Robbert Krebbers committed Jun 16, 2016 1 ``````From iris.algebra Require Export cmra. `````` Ralf Jung committed Sep 10, 2020 2 ``````From iris Require Import options. `````` Robbert Krebbers committed Jun 16, 2016 3 4 `````` (** * Frame preserving updates *) `````` Ralf Jung committed Jun 23, 2016 5 6 7 8 ``````(* This quantifies over [option A] for the frame. That is necessary to make the following hold: x ~~> P → Some c ~~> Some P *) `````` Robbert Krebbers committed Jun 16, 2016 9 10 ``````Definition cmra_updateP {A : cmraT} (x : A) (P : A → Prop) := ∀ n mz, ✓{n} (x ⋅? mz) → ∃ y, P y ∧ ✓{n} (y ⋅? mz). `````` Maxime Dénès committed Jan 24, 2019 11 ``````Instance: Params (@cmra_updateP) 1 := {}. `````` Robbert Krebbers committed Jun 16, 2016 12 13 14 15 16 ``````Infix "~~>:" := cmra_updateP (at level 70). Definition cmra_update {A : cmraT} (x y : A) := ∀ n mz, ✓{n} (x ⋅? mz) → ✓{n} (y ⋅? mz). Infix "~~>" := cmra_update (at level 70). `````` Maxime Dénès committed Jan 24, 2019 17 ``````Instance: Params (@cmra_update) 1 := {}. `````` Robbert Krebbers committed Jun 16, 2016 18 `````` `````` Robbert Krebbers committed Jul 25, 2016 19 ``````Section updates. `````` Robbert Krebbers committed Jun 16, 2016 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 ``````Context {A : cmraT}. Implicit Types x y : A. Global Instance cmra_updateP_proper : Proper ((≡) ==> pointwise_relation _ iff ==> iff) (@cmra_updateP A). Proof. rewrite /pointwise_relation /cmra_updateP=> x x' Hx P P' HP; split=> ? n mz; setoid_subst; naive_solver. Qed. Global Instance cmra_update_proper : Proper ((≡) ==> (≡) ==> iff) (@cmra_update A). Proof. rewrite /cmra_update=> x x' Hx y y' Hy; split=> ? n mz ?; setoid_subst; auto. Qed. `````` Robbert Krebbers committed Sep 19, 2019 35 ``````Lemma cmra_update_updateP x y : x ~~> y ↔ x ~~>: (y =.). `````` Robbert Krebbers committed Jun 16, 2016 36 37 38 39 40 41 42 43 44 ``````Proof. split=> Hup n z ?; eauto. destruct (Hup n z) as (?&<-&?); auto. Qed. Lemma cmra_updateP_id (P : A → Prop) x : P x → x ~~>: P. Proof. intros ? n mz ?; eauto. Qed. Lemma cmra_updateP_compose (P Q : A → Prop) x : x ~~>: P → (∀ y, P y → y ~~>: Q) → x ~~>: Q. Proof. intros Hx Hy n mz ?. destruct (Hx n mz) as (y&?&?); naive_solver. Qed. Lemma cmra_updateP_compose_l (Q : A → Prop) x y : x ~~> y → y ~~>: Q → x ~~>: Q. Proof. rewrite cmra_update_updateP. `````` Robbert Krebbers committed Sep 19, 2019 45 `````` intros; apply cmra_updateP_compose with (y =.); naive_solver. `````` Robbert Krebbers committed Jun 16, 2016 46 47 48 49 ``````Qed. Lemma cmra_updateP_weaken (P Q : A → Prop) x : x ~~>: P → (∀ y, P y → Q y) → x ~~>: Q. Proof. eauto using cmra_updateP_compose, cmra_updateP_id. Qed. `````` 50 51 52 53 54 ``````Lemma cmra_update_exclusive `{!Exclusive x} y: ✓ y → x ~~> y. Proof. move=>??[z|]=>[/exclusiveN_l[]|_]. by apply cmra_valid_validN. Qed. (** Updates form a preorder. *) `````` Robbert Krebbers committed Jun 16, 2016 55 56 57 58 59 60 61 62 63 64 65 66 67 68 ``````Global Instance cmra_update_preorder : PreOrder (@cmra_update A). Proof. split. - intros x. by apply cmra_update_updateP, cmra_updateP_id. - intros x y z. rewrite !cmra_update_updateP. eauto using cmra_updateP_compose with subst. Qed. Lemma cmra_updateP_op (P1 P2 Q : A → Prop) x1 x2 : x1 ~~>: P1 → x2 ~~>: P2 → (∀ y1 y2, P1 y1 → P2 y2 → Q (y1 ⋅ y2)) → x1 ⋅ x2 ~~>: Q. Proof. intros Hx1 Hx2 Hy n mz ?. destruct (Hx1 n (Some (x2 ⋅? mz))) as (y1&?