numbers.v 21.3 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2014, Robbert Krebbers. *)
2
(* This file is distributed under the terms of the BSD license. *)
3
4
5
(** This file collects some trivial facts on the Coq types [nat] and [N] for
natural numbers, and the type [Z] for integers. It also declares some useful
notations. *)
6
Require Export Eqdep PArith NArith ZArith NPeano.
7
Require Import QArith Qcanon.
8
Require Export base decidable.
9
Open Scope nat_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
10

11
12
Coercion Z.of_nat : nat >-> Z.

13
(** * Notations and properties of [nat] *)
14
15
16
17
Reserved Notation "x ≤ y ≤ z" (at level 70, y at next level).
Reserved Notation "x ≤ y < z" (at level 70, y at next level).
Reserved Notation "x < y < z" (at level 70, y at next level).
Reserved Notation "x < y ≤ z" (at level 70, y at next level).
18
19
Reserved Notation "x ≤ y ≤ z ≤ z'"
  (at level 70, y at next level, z at next level).
20

21
Infix "≤" := le : nat_scope.
22
23
24
25
Notation "x ≤ y ≤ z" := (x  y  y  z)%nat : nat_scope.
Notation "x ≤ y < z" := (x  y  y < z)%nat : nat_scope.
Notation "x < y < z" := (x < y  y < z)%nat : nat_scope.
Notation "x < y ≤ z" := (x < y  y  z)%nat : nat_scope.
26
Notation "x ≤ y ≤ z ≤ z'" := (x  y  y  z  z  z')%nat : nat_scope.
27
28
29
30
31
32
Notation "(≤)" := le (only parsing) : nat_scope.
Notation "(<)" := lt (only parsing) : nat_scope.

Infix "`div`" := NPeano.div (at level 35) : nat_scope.
Infix "`mod`" := NPeano.modulo (at level 35) : nat_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
33
Instance nat_eq_dec:  x y : nat, Decision (x = y) := eq_nat_dec.
34
35
Instance nat_le_dec:  x y : nat, Decision (x  y) := le_dec.
Instance nat_lt_dec:  x y : nat, Decision (x < y) := lt_dec.
36
Instance nat_inhabited: Inhabited nat := populate 0%nat.
37
38
39
40
Instance: Injective (=) (=) S.
Proof. by injection 1. Qed.
Instance: PartialOrder ().
Proof. repeat split; repeat intro; auto with lia. Qed.
41

42
43
44
45
46
47
Instance nat_le_pi:  x y : nat, ProofIrrel (x  y).
Proof.
  assert ( x y (p : x  y) y' (q : x  y'),
    y = y'  eq_dep nat (le x) y p y' q) as aux.
  { fix 3. intros x ? [|y p] ? [|y' q].
    * done.
48
49
    * clear nat_le_pi. intros; exfalso; auto with lia.
    * clear nat_le_pi. intros; exfalso; auto with lia.
50
51
52
53
54
55
56
    * injection 1. intros Hy. by case (nat_le_pi x y p y' q Hy). }
  intros x y p q.
  by apply (eq_dep_eq_dec (λ x y, decide (x = y))), aux.
Qed.
Instance nat_lt_pi:  x y : nat, ProofIrrel (x < y).
Proof. apply _. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
57
58
59
60
61
62
63
64
Definition sum_list_with {A} (f : A  nat) : list A  nat :=
  fix go l :=
  match l with
  | [] => 0
  | x :: l => f x + go l
  end.
Notation sum_list := (sum_list_with id).

65
66
67
Lemma Nat_lt_succ_succ n : n < S (S n).
Proof. auto with arith. Qed.
Lemma Nat_mul_split_l n x1 x2 y1 y2 :
68
69
  x2 < n  y2 < n  x1 * n + x2 = y1 * n + y2  x1 = y1  x2 = y2.
Proof.
70
  intros Hx2 Hy2 E. cut (x1 = y1); [intros; subst;lia |].
71
72
  revert y1 E. induction x1; simpl; intros [|?]; simpl; auto with lia.
Qed.
73
74
75
Lemma Nat_mul_split_r n x1 x2 y1 y2 :
  x1 < n  y1 < n  x1 + x2 * n = y1 + y2 * n  x1 = y1  x2 = y2.
Proof. intros. destruct (Nat_mul_split_l n x2 x1 y2 y1); auto with lia. Qed.
76

