numbers.v 21.1 KB
 Robbert Krebbers committed Nov 11, 2015 1 2 3 4 5 ``````(* Copyright (c) 2012-2015, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** 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. *) `````` Robbert Krebbers committed Feb 13, 2016 6 7 8 ``````From Coq Require Export Eqdep PArith NArith ZArith NPeano. From Coq Require Import QArith Qcanon. From prelude Require Export base decidable option. `````` Robbert Krebbers committed Nov 11, 2015 9 10 11 ``````Open Scope nat_scope. Coercion Z.of_nat : nat >-> Z. `````` Robbert Krebbers committed Feb 26, 2016 12 13 ``````Instance comparison_eq_dec (c1 c2 : comparison) : Decision (c1 = c2). Proof. solve_decision. Defined. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` (** * Notations and properties of [nat] *) Arguments minus !_ !_ /. 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). Reserved Notation "x ≤ y ≤ z ≤ z'" (at level 70, y at next level, z at next level). Infix "≤" := le : 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. Notation "x < y ≤ z" := (x < y ∧ y ≤ z)%nat : nat_scope. Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z')%nat : nat_scope. Notation "(≤)" := le (only parsing) : nat_scope. Notation "(<)" := lt (only parsing) : nat_scope. Infix "`div`" := Nat.div (at level 35) : nat_scope. Infix "`mod`" := Nat.modulo (at level 35) : nat_scope. Instance nat_eq_dec: ∀ x y : nat, Decision (x = y) := eq_nat_dec. Instance nat_le_dec: ∀ x y : nat, Decision (x ≤ y) := le_dec. Instance nat_lt_dec: ∀ x y : nat, Decision (x < y) := lt_dec. Instance nat_inhabited: Inhabited nat := populate 0%nat. `````` Robbert Krebbers committed Feb 11, 2016 40 ``````Instance: Inj (=) (=) S. `````` Robbert Krebbers committed Nov 11, 2015 41 42 43 44 45 46 47 48 49 ``````Proof. by injection 1. Qed. Instance: PartialOrder (≤). Proof. repeat split; repeat intro; auto with lia. Qed. 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]. `````` Robbert Krebbers committed Feb 17, 2016 50 51 52 53 `````` - 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). } `````` Robbert Krebbers committed Nov 11, 2015 54 `````` intros x y p q. `````` Robbert Krebbers committed Feb 13, 2016 55 `````` by apply (Eqdep_dec.eq_dep_eq_dec (λ x y, decide (x = y))), aux. `````` Robbert Krebbers committed Nov 11, 2015 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 ``````Qed. Instance nat_lt_pi: ∀ x y : nat, ProofIrrel (x < y). Proof. apply _. Qed. 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). 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 : x2 < n → y2 < n → x1 * n + x2 = y1 * n + y2 → x1 = y1 ∧ x2 = y2. Proof. intros Hx2 Hy2 E. cut (x1 = y1); [intros; subst;lia |]. revert y1 E. induction x1; simpl; intros [|?]; simpl; auto with lia. Qed. 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. Notation lcm := Nat.lcm. Notation divide := Nat.divide. Notation "( x | y )" := (divide x y) : nat_scope. Instance divide_dec x y : Decision (x | y). Proof. refine (cast_if (decide (lcm x y = y))); by rewrite Nat.divide_lcm_iff. Defined. 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. (** * Notations and properties of [positive] *) Open Scope positive_scope. Infix "≤" := Pos.le : 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" := (x < y ∧ y ≤ z) : positive_scope. Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : positive_scope. Notation "(≤)" := Pos.le (only parsing) : positive_scope. Notation "(<)" := Pos.lt (only parsing) : positive_scope. Notation "(~0)" := xO (only parsing) : positive_scope. Notation "(~1)" := xI (only parsing) : positive_scope. `````` Robbert Krebbers committed Feb 26, 2016 109 110 111 ``````Arguments Pos.of_nat : simpl never. Arguments Pmult : simpl never. `````` Robbert Krebbers committed Nov 11, 2015 112 113 114 ``````Instance positive_eq_dec: ∀ x y : positive, Decision (x = y) := Pos.eq_dec. Instance positive_inhabited: