frac.v 10.1 KB
 1 ``````From Coq.QArith Require Import Qcanon. `````` Robbert Krebbers committed Feb 26, 2016 2 ``````From algebra Require Export cmra. `````` Robbert Krebbers committed Mar 02, 2016 3 ``````From algebra Require Import upred. `````` Robbert Krebbers committed Feb 26, 2016 4 5 ``````Local Arguments validN _ _ _ !_ /. Local Arguments valid _ _ !_ /. `````` Ralf Jung committed Feb 29, 2016 6 ``````Local Arguments div _ _ !_ !_ /. `````` Robbert Krebbers committed Feb 26, 2016 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 `````` Inductive frac (A : Type) := | Frac : Qp → A → frac A | FracUnit : frac A. Arguments Frac {_} _ _. Arguments FracUnit {_}. Instance maybe_Frac {A} : Maybe2 (@Frac A) := λ x, match x with Frac q a => Some (q,a) | _ => None end. Instance: Params (@Frac) 2. Section cofe. Context {A : cofeT}. Implicit Types a b : A. Implicit Types x y : frac A. (* Cofe *) Inductive frac_equiv : Equiv (frac A) := | Frac_equiv q1 q2 a b : q1 = q2 → a ≡ b → Frac q1 a ≡ Frac q2 b | FracUnit_equiv : FracUnit ≡ FracUnit. Existing Instance frac_equiv. Inductive frac_dist : Dist (frac A) := | Frac_dist q1 q2 a b n : q1 = q2 → a ≡{n}≡ b → Frac q1 a ≡{n}≡ Frac q2 b | FracUnit_dist n : FracUnit ≡{n}≡ FracUnit. Existing Instance frac_dist. Global Instance Frac_ne q n : Proper (dist n ==> dist n) (@Frac A q). Proof. by constructor. Qed. Global Instance Frac_proper q : Proper ((≡) ==> (≡)) (@Frac A q). Proof. by constructor. Qed. Global Instance Frac_inj : Inj2 (=) (≡) (≡) (@Frac A). Proof. by inversion_clear 1. Qed. Global Instance Frac_dist_inj n : Inj2 (=) (dist n) (dist n) (@Frac A). Proof. by inversion_clear 1. Qed. Program Definition frac_chain (c : chain (frac A)) (q : Qp) (a : A) `````` Ralf Jung committed Feb 29, 2016 42 `````` (H : maybe2 Frac (c 0) = Some (q,a)) : chain A := `````` Robbert Krebbers committed Feb 26, 2016 43 44 `````` {| chain_car n := match c n return _ with Frac _ b => b | _ => a end |}. Next Obligation. `````` Ralf Jung committed Feb 29, 2016 45 46 47 `````` intros c q a ? n i ?; simpl. destruct (c 0) eqn:?; simplify_eq/=. by feed inversion (chain_cauchy c n i). `````` Robbert Krebbers committed Feb 26, 2016 48 49 ``````Qed. Instance frac_compl : Compl (frac A) := λ c, `````` Ralf Jung committed Feb 29, 2016 50 `````` match Some_dec (maybe2 Frac (c 0)) with `````` Robbert Krebbers committed Feb 26, 2016 51 `````` | inleft (exist (q,a) H) => Frac q (compl (frac_chain c q a H)) `````` Ralf Jung committed Feb 29, 2016 52 `````` | inright _ => c 0 `````` Robbert Krebbers committed Feb 26, 2016 53 54 55 56 57 58 59 60 61 62 63 64 65 66 `````` end. Definition frac_cofe_mixin : CofeMixin (frac A). Proof. split. - intros mx my; split. + by destruct 1; subst; constructor; try apply equiv_dist. + intros Hxy; feed inversion (Hxy 0); subst; constructor; try done. apply equiv_dist=> n; by feed inversion (Hxy n). - intros n; split. + by intros [q a|]; constructor. + by destruct 1; constructor. + destruct 1; inversion_clear 1; constructor; etrans; eauto. - by inversion_clear 