numbers.v 24.7 KB
Newer Older
1
(* Copyright (c) 2012-2017, Coq-std++ developers. *)
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
From Coq Require Export EqdepFacts PArith NArith ZArith NPeano.
7
8
From Coq Require Import QArith Qcanon.
From stdpp Require Export base decidable option.
9
Set Default Proof Using "Type".
10
Open Scope nat_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
11

12
Coercion Z.of_nat : nat >-> Z.
13
Instance comparison_eq_dec : EqDecision comparison.
14
Proof. solve_decision. Defined.
15

16
(** * Notations and properties of [nat] *)
17
Arguments minus !_ !_ / : assert.
18
19
20
21
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).
22
23
Reserved Notation "x ≤ y ≤ z ≤ z'"
  (at level 70, y at next level, z at next level).
24

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

Robbert Krebbers's avatar
Robbert Krebbers committed
33
34
Infix "`div`" := Nat.div (at level 35) : nat_scope.
Infix "`mod`" := Nat.modulo (at level 35) : nat_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
35
36
Infix "`max`" := Nat.max (at level 35) : nat_scope.
Infix "`min`" := Nat.min (at level 35) : nat_scope.
37

38
Instance nat_eq_dec: EqDecision nat := eq_nat_dec.
39
40
Instance nat_le_dec: RelDecision le := le_dec.
Instance nat_lt_dec: RelDecision lt := lt_dec.
41
Instance nat_inhabited: Inhabited nat := populate 0%nat.
42
Instance S_inj: Inj (=) (=) S.
43
Proof. by injection 1. Qed.
44
Instance nat_le_po: PartialOrder ().
45
Proof. repeat split; repeat intro; auto with lia. Qed.
46

47
48
49
50
51
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].
52
53
54
55
    - done.
    - clear nat_le_pi. intros; exfalso; auto with lia.
    - clear nat_le_pi. intros; exfalso; auto with lia.
    - injection 1. intros Hy. by case (nat_le_pi x y p y' q Hy). }
56
  intros x y p q.
57
  by apply (Eqdep_dec.eq_dep_eq_dec (λ x y, decide (x = y))), aux.
58
59
60
61
Qed.
Instance nat_lt_pi:  x y : nat, ProofIrrel (x < y).
Proof. apply _. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
62
63
Lemma nat_le_sum (x y : nat) : x  y   z, y = x + z.
Proof. split. exists (y - x); lia. intros [z ->]; lia. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
64

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
Notation lcm := Nat.lcm.
Notation divide := Nat.divide.
Notation "( x | y )" := (divide x y) : nat_scope.
80
Instance Nat_divide_dec : RelDecision Nat.divide.
81
Proof.
82
  refine (λ x y, cast_if (decide (lcm x y = y))); by rewrite Nat.divide_lcm_iff.
83
Defined.
84
85
86
87
88
89
90
91
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.

92
93
94
95
Lemma Nat_iter_S {A} n (f: A  A) x : Nat.iter (S n) f x = f (Nat.iter n f x).
Proof. done. Qed.
Lemma Nat_iter_S_r {A} n (f: A  A) x : Nat.iter (S n) f x = Nat.iter n f (f x).
Proof. induction n; f_equal/=; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
96
Lemma Nat_iter_ind {A} (P : A  Prop) f x k :
Robbert Krebbers's avatar
Robbert Krebbers committed
97
98
  P x  ( y, P y  P (f y))  P (Nat.iter k f x).
Proof. induction k; simpl; auto. Qed.
99

Robbert Krebbers's avatar
Robbert Krebbers committed
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
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).

Definition max_list_with {A} (f : A  nat) : list A  nat :=
  fix go l :=
  match l with
  | [] => 0
  | x :: l => f x `max` go l
  end.
Notation max_list := (max_list_with id).

116
117
118
(** * Notations and properties of [positive] *)
Open Scope positive_scope.

119
Infix "≤" := Pos.le : positive_scope.
120
121
122
123
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.
124
125
Notation "(≤)" := Pos.le (only parsing) : positive_scope.
Notation "(<)" := Pos.lt (only parsing) : positive_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
126
127
128
Notation "(~0)" := xO (only parsing) : positive_scope.
Notation "(~1)" := xI (only parsing) : positive_scope.

129
130
131
Arguments Pos.of_nat : simpl never.
Arguments Pmult : simpl never.

