numbers.v 21.2 KB
 Robbert Krebbers committed Feb 08, 2015 1 ``````(* Copyright (c) 2012-2015, Robbert Krebbers. *) `````` Robbert Krebbers committed Aug 29, 2012 2 ``````(* This file is distributed under the terms of the BSD license. *) `````` Robbert Krebbers committed Oct 19, 2012 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. *) `````` 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 stdpp Require Export base decidable option. `````` Robbert Krebbers committed Feb 19, 2013 9 ``````Open Scope nat_scope. `````` Robbert Krebbers committed Jun 11, 2012 10 `````` `````` Robbert Krebbers committed Feb 01, 2013 11 ``````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 Feb 01, 2013 14 `````` `````` Robbert Krebbers committed Feb 19, 2013 15 ``````(** * Notations and properties of [nat] *) `````` Robbert Krebbers committed Feb 01, 2017 16 ``````Arguments minus !_ !_ /. `````` Robbert Krebbers committed Nov 12, 2012 17 18 19 20 ``````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). `````` Robbert Krebbers committed May 02, 2014 21 22 ``````Reserved Notation "x ≤ y ≤ z ≤ z'" (at level 70, y at next level, z at next level). `````` Robbert Krebbers committed Nov 12, 2012 23 `````` `````` Robbert Krebbers committed Aug 21, 2012 24 ``````Infix "≤" := le : nat_scope. `````` Robbert Krebbers committed Nov 12, 2012 25 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. Notation "x < y ≤ z" := (x < y ∧ y ≤ z)%nat : nat_scope. `````` Robbert Krebbers committed May 02, 2014 29 ``````Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z')%nat : nat_scope. `````` Robbert Krebbers committed Nov 12, 2012 30 31 32 ``````Notation "(≤)" := le (only parsing) : nat_scope. Notation "(<)" := lt (only parsing) : nat_scope. `````` Robbert Krebbers committed Feb 01, 2017 33 34 ``````Infix "`div`" := Nat.div (at level 35) : nat_scope. Infix "`mod`" := Nat.modulo (at level 35) : nat_scope. `````` Robbert Krebbers committed Nov 12, 2012 35 `````` `````` Robbert Krebbers committed Jun 11, 2012 36 ``````Instance nat_eq_dec: ∀ x y : nat, Decision (x = y) := eq_nat_dec. `````` Robbert Krebbers committed Nov 12, 2012 37 38 ``````Instance nat_le_dec: ∀ x y : nat, Decision (x ≤ y) := le_dec. Instance nat_lt_dec: ∀ x y : nat, Decision (x < y) := lt_dec. `````` Robbert Krebbers committed Jan 05, 2013 39 ``````Instance nat_inhabited: Inhabited nat := populate 0%nat. `````` Robbert Krebbers committed Feb 11, 2016 40 ``````Instance: Inj (=) (=) S. `````` Robbert Krebbers committed Jun 17, 2013 41 42 43 ``````Proof. by injection 1. Qed. Instance: PartialOrder (≤). Proof. repeat split; repeat intro; auto with lia. Qed. `````` Robbert Krebbers committed Oct 19, 2012 44 `````` `````` Robbert Krebbers committed May 07, 2013 45 46 47 48 49 ``````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 May 07, 2013 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 May 07, 2013 56 57 58 59 ``````Qed. Instance nat_lt_pi: ∀ x y : nat, ProofIrrel (x < y). Proof. apply _. Qed. `````` Robbert Krebbers committed Jan 05, 2013 60 61 62 63 64 65 66 67 ``````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). `````` Robbert Krebbers committed Jun 17, 2013 68 69 70 ``````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 : `````` Robbert Krebbers committed May 07, 2013 71 72 `````` x2 < n → y2 < n → x1 * n + x2 = y1 * n + y2 → x1 = y1 ∧ x2 = y2. Proof. `````` Robbert Krebbers committed Jun 17, 2013 73 `````` intros Hx2 Hy2 E. cut (x1 = y1); [intros; subst;lia |]. `````` Robbert Krebbers committed May 07, 2013 74 75 `````` revert y1 E. induction x1; simpl; intros [|?]; simpl; auto with lia. Qed. `````` Robbert Krebbers committed Jun 17, 2013 76 77 78 ``````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. `````` Robbert Krebbers committed May 07, 2013 79 `````` `````` Robbert Krebbers committed May 02, 2014 80 81 82 ``````Notation lcm := Nat.lcm. Notation divide := Nat.divide. Notation "( x | y )" := (divide x y) : nat_scope. `````` Robbert Krebbers committed Nov 15, 2014 83 84 85 86 ``````Instance divide_dec x y : Decision (x | y). Proof. refine (cast_if (decide (lcm x y = y))); by rewrite Nat.divide_lcm_iff. Defined. `````` Robbert