numbers.v 23.4 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 Aug 22, 2016 6 ``````From Coq Require Export EqdepFacts PArith NArith ZArith NPeano. `````` Robbert Krebbers committed Feb 13, 2016 7 ``````From Coq Require Import QArith Qcanon. `````` Robbert Krebbers committed Mar 10, 2016 8 ``````From iris.prelude Require Export base decidable option. `````` Ralf Jung committed Jan 03, 2017 9 ``````Set Default Proof Using "Type*". `````` Robbert Krebbers committed Nov 11, 2015 10 11 12 ``````Open Scope nat_scope. Coercion Z.of_nat : nat >-> Z. `````` Robbert Krebbers committed Sep 20, 2016 13 ``````Instance comparison_eq_dec : EqDecision comparison. `````` Robbert Krebbers committed Feb 26, 2016 14 ``````Proof. solve_decision. Defined. `````` Robbert Krebbers committed Nov 11, 2015 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 `````` (** * 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. `````` Robbert Krebbers committed Jul 03, 2016 36 37 ``````Infix "`max`" := Nat.max (at level 35) : nat_scope. Infix "`min`" := Nat.min (at level 35) : nat_scope. `````` Robbert Krebbers committed Nov 11, 2015 38 `````` `````` Robbert Krebbers committed Sep 20, 2016 39 ``````Instance nat_eq_dec: EqDecision nat := eq_nat_dec. `````` Robbert Krebbers committed Nov 11, 2015 40 41 42 ``````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 Sep 20, 2016 43 ``````Instance S_inj: Inj (=) (=) S. `````` Robbert Krebbers committed Nov 11, 2015 44 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Sep 20, 2016 45 ``````Instance nat_le_po: PartialOrder (≤). `````` Robbert Krebbers committed Nov 11, 2015 46 47 48 49 50 51 52 ``````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 53 54 55 56 `````` - 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 57 `````` intros x y p q. `````` Robbert Krebbers committed Feb 13, 2016 58 `````` by apply (Eqdep_dec.eq_dep_eq_dec (λ x y, decide (x = y))), aux. `````` Robbert Krebbers committed Nov 11, 2015 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 ``````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. `````` Robbert Krebbers committed Aug 04, 2016 86 ``````Instance Nat_divide_dec x y : Decision (x | y). `````` Robbert Krebbers committed Nov 11, 2015 87 88 89 90 91 92 93 94 95 96 97 ``````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. `````` Robbert Krebbers committed Aug 04, 2016 98 99 100 101 102 ``````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 committed Nov 11, 2015 103 104 105 106 107 108 109 110 111 112 113 114 115 116 ``````(** * 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 117 118 119 ``````Arguments Pos.of_nat : simpl never. Arguments Pmult : simpl never. `````` Robbert Krebbers committed Sep 20, 2016 120 ``````Instance positive_eq_dec: EqDecision positive := Pos.eq_dec. `````` Robbert Krebbers committed Nov 11, 2015 121 122 ``````Instance positive_inhabited: Inhabited positive := populate 1. `````` Robbert Krebbers committed Dec 11, 2015 123 124 ``````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 125 ``````Instance: Inj (=) (=) (~0). `````` Robbert Krebbers committed Nov 11, 2015 126 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Feb 11, 2016 127 ``````Instance: Inj (=) (=) (~1). `````` Robbert Krebbers committed Nov 11, 2015 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 ``````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 153 ``````Proof. intros p. by induction p; intros; f_equal/=. Qed. `````` Robbert Krebbers committed Nov 11, 2015 154 155 ``````Global Instance: RightId (=) 1 (++). Proof. done. Qed. `````` Robbert Krebbers committed Feb 11, 2016 156 ``````Global Instance: Assoc (=) (++). `````` Robbert Krebbers committed Feb 17, 2016 157 ``````Proof. intros ?? p. by induction p; intros; f_equal/=. Qed. `````` Robbert Krebbers committed Feb 11, 2016 158 ``````Global Instance: ∀ p : positive, Inj (=) (=) (++ p). `````` Robbert Krebbers committed Feb 17, 2016 159 ``````Proof. intros p ???. induction p; simplify_eq; auto. Qed. `````` Robbert Krebbers committed Nov 11, 2015 160 161 162 163 `````` 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 164 165 166 167 `````` 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 168 169 `````` - apply (IH _ (_~1)). - apply (IH _ (_~0)). `````` Robbert Krebbers committed Nov 11, 2015 170 ``````Qed. `````` Robbert Krebbers committed Dec 08, 2015 171 ``````Lemma Preverse_app p1 p2 : Preverse (p1 ++ p2) = Preverse p2 ++ Preverse p1. `````` Robbert Krebbers committed Nov 11, 2015 172 173 174 175 176 177 178 179 ``````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 180 ``````Lemma Papp_length p1 p2 : Plength (p1 ++ p2) = (Plength p2 + Plength p1)%nat. `````` Robbert Krebbers committed Feb 17, 2016 181 ``````Proof. by induction p2; f_equal/=. Qed. `````` Robbert Krebbers committed Nov 11, 2015 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 `````` 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 199 ``````Instance: Inj (=) (=) Npos. `````` Robbert Krebbers committed Nov 11, 2015 200 201 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Sep 20, 2016 202 ``````Instance N_eq_dec: EqDecision N := N.eq_dec. `````` Robbert Krebbers committed Nov 11, 2015 203 ``````Program Instance N_le_dec (x y : N) : Decision (x ≤ y)%N := `````` Robbert Krebbers committed Jan 12, 2016 204 205 `````` match Ncompare x y with Gt => right _ | _ => left _ end. Solve Obligations with naive_solver. `````` Robbert Krebbers committed Nov 11, 2015 206 ``````Program Instance N_lt_dec (x y : N) : Decision (x < y)%N := `````` Robbert Krebbers committed Jan 12, 2016 207 208 `````` match Ncompare x y with Lt => left _ | _ => right _ end. Solve Obligations with naive_solver. `````` Robbert Krebbers committed Nov 11, 2015 209 ``````Instance N_inhabited: Inhabited N := populate 1%N. `````` Robbert Krebbers committed Sep 20, 2016 210 ``````Instance N_le_po: PartialOrder (≤)%N. `````` Robbert Krebbers committed Nov 11, 2015 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 ``````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 Aug 04, 2016 235 ``````Instance Zpos_inj : Inj (=) (=) Zpos. `````` Robbert Krebbers committed Nov 11, 2015 236 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Aug 04, 2016 237 ``````Instance Zneg_inj : Inj (=) (=) Zneg. `````` Robbert Krebbers committed Nov 11, 2015 238 239 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Aug 04, 2016 240 241 242 ``````Instance Z_of_nat_inj : Inj (=) (=) Z.of_nat. Proof. intros n1 n2. apply Nat2Z.inj. Qed. `````` Robbert Krebbers committed Sep 20, 2016 243 ``````Instance Z_eq_dec: EqDecision Z := Z.eq_dec. `````` Robbert Krebbers committed Nov 11, 2015 244 245 246 ``````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. `````` Robbert Krebbers committed Sep 20, 2016 247 ``````Instance Z_le_po : PartialOrder (≤). `````` Robbert Krebbers committed Nov 11, 2015 248 249 250 251 252 253 254 255 256 257 258 259 260 ``````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 261 `````` split. apply Z.quot_pos; lia. trans x; auto. apply Z.quot_lt; lia. `````` Robbert Krebbers committed Nov 11, 2015 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 ``````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 301 `````` - rewrite <-(Nat2Z.id m) at 2; intros [i ->]; exists (Z.to_nat i). `````` Robbert Krebbers committed Nov 11, 2015 302 303 304 `````` 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 305 `````` - intros [i ->]. exists (Z.of_nat i). by rewrite Nat2Z.inj_mul. `````` Robbert Krebbers committed Nov 11, 2015 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 ``````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. `````` Robbert Krebbers committed Sep 20, 2016 349 ``````Instance Qc_eq_dec: EqDecision Qc := Qc_eq_dec. `````` Robbert Krebbers committed Nov 11, 2015 350 351 ``````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 352 353 ``````Next Obligation. intros x y; apply Qclt_not_le. Qed. Next Obligation. done. Qed. `````` Robbert Krebbers committed Nov 11, 2015 354 355 ``````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 356 357 ``````Solve Obligations with done. Next Obligation. intros x y; apply Qcle_not_lt. Qed. `````` Robbert Krebbers committed Nov 11, 2015 358 359 360 361 362 363 364 365 366 367 368 369 370 `````` 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 371 372 ``````Lemma Qcplus_diag x : (x + x)%Qc = (2 * x)%Qc. Proof. ring. Qed. `````` Robbert Krebbers