77
78
79
80
81
82
83
84
85
86
87
Notation lcm := Nat.lcm.
Notation divide := Nat.divide.
Notation "( x | y )" := (divide x y) : nat_scope.
Instance: PartialOrder divide.
Proof.
  repeat split; try apply _. intros ??. apply Nat.divide_antisym_nonneg; lia.
Qed.
Hint Extern 0 (_ | _) => reflexivity.
Lemma Nat_divide_ne_0 x y : (x | y)  y  0  x  0.
Proof. intros Hxy Hy ->. by apply Hy, Nat.divide_0_l. Qed.

88
89
90
(** * Notations and properties of [positive] *)
Open Scope positive_scope.

91
Infix "≤" := Pos.le : positive_scope.
92
93
94
95
96
Notation "x ≤ y ≤ z" := (x  y  y  z) : positive_scope.
Notation "x ≤ y < z" := (x  y  y < z) : positive_scope.
Notation "x < y < z" := (x < y  y < z) : positive_scope.
Notation "x < y ≤ z" := (x < y  y  z) : positive_scope.
Notation "x ≤ y ≤ z ≤ z'" := (x  y  y  z  z  z') : positive_scope.
97
98
Notation "(≤)" := Pos.le (only parsing) : positive_scope.
Notation "(<)" := Pos.lt (only parsing) : positive_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
99
100
101
Notation "(~0)" := xO (only parsing) : positive_scope.
Notation "(~1)" := xI (only parsing) : positive_scope.

102
103
104
105
Arguments Pos.of_nat _ : simpl never.
Instance positive_eq_dec:  x y : positive, Decision (x = y) := Pos.eq_dec.
Instance positive_inhabited: Inhabited positive := populate 1.

106
Instance: Injective (=) (=) (~0).
Robbert Krebbers's avatar
Robbert Krebbers committed
107
Proof. by injection 1. Qed.
108
Instance: Injective (=) (=) (~1).
Robbert Krebbers's avatar
Robbert Krebbers committed
109
110
Proof. by injection 1. Qed.

111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
(** Since [positive] represents lists of bits, we define list operations
on it. These operations are in reverse, as positives are treated as snoc
lists instead of cons lists. *)
Fixpoint Papp (p1 p2 : positive) : positive :=
  match p2 with
  | 1 => p1
  | p2~0 => (Papp p1 p2)~0
  | p2~1 => (Papp p1 p2)~1
  end.
Infix "++" := Papp : positive_scope.
Notation "(++)" := Papp (only parsing) : positive_scope.
Notation "( p ++)" := (Papp p) (only parsing) : positive_scope.
Notation "(++ q )" := (λ p, Papp p q) (only parsing) : positive_scope.

Fixpoint Preverse_go (p1 p2 : positive) : positive :=
  match p2 with
  | 1 => p1
  | p2~0 => Preverse_go (p1~0) p2
  | p2~1 => Preverse_go (p1~1) p2
  end.
Definition Preverse : positive  positive := Preverse_go 1.

Global Instance: LeftId (=) 1 (++).
134
Proof. intros p. by induction p; intros; f_equal'. Qed.
135
136
137
Global Instance: RightId (=) 1 (++).
Proof. done. Qed.
Global Instance: Associative (=) (++).
138
Proof. intros ?? p. by induction p; intros; f_equal'. Qed.
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
Global Instance:  p : positive, Injective (=) (=) (++ p).
Proof. intros p ???. induction p; simplify_equality; auto. Qed.

Lemma Preverse_go_app_cont p1 p2 p3 :
  Preverse_go (p2 ++ p1) p3 = p2 ++ Preverse_go p1 p3.
Proof.
  revert p1. induction p3; simpl; intros.
  * apply (IHp3 (_~1)).
  * apply (IHp3 (_~0)).
  * done.
Qed.
Lemma Preverse_go_app p1 p2 p3 :
  Preverse_go p1 (p2 ++ p3) = Preverse_go p1 p3 ++ Preverse_go 1 p2.
Proof.
  revert p1. induction p3; intros p1; simpl; auto.
  by rewrite <-Preverse_go_app_cont.
Qed.
Lemma Preverse_app p1 p2 :
  Preverse (p1 ++ p2) = Preverse p2 ++ Preverse p1.
Proof. unfold Preverse. by rewrite Preverse_go_app. Qed.