Inhabited positive := populate 1. `````` Robbert Krebbers committed Dec 11, 2015 115 116 ``````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. `````` Robbert Krebbers committed Feb 11, 2016 117 ``````Instance: Inj (=) (=) (~0). `````` Robbert Krebbers committed Nov 11, 2015 118 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Feb 11, 2016 119 ``````Instance: Inj (=) (=) (~1). `````` Robbert Krebbers committed Nov 11, 2015 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 ``````Proof. by injection 1. Qed. (** 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 (++). `````` Robbert Krebbers committed Feb 17, 2016 145 ``````Proof. intros p. by induction p; intros; f_equal/=. Qed. `````` Robbert Krebbers committed Nov 11, 2015 146 147 ``````Global Instance: RightId (=) 1 (++). Proof. done. Qed. `````` Robbert Krebbers committed Feb 11, 2016 148 ``````Global Instance: Assoc (=) (++). `````` Robbert Krebbers committed Feb 17, 2016 149 ``````Proof. intros ?? p. by induction p; intros; f_equal/=. Qed. `````` Robbert Krebbers committed Feb 11, 2016 150 ``````Global Instance: ∀ p : positive, Inj (=) (=) (++ p). `````` Robbert Krebbers committed Feb 17, 2016 151 ``````Proof. intros p ???. induction p; simplify_eq; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 152 153 154 155 `````` Lemma Preverse_go_app p1 p2 p3 : Preverse_go p1 (p2 ++ p3) = Preverse_go p1 p3 ++ Preverse_go 1 p2. Proof. `````` Robbert Krebbers committed Dec 08, 2015 156 157 158 159 `````` 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. `````` Robbert Krebbers committed Feb 17, 2016 160 161 `````` - apply (IH _ (_~1)). - apply (IH _ (_~0)). `````` Robbert Krebbers committed Nov 11, 2015 162 ``````Qed. `````` Robbert Krebbers committed Dec 08, 2015 163 ``````Lemma Preverse_app p1 p2 : Preverse (p1 ++ p2) = Preverse p2 ++ Preverse p1. `````` Robbert Krebbers committed Nov 11, 2015 164 165 166 167 168 169 170 171 ``````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 := match p with 1 => 0%nat | p~0 | p~1 => S (Plength p) end. `````` Robbert Krebbers committed Dec 08, 2015 172 ``````Lemma Papp_length p1 p2 : Plength (p1 ++ p2) = (Plength p2 + Plength p1)%nat. `````` Robbert Krebbers committed Feb 17, 2016 173 ``````Proof. by induction p2; f_equal/=. Qed. `````` Robbert Krebbers committed Nov 11, 2015 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 `````` Close Scope positive_scope. (** * Notations and properties of [N] *) Infix "≤" := N.le : 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. Notation "x < y ≤ z" := (x < y ∧ y ≤ z)%N : N_scope. Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z')%N : N_scope. Notation "(≤)" := N.le (only parsing) : N_scope. Notation "(<)" := N.lt (only parsing) : N_scope. Infix "`div`" := N.div (at level 35) : N_scope. Infix "`mod`" := N.modulo (at level 35) : N_scope. Arguments N.add _ _ : simpl never. `````` Robbert Krebbers committed Feb 11, 2016 191 ``````Instance: Inj (=) (=) Npos. `````` Robbert Krebbers committed Nov 11, 2015 192 193 194 195 ``````Proof. by injection 1. Qed. 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 := `````` Robbert Krebbers committed Jan 12, 2016 196 197 `````` match Ncompare x y with Gt => right _ | _ => left _ end. Solve Obligations with naive_solver. `````` Robbert Krebbers committed Nov 11, 2015 198 ``````Program Instance N_lt_dec (x y : N) : Decision (x < y)%N := `````` Robbert Krebbers committed Jan 12, 2016 199 200 `````` match Ncompare x y with Lt => left _ | _ => right _ end. Solve Obligations with naive_solver. `````` Robbert Krebbers committed Nov 11, 2015 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 ``````Instance N_inhabited: Inhabited N := populate 1%N. 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. (** * Notations and properties of [Z] *) Open Scope Z_scope. Infix "≤" := Z.le : 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. Notation "x < y ≤ z" := (x < y ∧ y ≤ z) : Z_scope. Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : Z_scope. Notation "(≤)" := Z.le (only parsing) : Z_scope. Notation "(<)" := Z.lt (only parsing) : Z_scope. Infix "`div`" := Z.div (at level 35) : Z_scope. Infix "`mod`" := Z.modulo (at level 35) : Z_scope. Infix "`quot`" := Z.quot (at level 35) : Z_scope. Infix "`rem`" := Z.rem (at level 35) : Z_scope. Infix "≪" := Z.shiftl (at level 35) : Z_scope. Infix "≫" := Z.shiftr (at level 35) : Z_scope. `````` Robbert Krebbers committed Feb 11, 2016 227 ``````Instance: Inj (=) (=) Zpos. `````` Robbert Krebbers committed Nov 11, 2015 228 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Feb 11, 2016 229 ``````Instance: Inj (=) (=) Zneg. `````` Robbert Krebbers committed Nov 11, 2015 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 ``````Proof. by injection 1. Qed. Instance Z_eq_dec: ∀ x y : Z, Decision (x = y) := Z.eq_dec. 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. Instance: PartialOrder (≤). Proof. repeat split; red. apply Z.le_refl. apply Z.le_trans. apply Z.le_antisymm. Qed. 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 |]. `````` Ralf Jung committed Feb 20, 2016 250 `````` split. apply Z.quot_pos; lia. trans x; auto. apply Z.quot_lt; lia. `````` Robbert Krebbers committed Nov 11, 2015 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 ``````Qed. (* Note that we cannot disable simpl for [Z.of_nat] as that would break tactics as [lia]. *) 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. 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. 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. Hint Resolve Z.pow_pos_nonneg Z.pow_nonneg: zpos. Hint Resolve Z_mod_pos Z.div_pos : zpos. Hint Extern 1000 => lia : zpos. 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. 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 Nat2Z_divide n m : (Z.of_nat n | Z.of_nat m) ↔ (n | m)%nat. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 290 `````` - rewrite <-(Nat2Z.id m) at 2; intros [i ->]; exists (Z.to_nat i). `````` Robbert Krebbers committed Nov 11, 2015 291 292 293 `````` 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. `````` Robbert Krebbers committed Feb 17, 2016 294 `````` - intros [i ->]. exists (Z.of_nat i). by rewrite Nat2Z.inj_mul. `````` Robbert Krebbers committed Nov 11, 2015 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 ``````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. 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. (** * Notations and properties of [Qc] *) Open Scope Qc_scope. Delimit Scope Qc_scope with Qc. Notation "1" := (Q2Qc 1) : Qc_scope. Notation "2" := (1+1) : Qc_scope. 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. Infix "≤" := Qcle : 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. Notation "x < y ≤ z" := (x < y ∧ y ≤ z) : Qc_scope. Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : Qc_scope. Notation "(≤)" := Qcle (only parsing) : Qc_scope. Notation "(<)" := Qclt (only parsing) : Qc_scope. Hint Extern 1 (_ ≤ _) => reflexivity || discriminate. Arguments Qred _ : simpl never. Instance Qc_eq_dec: ∀ x y : Qc, Decision (x = y) := Qc_eq_dec. Program Instance Qc_le_dec (x y : Qc) : Decision (x ≤ y) := if Qclt_le_dec y x then right _ else left _. `````` Robbert Krebbers committed Jan 12, 2016 341 342 ``````Next Obligation. intros x y; apply Qclt_not_le. Qed. Next Obligation. done. Qed. `````` Robbert Krebbers committed Nov 11, 2015 343 344 ``````Program Instance Qc_lt_dec (x y : Qc) : Decision (x < y) := if Qclt_le_dec x y then left _ else right _. `````` Robbert Krebbers committed Jan 12, 2016 345 346 ``````Solve Obligations with done. Next Obligation. intros x y; apply Qcle_not_lt. Qed. `````` Robbert Krebbers committed Nov 11, 2015 347 348 349 350 351 352 353 354 355 356 357 358 359 `````` 