132
Instance positive_eq_dec: EqDecision positive := Pos.eq_dec.
133
134
135
136
Instance positive_le_dec: RelDecision Pos.le.
Proof. refine (λ x y, decide ((x ?= y)  Gt)). Defined.
Instance positive_lt_dec: RelDecision Pos.lt.
Proof. refine (λ x y, decide ((x ?= y) = Lt)). Defined.
137
138
Instance positive_inhabited: Inhabited positive := populate 1.

139
Instance maybe_xO : Maybe xO := λ p, match p with p~0 => Some p | _ => None end.
Robbert Krebbers's avatar
Robbert Krebbers committed
140
141
Instance maybe_xI : Maybe xI := λ p, match p with p~1 => Some p | _ => None end.
Instance xO_inj : Inj (=) (=) (~0).
Robbert Krebbers's avatar
Robbert Krebbers committed
142
Proof. by injection 1. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
143
Instance xI_inj : Inj (=) (=) (~1).
Robbert Krebbers's avatar
Robbert Krebbers committed
144
145
Proof. by injection 1. Qed.

146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
(** 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.

Robbert Krebbers's avatar
Robbert Krebbers committed
168
Global Instance Papp_1_l : LeftId (=) 1 (++).
169
Proof. intros p. by induction p; intros; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
170
Global Instance Papp_1_r : RightId (=) 1 (++).
171
Proof. done. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
172
Global Instance Papp_assoc : Assoc (=) (++).
173
Proof. intros ?? p. by induction p; intros; f_equal/=. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
174
175
Global Instance Papp_inj p : Inj (=) (=) (++ p).
Proof. intros ???. induction p; simplify_eq; auto. Qed.
176
177
178
179

Lemma Preverse_go_app p1 p2 p3 :
  Preverse_go p1 (p2 ++ p3) = Preverse_go p1 p3 ++ Preverse_go 1 p2.
Proof.
180
181
182
183
  revert p3 p1 p2.
  cut ( p1 p2 p3, Preverse_go (p2 ++ p3) p1 = p2 ++ Preverse_go p3 p1).
  { by intros go p3; induction p3; intros p1 p2; simpl; auto; rewrite <-?go. }
  intros p1; induction p1 as [p1 IH|p1 IH|]; intros p2 p3; simpl; auto.
184
185
  - apply (IH _ (_~1)).
  - apply (IH _ (_~0)).
186
Qed.
187
Lemma Preverse_app p1 p2 : Preverse (p1 ++ p2) = Preverse p2 ++ Preverse p1.
188
189
190
191
192
193
194
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 :=
195
  match p with 1 => 0%nat | p~0 | p~1 => S (Plength p) end.
196
Lemma Papp_length p1 p2 : Plength (p1 ++ p2) = (Plength p2 + Plength p1)%nat.
197
Proof. by induction p2; f_equal/=. Qed.
198

Robbert Krebbers's avatar
Robbert Krebbers committed
199
200
201
202
203
204
205
Lemma Plt_sum (x y : positive) : x < y   z, y = x + z.
Proof.
  split.
  - exists (y - x)%positive. symmetry. apply Pplus_minus. lia.
  - intros [z ->]. lia.
Qed.

206
207
208
Close Scope positive_scope.

(** * Notations and properties of [N] *)
Robbert Krebbers's avatar
Robbert Krebbers committed
209
Infix "≤" := N.le : N_scope.
210
211
212
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.
213
Notation "x ≤ y ≤ z ≤ z'" := (x  y  y  z  z  z')%N : N_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
214
Notation "(≤)" := N.le (only parsing) : N_scope.
215
Notation "(<)" := N.lt (only parsing) : N_scope.
216
217
218
Infix "`div`" := N.div (at level 35) : N_scope.
Infix "`mod`" := N.modulo (at level 35) : N_scope.

219
Arguments N.add : simpl never.
220

Robbert Krebbers's avatar
Robbert Krebbers committed
221
Instance Npos_inj : Inj (=) (=) Npos.
Robbert Krebbers's avatar
Robbert Krebbers committed
222
223
Proof. by injection 1. Qed.

224
Instance N_eq_dec: EqDecision N := N.eq_dec.
225
Program Instance N_le_dec : RelDecision N.le := λ x y,
226
227
  match Ncompare x y with Gt => right _ | _ => left _ end.
Solve Obligations with naive_solver.
228
Program Instance N_lt_dec : RelDecision N.lt := λ x y,
229
230
  match Ncompare x y with Lt => left _ | _ => right _ end.
Solve Obligations with naive_solver.
231
Instance N_inhabited: Inhabited N := populate 1%N.
232
Instance N_le_po: PartialOrder ()%N.
233
234
235
236
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
237

238
(** * Notations and properties of [Z] *)
239
240
Open Scope Z_scope.

