numbers.v 18.6 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2015, 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 prelude.base prelude.decidable prelude.option.
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] *)
Robbert Krebbers's avatar
Robbert Krebbers committed
14
Arguments minus !_ !_ /.
15 16 17 18
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).
19 20
Reserved Notation "x ≤ y ≤ z ≤ z'"
  (at level 70, y at next level, z at next level).
21

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

Robbert Krebbers's avatar
Robbert Krebbers committed
31 32
Infix "`div`" := Nat.div (at level 35) : nat_scope.
Infix "`mod`" := Nat.modulo (at level 35) : nat_scope.
33

Robbert Krebbers's avatar
Robbert Krebbers committed
34
Instance nat_eq_dec:  x y : nat, Decision (x = y) := eq_nat_dec.
35 36
Instance nat_le_dec:  x y : nat, Decision (x  y) := le_dec.
Instance nat_lt_dec:  x y : nat, Decision (x < y) := lt_dec.
37
Instance nat_inhabited: Inhabited nat := populate 0%nat.
38 39 40 41
Instance: Injective (=) (=) S.
Proof. by injection 1. Qed.
Instance: PartialOrder ().
Proof. repeat split; repeat intro; auto with lia. Qed.
42

43 44 45 46 47 48
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.
49 50
    * clear nat_le_pi. intros; exfalso; auto with lia.
    * clear nat_le_pi. intros; exfalso; auto with lia.
51 52 53 54 55 56 57
    * 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
58 59 60 61 62 63 64 65
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).

66 67 68
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 :
69 70
  x2 < n  y2 < n  x1 * n + x2 = y1 * n + y2  x1 = y1  x2 = y2.
Proof.
71
  intros Hx2 Hy2 E. cut (x1 = y1); [intros; subst;lia |].
72 73
  revert y1 E. induction x1; simpl; intros [|?]; simpl; auto with lia.
Qed.
74 75 76
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.
77

78 79 80
Notation lcm := Nat.lcm.
Notation divide := Nat.divide.
Notation "( x | y )" := (divide x y) : nat_scope.
81 82 83 84
Instance divide_dec x y : Decision (x | y).
Proof.
  refine (cast_if (decide (lcm x y = y))); by rewrite Nat.divide_lcm_iff.
Defined.
85 86 87 88 89 90 91 92
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.

93 94 95
(** * Notations and properties of [positive] *)
Open Scope positive_scope.

96
Infix "≤" := Pos.le : positive_scope.
97 98 99 100 101
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.
102 103
Notation "(≤)" := Pos.le (only parsing) : positive_scope.
Notation "(<)" := Pos.lt (only parsing) : positive_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
104 105 106
Notation "(~0)" := xO (only parsing) : positive_scope.
Notation "(~1)" := xI (only parsing) : positive_scope.

107 108 109 110
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.

111 112
Instance maybe_xO : Maybe xO := λ p, match p with p~0 => Some p | _ => None end.
Instance maybe_x1 : Maybe xI := λ p, match p with p~1 => Some p | _ => None end.
113
Instance: Injective (=) (=) (~0).
Robbert Krebbers's avatar
Robbert Krebbers committed
114
Proof. by injection 1. Qed.
115
Instance: Injective (=) (=) (~1).
Robbert Krebbers's avatar
Robbert Krebbers committed
116 117
Proof. by injection 1. Qed.

118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
(** 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 (++).
141
Proof. intros p. by induction p; intros; f_equal'. Qed.
142 143 144
Global Instance: RightId (=) 1 (++).
Proof. done. Qed.
Global Instance: Associative (=) (++).
145
Proof. intros ?? p. by induction p; intros; f_equal'. Qed.
146 147 148 149 150 151
Global Instance:  p : positive, Injective (=) (=) (++ p).
Proof. intros p ???. induction p; simplify_equality; auto. Qed.

Lemma Preverse_go_app p1 p2 p3 :
  Preverse_go p1 (p2 ++ p3) = Preverse_go p1 p3 ++ Preverse_go 1 p2.
Proof.
152 153 154 155 156 157
  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.
  * apply (IH _ (_~1)).
  * apply (IH _ (_~0)).
158
Qed.
159
Lemma Preverse_app p1 p2 : Preverse (p1 ++ p2) = Preverse p2 ++ Preverse p1.
160 161 162 163 164 165 166
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 :=
167
  match p with 1 => 0%nat | p~0 | p~1 => S (Plength p) end.
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.