Krebbers committed May 02, 2014 87 88 89 90 91 92 93 94 ``````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 Feb 19, 2013 95 96 97 ``````(** * Notations and properties of [positive] *) Open Scope positive_scope. `````` Robbert Krebbers committed Jun 17, 2013 98 ``````Infix "≤" := Pos.le : positive_scope. `````` Robbert Krebbers committed May 02, 2014 99 100 101 102 103 ``````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. `````` Robbert Krebbers committed Jun 17, 2013 104 105 ``````Notation "(≤)" := Pos.le (only parsing) : positive_scope. Notation "(<)" := Pos.lt (only parsing) : positive_scope. `````` Robbert Krebbers committed Jun 11, 2012 106 107 108 ``````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 Jun 17, 2013 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 Jan 05, 2013 118 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Feb 11, 2016 119 ``````Instance: Inj (=) (=) (~1). `````` Robbert Krebbers committed Jan 05, 2013 120 121 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Feb 19, 2013 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 ``````(** 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 Feb 19, 2013 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 Feb 19, 2013 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 Feb 19, 2013 162 ``````Qed. `````` Robbert Krebbers committed Dec 08, 2015 163 ``````Lemma Preverse_app p1 p2 : Preverse (p1 ++ p2) = Preverse p2 ++ Preverse p1. `````` Robbert Krebbers committed Feb 19, 2013 164 165 166 167 168 169 170 ``````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 := `````` Robbert Krebbers committed May 02, 2014 171 `````` 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 Feb 19, 2013 174 175 176 177 `````` Close Scope positive_scope. (** * Notations and properties of [N] *) `````` Robbert Krebbers committed Jun 11, 2012 178 ``````Infix "≤" := N.le : N_scope. `````` Robbert Krebbers committed Nov 12, 2012 179 180 181 182 ``````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. `````` Robbert Krebbers committed May 02, 2014 183 ``````Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z')%N : N_scope. `````` Robbert Krebbers committed Jun 11, 2012 184 ``````Notation "(≤)" := N.le (only parsing) : N_scope. `````` Robbert Krebbers committed Oct 19, 2012 185 ``````Notation "(<)" := N.lt (only parsing) : N_scope. `````` Robbert Krebbers committed Nov 12, 2012 186 187 188 ``````Infix "`div`" := N.div (at level 35) : N_scope. Infix "`mod`" := N.modulo (at level 35) : N_scope. `````` Robbert Krebbers committed Jun 16, 2014 189 190 ``````Arguments N.add _ _ : simpl never. `````` Robbert Krebbers committed Feb 11, 2016 191 ``````Instance: Inj (=) (=) Npos. `````` Robbert Krebbers committed Jan 05, 2013 192 193 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Jun 11, 2012 194 195 ``````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 Oct 19, 2012 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 Jan 05, 2013 201 ``````Instance N_inhabited: Inhabited N := populate 1%N. `````` Robbert Krebbers committed Aug 12, 2013 202 203 204 205 206 ``````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 committed Jun 11, 2012 207 `````` `````` Robbert Krebbers committed Feb 19, 2013 208 ``````(** * Notations and properties of [Z] *) `````` Robbert Krebbers committed Jun 17, 2013 209 210 ``````Open Scope Z_scope. `````` Robbert Krebbers committed Jun 11, 2012 211 ``````Infix "≤" := Z.le : Z_scope. `````` Robbert Krebbers committed Jun 17, 2013 212 213 214 215 ``````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. `````` Robbert Krebbers committed May 02, 2014 216 ``````Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : Z_scope. `````` Robbert Krebbers committed Jun 11, 2012 217 ``````Notation "(≤)" := Z.le (only parsing) : Z_scope. `````` Robbert Krebbers committed Oct 19, 2012 218 ``````Notation "(<)" := Z.lt (only parsing) : Z_scope. `````` Robbert Krebbers committed Nov 12, 2012 219 `````` `````` Robbert Krebbers committed Jan 05, 2013 220 221 ``````Infix "`div`" := Z.div (at level 35) : Z_scope. Infix "`mod`" := Z.modulo (at level 35) : Z_scope. `````` Robbert Krebbers