numbers.v 22.5 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 Aug 22, 2016 6 ``````From Coq Require Export EqdepFacts PArith NArith ZArith NPeano. `````` Robbert Krebbers committed Feb 13, 2016 7 8 ``````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 Jul 03, 2016 35 36 ``````Infix "`max`" := Nat.max (at level 35) : nat_scope. Infix "`min`" := Nat.min (at level 35) : nat_scope. `````` Robbert Krebbers committed Nov 12, 2012 37 `````` `````` Robbert Krebbers committed Jun 11, 2012 38 ``````Instance nat_eq_dec: ∀ x y : nat, Decision (x = y) := eq_nat_dec. `````` Robbert Krebbers committed Nov 12, 2012 39 40 ``````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 41 ``````Instance nat_inhabited: Inhabited nat := populate 0%nat. `````` Robbert Krebbers committed Feb 11, 2016 42 ``````Instance: Inj (=) (=) S. `````` Robbert Krebbers committed Jun 17, 2013 43 44 45 ``````Proof. by injection 1. Qed. Instance: PartialOrder (≤). Proof. repeat split; repeat intro; auto with lia. Qed. `````` Robbert Krebbers committed Oct 19, 2012 46 `````` `````` Robbert Krebbers committed May 07, 2013 47 48 49 50 51 ``````Instance nat_le_pi: ∀ x y : nat, ProofIrrel (x ≤ y). Proof. assert (∀ x y (p : x ≤ y) y' (q : x ≤ y'), y = y' → eq_dep nat (le x) y p y' q) as aux. { fix 3. intros x ? [|y p] ? [|y' q]. `````` Robbert Krebbers committed Feb 17, 2016 52 53 54 55 `````` - done. - clear nat_le_pi. intros; exfalso; auto with lia. - clear nat_le_pi. intros; exfalso; auto with lia. - injection 1. intros Hy. by case (nat_le_pi x y p y' q Hy). } `````` Robbert Krebbers committed May 07, 2013 56 `````` intros x y p q. `````` Robbert Krebbers committed Feb 13, 2016 57 `````` by apply (Eqdep_dec.eq_dep_eq_dec (λ x y, decide (x = y))), aux. `````` Robbert Krebbers committed May 07, 2013 58 59 60 61 ``````Qed. Instance nat_lt_pi: ∀ x y : nat, ProofIrrel (x < y). Proof. apply _. Qed. `````` Robbert Krebbers committed Jan 05, 2013 62 63 64 65 66 67 68 69 ``````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 70 71 72 ``````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 73 74 `````` x2 < n → y2 < n → x1 * n + x2 = y1 * n + y2 → x1 = y1 ∧ x2 = y2. Proof. `````` Robbert Krebbers committed Jun 17, 2013 75 `````` intros Hx2 Hy2 E. cut (x1 = y1); [intros; subst;lia |]. `````` Robbert Krebbers committed May 07, 2013 76 77 `````` revert y1 E. induction x1; simpl; intros [|?]; simpl; auto with lia. Qed. `````` Robbert Krebbers committed Jun 17, 2013 78 79 80 ``````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 81 `````` `````` Robbert Krebbers committed May 02, 2014 82 83 84 ``````Notation lcm := Nat.lcm. Notation divide := Nat.divide. Notation "( x | y )" := (divide x y) : nat_scope. `````` Robbert Krebbers committed Aug 04, 2016 85 ``````Instance Nat_divide_dec x y : Decision (x | y). `````` Robbert Krebbers committed Nov 15, 2014 86 87 88 ``````Proof. refine (cast_if (decide (lcm x y = y))); by rewrite Nat.divide_lcm_iff. Defined. `````` Robbert Krebbers committed May 02, 2014 89 90 91 92 93 94 95 96 ``````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 97 98 99 100 101 ``````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 Feb 19, 2013 102 103 104 ``````(** * Notations and properties of [positive] *) Open Scope positive_scope. `````` Robbert Krebbers committed Jun 17, 2013 105 ``````Infix "≤" := Pos.le : positive_scope. `````` Robbert Krebbers committed May 02, 2014 106 107 108 109 110 ``````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 111 112 ``````Notation "(≤)" := Pos.le (only parsing) : positive_scope. Notation "(<)" := Pos.lt (only parsing) : positive_scope. `````` Robbert Krebbers committed Jun 11, 2012 113 114 115 ``````Notation "(~0)" := xO (only parsing) : positive_scope. Notation "(~1)" := xI (only parsing) : positive_scope. `````` Robbert Krebbers committed Feb 26, 2016 116 117 118 ``````Arguments Pos.of_nat : simpl never. Arguments Pmult : simpl never. `````` Robbert Krebbers committed Jun 17, 2013 119 120 121 ``````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 122 123 ``````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 124 ``````Instance: Inj (=) (=) (~0). `````` Robbert Krebbers committed Jan 05, 2013 125 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Feb 11, 2016 126 ``````Instance: Inj (=) (=) (~1). `````` Robbert Krebbers committed Jan 05, 2013 127 128 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Feb 19, 2013 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 ``````(** 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 152 ``````Proof. intros p. by induction p; intros; f_equal/=. Qed. `````` Robbert Krebbers committed Feb 19, 2013 153 154 ``````Global Instance: RightId (=) 1 (++). Proof. done. Qed. `````` Robbert Krebbers committed Feb 11, 2016 155 ``````Global Instance: Assoc (=) (++). `````` Robbert Krebbers committed Feb 17, 2016 156 ``````Proof. intros ?? p. by induction p; intros; f_equal/=. Qed. `````` Robbert Krebbers committed Feb 11, 2016 157 ``````Global Instance: ∀ p : positive, Inj (=) (=) (++ p). `````` Robbert Krebbers committed Feb 17, 2016 158 ``````Proof. intros p ???. induction p; simplify_eq; auto. Qed. `````` Robbert Krebbers committed Feb 19, 2013 159 160 161 162 `````` 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 163 164 165 166 `````` 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 167 168 `````` - apply (IH _ (_~1)). - apply (IH _ (_~0)). `````` Robbert Krebbers committed Feb 19, 2013 169 ``````Qed. `````` Robbert Krebbers committed Dec 08, 2015 170 ``````Lemma Preverse_app p1 p2 : Preverse (p1 ++ p2) = Preverse p2 ++ Preverse p1. `````` Robbert Krebbers committed Feb 19, 2013 171 172 173 174 175 176 177 ``````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 178 `````` match p with 1 => 0%nat | p~0 | p~1 => S (Plength p) end. `````` Robbert Krebbers committed Dec 08, 2015 179 ``````Lemma Papp_length p1 p2 : Plength (p1 ++ p2) = (Plength p2 + Plength p1)%nat. `````` Robbert Krebbers committed Feb 17, 2016 180 ``````Proof. by induction p2; f_equal/=. Qed. `````` Robbert Krebbers committed Feb 19, 2013 181 182 183 184 `````` Close Scope positive_scope. (** * Notations and properties of [N] *) `````` Robbert Krebbers committed Jun 11, 2012 185 ``````Infix "≤" := N.le : N_scope. `````` Robbert Krebbers committed Nov 12, 2012 186 187 188 189 ``````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 190 ``````Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z')%N : N_scope. `````` Robbert Krebbers committed Jun 11, 2012 191 ``````Notation "(≤)" := N.le (only parsing) : N_scope. `````` Robbert Krebbers committed Oct 19, 2012 192 ``````Notation "(<)" := N.lt (only parsing) : N_scope. `````` Robbert Krebbers committed Nov 12, 2012 193 194 195 ``````Infix "`div`" := N.div (at level 35) : N_scope. Infix "`mod`" := N.modulo (at level 35) : N_scope. `````` Robbert Krebbers committed Jun 16, 2014 196 197 ``````Arguments N.add _ _ : simpl never. `````` Robbert Krebbers committed Feb 11, 2016 198 ``````Instance: Inj (=) (=) Npos. `````` Robbert Krebbers committed Jan 05, 2013 199 200 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Jun 11, 2012 201 202 ``````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 203 204 `````` match Ncompare x y with Gt => right _ | _ => left _ end. Solve Obligations with naive_solver. `````` Robbert Krebbers committed Oct 19, 2012 205 ``````Program Instance N_lt_dec (x y : N) : Decision (x < y)%N := `````` Robbert Krebbers committed Jan 12, 2016 206 207 `````` match Ncompare x y with Lt => left _ | _ => right _ end. Solve Obligations with naive_solver. `````` Robbert Krebbers committed Jan 05, 2013 208 ``````Instance N_inhabited: Inhabited N := populate 1%N. `````` Robbert Krebbers