Lemma Preverse_xO p : Preverse (p~0) = (1~0) ++ Preverse p.
Proof Preverse_app p (1~0).
Lemma Preverse_xI p : Preverse (p~1) = (1~1) ++ Preverse p.
Proof Preverse_app p (1~1).

Fixpoint Plength (p : positive) : nat :=
166
  match p with 1 => 0%nat | p~0 | p~1 => S (Plength p) end.
167
168
Lemma Papp_length p1 p2 :
  Plength (p1 ++ p2) = (Plength p2 + Plength p1)%nat.
169
Proof. by induction p2; f_equal'. Qed.
170
171
172
173

Close Scope positive_scope.

(** * Notations and properties of [N] *)
Robbert Krebbers's avatar
Robbert Krebbers committed
174
Infix "≤" := N.le : N_scope.
175
176
177
178
Notation "x ≤ y ≤ z" := (x  y  y  z)%N : N_scope.
Notation "x ≤ y < z" := (x  y  y < z)%N : N_scope.
Notation "x < y < z" := (x < y  y < z)%N : N_scope.
Notation "x < y ≤ z" := (x < y  y  z)%N : N_scope.
179
Notation "x ≤ y ≤ z ≤ z'" := (x  y  y  z  z  z')%N : N_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
180
Notation "(≤)" := N.le (only parsing) : N_scope.
181
Notation "(<)" := N.lt (only parsing) : N_scope.
182
183
184
Infix "`div`" := N.div (at level 35) : N_scope.
Infix "`mod`" := N.modulo (at level 35) : N_scope.

185
186
Arguments N.add _ _ : simpl never.

Robbert Krebbers's avatar
Robbert Krebbers committed
187
188
189
Instance: Injective (=) (=) Npos.
Proof. by injection 1. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
190
191
192
193
194
195
196
Instance N_eq_dec:  x y : N, Decision (x = y) := N.eq_dec.
Program Instance N_le_dec (x y : N) : Decision (x  y)%N :=
  match Ncompare x y with
  | Gt => right _
  | _ => left _
  end.
Next Obligation. congruence. Qed.
197
198
199
200
201
202
Program Instance N_lt_dec (x y : N) : Decision (x < y)%N :=
  match Ncompare x y with
  | Lt => left _
  | _ => right _
  end.
Next Obligation. congruence. Qed.
203
Instance N_inhabited: Inhabited N := populate 1%N.
204
205
206
207
208
Instance: PartialOrder ()%N.
Proof.
  repeat split; red. apply N.le_refl. apply N.le_trans. apply N.le_antisymm.
Qed.
Hint Extern 0 (_  _)%N => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
209

210
(** * Notations and properties of [Z] *)
211
212
Open Scope Z_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
213
Infix "≤" := Z.le : Z_scope.
214
215
216
217
Notation "x ≤ y ≤ z" := (x  y  y  z) : Z_scope.
Notation "x ≤ y < z" := (x  y  y < z) : Z_scope.
Notation "x < y < z" := (x < y  y < z) : Z_scope.
Notation "x < y ≤ z" := (x < y  y  z) : Z_scope.
218
Notation "x ≤ y ≤ z ≤ z'" := (x  y  y  z  z  z') : Z_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
219
Notation "(≤)" := Z.le (only parsing) : Z_scope.
220
Notation "(<)" := Z.lt (only parsing) : Z_scope.
221

Robbert Krebbers's avatar
Robbert Krebbers committed
222
223
Infix "`div`" := Z.div (at level 35) : Z_scope.
Infix "`mod`" := Z.modulo (at level 35) : Z_scope.
224
225
Infix "`quot`" := Z.quot (at level 35) : Z_scope.
Infix "`rem`" := Z.rem (at level 35) : Z_scope.
226
227
Infix "≪" := Z.shiftl (at level 35) : Z_scope.
Infix "≫" := Z.shiftr (at level 35) : Z_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
228