1; constructor; done || apply dist_S. - intros n c; unfold compl, frac_compl. `````` Ralf Jung committed Feb 29, 2016 67 68 69 70 `````` destruct (Some_dec (maybe2 Frac (c 0))) as [[[q a] Hx]|]. { assert (c 0 = Frac q a) by (by destruct (c 0); simplify_eq/=). assert (∃ b, c n = Frac q b) as [y Hy]. { feed inversion (chain_cauchy c 0 n); `````` Robbert Krebbers committed Feb 26, 2016 71 72 73 `````` eauto with lia congruence f_equal. } rewrite Hy; constructor; auto. by rewrite (conv_compl n (frac_chain c q a Hx)) /= Hy. } `````` Ralf Jung committed Feb 29, 2016 74 75 `````` feed inversion (chain_cauchy c 0 n); first lia; constructor; destruct (c 0); simplify_eq/=. `````` Robbert Krebbers committed Feb 26, 2016 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 ``````Qed. Canonical Structure fracC : cofeT := CofeT frac_cofe_mixin. Global Instance frac_discrete : Discrete A → Discrete fracC. Proof. by inversion_clear 2; constructor; done || apply (timeless _). Qed. Global Instance frac_leibniz : LeibnizEquiv A → LeibnizEquiv (frac A). Proof. by destruct 2; f_equal; done || apply leibniz_equiv. Qed. Global Instance Frac_timeless q (a : A) : Timeless a → Timeless (Frac q a). Proof. by inversion_clear 2; constructor; done || apply (timeless _). Qed. Global Instance FracUnit_timeless : Timeless (@FracUnit A). Proof. by inversion_clear 1; constructor. Qed. End cofe. Arguments fracC : clear implicits. (* Functor on COFEs *) Definition frac_map {A B} (f : A → B) (x : frac A) : frac B := match x with | Frac q a => Frac q (f a) | FracUnit => FracUnit end. Instance: Params (@frac_map) 2. Lemma frac_map_id {A} (x : frac A) : frac_map id x = x. Proof. by destruct x. Qed. Lemma frac_map_compose {A B C} (f : A → B) (g : B → C) (x : frac A) : frac_map (g ∘ f) x = frac_map g (frac_map f x). Proof. by destruct x. Qed. Lemma frac_map_ext {A B : cofeT} (f g : A → B) x : (∀ x, f x ≡ g x) → frac_map f x ≡ frac_map g x. Proof. by destruct x; constructor. Qed. Instance frac_map_cmra_ne {A B : cofeT} n : Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@frac_map A B). Proof. intros f f' Hf; destruct 1; constructor; by try apply Hf. Qed. Definition fracC_map {A B} (f : A -n> B) : fracC A -n> fracC B := CofeMor (frac_map f). Instance fracC_map_ne A B n : Proper (dist n ==> dist n) (@fracC_map A B). Proof. intros f f' Hf []; constructor; by try apply Hf. Qed. Section cmra. Context {A : cmraT}. Implicit Types a b : A. Implicit Types x y : frac A. (* CMRA *) Instance frac_valid : Valid (frac A) := λ x, match x with Frac q a => (q ≤ 1)%Qc ∧ ✓ a | FracUnit => True end. Instance frac_validN : ValidN (frac A) := λ n x, match x with Frac q a => (q ≤ 1)%Qc ∧ ✓{n} a | FracUnit => True end. Global Instance frac_empty : Empty (frac A) := FracUnit. Instance frac_unit : Unit (frac A) := λ _, ∅. Instance frac_op : Op (frac A) := λ x y, match x, y with | Frac q1 a, Frac q2 b => Frac (q1 + q2) (a ⋅ b) | Frac q a, FracUnit | FracUnit, Frac q a => Frac q a | FracUnit, FracUnit => FracUnit end. `````` Ralf Jung committed Feb 29, 