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
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
|
(************************************************************************)
(* 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 *)
(************************************************************************)
(* Evgeny Makarov, INRIA, 2007 *)
(************************************************************************)
Require Import Setoid.
Require Import Ring.
(** Generic properties of ordered rings on a setoid equality *)
Set Implicit Arguments.
Module Import OrderedRingSyntax.
Export RingSyntax.
Reserved Notation "x ~= y" (at level 70, no associativity).
Reserved Notation "x [=] y" (at level 70, no associativity).
Reserved Notation "x [~=] y" (at level 70, no associativity).
Reserved Notation "x [<] y" (at level 70, no associativity).
Reserved Notation "x [<=] y" (at level 70, no associativity).
End OrderedRingSyntax.
Section DEFINITIONS.
Variable R : Type.
Variable (rO rI : R) (rplus rtimes rminus: R -> R -> R) (ropp : R -> R).
Variable req rle rlt : R -> R -> Prop.
Notation "0" := rO.
Notation "1" := rI.
Notation "x + y" := (rplus x y).
Notation "x * y " := (rtimes x y).
Notation "x - y " := (rminus x y).
Notation "- x" := (ropp x).
Notation "x == y" := (req x y).
Notation "x ~= y" := (~ req x y).
Notation "x <= y" := (rle x y).
Notation "x < y" := (rlt x y).
Record SOR : Type := mk_SOR_theory {
SORsetoid : Setoid_Theory R req;
SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
SORopp_wd : forall x1 x2, x1 == x2 -> -x1 == -x2;
SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
SORle_refl : forall n : R, n <= n;
SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
SORlt_trichotomy : forall n m : R, n < m \/ n == m \/ m < n;
SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
SORneq_0_1 : 0 ~= 1
}.
(* We cannot use Relation_Definitions.order.ord_antisym and
Relations_1.Antisymmetric because they refer to Leibniz equality *)
End DEFINITIONS.
Section STRICT_ORDERED_RING.
Variable R : Type.
Variable (rO rI : R) (rplus rtimes rminus: R -> R -> R) (ropp : R -> R).
Variable req rle rlt : R -> R -> Prop.
Variable sor : SOR rO rI rplus rtimes rminus ropp req rle rlt.
Notation "0" := rO.
Notation "1" := rI.
Notation "x + y" := (rplus x y).
Notation "x * y " := (rtimes x y).
Notation "x - y " := (rminus x y).
Notation "- x" := (ropp x).
Notation "x == y" := (req x y).
Notation "x ~= y" := (~ req x y).
Notation "x <= y" := (rle x y).
Notation "x < y" := (rlt x y).
Add Relation R req
reflexivity proved by sor.(SORsetoid).(@Equivalence_Reflexive _ _ )
symmetry proved by sor.(SORsetoid).(@Equivalence_Symmetric _ _ )
transitivity proved by sor.(SORsetoid).(@Equivalence_Transitive _ _ )
as sor_setoid.
Add Morphism rplus with signature req ==> req ==> req as rplus_morph.
Proof.
exact sor.(SORplus_wd).
Qed.
Add Morphism rtimes with signature req ==> req ==> req as rtimes_morph.
Proof.
exact sor.(SORtimes_wd).
Qed.
Add Morphism ropp with signature req ==> req as ropp_morph.
Proof.
exact sor.(SORopp_wd).
Qed.
Add Morphism rle with signature req ==> req ==> iff as rle_morph.
Proof.
exact sor.(SORle_wd).
Qed.
Add Morphism rlt with signature req ==> req ==> iff as rlt_morph.
Proof.
exact sor.(SORlt_wd).
Qed.
Add Ring SOR : sor.(SORrt).
Add Morphism rminus with signature req ==> req ==> req as rminus_morph.
Proof.
intros x1 x2 H1 y1 y2 H2.
rewrite (sor.(SORrt).(Rsub_def) x1 y1).
rewrite (sor.(SORrt).(Rsub_def) x2 y2).
rewrite H1; now rewrite H2.
Qed.
Theorem Rneq_symm : forall n m : R, n ~= m -> m ~= n.
Proof.
intros n m H1 H2; rewrite H2 in H1; now apply H1.
Qed.
(* Propeties of plus, minus and opp *)
Theorem Rplus_0_l : forall n : R, 0 + n == n.
Proof.
intro; ring.
Qed.
Theorem Rplus_0_r : forall n : R, n + 0 == n.
Proof.
intro; ring.
Qed.
Theorem Rtimes_0_r : forall n : R, n * 0 == 0.