229
230
231
232
233
Instance: Injective (=) (=) Zpos.
Proof. by injection 1. Qed.
Instance: Injective (=) (=) Zneg.
Proof. by injection 1. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
234
Instance Z_eq_dec:  x y : Z, Decision (x = y) := Z.eq_dec.
235
236
237
Instance Z_le_dec:  x y : Z, Decision (x  y) := Z_le_dec.
Instance Z_lt_dec:  x y : Z, Decision (x < y) := Z_lt_dec.
Instance Z_inhabited: Inhabited Z := populate 1.
238
239
240
241
Instance: PartialOrder ().
Proof.
  repeat split; red. apply Z.le_refl. apply Z.le_trans. apply Z.le_antisymm.
Qed.
242
243
244
245
246
247
248
249
250
251
252
253

Lemma Z_pow_pred_r n m : 0 < m  n * n ^ (Z.pred m) = n ^ m.
Proof.
  intros. rewrite <-Z.pow_succ_r, Z.succ_pred. done. by apply Z.lt_le_pred.
Qed.
Lemma Z_quot_range_nonneg k x y : 0  x < k  0 < y  0  x `quot` y < k.
Proof.
  intros [??] ?.
  destruct (decide (y = 1)); subst; [rewrite Z.quot_1_r; auto |].
  destruct (decide (x = 0)); subst; [rewrite Z.quot_0_l; auto with lia |].
  split. apply Z.quot_pos; lia. transitivity x; auto. apply Z.quot_lt; lia.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
254

255
(* Note that we cannot disable simpl for [Z.of_nat] as that would break
256
tactics as [lia]. *)
257
258
259
260
261
262
263
264
265
266
Arguments Z.to_nat _ : simpl never.
Arguments Z.mul _ _ : simpl never.
Arguments Z.add _ _ : simpl never.
Arguments Z.opp _ : simpl never.
Arguments Z.pow _ _ : simpl never.
Arguments Z.div _ _ : simpl never.
Arguments Z.modulo _ _ : simpl never.
Arguments Z.quot _ _ : simpl never.
Arguments Z.rem _ _ : simpl never.

267
Lemma Z_mod_pos a b : 0 < b  0  a `mod` b.
268
269
270
271
272
Proof. apply Z.mod_pos_bound. Qed.

Hint Resolve Z.lt_le_incl : zpos.
Hint Resolve Z.add_nonneg_pos Z.add_pos_nonneg Z.add_nonneg_nonneg : zpos.
Hint Resolve Z.mul_nonneg_nonneg Z.mul_pos_pos : zpos.
273
274
Hint Resolve Z.pow_pos_nonneg Z.pow_nonneg: zpos.
Hint Resolve Z_mod_pos Z.div_pos : zpos.
275
276
Hint Extern 1000 => lia : zpos.

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
Lemma Z2Nat_inj_pow (x y : nat) : Z.of_nat (x ^ y) = x ^ y.
Proof.
  induction y as [|y IH].
  * by rewrite Z.pow_0_r, Nat.pow_0_r.
  * by rewrite Nat.pow_succ_r, Nat2Z.inj_succ, Z.pow_succ_r,
      Nat2Z.inj_mul, IH by auto with zpos.
Qed.
Lemma Z2Nat_inj_div x y : Z.of_nat (x `div` y) = x `div` y.
Proof.
  destruct (decide (y = 0%nat)); [by subst; destruct x |].
  apply Z.div_unique with (x `mod` y)%nat.
  { left. rewrite <-(Nat2Z.inj_le 0), <-Nat2Z.inj_lt.
    apply Nat.mod_bound_pos; lia. }
  by rewrite <-Nat2Z.inj_mul, <-Nat2Z.inj_add, <-Nat.div_mod.
Qed.
Lemma Z2Nat_inj_mod x y : Z.of_nat (x `mod` y) = x `mod` y.
Proof.
  destruct (decide (y = 0%nat)); [by subst; destruct x |].
  apply Z.mod_unique with (x `div` y)%nat.
  { left. rewrite <-(Nat2Z.inj_le 0), <-Nat2Z.inj_lt.
    apply Nat.mod_bound_pos; lia. }
  by rewrite <-Nat2Z.inj_mul, <-Nat2Z.inj_add, <-Nat.div_mod.
Qed.
Close Scope Z_scope.