Robbert Krebbers's avatar
Robbert Krebbers committed
241
Infix "≤" := Z.le : Z_scope.
242
243
244
245
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.
246
Notation "x ≤ y ≤ z ≤ z'" := (x  y  y  z  z  z') : Z_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
247
Notation "(≤)" := Z.le (only parsing) : Z_scope.
248
Notation "(<)" := Z.lt (only parsing) : Z_scope.
249

Robbert Krebbers's avatar
Robbert Krebbers committed
250
251
Infix "`div`" := Z.div (at level 35) : Z_scope.
Infix "`mod`" := Z.modulo (at level 35) : Z_scope.
252
253
Infix "`quot`" := Z.quot (at level 35) : Z_scope.
Infix "`rem`" := Z.rem (at level 35) : Z_scope.
254
255
Infix "≪" := Z.shiftl (at level 35) : Z_scope.
Infix "≫" := Z.shiftr (at level 35) : Z_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
256

257
Instance Zpos_inj : Inj (=) (=) Zpos.
258
Proof. by injection 1. Qed.
259
Instance Zneg_inj : Inj (=) (=) Zneg.
260
261
Proof. by injection 1. Qed.

262
263
264
Instance Z_of_nat_inj : Inj (=) (=) Z.of_nat.
Proof. intros n1 n2. apply Nat2Z.inj. Qed.

265
Instance Z_eq_dec: EqDecision Z := Z.eq_dec.
266
267
Instance Z_le_dec: RelDecision Z.le := Z_le_dec.
Instance Z_lt_dec: RelDecision Z.lt := Z_lt_dec.
268
Instance Z_inhabited: Inhabited Z := populate 1.
269
Instance Z_le_po : PartialOrder ().
270
271
272
Proof.
  repeat split; red. apply Z.le_refl. apply Z.le_trans. apply Z.le_antisymm.
Qed.
273
274
275
276
277
278
279
280
281
282

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 |].
283
  split. apply Z.quot_pos; lia. trans x; auto. apply Z.quot_lt; lia.
284
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
285

286
(* Note that we cannot disable simpl for [Z.of_nat] as that would break
287
tactics as [lia]. *)
288
289
290
291
292
293
294
295
296
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.
297

298
299
300
301
302
Lemma Z_to_nat_neq_0_pos x : Z.to_nat x  0%nat  0 < x.
Proof. by destruct x. Qed.
Lemma Z_to_nat_neq_0_nonneg x : Z.to_nat x  0%nat  0  x.
Proof. by destruct x. Qed.
Lemma Z_mod_pos x y : 0 < y  0  x `mod` y.
303
304
305
306
307
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.
308
309
Hint Resolve Z.pow_pos_nonneg Z.pow_nonneg: zpos.
Hint Resolve Z_mod_pos Z.div_pos : zpos.
310
311
Hint Extern 1000 => lia : zpos.

Robbert Krebbers's avatar
Robbert Krebbers committed
312
313
Lemma Z_to_nat_nonpos x : x  0  Z.to_nat x = 0%nat.
Proof. destruct x; simpl; auto using Z2Nat.inj_neg. by intros []. Qed.
314
315
Lemma Z2Nat_inj_pow (x y : nat) : Z.of_nat (x ^ y) = x ^ y.
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
316
317
318
  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.
319
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
320
321
322
Lemma Nat2Z_divide n m : (Z.of_nat n | Z.of_nat m)  (n | m)%nat.
Proof.
  split.
323
  - rewrite <-(Nat2Z.id m) at 2; intros [i ->]; exists (Z.to_nat i).
Robbert Krebbers's avatar
Robbert Krebbers committed
324
325
326
    destruct (decide (0  i)%Z).
    { by rewrite Z2Nat.inj_mul, Nat2Z.id by lia. }
    by rewrite !Z_to_nat_nonpos by auto using Z.mul_nonpos_nonneg with lia.
327
  - intros [i ->]. exists (Z.of_nat i). by rewrite Nat2Z.inj_mul.
Robbert Krebbers's avatar
Robbert Krebbers committed
328
329
330
331
Qed.
Lemma Z2Nat_divide n m :
  0  n  0  m  (Z.to_nat n | Z.to_nat m)%nat  (n | m).
Proof. intros. by rewrite <-Nat2Z_divide, !Z2Nat.id by done. Qed.
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
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.