Proof.
intro; ring.
Qed.
Theorem Rplus_comm : forall n m : R, n + m == m + n.
Proof.
intros; ring.
Qed.
Theorem Rtimes_0_l : forall n : R, 0 * n == 0.
Proof.
intro; ring.
Qed.
Theorem Rtimes_comm : forall n m : R, n * m == m * n.
Proof.
intros; ring.
Qed.
Theorem Rminus_eq_0 : forall n m : R, n - m == 0 <-> n == m.
Proof.
intros n m.
split; intro H. setoid_replace n with ((n - m) + m) by ring. rewrite H.
now rewrite Rplus_0_l.
rewrite H; ring.
Qed.
Theorem Rplus_cancel_l : forall n m p : R, p + n == p + m <-> n == m.
Proof.
intros n m p; split; intro H.
setoid_replace n with (- p + (p + n)) by ring.
setoid_replace m with (- p + (p + m)) by ring. now rewrite H.
now rewrite H.
Qed.
(* Relations *)
Theorem Rle_refl : forall n : R, n <= n.
Proof sor.(SORle_refl).
Theorem Rle_antisymm : forall n m : R, n <= m -> m <= n -> n == m.
Proof sor.(SORle_antisymm).
Theorem Rle_trans : forall n m p : R, n <= m -> m <= p -> n <= p.
Proof sor.(SORle_trans).
Theorem Rlt_trichotomy : forall n m : R, n < m \/ n == m \/ m < n.
Proof sor.(SORlt_trichotomy).
Theorem Rlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m.
Proof sor.(SORlt_le_neq).
Theorem Rneq_0_1 : 0 ~= 1.
Proof sor.(SORneq_0_1).
Theorem Req_em : forall n m : R, n == m \/ n ~= m.
Proof.
intros n m. destruct (Rlt_trichotomy n m) as [H | [H | H]]; try rewrite Rlt_le_neq in H.
right; now destruct H.
now left.
right; apply Rneq_symm; now destruct H.
Qed.
Theorem Req_dne : forall n m : R, ~ ~ n == m <-> n == m.
Proof.
intros n m; destruct (Req_em n m) as [H | H].
split; auto.
split. intro H1; false_hyp H H1. auto.
Qed.
Theorem Rle_lt_eq : forall n m : R, n <= m <-> n < m \/ n == m.
Proof.
intros n m; rewrite Rlt_le_neq.
split; [intro H | intros [[H1 H2] | H]].
destruct (Req_em n m) as [H1 | H1]. now right. left; now split.
assumption.
rewrite H; apply Rle_refl.
Qed.
Ltac le_less := rewrite Rle_lt_eq; left; try assumption.
Ltac le_equal := rewrite Rle_lt_eq; right; try reflexivity; try assumption.
Ltac le_elim H := rewrite Rle_lt_eq in H; destruct H as [H | H].
Theorem Rlt_trans : forall n m p : R, n < m -> m < p -> n < p.
Proof.
intros n m p; repeat rewrite Rlt_le_neq; intros [H1 H2] [H3 H4]; split.
now apply Rle_trans with m.
intro H. rewrite H in H1. pose proof (Rle_antisymm H3 H1). now apply H4.
Qed.
Theorem Rle_lt_trans : forall n m p : R, n <= m -> m < p -> n < p.
Proof.
intros n m p H1 H2; le_elim H1.
now apply Rlt_trans with (m := m). now rewrite H1.
Qed.
Theorem Rlt_le_trans : forall n m p : R, n < m -> m <= p -> n < p.
Proof.
intros n m p H1 H2; le_elim H2.
now apply Rlt_trans with (m := m). now rewrite <- H2.
Qed.
Theorem Rle_gt_cases : forall n m : R, n <= m \/ m < n.
Proof.
intros n m; destruct (Rlt_trichotomy n m) as [H | [H | H]].
left; now le_less. left; now le_equal. now right.
Qed.
Theorem Rlt_neq : forall n m : R, n < m -> n ~= m.
Proof.
intros n m; rewrite Rlt_le_neq; now intros [_ H].
Qed.
Theorem Rle_ngt : forall n m : R, n <= m <-> ~ m < n.
Proof.
intros n m; split.
intros H H1; assert (H2 : n < n) by now apply Rle_lt_trans with m. now apply (Rlt_neq H2).
intro H. destruct (Rle_gt_cases n m) as [H1 | H1]. assumption. false_hyp H1 H.
Qed.
Theorem Rlt_nge : forall n m : R, n < m <-> ~ m <= n.