302
(** * Notations and properties of [Qc] *)
303
Open Scope Qc_scope.
304
305
Delimit Scope Qc_scope with Qc.
Notation "1" := (Q2Qc 1) : Qc_scope.
306
Notation "2" := (1+1) : Qc_scope.
307
308
309
310
Notation "- 1" := (Qcopp 1) : Qc_scope.
Notation "- 2" := (Qcopp 2) : Qc_scope.
Notation "x - y" := (x + -y) : Qc_scope.
Notation "x / y" := (x * /y) : Qc_scope.
311
Infix "≤" := Qcle : Qc_scope.
312
313
314
315
Notation "x ≤ y ≤ z" := (x  y  y  z) : Qc_scope.
Notation "x ≤ y < z" := (x  y  y < z) : Qc_scope.
Notation "x < y < z" := (x < y  y < z) : Qc_scope.
Notation "x < y ≤ z" := (x < y  y  z) : Qc_scope.
316
Notation "x ≤ y ≤ z ≤ z'" := (x  y  y  z  z  z') : Qc_scope.
317
318
319
Notation "(≤)" := Qcle (only parsing) : Qc_scope.
Notation "(<)" := Qclt (only parsing) : Qc_scope.

320
321
322
Hint Extern 1 (_  _) => reflexivity || discriminate.
Arguments Qred _ : simpl never.

323
Instance Qc_eq_dec:  x y : Qc, Decision (x = y) := Qc_eq_dec.
324
Program Instance Qc_le_dec (x y : Qc) : Decision (x  y) :=
325
326
  if Qclt_le_dec y x then right _ else left _.
Next Obligation. by apply Qclt_not_le. Qed.
327
Program Instance Qc_lt_dec (x y : Qc) : Decision (x < y) :=
328
329
330
  if Qclt_le_dec x y then left _ else right _.
Next Obligation. by apply Qcle_not_lt. Qed.