2016 132 ``````Instance frac_div : Div (frac A) := λ x y, `````` Robbert Krebbers committed Feb 26, 2016 133 134 135 `````` match x, y with | _, FracUnit => x | Frac q1 a, Frac q2 b => `````` Ralf Jung committed Feb 29, 2016 136 `````` match q1 - q2 with Some q => Frac q (a ÷ b) | None => FracUnit end%Qp `````` Robbert Krebbers committed Feb 26, 2016 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 `````` | FracUnit, _ => FracUnit end. Lemma Frac_op q1 q2 a b : Frac q1 a ⋅ Frac q2 b = Frac (q1 + q2) (a ⋅ b). Proof. done. Qed. Definition frac_cmra_mixin : CMRAMixin (frac A). Proof. split. - intros n []; destruct 1; constructor; by cofe_subst. - constructor. - do 2 destruct 1; split; by cofe_subst. - do 2 destruct 1; simplify_eq/=; try case_match; constructor; by cofe_subst. - intros [q a|]; rewrite /= ?cmra_valid_validN; naive_solver eauto using O. - intros n [q a|]; destruct 1; split; auto using cmra_validN_S. - intros [q1 a1|] [q2 a2|] [q3 a3|]; constructor; by rewrite ?assoc. - intros [q1 a1|] [q2 a2|]; constructor; by rewrite 1?comm ?[(q1+_)%Qp]comm. - intros []; by constructor. - done. - by exists FracUnit. - intros n [q1 a1|] [q2 a2|]; destruct 1; split; eauto using cmra_validN_op_l. trans (q1 + q2)%Qp; simpl; last done. rewrite -{1}(Qcplus_0_r q1) -Qcplus_le_mono_l; auto using Qclt_le_weak. - intros [q1 a1|] [q2 a2|] [[q3 a3|] Hx]; inversion_clear Hx; simplify_eq/=; auto. `````` Ralf Jung committed Feb 29, 2016 162 `````` + rewrite Qp_op_minus. by constructor; [|apply cmra_op_div; exists a3]. `````` Robbert Krebbers committed Feb 26, 2016 163 164 165 166 167 168 169 170 171 172 173 `````` + rewrite Qp_minus_diag. by constructor. - intros n [q a|] y1 y2 Hx Hx'; last first. { by exists (∅, ∅); destruct y1, y2; inversion_clear Hx'. } destruct Hx, y1 as [q1 b1|], y2 as [q2 b2|]. + apply (inj2 Frac) in Hx'; destruct Hx' as [-> ?]. destruct (cmra_extend n a b1 b2) as ([z1 z2]&?&?&?); auto. exists (Frac q1 z1,Frac q2 z2); by repeat constructor. + exists (Frac q a, ∅); inversion_clear Hx'; by repeat constructor. + exists (∅, Frac q a); inversion_clear Hx'; by repeat constructor. + exfalso; inversion_clear Hx'. Qed. `````` Robbert Krebbers committed Mar 01, 2016 174 175 ``````Canonical Structure fracR : cmraT := CMRAT frac_cofe_mixin frac_cmra_mixin. Global Instance frac_cmra_identity : CMRAIdentity fracR. `````` Robbert Krebbers committed Feb 26, 2016 176 ``````Proof. split. done. by intros []. apply _. Qed. `````` Robbert Krebbers committed Mar 01, 2016 177 ``````Global Instance frac_cmra_discrete : CMRADiscrete A → CMRADiscrete fracR. `````` Robbert Krebbers committed Feb 26, 2016 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 ``````Proof. split; first apply _. intros [q a|]; destruct 1; split; auto using cmra_discrete_valid. Qed. Lemma frac_validN_inv_l n y a : ✓{n} (Frac 1 a ⋅ y) → y = ∅. Proof. destruct y as [q b|]; [|done]=> -[Hq ?]; destruct (Qcle_not_lt _ _ Hq). by rewrite -{1}(Qcplus_0_r 1) -Qcplus_lt_mono_l. Qed. Lemma frac_valid_inv_l y a : ✓ (Frac 1 a ⋅ y) → y = ∅. Proof. intros. by