Proof.
intros n m; split.
intros H H1; assert (H2 : n < n) by now apply Rlt_le_trans with m. now apply (Rlt_neq H2).
intro H. destruct (Rle_gt_cases m n) as [H1 | H1]. false_hyp H1 H. assumption.
Qed.
(* Plus, minus and order *)
Theorem Rplus_le_mono_l : forall n m p : R, n <= m <-> p + n <= p + m.
Proof.
intros n m p; split.
apply sor.(SORplus_le_mono_l).
intro H. apply (sor.(SORplus_le_mono_l) (p + n) (p + m) (- p)) in H.
setoid_replace (- p + (p + n)) with n in H by ring.
setoid_replace (- p + (p + m)) with m in H by ring. assumption.
Qed.
Theorem Rplus_le_mono_r : forall n m p : R, n <= m <-> n + p <= m + p.
Proof.
intros n m p; rewrite (Rplus_comm n p); rewrite (Rplus_comm m p).
apply Rplus_le_mono_l.
Qed.
Theorem Rplus_lt_mono_l : forall n m p : R, n < m <-> p + n < p + m.
Proof.
intros n m p; do 2 rewrite Rlt_le_neq. rewrite Rplus_cancel_l.
now rewrite <- Rplus_le_mono_l.
Qed.
Theorem Rplus_lt_mono_r : forall n m p : R, n < m <-> n + p < m + p.
Proof.
intros n m p.
rewrite (Rplus_comm n p); rewrite (Rplus_comm m p); apply Rplus_lt_mono_l.
Qed.
Theorem Rplus_lt_mono : forall n m p q : R, n < m -> p < q -> n + p < m + q.
Proof.
intros n m p q H1 H2.
apply Rlt_trans with (m + p); [now apply -> Rplus_lt_mono_r | now apply -> Rplus_lt_mono_l].
Qed.
Theorem Rplus_le_mono : forall n m p q : R, n <= m -> p <= q -> n + p <= m + q.
Proof.
intros n m p q H1 H2.
apply Rle_trans with (m + p); [now apply -> Rplus_le_mono_r | now apply -> Rplus_le_mono_l].
Qed.
Theorem Rplus_lt_le_mono : forall n m p q : R, n < m -> p <= q -> n + p < m + q.
Proof.
intros n m p q H1 H2.
apply Rlt_le_trans with (m + p); [now apply -> Rplus_lt_mono_r | now apply -> Rplus_le_mono_l].
Qed.
Theorem Rplus_le_lt_mono : forall n m p q : R, n <= m -> p < q -> n + p < m + q.
Proof.
intros n m p q H1 H2.
apply Rle_lt_trans with (m + p); [now apply -> Rplus_le_mono_r | now apply -> Rplus_lt_mono_l].
Qed.
Theorem Rplus_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n + m.
Proof.
intros n m H1 H2. rewrite <- (Rplus_0_l 0). now apply Rplus_lt_mono.
Qed.
Theorem Rplus_pos_nonneg : forall n m : R, 0 < n -> 0 <= m -> 0 < n + m.
Proof.
intros n m H1 H2. rewrite <- (Rplus_0_l 0). now apply Rplus_lt_le_mono.
Qed.
Theorem Rplus_nonneg_pos : forall n m : R, 0 <= n -> 0 < m -> 0 < n + m.
Proof.
intros n m H1 H2. rewrite <- (Rplus_0_l 0). now apply Rplus_le_lt_mono.
Qed.
Theorem Rplus_nonneg_nonneg : forall n m : R, 0 <= n -> 0 <= m -> 0 <= n + m.
Proof.
intros n m H1 H2. rewrite <- (Rplus_0_l 0). now apply Rplus_le_mono.
Qed.
Theorem Rle_le_minus : forall n m : R, n <= m <-> 0 <= m - n.
Proof.
intros n m. rewrite (@Rplus_le_mono_r n m (- n)).
setoid_replace (n + - n) with 0 by ring.
now setoid_replace (m + - n) with (m - n) by ring.
Qed.
Theorem Rlt_lt_minus : forall n m : R, n < m <-> 0 < m - n.
Proof.
intros n m. rewrite (@Rplus_lt_mono_r n m (- n)).
setoid_replace (n + - n) with 0 by ring.
now setoid_replace (m + - n) with (m - n) by ring.
Qed.
Theorem Ropp_lt_mono : forall n m : R, n < m <-> - m < - n.