331
332
333
334
335
336
337
338
Instance: PartialOrder ().
Proof.
  repeat split; red. apply Qcle_refl. apply Qcle_trans. apply Qcle_antisym.
Qed.
Instance: StrictOrder (<).
Proof.
  split; red. intros x Hx. by destruct (Qclt_not_eq x x). apply Qclt_trans.
Qed.
339
340
341
342
Lemma Qcmult_0_l x : 0 * x = 0.
Proof. ring. Qed.
Lemma Qcmult_0_r x : x * 0 = 0.
Proof. ring. Qed.
343
Lemma Qcle_ngt (x y : Qc) : x  y  ¬y < x.
344
Proof. split; auto using Qcle_not_lt, Qcnot_lt_le. Qed.
345
Lemma Qclt_nge (x y : Qc) : x < y  ¬y  x.
346
Proof. split; auto using Qclt_not_le, Qcnot_le_lt. Qed.
347
Lemma Qcplus_le_mono_l (x y z : Qc) : x  y  z + x  z + y.
348
349
350
Proof.
  split; intros.
  * by apply Qcplus_le_compat.
351
352
  * replace x with ((0 - z) + (z + x)) by ring.
    replace y with ((0 - z) + (z + y)) by ring.
353
354
    by apply Qcplus_le_compat.
Qed.
355
Lemma Qcplus_le_mono_r (x y z : Qc) : x  y  x + z  y + z.
356
Proof. rewrite !(Qcplus_comm _ z). apply Qcplus_le_mono_l. Qed.
357
Lemma Qcplus_lt_mono_l (x y z : Qc) : x < y  z + x < z + y.
358
Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_l. Qed.
359
Lemma Qcplus_lt_mono_r (x y z : Qc) : x < y  x + z < y + z.
360
Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_r. Qed.
361
362
363
364
Instance: Injective (=) (=) Qcopp.
Proof.
  intros x y H. by rewrite <-(Qcopp_involutive x), H, Qcopp_involutive.
Qed.
365
Instance:  z, Injective (=) (=) (Qcplus z).
366
367
368
369
Proof.
  intros z x y H. by apply (anti_symmetric ());
    rewrite (Qcplus_le_mono_l _ _ z), H.
Qed.
370
371
372
373
374
Instance:  z, Injective (=) (=) (λ x, x + z).
Proof.
  intros z x y H. by apply (anti_symmetric ());
    rewrite (Qcplus_le_mono_r _ _ z), H.
Qed.
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
Lemma Qcplus_pos_nonneg (x y : Qc) : 0 < x  0  y  0 < x + y.
Proof.
  intros. apply Qclt_le_trans with (x + 0); [by rewrite Qcplus_0_r|].
  by apply Qcplus_le_mono_l.
Qed.
Lemma Qcplus_nonneg_pos (x y : Qc) : 0  x  0 < y  0 < x + y.
Proof. rewrite (Qcplus_comm x). auto using Qcplus_pos_nonneg. Qed. 
Lemma Qcplus_pos_pos (x y : Qc) : 0 < x  0 < y  0 < x + y.
Proof. auto using Qcplus_pos_nonneg, Qclt_le_weak. Qed.
Lemma Qcplus_nonneg_nonneg (x y : Qc) : 0  x  0  y  0  x + y.
Proof.
  intros. transitivity (x + 0); [by rewrite Qcplus_0_r|].
  by apply Qcplus_le_mono_l.
Qed.
Lemma Qcplus_neg_nonpos (x y : Qc) : x < 0  y  0  x + y < 0.
Proof.
  intros. apply Qcle_lt_trans with (x + 0); [|by rewrite Qcplus_0_r].
  by apply Qcplus_le_mono_l.
Qed.
Lemma Qcplus_nonpos_neg (x y : Qc) : x  0  y < 0  x + y < 0.
Proof. rewrite (Qcplus_comm x). auto using Qcplus_neg_nonpos. Qed.
Lemma Qcplus_neg_neg (x y : Qc) : x < 0  y < 0  x + y < 0.
Proof. auto using Qcplus_nonpos_neg, Qclt_le_weak. Qed.
Lemma Qcplus_nonpos_nonpos (x y : Qc) : x  0  y  0  x + y  0.
Proof.
  intros. transitivity (x + 0); [|by rewrite Qcplus_0_r].
  by apply Qcplus_le_mono_l.
Qed.
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
Lemma Qcmult_le_mono_nonneg_l x y z : 0  z  x  y  z * x  z * y.
Proof. intros. rewrite !(Qcmult_comm z). by apply Qcmult_le_compat_r. Qed.
Lemma Qcmult_le_mono_nonneg_r x y z : 0  z  x  y  x * z  y * z.
Proof. intros. by apply Qcmult_le_compat_r. Qed.
Lemma Qcmult_le_mono_pos_l x y z : 0 < z  x  y  z * x  z * y.
Proof.
  split; auto using Qcmult_le_mono_nonneg_l, Qclt_le_weak.
  rewrite !Qcle_ngt, !(Qcmult_comm z).
  intuition auto using Qcmult_lt_compat_r.
Qed.
Lemma Qcmult_le_mono_pos_r x y z : 0 < z  x  y  x * z  y * z.
Proof. rewrite !(Qcmult_comm _ z). by apply Qcmult_le_mono_pos_l. Qed.
Lemma Qcmult_lt_mono_pos_l x y z : 0 < z  x < y  z * x < z * y.
Proof. intros. by rewrite !Qclt_nge, <-Qcmult_le_mono_pos_l. Qed.
Lemma Qcmult_lt_mono_pos_r x y z : 0 < z  x < y  x * z < y * z.
Proof. intros. by rewrite !Qclt_nge, <-Qcmult_le_mono_pos_r. Qed.
Lemma Qcmult_pos_pos x y : 0 < x  0 < y  0 < x * y.
Proof.
  intros. apply Qcle_lt_trans with (0 * y); [by rewrite Qcmult_0_l|].
  by apply Qcmult_lt_mono_pos_r.
Qed.
Lemma Qcmult_nonneg_nonneg x y : 0  x  0  y  0  x * y.
Proof.
  intros. transitivity (0 * y); [by rewrite Qcmult_0_l|].
  by apply Qcmult_le_mono_nonneg_r.
Qed.