committed Nov 11, 2015 373 374 375 376 377 378 379 ``````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 380 381 `````` - by apply Qcplus_le_compat. - replace x with ((0 - z) + (z + x)) by ring. `````` Robbert Krebbers committed Nov 11, 2015 382 383 384 385 386 387 388 389 390 `````` 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 391 ``````Instance: Inj (=) (=) Qcopp. `````` Robbert Krebbers committed Nov 11, 2015 392 393 394 ``````Proof. intros x y H. by rewrite <-(Qcopp_involutive x), H, Qcopp_involutive. Qed. `````` Robbert Krebbers committed Feb 11, 2016 395 ``````Instance: ∀ z, Inj (=) (=) (Qcplus z). `````` Robbert Krebbers committed Nov 11, 2015 396 ``````Proof. `````` Robbert Krebbers committed Feb 11, 2016 397 `````` intros z x y H. by apply (anti_symm (≤)); `````` Robbert Krebbers committed Nov 11, 2015 398 399 `````` rewrite (Qcplus_le_mono_l _ _ z), H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 400 ``````Instance: ∀ z, Inj (=) (=) (λ x, x + z). `````` Robbert Krebbers committed Nov 11, 2015 401 ``````Proof. `````` Robbert Krebbers committed Feb 11, 2016 402 `````` intros z x y H. by apply (anti_symm (≤)); `````` Robbert Krebbers committed Nov 11, 2015 403 404 405 406 407 408 409 410 411 412 413 414 415 `````` 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 416 `````` intros. trans (x + 0); [by rewrite Qcplus_0_r|]. `````` Robbert Krebbers committed Nov 11, 2015 417 418 419 420 421 422 423 424 425 426 427 428 429 `````` 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 430 `````` intros. trans (x + 0); [|by rewrite Qcplus_0_r]. `````` Robbert Krebbers committed Nov 11, 2015 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 `````` 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 456 `````` intros. trans (0 * y); [by rewrite Qcmult_0_l|]. `````` Robbert Krebbers committed Nov 11, 2015 457 458 459 460 461 462 463 464 `````` 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 465 466 467 468 ``````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 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 ``````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 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 `````` (** * 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. `````` Jacques-Henri Jourdan committed Nov 07, 2016 505 506 ``````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. `````` Robbert Krebbers committed Feb 26, 2016 507 508 509 510 511 512 513 514 515 516 517 ``````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. `````` Jacques-Henri Jourdan committed Nov 07, 2016 518 ``````Infix "*" := Qp_mult : Qp_scope. `````` Robbert Krebbers committed Feb 26, 2016 519 520 521 522 523 524 525 ``````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 committed Nov 16, 2016 526 527 528 529 530 531 532 `````` 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. `````` Robbert Krebbers committed Feb 26, 2016 533 534 535 536 537 538 539 540 541 542 543 544 545 546 ``````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. `````` Jacques-Henri Jourdan committed Nov 07, 2016 547 548 549 550 551 552 553 554 555 556 557 558 559 ``````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. `````` Robbert Krebbers committed Feb 26, 2016 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 ``````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. `````` Jacques-Henri Jourdan committed Sep 09, 2016 575 `````` `````` Robbert Krebbers committed Sep 09, 2016 576 577 578 579 580 581 582 583 584 585 586 587 588 589 ``````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. `````` Jacques-Henri Jourdan committed Sep 09, 2016 590 ``````Qed. `````` Zhen Zhang committed Oct 04, 2016 591 `````` `````` Zhen Zhang committed Oct 04, 2016 592 ``````Lemma Qp_not_plus_q_ge_1 (q: Qp): ¬ ((1 + q)%Qp ≤ 1%Qp)%Qc. `````` Zhen Zhang committed Oct 04, 2016 593 594 595 ``````Proof. intros Hle. apply (Qcplus_le_mono_l q 0 1) in Hle. `````` Zhen Zhang committed Oct 04, 2016 596 `````` apply Qcle_ngt in Hle. apply Hle, Qp_prf. `````` Zhen Zhang committed Oct 04, 2016 597 ``````Qed. `````` Zhen Zhang committed Oct 04, 2016 598 599 600 `````` Lemma Qp_ge_0 (q: Qp): (0 ≤ q)%Qc. Proof. apply Qclt_le_weak, Qp_prf. Qed.``````