Proof.
intros n m. split; intro H.
apply -> (@Rplus_lt_mono_l n m (- n - m)) in H.
setoid_replace (- n - m + n) with (- m) in H by ring.
now setoid_replace (- n - m + m) with (- n) in H by ring.
apply -> (@Rplus_lt_mono_l (- m) (- n) (n + m)) in H.
setoid_replace (n + m + - m) with n in H by ring.
now setoid_replace (n + m + - n) with m in H by ring.
Qed.
Theorem Ropp_pos_neg : forall n : R, 0 < - n <-> n < 0.
Proof.
intro n; rewrite (Ropp_lt_mono n 0). now setoid_replace (- 0) with 0 by ring.
Qed.
(* Times and order *)
Theorem Rtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m.
Proof sor.(SORtimes_pos_pos).
Theorem Rtimes_nonneg_nonneg : forall n m : R, 0 <= n -> 0 <= m -> 0 <= n * m.
Proof.
intros n m H1 H2.
le_elim H1. le_elim H2.
le_less; now apply Rtimes_pos_pos.
rewrite <- H2; rewrite Rtimes_0_r; le_equal.
rewrite <- H1; rewrite Rtimes_0_l; le_equal.
Qed.
Theorem Rtimes_pos_neg : forall n m : R, 0 < n -> m < 0 -> n * m < 0.
Proof.
intros n m H1 H2. apply -> Ropp_pos_neg.
setoid_replace (- (n * m)) with (n * (- m)) by ring.
apply Rtimes_pos_pos. assumption. now apply <- Ropp_pos_neg.
Qed.
Theorem Rtimes_neg_neg : forall n m : R, n < 0 -> m < 0 -> 0 < n * m.
Proof.
intros n m H1 H2.
setoid_replace (n * m) with ((- n) * (- m)) by ring.
apply Rtimes_pos_pos; now apply <- Ropp_pos_neg.
Qed.
Theorem Rtimes_square_nonneg : forall n : R, 0 <= n * n.
Proof.
intro n; destruct (Rlt_trichotomy 0 n) as [H | [H | H]].
le_less; now apply Rtimes_pos_pos.
rewrite <- H, Rtimes_0_l; le_equal.
le_less; now apply Rtimes_neg_neg.
Qed.
Theorem Rtimes_neq_0 : forall n m : R, n ~= 0 /\ m ~= 0 -> n * m ~= 0.
Proof.
intros n m [H1 H2].
destruct (Rlt_trichotomy n 0) as [H3 | [H3 | H3]];
destruct (Rlt_trichotomy m 0) as [H4 | [H4 | H4]];
try (false_hyp H3 H1); try (false_hyp H4 H2).
apply Rneq_symm. apply Rlt_neq. now apply Rtimes_neg_neg.
apply Rlt_neq. rewrite Rtimes_comm. now apply Rtimes_pos_neg.
apply Rlt_neq. now apply Rtimes_pos_neg.
apply Rneq_symm. apply Rlt_neq. now apply Rtimes_pos_pos.
Qed.
(* The following theorems are used to build a morphism from Z to R and
prove its properties in ZCoeff.v. They are not used in RingMicromega.v. *)
(* Surprisingly, multilication is needed to prove the following theorem *)
Theorem Ropp_neg_pos : forall n : R, - n < 0 <-> 0 < n.
Proof.
intro n; setoid_replace n with (- - n) by ring. rewrite Ropp_pos_neg.
now setoid_replace (- - n) with n by ring.
Qed.
Theorem Rlt_0_1 : 0 < 1.
Proof.
apply <- Rlt_le_neq. split.
setoid_replace 1 with (1 * 1) by ring. apply Rtimes_square_nonneg.
apply Rneq_0_1.
Qed.
Theorem Rlt_succ_r : forall n : R, n < 1 + n.
Proof.
intro n. rewrite <- (Rplus_0_l n); setoid_replace (1 + (0 + n)) with (1 + n) by ring.
apply -> Rplus_lt_mono_r. apply Rlt_0_1.
Qed.
Theorem Rlt_lt_succ : forall n m : R, n < m -> n < 1 + m.
Proof.
intros n m H; apply Rlt_trans with m. assumption. apply Rlt_succ_r.
Qed.
(*Theorem Rtimes_lt_mono_pos_l : forall n m p : R, 0 < p -> n < m -> p * n < p * m.
Proof.
intros n m p H1 H2. apply <- Rlt_lt_minus.
setoid_replace (p * m - p * n) with (p * (m - n)) by ring.
apply Rtimes_pos_pos. assumption. now apply -> Rlt_lt_minus.
Qed.*)
End STRICT_ORDERED_RING.
|