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. Lemma Qcmult_0_l x : 0 * x = 0. Proof. ring. Qed. Lemma Qcmult_0_r x : x * 0 = 0. Proof. ring. Qed. `````` Robbert Krebbers committed Feb 26, 2016 360 361 ``````Lemma Qcplus_diag x : (x + x)%Qc = (2 * x)%Qc. Proof. ring. Qed. `````` Robbert Krebbers committed Nov 11, 2015 362 363 364 365 366 367 368 ``````Lemma Qcle_ngt (x y : Qc) : x ≤ y ↔ ¬y < x. Proof. split; auto using Qcle_not_lt, Qcnot_lt_le. Qed. Lemma Qclt_nge (x y : Qc) : x < y ↔ ¬y ≤ x. Proof. split; auto using Qclt_not_le, Qcnot_le_lt. Qed. Lemma Qcplus_le_mono_l (x y z : Qc) : x ≤ y ↔ z + x ≤ z + y. Proof. split; intros. `````` Robbert Krebbers committed Feb 17, 2016 369 370 `````` - by apply Qcplus_le_compat. - replace x with ((0 - z) + (z + x)) by ring. `````` Robbert Krebbers committed Nov 11, 2015 371 372 373 374 375 376 377 378 379 `````` replace y with ((0 - z) + (z + y)) by ring. by apply Qcplus_le_compat. Qed. Lemma Qcplus_le_mono_r (x y z : Qc) : x ≤ y ↔ x + z ≤ y + z. Proof. rewrite !(Qcplus_comm _ z). apply Qcplus_le_mono_l. Qed. Lemma Qcplus_lt_mono_l (x y z : Qc) : x < y ↔ z + x < z + y. Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_l. Qed. Lemma Qcplus_lt_mono_r (x y z : Qc) : x < y ↔ x + z < y + z. Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_r. Qed. `````` Robbert Krebbers committed Feb 11, 2016 380 ``````Instance: Inj (=) (=) Qcopp. `````` Robbert Krebbers committed Nov 11, 2015 381 382 383 ``````Proof. intros x y H. by rewrite <-(Qcopp_involutive x), H, Qcopp_involutive. Qed. `````` Robbert Krebbers committed Feb 11, 2016 384 ``````Instance: ∀ z, Inj (=) (=) (Qcplus z). `````` Robbert Krebbers committed Nov 11, 2015 385 ``````Proof. `````` Robbert Krebbers committed Feb 11, 2016 386 `````` intros z x y H. by apply (anti_symm (≤)); `````` Robbert Krebbers committed Nov 11, 2015 387 388 `````` rewrite (Qcplus_le_mono_l _ _ z), H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 389 ``````Instance: ∀ z, Inj (=) (=) (λ x, x + z). `````` Robbert Krebbers committed Nov 11, 2015 390 ``````Proof. `````` Robbert Krebbers committed Feb 11, 2016 391 `````` intros z x y H. by apply (anti_symm (≤)); `````` Robbert Krebbers committed Nov 11, 2015 392 393 394 395 396 397 398 399 400 401 402 403 404 `````` rewrite (Qcplus_le_mono_r _ _ z), H. Qed. 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. `````` Ralf Jung committed Feb 20, 2016 405 `````` intros. trans (x + 0); [by rewrite Qcplus_0_r|]. `````` Robbert Krebbers committed Nov 11, 2015 406 407 408 409 410 411 412 413 414 415 416 417 418 `````` 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. `````` Ralf Jung committed Feb 20, 2016 419 `````` intros. trans (x + 0); [|by rewrite Qcplus_0_r]. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` by apply Qcplus_le_mono_l. Qed. 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. `````` Ralf Jung committed Feb 20, 2016 445 `````` intros. trans (0 * y); [by rewrite Qcmult_0_l|]. `````` Robbert Krebbers committed Nov 11, 2015 446 447 448 449 450 451 452 453 `````` 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. `````` Robbert Krebbers committed Feb 26, 2016 454 455 456 457 ``````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. `````` Robbert Krebbers committed Nov 11, 2015 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 ``````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. Close Scope Qc_scope. `````` Robbert Krebbers committed Feb 26, 2016 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 `````` (** * 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. Arguments Qp_car _%Qp. 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. 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. 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. 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. 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.``````