350
351
352
353
354
355
356
357
(** * Injectivity of casts *)
Instance N_of_nat_inj: Inj (=) (=) N.of_nat := Nat2N.inj.
Instance nat_of_N_inj: Inj (=) (=) N.to_nat := N2Nat.inj.
Instance nat_of_pos_inj: Inj (=) (=) Pos.to_nat := Pos2Nat.inj.
Instance pos_of_Snat_inj: Inj (=) (=) Pos.of_succ_nat := SuccNat2Pos.inj.
Instance Z_of_N_inj: Inj (=) (=) Z.of_N := N2Z.inj.
(* Add others here. *)

358
(** * Notations and properties of [Qc] *)
359
Open Scope Qc_scope.
360
361
Delimit Scope Qc_scope with Qc.
Notation "1" := (Q2Qc 1) : Qc_scope.
362
Notation "2" := (1+1) : Qc_scope.
363
364
365
366
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.
367
Infix "≤" := Qcle : Qc_scope.
368
369
370
371
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.
372
Notation "x ≤ y ≤ z ≤ z'" := (x  y  y  z  z  z') : Qc_scope.
373
374
375
Notation "(≤)" := Qcle (only parsing) : Qc_scope.
Notation "(<)" := Qclt (only parsing) : Qc_scope.

376
Hint Extern 1 (_  _) => reflexivity || discriminate.
377
Arguments Qred : simpl never.
378

379
Instance Qc_eq_dec: EqDecision Qc := Qc_eq_dec.
380
Program Instance Qc_le_dec: RelDecision Qcle := λ x y,
381
  if Qclt_le_dec y x then right _ else left _.
382
383
Next Obligation. intros x y; apply Qclt_not_le. Qed.
Next Obligation. done. Qed.
384
Program Instance Qc_lt_dec: RelDecision Qclt := λ x y,
385
  if Qclt_le_dec x y then left _ else right _.
386
387
Solve Obligations with done.
Next Obligation. intros x y; apply Qcle_not_lt. Qed.
388

389
390
391
392
393
394
395
396
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.
397
398
399
400
Lemma Qcmult_0_l x : 0 * x = 0.
Proof. ring. Qed.
Lemma Qcmult_0_r x : x * 0 = 0.
Proof. ring. Qed.
401
402
Lemma Qcplus_diag x : (x + x)%Qc = (2 * x)%Qc.
Proof. ring. Qed.
403
Lemma Qcle_ngt (x y : Qc) : x  y  ¬y < x.
404
Proof. split; auto using Qcle_not_lt, Qcnot_lt_le. Qed.
405
Lemma Qclt_nge (x y : Qc) : x < y  ¬y  x.
406
Proof. split; auto using Qclt_not_le, Qcnot_le_lt. Qed.
407
Lemma Qcplus_le_mono_l (x y z : Qc) : x  y  z + x  z + y.
408
409
Proof.
  split; intros.
410
411
  - by apply Qcplus_le_compat.
  - replace x with ((0 - z) + (z + x)) by ring.
412
    replace y with ((0 - z) + (z + y)) by ring.
413
414
    by apply Qcplus_le_compat.
Qed.
415
Lemma Qcplus_le_mono_r (x y z : Qc) : x  y  x + z  y + z.
416
Proof. rewrite !(Qcplus_comm _ z). apply Qcplus_le_mono_l. Qed.
417
Lemma Qcplus_lt_mono_l (x y z : Qc) : x < y  z + x < z + y.
418
Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_l. Qed.
419
Lemma Qcplus_lt_mono_r (x y z : Qc) : x < y  x + z < y + z.
420
Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_r. Qed.
421
Instance: Inj (=) (=) Qcopp.
422
423
424
Proof.
  intros x y H. by rewrite <-(Qcopp_involutive x), H, Qcopp_involutive.
Qed.
425
Instance:  z, Inj (=) (=) (Qcplus z).
426
Proof.
427
  intros z x y H. by apply (anti_symm ());
428
429
    rewrite (Qcplus_le_mono_l _ _ z), H.
Qed.
430
Instance:  z, Inj (=) (=) (λ x, x + z).
431
Proof.
432
  intros z x y H. by apply (anti_symm ());
433
434
    rewrite (Qcplus_le_mono_r _ _ z), H.
Qed.
435
436
437
438
439
440
441
442
443
444
445
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.
446
  intros. trans (x + 0); [by rewrite Qcplus_0_r|].
447
448
449
450
451
452
453
454
455
456
457
458
459
  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.
460
  intros. trans (x + 0); [|by rewrite Qcplus_0_r].
461
462
  by apply Qcplus_le_mono_l.
Qed.
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
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.
486
  intros. trans (0 * y); [by rewrite Qcmult_0_l|].
487
488
489
490
491
492
493
494
  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.
495
496
497
498
Lemma Z2Qc_inj_1 : Qc_of_Z 1 = 1.
Proof. by apply Qc_is_canon. Qed.
Lemma Z2Qc_inj_2 : Qc_of_Z 2 = 2.
Proof. by apply Qc_is_canon. Qed.
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
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.
518
Close Scope Qc_scope.
519
520
521
522
523
524
525
526