committed Aug 12, 2013 209 210 211 212 213 ``````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 214 `````` `````` Robbert Krebbers committed Feb 19, 2013 215 ``````(** * Notations and properties of [Z] *) `````` Robbert Krebbers committed Jun 17, 2013 216 217 ``````Open Scope Z_scope. `````` Robbert Krebbers committed Jun 11, 2012 218 ``````Infix "≤" := Z.le : Z_scope. `````` Robbert Krebbers committed Jun 17, 2013 219 220 221 222 ``````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 223 ``````Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : Z_scope. `````` Robbert Krebbers committed Jun 11, 2012 224 ``````Notation "(≤)" := Z.le (only parsing) : Z_scope. `````` Robbert Krebbers committed Oct 19, 2012 225 ``````Notation "(<)" := Z.lt (only parsing) : Z_scope. `````` Robbert Krebbers committed Nov 12, 2012 226 `````` `````` Robbert Krebbers committed Jan 05, 2013 227 228 ``````Infix "`div`" := Z.div (at level 35) : Z_scope. Infix "`mod`" := Z.modulo (at level 35) : Z_scope. `````` Robbert Krebbers committed Mar 14, 2013 229 230 ``````Infix "`quot`" := Z.quot (at level 35) : Z_scope. Infix "`rem`" := Z.rem (at level 35) : Z_scope. `````` Robbert Krebbers committed Jun 17, 2013 231 232 ``````Infix "≪" := Z.shiftl (at level 35) : Z_scope. Infix "≫" := Z.shiftr (at level 35) : Z_scope. `````` Robbert Krebbers committed Jan 05, 2013 233 `````` `````` Robbert Krebbers committed Aug 04, 2016 234 ``````Instance Zpos_inj : Inj (=) (=) Zpos. `````` Robbert Krebbers committed May 02, 2014 235 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Aug 04, 2016 236 ``````Instance Zneg_inj : Inj (=) (=) Zneg. `````` Robbert Krebbers committed May 02, 2014 237 238 ``````Proof. by injection 1. Qed. `````` Robbert Krebbers committed Aug 04, 2016 239 240 241 ``````Instance Z_of_nat_inj : Inj (=) (=) Z.of_nat. Proof. intros n1 n2. apply Nat2Z.inj. Qed. `````` Robbert Krebbers committed Jun 11, 2012 242 ``````Instance Z_eq_dec: ∀ x y : Z, Decision (x = y) := Z.eq_dec. `````` Robbert Krebbers committed Jun 17, 2013 243 244 245 ``````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 04, 2016 246 ``````Instance Z_le_order : PartialOrder (≤). `````` Robbert Krebbers committed Aug 12, 2013 247 248 249 ``````Proof. repeat split; red. apply Z.le_refl. apply Z.le_trans. apply Z.le_antisymm. Qed. `````` Robbert Krebbers committed Jun 17, 2013 250 251 252 253 254 255 256 257 258 259 `````` 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 260 `````` split. apply Z.quot_pos; lia. trans x; auto. apply Z.quot_lt; lia. `````` Robbert Krebbers committed Jun 17, 2013 261 ``````Qed. `````` Robbert Krebbers committed Jun 11, 2012 262 `````` `````` Robbert Krebbers committed Mar 14, 2013 263 ``````(* Note that we cannot disable simpl for [Z.of_nat] as that would break `````` Robbert Krebbers committed Jun 17, 2013 264 ``````tactics as [lia]. *) `````` Robbert Krebbers committed Mar 14, 2013 265 266 267 268 269 270 271 272 273 274 ``````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 275 276 277 278 279 ``````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 280 281 282 283 284 ``````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 285 286 ``````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 287 288 ``````Hint Extern 1000 => lia : zpos. `````` Robbert Krebbers committed Feb 01, 2015 289 290 ``````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 291 292 ``````Lemma Z2Nat_inj_pow (x y : nat) : Z.of_nat (x ^ y) = x ^ y. Proof. `````` Robbert Krebbers committed Feb 01, 2015 293 294 295 `````` 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 296 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2015 297 298 299 ``````Lemma Nat2Z_divide n m : (Z.of_nat n | Z.of_nat m) ↔ (n | m)%nat. Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 300 `````` - rewrite <-(Nat2Z.id m) at 2; intros [i ->]; exists (Z.to_nat i). `````` Robbert Krebbers committed Feb 01, 2015 301 302 303 `````` 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 304 `````` - intros [i ->]. exists (Z.of_nat i). by rewrite Nat2Z.inj_mul. `````` Robbert Krebbers committed Feb 01, 2015 305 306 307 308 ``````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 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 ``````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 327 ``````(** * Notations and properties of [Qc] *) `````` Robbert Krebbers committed Aug 21, 2013 328 ``````Open Scope Qc_scope. `````` Robbert Krebbers committed May 02, 2014 329 330 ``````Delimit Scope Qc_scope with Qc. Notation "1" := (Q2Qc 1) : Qc_scope. `````` Robbert Krebbers committed Aug 21, 2013 331 ``````Notation "2" := (1+1) : Qc_scope. `````` Robbert Krebbers committed May 02, 2014 332 333 334 335 ``````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 336 ``````Infix "≤" := Qcle : Qc_scope. `````` Robbert Krebbers committed Aug 21, 2013 337 338 339 340 ``````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 341 ``````Notation "x ≤ y ≤ z ≤ z'" := (x ≤ y ∧ y ≤ z ∧ z ≤ z') : Qc_scope. `````` Robbert Krebbers committed Feb 19, 2013 342 343 344 ``````Notation "(≤)" := Qcle (only parsing) : Qc_scope. Notation "(<)" := Qclt (only parsing) : Qc_scope. `````` Robbert Krebbers committed May 02, 2014 345 346 347 ``````Hint Extern 1 (_ ≤ _) => reflexivity || discriminate. Arguments Qred _ : simpl never. `````` Robbert Krebbers committed Feb 19, 2013 348 ``````Instance Qc_eq_dec: ∀ x y : Qc, Decision (x = y) := Qc_eq_dec. `````` Robbert Krebbers committed Aug 21, 2013 349 ``````Program Instance Qc_le_dec (x y : Qc) : Decision (x ≤ y) := `````` Robbert Krebbers committed Feb 19, 2013 350 `````` if Qclt_le_dec y x then right _ else left _. `````` Robbert Krebbers committed Jan 12, 2016 351 352 ``````Next Obligation. intros x y; apply Qclt_not_le. Qed. Next Obligation. done. Qed. `````` Robbert Krebbers committed Aug 21, 2013 353 ``````Program Instance Qc_lt_dec (x y : Qc) : Decision (x < y) := `````` Robbert Krebbers committed Feb 19, 2013 354 `````` if Qclt_le_dec x y then left _ else right _. `````` Robbert Krebbers committed Jan 12, 2016 355 356 ``````Solve Obligations with done. Next Obligation. intros x y; apply Qcle_not_lt. Qed. `````` Robbert Krebbers committed Feb 19, 2013 357 `````` `````` Robbert Krebbers committed Aug 21, 2013 358 359 360 361 362 363 364 365 ``````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 366 367 368 369 ``````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 370 371 ``````Lemma Qcplus_diag x : (x + x)%Qc = (2 * x)%Qc. Proof. ring. Qed. `````` Robbert Krebbers committed Aug 21, 2013 372 ``````Lemma Qcle_ngt (x y : Qc) : x ≤ y ↔ ¬y < x. `````` Robbert Krebbers committed Feb 19, 2013 373 ``````Proof. split; auto using Qcle_not_lt, Qcnot_lt_le. Qed. `````` Robbert Krebbers committed Aug 21, 2013 374 ``````Lemma Qclt_nge (x y : Qc) : x < y ↔ ¬y ≤ x. `````` Robbert Krebbers committed Feb 19, 2013 375 ``````Proof. split; auto using Qclt_not_le, Qcnot_le_lt. Qed. `````` Robbert Krebbers committed Aug 21, 2013 376 ``````Lemma Qcplus_le_mono_l (x y z : Qc) : x ≤ y ↔ z + x ≤ z + y. `````` Robbert Krebbers committed Feb 19, 2013 377 378 ``````Proof. split; intros. `````` Robbert Krebbers committed Feb 17, 2016 379 380 `````` - by apply Qcplus_le_compat. - replace x with ((0 - z) + (z + x)) by ring. `````` Robbert Krebbers committed Aug 21, 2013 381 `````` replace y with ((0 - z) + (z + y)) by ring. `````` Robbert Krebbers committed Feb 19, 2013 382 383 `````` by apply Qcplus_le_compat. Qed. `````` Robbert Krebbers committed Aug 21, 2013 384 ``````Lemma Qcplus_le_mono_r (x y z : Qc) : x ≤ y ↔ x + z ≤ y + z. `````` Robbert Krebbers committed Feb 19, 2013 385 ``````Proof. rewrite !