&?). `````` 69 `````` { by rewrite /= -cmra_op_opM_assoc. } `````` Robbert Krebbers committed Jun 16, 2016 70 `````` destruct (Hx2 n (Some (y1 ⋅? mz))) as (y2&?&?). `````` 71 72 `````` { by rewrite /= -cmra_op_opM_assoc (comm _ x2) cmra_op_opM_assoc. } exists (y1 ⋅ y2); split; last rewrite (comm _ y1) cmra_op_opM_assoc; auto. `````` Robbert Krebbers committed Jun 16, 2016 73 74 75 76 77 78 79 80 81 ``````Qed. Lemma cmra_updateP_op' (P1 P2 : A → Prop) x1 x2 : x1 ~~>: P1 → x2 ~~>: P2 → x1 ⋅ x2 ~~>: λ y, ∃ y1 y2, y = y1 ⋅ y2 ∧ P1 y1 ∧ P2 y2. Proof. eauto 10 using cmra_updateP_op. Qed. Lemma cmra_update_op x1 x2 y1 y2 : x1 ~~> y1 → x2 ~~> y2 → x1 ⋅ x2 ~~> y1 ⋅ y2. Proof. rewrite !cmra_update_updateP; eauto using cmra_updateP_op with congruence. Qed. `````` Robbert Krebbers committed Dec 01, 2017 82 83 `````` Lemma cmra_update_op_l x y : x ⋅ y ~~> x. `````` 84 ``````Proof. intros n mz. rewrite comm cmra_op_opM_assoc. apply cmra_validN_op_r. Qed. `````` Robbert Krebbers committed Dec 01, 2017 85 86 87 ``````Lemma cmra_update_op_r x y : x ⋅ y ~~> y. Proof. rewrite comm. apply cmra_update_op_l. Qed. `````` Robbert Krebbers committed Oct 06, 2016 88 ``````Lemma cmra_update_valid0 x y : (✓{0} x → x ~~> y) → x ~~> y. `````` 89 90 91 ``````Proof. intros H n mz Hmz. apply H, Hmz. apply (cmra_validN_le n); last lia. `````` Tej Chajed committed Nov 03, 2020 92 93 94 `````` destruct mz. - eapply cmra_validN_op_l, Hmz. - apply Hmz. `````` 95 ``````Qed. `````` Robbert Krebbers committed Jun 16, 2016 96 97 98 `````` (** ** Frame preserving updates for total CMRAs *) Section total_updates. `````` Ralf Jung committed Jan 25, 2017 99 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Oct 25, 2017 100 `````` Context `{CmraTotal A}. `````` Robbert Krebbers committed Jun 16, 2016 101 102 103 104 105 106 107 108 109 110 111 112 `````` Lemma cmra_total_updateP x (P : A → Prop) : x ~~>: P ↔ ∀ n z, ✓{n} (x ⋅ z) → ∃ y, P y ∧ ✓{n} (y ⋅ z). Proof. split=> Hup; [intros n z; apply (Hup n (Some z))|]. intros n [z|] ?; simpl; [by apply Hup|]. destruct (Hup n (core x)) as (y&?&?); first by rewrite cmra_core_r. eauto using cmra_validN_op_l. Qed. Lemma cmra_total_update x y : x ~~> y ↔ ∀ n z, ✓{n} (x ⋅ z) → ✓{n} (y ⋅ z). Proof. rewrite cmra_update_updateP cmra_total_updateP. naive_solver. Qed. `````` Robbert Krebbers committed Oct 25, 2017 113 `````` Context `{CmraDiscrete A}. `````` Robbert Krebbers committed Jun 16, 2016 114 115 116 117 118 `````` Lemma cmra_discrete_updateP (x : A) (P : A → Prop) : x ~~>: P ↔ ∀ z, ✓ (x ⋅ z) → ∃ y, P y ∧ ✓ (y ⋅ z). Proof. rewrite cmra_total_updateP; setoid_rewrite <-cmra_discrete_valid_iff. `````` Robbert Krebbers committed Jun 19, 2020 119 `````` naive_solver eauto using O. `````` Robbert Krebbers committed Jun 16, 2016 120 `````` Qed. `````` 121 `````` Lemma cmra_discrete_update (x y : A) : `````` Robbert Krebbers committed Jun 16, 2016 122 123 124 `````` x ~~> y ↔ ∀ z, ✓ (x ⋅ z) → ✓ (y ⋅ z). Proof. rewrite cmra_total_update; setoid_rewrite <-cmra_discrete_valid_iff. `````` Robbert Krebbers committed Jun 19, 2020 125 `````` naive_solver eauto using O. `````` Robbert Krebbers committed Jun 16, 2016 126 127 `````` Qed. End