(** * Positive rationals *)
(** The theory of positive rationals is very incomplete. We merely provide
some operations and theorems that are relevant for fractional permissions. *)
Record Qp := mk_Qp { Qp_car :> Qc ; Qp_prf : (0 < Qp_car)%Qc }.
Hint Resolve Qp_prf.
Delimit Scope Qp_scope with Qp.
Bind Scope Qp_scope with Qp.
527
Arguments Qp_car _%Qp : assert.
528
529
530
531
532
533
534

Definition Qp_one : Qp := mk_Qp 1 eq_refl.
Program Definition Qp_plus (x y : Qp) : Qp := mk_Qp (x + y) _.
Next Obligation. by intros x y; apply Qcplus_pos_pos. Qed.
Definition Qp_minus (x y : Qp) : option Qp :=
  let z := (x - y)%Qc in
  match decide (0 < z)%Qc with left Hz => Some (mk_Qp z Hz) | _ => None end.
535
536
Program Definition Qp_mult (x y : Qp) : Qp := mk_Qp (x * y) _.
Next Obligation. intros x y. apply Qcmult_pos_pos; apply Qp_prf. Qed.
537
538
539
540
541
542
543
544
545
546
547
Program Definition Qp_div (x : Qp) (y : positive) : Qp := mk_Qp (x / ('y)%Z) _.  
Next Obligation.
  intros x y. assert (0 < ('y)%Z)%Qc.
  { apply (Z2Qc_inj_lt 0%Z (' y)), Pos2Z.is_pos. }
  by rewrite (Qcmult_lt_mono_pos_r _ _ ('y)%Z), Qcmult_0_l,
    <-Qcmult_assoc, Qcmult_inv_l, Qcmult_1_r.
Qed.

Notation "1" := Qp_one : Qp_scope.
Infix "+" := Qp_plus : Qp_scope.
Infix "-" := Qp_minus : Qp_scope.
548
Infix "*" := Qp_mult : Qp_scope.
549
550
551
552
553
554
555
Infix "/" := Qp_div : Qp_scope.

Lemma Qp_eq x y : x = y  Qp_car x = Qp_car y.
Proof.
  split; [by intros ->|].
  destruct x, y; intros; simplify_eq/=; f_equal; apply (proof_irrel _).
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
556
557
558
559
560
561
562

Instance Qp_inhabited : Inhabited Qp := populate 1%Qp.
Instance Qp_eq_dec : EqDecision Qp.
Proof.
 refine (λ x y, cast_if (decide (Qp_car x = Qp_car y))); by rewrite Qp_eq.
Defined.

563
564
565
566
567
568
569
570
571
572
573
574
575
576
Instance Qp_plus_assoc : Assoc (=) Qp_plus.
Proof. intros x y z; apply Qp_eq, Qcplus_assoc. Qed.
Instance Qp_plus_comm : Comm (=) Qp_plus.
Proof. intros x y; apply Qp_eq, Qcplus_comm. Qed.

Lemma Qp_minus_diag x : (x - x)%Qp = None.
Proof. unfold Qp_minus. by rewrite Qcplus_opp_r. Qed.
Lemma Qp_op_minus x y : ((x + y) - x)%Qp = Some y.
Proof.
  unfold Qp_minus; simpl.
  rewrite (Qcplus_comm x), <- Qcplus_assoc, Qcplus_opp_r, Qcplus_0_r.
  destruct (decide _) as [|[]]; auto. by f_equal; apply Qp_eq.
Qed.