267 268 269 270 271
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.
272 273 274 275 276
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.
277 278
Hint Resolve Z.pow_pos_nonneg Z.pow_nonneg: zpos.
Hint Resolve Z_mod_pos Z.div_pos : zpos.
279 280
Hint Extern 1000 => lia : zpos.

Robbert Krebbers's avatar
Robbert Krebbers committed
281 282
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.
283 284
Lemma Z2Nat_inj_pow (x y : nat) : Z.of_nat (x ^ y) = x ^ y.
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
285 286 287
  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.
288
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
289 290 291 292 293 294 295 296 297 298 299 300
Lemma Nat2Z_divide n m : (Z.of_nat n | Z.of_nat m)  (n | m)%nat.
Proof.
  split.
  * rewrite <-(Nat2Z.id m) at 2; intros [i ->]; exists (Z.to_nat i).
    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.
  * intros [i ->]. exists (Z.of_nat i). by rewrite Nat2Z.inj_mul.
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.
301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318
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.

319
(** * Notations and properties of [Qc] *)
320
Open Scope Qc_scope.
321 322
Delimit Scope Qc_scope with Qc.
Notation "1" := (Q2Qc 1) : Qc_scope.
323
Notation "2" := (1+1) : Qc_scope.
324 325 326 327
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.
328
Infix "≤" := Qcle : Qc_scope.
329 330 331 332
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.
333
Notation "x ≤ y ≤ z ≤ z'" := (x  y  y  z  z  z') : Qc_scope.
334 335 336
Notation "(≤)" := Qcle (only parsing) : Qc_scope.
Notation "(<)" := Qclt (only parsing) : Qc_scope.

337 338 339
Hint Extern 1 (_  _) => reflexivity || discriminate.
Arguments Qred _ : simpl never.

340
Instance Qc_eq_dec:  x y : Qc, Decision (x = y) := Qc_eq_dec.
341
Program Instance Qc_le_dec (x y : Qc) : Decision (x  y) :=
342 343
  if Qclt_le_dec y x then right _ else left _.
Next Obligation. by apply Qclt_not_le. Qed.
344
Program Instance Qc_lt_dec (x y : Qc) : Decision (x < y) :=
345 346 347
  if Qclt_le_dec x y then left _ else right _.
Next Obligation. by apply Qcle_not_lt. Qed.

348 349 350 351 352 353 354 355
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.
356 357 358 359
Lemma Qcmult_0_l x : 0 * x = 0.
Proof. ring. Qed.
Lemma Qcmult_0_r x : x * 0 = 0.
Proof. ring. Qed.
360
Lemma Qcle_ngt (x y : Qc) : x  y  ¬y < x.
361
Proof. split; auto using Qcle_not_lt, Qcnot_lt_le. Qed.
362
Lemma Qclt_nge (x y : Qc) : x < y  ¬y  x.
363
Proof. split; auto using Qclt_not_le, Qcnot_le_lt. Qed.
364
Lemma Qcplus_le_mono_l (x y z : Qc) : x  y  z + x  z + y.
365 366 367
Proof.
  split; intros.
  * by apply Qcplus_le_compat.
368 369
  * replace x with ((0 - z) + (z + x)) by ring.
    replace y with ((0 - z) + (z + y)) by ring.
370 371
    by apply Qcplus_le_compat.
Qed.
372
Lemma Qcplus_le_mono_r (x y z : Qc) : x  y  x + z  y + z.
373
Proof. rewrite !(Qcplus_comm _ z). apply Qcplus_le_mono_l. Qed.
374
Lemma Qcplus_lt_mono_l (x y z : Qc) : x < y  z + x < z + y.
375
Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_l. Qed.
376
Lemma Qcplus_lt_mono_r (x y z : Qc) : x < y  x + z < y + z.
377
Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_r. Qed.
378 379 380 381
Instance: Injective (=) (=) Qcopp.
Proof.
  intros x y H. by rewrite <-(Qcopp_involutive x), H, Qcopp_involutive.
Qed.
382
Instance:  z, Injective (=) (=) (Qcplus z).
383 384 385 386
Proof.
  intros z x y H. by apply (anti_symmetric ());
    rewrite (Qcplus_le_mono_l _ _ z), H.
Qed.
387 388 389 390 391
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.
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
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.
420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470
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.
471
Close Scope Qc_scope.