committed Mar 14, 2013 222 223 ``````Infix "`quot`" := Z.quot (at level 35) : Z_scope. Infix "`rem`" := Z.rem (at level 35) : Z_scope. `````` Robbert Krebbers committed Jun 17, 2013 224 225 ``````Infix "≪" := Z.shiftl (at level 35) : Z_scope. Infix "≫" := Z.shiftr (at level 35) : Z_scope. `````` Robbert Krebbers committed Jan 05, 2013 226 `````` `````` Robbert Krebbers committed Feb 11, 2016 227 ``````Instance: Inj (=) (=) Zpos. `````` Robbert Krebbers committed May 02, 2014 228 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Feb 11, 2016 229 ``````Instance: Inj (=) (=) Zneg. `````` Robbert Krebbers committed May 02, 2014 230 231 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Jun 11, 2012 232 ``````Instance Z_eq_dec: ∀ x y : Z, Decision (x = y) := Z.eq_dec. `````` Robbert Krebbers committed Jun 17, 2013 233 234 235 ``````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 Aug 12, 2013 236 237 238 239 ``````Instance: PartialOrder (≤). Proof. repeat split; red. apply Z.le_refl. apply Z.le_trans. apply Z.le_antisymm. Qed. `````` Robbert Krebbers committed Jun 17, 2013 240 241 242 243 244 245 246 247 248 249 `````` 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 Jun 17, 2013 251 ``````Qed. `````` Robbert Krebbers committed Jun 11, 2012 252 `````` `````` Robbert Krebbers committed Mar 14, 2013 253 ``````(* Note that we cannot disable simpl for [Z.of_nat] as that would break `````` Robbert Krebbers committed Jun 17, 2013 254 ``````tactics as [lia]. *) `````` Robbert Krebbers committed Mar 14, 2013 255 256 257 258 259 260 261 262 263 264 ``````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. `````` Robbert Krebbers committed Aug 26, 2014 265 266 267 268 269 ``````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. `````` Robbert Krebbers committed May 07, 2013 270 271 272 273 274 ``````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. `````` Robbert Krebbers committed Jun 17, 2013 275 276 ``````Hint Resolve Z.pow_pos_nonneg Z.pow_nonneg: zpos. Hint Resolve Z_mod_pos Z.div_pos : zpos. `````` Robbert Krebbers committed May 07, 2013 277 278 ``````Hint Extern 1000 => lia : zpos. `````` Robbert Krebbers committed Feb 01, 2015 279 280 ``````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. `````` Robbert Krebbers committed Jun 17, 2013 281 282 ``````Lemma Z2Nat_inj_pow (x y : nat) : Z.of_nat (x ^ y) = x ^ y. Proof. `````` Robbert Krebbers committed Feb 01, 2015 283 284 285 `````` 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. `````` Robbert Krebbers committed Jun 17, 2013 286 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2015 287 288 289 ``````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 Feb 01, 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 Feb 01, 2015 295 296 297 298 ``````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. `````` Robbert Krebbers committed Jun 17, 2013 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 ``````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. `````` Robbert Krebbers committed Feb 19, 2013 317 ``````(** * Notations and properties of [Qc] *) `````` Robbert Krebbers committed Aug 21, 2013 318 ``````Open Scope Qc_scope. `````` Robbert Krebbers committed May 02, 2014 319 320 ``````Delimit Scope Qc_scope with Qc. Notation "1" := (Q2Qc 1) : Qc_scope. `````` Robbert Krebbers committed Aug 21, 2013 321 ``````Notation "2" := (1+1) : Qc_scope. `````` Robbert Krebbers committed May 02, 2014 322 323 324 325 ``````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. `````` Robbert Krebbers committed Feb 19, 2013 326 ``````Infix "≤" := Qcle : Qc_scope. `````` Robbert Krebbers committed Aug 21, 2013 327 328 329 330 ``````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. `````` Robbert Krebbers committed May 02, 2014 331 ``````Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : Qc_scope. `````` Robbert Krebbers committed Feb 19, 2013 332 333 334 ``````Notation "(≤)" := Qcle (only parsing) : Qc_scope. Notation "(<)" := Qclt (only parsing) : Qc_scope. `````` Robbert Krebbers committed May 02, 2014 335 336 337 ``````Hint Extern 1 (_ ≤ _) => reflexivity || discriminate. Arguments Qred _ : simpl never. `````` Robbert