apply frac_validN_inv_l with 0 a, cmra_valid_validN. Qed. (** Internalized properties *) Lemma frac_equivI {M} (x y : frac A) : (x ≡ y)%I ≡ (match x, y with | Frac q1 a, Frac q2 b => q1 = q2 ∧ a ≡ b | FracUnit, FracUnit => True | _, _ => False end : uPred M)%I. Proof. uPred.unseal; do 2 split; first by destruct 1. by destruct x, y; destruct 1; try constructor. Qed. Lemma frac_validI {M} (x : frac A) : (✓ x)%I ≡ (if x is Frac q a then ■ (q ≤ 1)%Qc ∧ ✓ a else True : uPred M)%I. Proof. uPred.unseal. by destruct x. Qed. (** ** Local updates *) Global Instance frac_local_update_full p a : LocalUpdate (λ x, if x is Frac q _ then q = 1%Qp else False) (λ _, Frac p a). Proof. split; first by intros ???. by intros n [q b|] y; [|done]=> -> /frac_validN_inv_l ->. Qed. Global Instance frac_local_update `{!LocalUpdate Lv L} : LocalUpdate (λ x, if x is Frac _ a then Lv a else False) (frac_map L). Proof. split; first apply _. intros n [p a|] [q b|]; simpl; try done. intros ? [??]; constructor; [done|by apply (local_updateN L)]. Qed. (** Updates *) Lemma frac_update_full (a1 a2 : A) : ✓ a2 → Frac 1 a1 ~~> Frac 1 a2. Proof. move=> ? n y /frac_validN_inv_l ->. split. done. by apply cmra_valid_validN. Qed. Lemma frac_update (a1 a2 : A) p : a1 ~~> a2 → Frac p a1 ~~> Frac p a2. Proof. intros Ha n [q b|] [??]; split; auto. apply cmra_validN_op_l with (unit a1), Ha. by rewrite cmra_unit_r. Qed. End cmra. `````` Robbert Krebbers committed Mar 01, 2016 232 ``````Arguments fracR : clear implicits. `````` Robbert Krebbers committed Feb 26, 2016 233 234 235 236 237 238 239 240 241 242 243 244 245 246 `````` (* Functor *) Instance frac_map_cmra_monotone {A B : cmraT} (f : A → B) : CMRAMonotone f → CMRAMonotone (frac_map f). Proof. split; try apply _. - intros n [p a|]; destruct 1; split; auto using validN_preserving. - intros [q1 a1|] [q2 a2|] [[q3 a3|] Hx]; inversion Hx; setoid_subst; try apply: cmra_empty_least; auto. destruct (included_preserving f a1 (a1 ⋅ a3)) as [b ?]. { by apply cmra_included_l. } by exists (Frac q3 b); constructor. Qed. `````` Robbert Krebbers committed Mar 02, 2016 247 248 249 ``````Program Definition fracRF (F : rFunctor) : rFunctor := {| rFunctor_car A B := fracR (rFunctor_car F A B); rFunctor_map A1 A2 B1 B2 fg := fracC_map (rFunctor_map F fg) `````` Robbert Krebbers committed Feb 26, 2016 250 251 ``````|}. Next Obligation. `````` Robbert Krebbers committed Mar 06, 2016 252 `````` by intros F A1 A2 B1 B2 n f g Hfg; apply fracC_map_ne, rFunctor_contractive. `````` Robbert Krebbers committed Feb 26, 2016 253 254 ``````Qed. Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 255 256 `````` intros F A B x. rewrite /= -{2}(frac_map_id x). apply frac_map_ext=>y; apply rFunctor_id. `````` Robbert Krebbers committed Feb 26, 2016 257 258 ``````Qed. Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 259 260 `````` intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -frac_map_compose. apply frac_map_ext=>y; apply rFunctor_compose. `````` Robbert Krebbers committed Feb 26, 2016 261 ``Qed.``