros a b. zify. intros Hb.
now rewrite spec_pow_N, Z_of_N_abs, Zabs_eq.
Qed.
Lemma pow_pos_N : forall a p, pow_pos a p == pow_N a (Npos p).
Proof.
intros a b. red. now rewrite spec_pow_N, spec_pow_pos.
Qed.
(** Sqrt *)
Program Instance sqrt_wd : Proper (eq==>eq) sqrt.
Lemma sqrt_spec : forall n, 0<=n ->
(sqrt n)*(sqrt n) <= n /\ n < (succ (sqrt n))*(succ (sqrt n)).
Proof.
intros n. zify. apply Zsqrt_spec.
Qed.
Lemma sqrt_neg : forall n, n<0 -> sqrt n == 0.
Proof.
intros n. zify. apply Zsqrt_neg.
Qed.
(** Log2 *)
Program Instance log2_wd : Proper (eq==>eq) log2.
Lemma log2_spec : forall n, 0<n ->
2^(log2 n) <= n /\ n < 2^(succ (log2 n)).
Proof.
intros n. zify. apply Zlog2_spec.
Qed.
Lemma log2_nonpos : forall n, n<=0 -> log2 n == 0.
Proof.
intros n. zify. apply Zlog2_nonpos.
Qed.
(** Even / Odd *)
Definition Even n := exists m, n == 2*m.
Definition Odd n := exists m, n == 2*m+1.
Lemma even_spec : forall n, even n = true <-> Even n.
Proof.
intros n. unfold Even. zify.
rewrite Zeven_bool_iff, Zeven_ex_iff.
split; intros (m,Hm).
exists (of_Z m). now zify.
exists [m]. revert Hm. now zify.
Qed.
Lemma odd_spec : forall n, odd n = true <-> Odd n.
Proof.
intros n. unfold Odd. zify.
rewrite Zodd_bool_iff, Zodd_ex_iff.
split; intros (m,Hm).
exists (of_Z m). now zify.
exists [m]. revert Hm. now zify.
Qed.
(** Div / Mod *)
Program Instance div_wd : Proper (eq==>eq==>eq) div.
Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.
Theorem div_mod : forall a b, ~b==0 -> a == b*(div a b) + (modulo a b).
Proof.
intros a b. zify. intros. apply Z_div_mod_eq_full; auto.
Qed.
Theorem mod_pos_bound :
forall a b, 0 < b -> 0 <= modulo a b /\ modulo a b < b.
Proof.
intros a b. zify. intros. apply Z_mod_lt; auto with zarith.
Qed.
Theorem mod_neg_bound :
forall a b, b < 0 -> b < modulo a b /\ modulo a b <= 0.
Proof.
intros a b. zify. intros. apply Z_mod_neg; auto with zarith.
Qed.
End ZTypeIsZAxioms.
Module ZType_ZAxioms (Z : ZType)
<: ZAxiomsSig <: ZDivSig <: HasCompare Z <: HasEqBool Z <: HasMinMax Z
:= Z <+ ZTypeIsZAxioms.
|