Krebbers committed Feb 19, 2013 338 ``````Instance Qc_eq_dec: ∀ x y : Qc, Decision (x = y) := Qc_eq_dec. `````` Robbert Krebbers committed Aug 21, 2013 339 ``````Program Instance Qc_le_dec (x y : Qc) : Decision (x ≤ y) := `````` Robbert Krebbers committed Feb 19, 2013 340 `````` 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 Aug 21, 2013 343 ``````Program Instance Qc_lt_dec (x y : Qc) : Decision (x < y) := `````` Robbert Krebbers committed Feb 19, 2013 344 `````` 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 Feb 19, 2013 347 `````` `````` Robbert Krebbers committed Aug 21, 2013 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. `````` Robbert Krebbers committed May 02, 2014 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. `````` Robbert Krebbers committed Feb 26, 2016 360 361 ``````Lemma Qcplus_diag x : (x + x)%Qc = (2 * x)%Qc. Proof. ring. Qed. `````` Robbert Krebbers committed Aug 21, 2013 362 ``````Lemma Qcle_ngt (x y : Qc) : x ≤ y ↔ ¬y < x. `````` Robbert Krebbers committed Feb 19, 2013 363 ``````Proof. split; auto using Qcle_not_lt, Qcnot_lt_le. Qed. `````` Robbert Krebbers committed Aug 21, 2013 364 ``````Lemma Qclt_nge (x y : Qc) : x < y ↔ ¬y ≤ x. `````` Robbert Krebbers committed Feb 19, 2013 365 ``````Proof. split; auto using Qclt_not_le, Qcnot_le_lt. Qed. `````` Robbert Krebbers committed Aug 21, 2013 366 ``````Lemma Qcplus_le_mono_l (x y z : Qc) : x ≤ y ↔ z + x ≤ z + y. `````` Robbert Krebbers committed Feb 19, 2013 367 368 ``````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 Aug 21, 2013 371 `````` replace y with ((0 - z) + (z + y)) by ring. `````` Robbert Krebbers committed Feb 19, 2013 372 373 `````` by apply Qcplus_le_compat. Qed. `````` Robbert Krebbers committed Aug 21, 2013 374 ``````Lemma Qcplus_le_mono_r (x y z : Qc) : x ≤ y ↔ x + z ≤ y + z. `````` Robbert Krebbers committed Feb 19, 2013 375 ``````Proof. rewrite !(Qcplus_comm _ z). apply Qcplus_le_mono_l. Qed. `````` Robbert Krebbers committed Aug 21, 2013 376 ``````Lemma Qcplus_lt_mono_l (x y z : Qc) : x < y ↔ z + x < z + y. `````` Robbert Krebbers committed Feb 19, 2013 377 ``````Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_l. Qed. `````` Robbert Krebbers committed Aug 21, 2013 378 ``````Lemma Qcplus_lt_mono_r (x y z : Qc) : x < y ↔ x + z < y + z. `````` Robbert Krebbers committed Feb 19, 2013 379 ``````Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_r. Qed. `````` Robbert Krebbers committed Feb 11, 2016 380 ``````Instance: Inj (=) (=) Qcopp. `````` Robbert Krebbers committed Aug 21, 2013 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 Aug 21, 2013 385 ``````Proof. `````` Robbert Krebbers committed Feb 11, 2016 386 `````` intros z x y H. by apply (anti_symm (≤)); `````` Robbert Krebbers committed Aug 21, 2013 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 May 02, 2014 390 ``````Proof. `````` Robbert Krebbers committed Feb 11, 2016 391 `````` intros z x y H. by apply (anti_symm (≤)); `````` Robbert Krebbers committed May 02, 2014 392 393 `````` rewrite (Qcplus_le_mono_r _ _ z), H. Qed. `````` Robbert Krebbers committed Aug 21, 2013 394 395 396 397 398 399 400 401 402 403 404 ``````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 Aug 21, 2013 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 Aug 21, 2013 420 421 `````` by apply Qcplus_le_mono_l. Qed. `````` Robbert Krebbers committed May 02, 2014 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 ``````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 May 02, 2014 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 May 02, 2014 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 ``````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. `````` Robbert Krebbers committed Aug 21, 2013 477 ``````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 `````` (** * 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. `````` Robbert Krebbers committed Jul 03, 2016 507 508 ``````Instance Qp_inhabited : Inhabited Qp := populate 1%Qp. `````` Robbert Krebbers committed Feb 26, 2016 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 ``````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.``````