total_updates. `````` Robbert Krebbers committed Jul 25, 2016 128 ``````End updates. `````` Robbert Krebbers committed Jun 16, 2016 129 `````` `````` Robbert Krebbers committed Jun 16, 2016 130 ``````(** * Transport *) `````` Robbert Krebbers committed Jun 16, 2016 131 132 133 134 135 136 137 138 139 140 141 ``````Section cmra_transport. Context {A B : cmraT} (H : A = B). Notation T := (cmra_transport H). Lemma cmra_transport_updateP (P : A → Prop) (Q : B → Prop) x : x ~~>: P → (∀ y, P y → Q (T y)) → T x ~~>: Q. Proof. destruct H; eauto using cmra_updateP_weaken. Qed. Lemma cmra_transport_updateP' (P : A → Prop) x : x ~~>: P → T x ~~>: λ y, ∃ y', y = cmra_transport H y' ∧ P y'. Proof. eauto using cmra_transport_updateP. Qed. End cmra_transport. `````` Robbert Krebbers committed Oct 04, 2020 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 ``````(** * Isomorphism *) Section iso_cmra. Context {A B : cmraT} (f : A → B) (g : B → A). Lemma iso_cmra_updateP (P : B → Prop) (Q : A → Prop) y (gf : ∀ x, g (f x) ≡ x) (g_op : ∀ y1 y2, g (y1 ⋅ y2) ≡ g y1 ⋅ g y2) (g_validN : ∀ n y, ✓{n} (g y) ↔ ✓{n} y) : y ~~>: P → (∀ y', P y' → Q (g y')) → g y ~~>: Q. Proof. intros Hup Hx n mz Hmz. destruct (Hup n (f <\$> mz)) as (y'&HPy'&Hy'%g_validN). { apply g_validN. destruct mz as [z|]; simpl in *; [|done]. by rewrite g_op gf. } exists (g y'); split; [by eauto|]. destruct mz as [z|]; simpl in *; [|done]. revert Hy'. by rewrite g_op gf. Qed. Lemma iso_cmra_updateP' (P : B → Prop) y (gf : ∀ x, g (f x) ≡ x) (g_op : ∀ y1 y2, g (y1 ⋅ y2) ≡ g y1 ⋅ g y2) (g_validN : ∀ n y, ✓{n} (g y) ↔ ✓{n} y) : y ~~>: P → g y ~~>: λ x, ∃ y, x = g y ∧ P y. Proof. eauto using iso_cmra_updateP. Qed. End iso_cmra. `````` Robbert Krebbers committed Jun 16, 2016 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 ``````(** * Product *) Section prod. Context {A B : cmraT}. Implicit Types x : A * B. Lemma prod_updateP P1 P2 (Q : A * B → Prop) x : x.1 ~~>: P1 → x.2 ~~>: P2 → (∀ a b, P1 a → P2 b → Q (a,b)) → x ~~>: Q. Proof. intros Hx1 Hx2 HP n mz [??]; simpl in *. destruct (Hx1 n (fst <\$> mz)) as (a&?&?); first by destruct mz. destruct (Hx2 n (snd <\$> mz)) as (b&?&?); first by destruct mz. exists (a,b); repeat split; destruct mz; auto. Qed. Lemma prod_updateP' P1 P2 x : x.1 ~~>: P1 → x.2 ~~>: P2 → x ~~>: λ y, P1 (y.1) ∧ P2 (y.2). Proof. eauto using prod_updateP. Qed. Lemma prod_update x y : x.1 ~~> y.1 → x.2 ~~> y.2 → x ~~> y. Proof. rewrite !cmra_update_updateP. destruct x, y; eauto using prod_updateP with subst. Qed. End prod. (** * Option *) Section option. Context {A : cmraT}. Implicit Types x y : A. Lemma option_updateP (P : A → Prop) (Q : option A → Prop) x : x ~~>: P → (∀ y, P y → Q (Some y)) → Some x ~~>: Q. Proof. intros Hx Hy; apply cmra_total_updateP=> n [y|] ?. { destruct (Hx n (Some y)) as (y'&?&?); auto. exists (Some y'); auto. } destruct (Hx n None) as (y'&?&?); rewrite ?cmra_core_r; auto. by exists (Some y'); auto. Qed. Lemma option_updateP' (P : A → Prop) x : x ~~>: P → Some x ~~>: from_option P False. Proof. eauto using option_updateP. Qed. Lemma option_update x y : x ~~> y → Some x ~~> Some y. `````` Robbert Krebbers committed Jun 16, 2016 212 `````` Proof. rewrite !cmra_update_updateP; eauto using option_updateP with subst. Qed. `````` Robbert Krebbers committed Jun 16, 2016 213 ``End option.``