(Qcplus_comm _ z). apply Qcplus_le_mono_l. Qed. `````` Robbert Krebbers committed Aug 21, 2013 386 ``````Lemma Qcplus_lt_mono_l (x y z : Qc) : x < y ↔ z + x < z + y. `````` Robbert Krebbers committed Feb 19, 2013 387 ``````Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_l. Qed. `````` Robbert Krebbers committed Aug 21, 2013 388 ``````Lemma Qcplus_lt_mono_r (x y z : Qc) : x < y ↔ x + z < y + z. `````` Robbert Krebbers committed Feb 19, 2013 389 ``````Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_r. Qed. `````` Robbert Krebbers committed Feb 11, 2016 390 ``````Instance: Inj (=) (=) Qcopp. `````` Robbert Krebbers committed Aug 21, 2013 391 392 393 ``````Proof. intros x y H. by rewrite <-(Qcopp_involutive x), H, Qcopp_involutive. Qed. `````` Robbert Krebbers committed Feb 11, 2016 394 ``````Instance: ∀ z, Inj (=) (=) (Qcplus z). `````` Robbert Krebbers committed Aug 21, 2013 395 ``````Proof. `````` Robbert Krebbers committed Feb 11, 2016 396 `````` intros z x y H. by apply (anti_symm (≤)); `````` Robbert Krebbers committed Aug 21, 2013 397 398 `````` rewrite (Qcplus_le_mono_l _ _ z), H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 399 ``````Instance: ∀ z, Inj (=) (=) (λ x, x + z). `````` Robbert Krebbers committed May 02, 2014 400 ``````Proof. `````` Robbert Krebbers committed Feb 11, 2016 401 `````` intros z x y H. by apply (anti_symm (≤)); `````` Robbert Krebbers committed May 02, 2014 402 403 `````` rewrite (Qcplus_le_mono_r _ _ z), H. Qed. `````` Robbert Krebbers committed Aug 21, 2013 404 405 406 407 408 409 410 411 412 413 414 ``````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 415 `````` intros. trans (x + 0); [by rewrite Qcplus_0_r|]. `````` Robbert Krebbers committed Aug 21, 2013 416 417 418 419 420 421 422 423 424 425 426 427 428 `````` 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 429 `````` intros. trans (x + 0); [|by rewrite Qcplus_0_r]. `````` Robbert Krebbers committed Aug 21, 2013 430 431 `````` by apply Qcplus_le_mono_l. Qed. `````` Robbert Krebbers committed May 02, 2014 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 ``````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 455 `````` intros. trans (0 * y); [by rewrite Qcmult_0_l|]. `````` Robbert Krebbers committed May 02, 2014 456 457 458 459 460 461 462 463 `````` 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 464 465 466 467 ``````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 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 ``````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 487 ``````Close Scope Qc_scope. `````` Robbert Krebbers committed Feb 26, 2016 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 `````` (** * 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 517 518 ``````Instance Qp_inhabited : Inhabited Qp := populate 1%Qp. `````` Robbert Krebbers committed Feb 26, 2016 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 ``````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. `````` Jacques-Henri Jourdan committed Sep 09, 2016 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 `````` Lemma Qp_lower_bound q1 q2: ∃ q q1' q2', (q1 = q + q1' ∧ q2 = q + q2')%Qp. Proof. assert (Hdiff : ∀ a b:Qp, (a ≤ b)%Qc → ∃ c, (b - a / 2)%Qp = Some c ∧ (a/2 + c)%Qp = b). { intros a b Hab. unfold Qp_minus. destruct decide as [|[]]. - eexists. split. done. apply Qp_eq. simpl. ring. - eapply Qclt_le_trans; [|by apply Qcplus_le_mono_r, Hab]. change (0 < a - a/2)%Qc. replace (a - a / 2)%Qc with (a * (1 - 1/2))%Qc by ring. replace 0%Qc with (0 * (1-1/2))%Qc by ring. by apply Qcmult_lt_compat_r. } destruct (Qc_le_dec q1 q2) as [LE|LE%Qclt_nge%Qclt_le_weak]. - destruct (Hdiff _ _ LE) as [q2' [EQ <-]]. exists (q1 / 2)%Qp, (q1 / 2)%Qp, q2'. split; apply Qp_eq. by rewrite Qp_div_2. ring. - destruct (Hdiff _ _ LE) as [q1' [EQ <-]]. exists (q2 / 2)%Qp, q1', (q2 / 2)%Qp. split; apply Qp_eq. ring. by rewrite Qp_div_2. Qed.``````