Lemma inject_Z_Qred n : Qred (inject_Z n) = inject_Z n.
Proof. apply Qred_identity; auto using Z.gcd_1_r. Qed.
Coercion Qc_of_Z (n : Z) : Qc := Qcmake _ (inject_Z_Qred n).
Lemma Z2Qc_inj_0 : Qc_of_Z 0 = 0.
Proof. by apply Qc_is_canon. Qed.
Lemma Z2Qc_inj n m : Qc_of_Z n = Qc_of_Z m  n = m.
Proof. by injection 1. Qed.
Lemma Z2Qc_inj_iff n m : Qc_of_Z n = Qc_of_Z m  n = m.
Proof. split. auto using Z2Qc_inj. by intros ->. Qed.
Lemma Z2Qc_inj_le n m : (n  m)%Z  Qc_of_Z n  Qc_of_Z m.
Proof. by rewrite Zle_Qle. Qed.
Lemma Z2Qc_inj_lt n m : (n < m)%Z  Qc_of_Z n < Qc_of_Z m.
Proof. by rewrite Zlt_Qlt. Qed.
Lemma Z2Qc_inj_add n m : Qc_of_Z (n + m) = Qc_of_Z n + Qc_of_Z m.
Proof. apply Qc_is_canon; simpl. by rewrite Qred_correct, inject_Z_plus. Qed.
Lemma Z2Qc_inj_mul n m : Qc_of_Z (n * m) = Qc_of_Z n * Qc_of_Z m.
Proof. apply Qc_is_canon; simpl. by rewrite Qred_correct, inject_Z_mult. Qed.
Lemma Z2Qc_inj_opp n : Qc_of_Z (-n) = -Qc_of_Z n.
Proof. apply Qc_is_canon; simpl. by rewrite Qred_correct, inject_Z_opp. Qed.
Lemma Z2Qc_inj_sub n m : Qc_of_Z (n - m) = Qc_of_Z n - Qc_of_Z m.
Proof.
  apply Qc_is_canon; simpl.
  by rewrite !Qred_correct, <-inject_Z_opp, <-inject_Z_plus.
Qed.
454
Close Scope Qc_scope.
455

456
(** * Conversions *)
457
Lemma Z_to_nat_nonpos x : (x  0)%Z  Z.to_nat x = 0.
458
Proof. destruct x; simpl; auto using Z2Nat.inj_neg. by intros []. Qed.
459

460
461
(** The function [Z_to_option_N] converts an integer [x] into a natural number
by giving [None] in case [x] is negative. *)
462
Definition Z_to_option_N (x : Z) : option N :=
Robbert Krebbers's avatar
Robbert Krebbers committed
463
  match x with
464
  | Z0 => Some N0 | Zpos p => Some (Npos p) | Zneg _ => None
Robbert Krebbers's avatar
Robbert Krebbers committed
465
  end.
466
467
Definition Z_to_option_nat (x : Z) : option nat :=
  match x with
468
  | Z0 => Some 0 | Zpos p => Some (Pos.to_nat p) | Zneg _ => None
469
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
470

471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
Lemma Z_to_option_N_Some x y :
  Z_to_option_N x = Some y  (0  x)%Z  y = Z.to_N x.
Proof.
  split.
  * intros. by destruct x; simpl in *; simplify_equality;
      auto using Zle_0_pos.
  * intros [??]. subst. destruct x; simpl; auto; lia.
Qed.
Lemma Z_to_option_N_Some_alt x y :
  Z_to_option_N x = Some y  (0  x)%Z  x = Z.of_N y.
Proof.
  rewrite Z_to_option_N_Some.
  split; intros [??]; subst; auto using N2Z.id, Z2N.id, eq_sym.
Qed.