577
578
579
580
581
582
583
584
585
586
587
588
589
Instance Qp_mult_assoc : Assoc (=) Qp_mult.
Proof. intros x y z; apply Qp_eq, Qcmult_assoc. Qed.
Instance Qp_mult_comm : Comm (=) Qp_mult.
Proof. intros x y; apply Qp_eq, Qcmult_comm. Qed.
Lemma Qp_mult_plus_distr_r x y z: (x * (y + z) = x * y + x * z)%Qp.
Proof. apply Qp_eq, Qcmult_plus_distr_r. Qed.
Lemma Qp_mult_plus_distr_l x y z: ((x + y) * z = x * z + y * z)%Qp.
Proof. apply Qp_eq, Qcmult_plus_distr_l. Qed.
Lemma Qp_mult_1_l x: (1 * x)%Qp = x.
Proof. apply Qp_eq, Qcmult_1_l. Qed.
Lemma Qp_mult_1_r x: (x * 1)%Qp = x.
Proof. apply Qp_eq, Qcmult_1_r. Qed.

590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
Lemma Qp_div_1 x : (x / 1 = x)%Qp.
Proof.
  apply Qp_eq; simpl.
  rewrite <-(Qcmult_div_r x 1) at 2 by done. by rewrite Qcmult_1_l.
Qed.
Lemma Qp_div_S x y : (x / (2 * y) + x / (2 * y) = x / y)%Qp.
Proof.
  apply Qp_eq; simpl.
  rewrite <-Qcmult_plus_distr_l, Pos2Z.inj_mul, Z2Qc_inj_mul, Z2Qc_inj_2.
  rewrite Qcplus_diag. by field_simplify.
Qed.
Lemma Qp_div_2 x : (x / 2 + x / 2 = x)%Qp.
Proof.
  change 2%positive with (2 * 1)%positive. by rewrite Qp_div_S, Qp_div_1.
Qed.
605

Robbert Krebbers's avatar
Robbert Krebbers committed
606
607
608
609
610
611
612
613
614
Lemma Qp_lt_sum (x y : Qp) : (x < y)%Qc   z, y = (x + z)%Qp.
Proof.
  split.
  - intros Hlt%Qclt_minus_iff. exists (mk_Qp (y - x) Hlt). apply Qp_eq; simpl.
    by rewrite (Qcplus_comm y), Qcplus_assoc, Qcplus_opp_r, Qcplus_0_l.
  - intros [z ->]; simpl.
    rewrite <-(Qcplus_0_r x) at 1. apply Qcplus_lt_mono_l, Qp_prf.
Qed.

615
616
617
618
619
620
621
622
623
624
625
626
627
628
Lemma Qp_lower_bound q1 q2 :  q q1' q2', (q1 = q + q1'  q2 = q + q2')%Qp.
Proof.
  revert q1 q2. cut ( q1 q2 : Qp, (q1  q2)%Qc 
     q q1' q2', (q1 = q + q1'  q2 = q + q2')%Qp).
  { intros help q1 q2.
    destruct (Qc_le_dec q1 q2) as [LE|LE%Qclt_nge%Qclt_le_weak]; [by eauto|].
    destruct (help q2 q1) as (q&q1'&q2'&?&?); eauto. }
  intros q1 q2 Hq. exists (q1 / 2)%Qp, (q1 / 2)%Qp.
  assert (0 < q2 - q1 / 2)%Qc as Hq2'.
  { eapply Qclt_le_trans; [|by apply Qcplus_le_mono_r, Hq].
    replace (q1 - q1 / 2)%Qc with (q1 * (1 - 1/2))%Qc by ring.
    replace 0%Qc with (0 * (1-1/2))%Qc by ring. by apply Qcmult_lt_compat_r. }
  exists (mk_Qp (q2 - q1 / 2%Z) Hq2'). split; [by rewrite Qp_div_2|].
  apply Qp_eq; simpl. ring.
629
Qed.
Zhen Zhang's avatar
Zhen Zhang committed
630

Zhen Zhang's avatar
Zhen Zhang committed
631
Lemma Qp_not_plus_q_ge_1 (q: Qp): ¬ ((1 + q)%Qp  1%Qp)%Qc.
Zhen Zhang's avatar
Zhen Zhang committed
632
633
634
Proof.
  intros Hle.
  apply (Qcplus_le_mono_l q 0 1) in Hle.
Zhen Zhang's avatar
Zhen Zhang committed
635
  apply Qcle_ngt in Hle. apply Hle, Qp_prf.
Zhen Zhang's avatar
Zhen Zhang committed
636
Qed.
Zhen Zhang's avatar
Zhen Zhang committed
637
638
639

Lemma Qp_ge_0 (q: Qp): (0  q)%Qc.
Proof. apply Qclt_le_weak, Qp_prf. Qed.