Lemma Z_to_option_nat_Some x y :
  Z_to_option_nat x = Some y  (0  x)%Z  y = Z.to_nat x.
Proof.
  split.
  * intros. by destruct x; simpl in *; simplify_equality;
      auto using Zle_0_pos.
  * intros [??]. subst. destruct x; simpl; auto; lia.
Qed.
Lemma Z_to_option_nat_Some_alt x y :
  Z_to_option_nat x = Some y  (0  x)%Z  x = Z.of_nat y.
Proof.
  rewrite Z_to_option_nat_Some.
  split; intros [??]; subst; auto using Nat2Z.id, Z2Nat.id, eq_sym.
Qed.
500
Lemma Z_to_option_of_nat x : Z_to_option_nat (Z.of_nat x) = Some x.
501
502
Proof. apply Z_to_option_nat_Some_alt. auto using Nat2Z.is_nonneg. Qed.

503
504
505
506
507
508
509
510
511
512
513
514
515
(** Some correspondence lemmas between [nat] and [N] that are not part of the
standard library. We declare a hint database [natify] to rewrite a goal
involving [N] into a corresponding variant involving [nat]. *)
Lemma N_to_nat_lt x y : N.to_nat x < N.to_nat y  (x < y)%N.
Proof. by rewrite <-N.compare_lt_iff, nat_compare_lt, N2Nat.inj_compare. Qed.
Lemma N_to_nat_le x y : N.to_nat x  N.to_nat y  (x  y)%N.
Proof. by rewrite <-N.compare_le_iff, nat_compare_le, N2Nat.inj_compare. Qed.
Lemma N_to_nat_0 : N.to_nat 0 = 0.
Proof. done. Qed.
Lemma N_to_nat_1 : N.to_nat 1 = 1.
Proof. done. Qed.
Lemma N_to_nat_div x y : N.to_nat (x `div` y) = N.to_nat x `div` N.to_nat y.
Proof.
516
517
  destruct (decide (y = 0%N)); [by subst; destruct x |].
  apply Nat.div_unique with (N.to_nat (x `mod` y)).
518
519
520
521
522
523
  { by apply N_to_nat_lt, N.mod_lt. }
  rewrite (N.div_unique_exact (x * y) y x), N.div_mul by lia.
  by rewrite <-N2Nat.inj_mul, <-N2Nat.inj_add, <-N.div_mod.
Qed.
(* We have [x `mod` 0 = 0] on [nat], and [x `mod` 0 = x] on [N]. *)
Lemma N_to_nat_mod x y :
524
  y  0%N  N.to_nat (x `mod` y) = N.to_nat x `mod` N.to_nat y.
525
Proof.
526
  intros. apply Nat.mod_unique with (N.to_nat (x `div` y)).
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
  { by apply N_to_nat_lt, N.mod_lt. }
  rewrite (N.div_unique_exact (x * y) y x), N.div_mul by lia.
  by rewrite <-N2Nat.inj_mul, <-N2Nat.inj_add, <-N.div_mod.
Qed.

Hint Rewrite <-N2Nat.inj_iff : natify.
Hint Rewrite <-N_to_nat_lt : natify.
Hint Rewrite <-N_to_nat_le : natify.
Hint Rewrite Nat2N.id : natify.
Hint Rewrite N2Nat.inj_add : natify.
Hint Rewrite N2Nat.inj_mul : natify.
Hint Rewrite N2Nat.inj_sub : natify.
Hint Rewrite N2Nat.inj_succ : natify.
Hint Rewrite N2Nat.inj_pred : natify.
Hint Rewrite N_to_nat_div : natify.
Hint Rewrite N_to_nat_0 : natify.
Hint Rewrite N_to_nat_1 : natify.
Ltac natify := repeat autorewrite with natify in *.

Hint Extern 100 (Nlt _ _) => natify : natify.
Hint Extern 100 (Nle _ _) => natify : natify.
Hint Extern 100 (@eq N _ _) => natify : natify.
Hint Extern 100 (lt _ _) => natify : natify.
Hint Extern 100 (le _ _) => natify : natify.
Hint Extern 100 (@eq nat _ _) => natify : natify.

Instance:  x, PropHolds (0 < x)%N  PropHolds (0 < N.to_nat x).
Proof. unfold PropHolds. intros. by natify. Qed.
Instance:  x, PropHolds (0  x)%N  PropHolds (0  N.to_nat x).
Proof. unfold PropHolds. intros. by natify. Qed.