From fri.algebra Require Export cmra. (** * Frame preserving updates *) (* This quantifies over [option A] for the frame. That is necessary to make the following hold: x ~~> P → Some c ~~> Some P *) Definition cmra_updateP {A : cmraT} (x : A) (P : A → Prop) := ∀ n mz, ✓{n} (x ⋅? mz) → ∃ y, P y ∧ ✓{n} (y ⋅? mz). Instance: Params (@cmra_updateP) 1 := {}. 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). Instance: Params (@cmra_update) 1 := {}. Definition cmra_step_updateP {A : cmraT} (x : A) (xl: A) (P : A → A → Prop) := ∀ n mz, ✓{n} (x ⋅ xl ⋅? mz) → ∃ y yl, P y yl ∧ ✓{n} (y ⋅ yl ⋅? mz) ∧ xl ⤳_(n) yl. Notation "x # y ~~>>: P" := (cmra_step_updateP x y P) (at level 70). Instance: Params (@cmra_step_updateP) 1 := {}. Definition cmra_step_update {A : cmraT} (x xl y yl : A) := ∀ n mz, ✓{n} (x ⋅ xl ⋅? mz) → ✓{n} (y ⋅ yl ⋅? mz) ∧ xl ⤳_(n) yl. Notation "x # xl ~~>> y #' yl " := (cmra_step_update x xl y yl) (at level 70). Instance: Params (@cmra_step_update) 1 := {}. Section updates. 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. Lemma cmra_update_updateP x y : x ~~> y ↔ x ~~>: (y =.). 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. intros; apply cmra_updateP_compose with (y =.); naive_solver. 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. 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_update_exclusive `{!Exclusive x} y: ✓ y → x ~~> y. Proof. move=>??[z|]=>[/exclusiveN_l[]|_]. by apply cmra_valid_validN. 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&?&?). { by rewrite /= -cmra_opM_assoc. } destruct (Hx2 n (Some (y1 ⋅? mz))) as (y2&?&?). { by rewrite /= -cmra_opM_assoc (comm _ x2) cmra_opM_assoc. } exists (y1 ⋅ y2); split; last rewrite (comm _ y1) cmra_opM_assoc; auto. 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. Lemma cmra_update_valid0 x y : (✓{0} x → x ~~> y) → x ~~> y. Proof. intros H n mz Hmz. apply H, Hmz. apply (cmra_validN_le n); last lia. destruct mz. eapply cmra_validN_op_l, Hmz. apply Hmz. Qed. Lemma cmra_step_stepP (x xl y yl: A) : x # xl ~~>> y #' yl ↔ (x # xl ~~>>: (λ x xl, y = x ∧ yl = xl)). Proof. split. - by intros Hx n z ?; exists y, yl; destruct (Hx n z) as (?&?); auto. - intros Hx n z ?. destruct (Hx n z) as (?&?&(<-&<-)&?); auto. Qed. Lemma cmra_step_updateP_weaken (P Q : A → A → Prop) x xl: x # xl ~~>>: P → (∀ y yl, P y yl → Q y yl) → x # xl ~~>>: Q. Proof. intros Hs HPQ n z Hval. edestruct (Hs n z Hval) as (y&yl&HP&Hval'&Hs'). exists y, yl. split_and!; eauto. Qed. (** ** Frame preserving updates for total CMRAs *) Section total_updates. Context `{CMRATotal A}. 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. Lemma cmra_total_step_updateP (x : A) (xl: A) (P : A → A → Prop) : x # xl ~~>>: P ↔ ∀ n z, ✓{n} (x ⋅ xl ⋅ z) → ∃ y yl, P y yl ∧ ✓{n} (y ⋅ yl ⋅ z) ∧ xl ⤳_(n) yl. Proof. split=> Hup; [intros n z; apply (Hup n (Some z))|]. intros n [z|] ?; simpl; [by apply Hup|]. destruct (Hup n (core (x ⋅ xl))) as (y&yl&?&?&?); first by rewrite cmra_core_r. do 2 eexists. split_and!; eauto using cmra_validN_op_l. Qed. Lemma cmra_total_step_update (x xl y yl : A) : (x # xl ~~>> y #' yl) ↔ ∀ n z, ✓{n} (x ⋅ xl ⋅ z) → ✓{n} (y ⋅ yl ⋅ z) ∧ xl ⤳_(n) yl. Proof. rewrite cmra_step_stepP cmra_total_step_updateP. naive_solver. Qed. Context `{CMRADiscrete A}. 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. naive_solver eauto using 0. Qed. Lemma cmra_discrete_update `{CMRADiscrete A} (x y : A) : x ~~> y ↔ ∀ z, ✓ (x ⋅ z) → ✓ (y ⋅ z). Proof. rewrite cmra_total_update; setoid_rewrite <-cmra_discrete_valid_iff. naive_solver eauto using 0. Qed. End total_updates. End updates. Section unit_updates. Context {A: ucmraT}. Implicit Types x y : A. Lemma ucmra_update_unit x : x ~~> ∅. Proof. apply cmra_total_update=>n z. rewrite left_id; apply cmra_validN_op_r. Qed. Lemma ucmra_update_unit_alt y : ∅ ~~> y ↔ ∀ x, x ~~> y. Proof. split; [intros; trans (∅ : A)|]; auto using ucmra_update_unit. Qed. End unit_updates. (** * Transport *) 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. Lemma cmra_transport_update x y : x ~~> y → T x ~~> T y. Proof. destruct H; eauto using cmra_updateP_weaken. Qed. Lemma cmra_transport_step_updateP (P : A → A → Prop) (Q : B → B → Prop) x xl : x # xl ~~>>: P → (∀ y yl, P y yl → Q (T y) (T yl)) → T x # T xl ~~>>: Q. Proof. destruct H. eauto using cmra_step_updateP_weaken. Qed. Lemma cmra_transport_step_updateP' (P : A → A → Prop) (Q : B → B → Prop) x xl : x # xl ~~>>: P → T x # T xl ~~>>: (λ y yl, ∃ y' yl', y = cmra_transport H y' ∧ yl = cmra_transport H yl' ∧ P y' yl'). Proof. intros. eapply cmra_transport_step_updateP; eauto. Qed. Lemma cmra_transport_step_update x xl y yl : x # xl ~~>> y #' yl → T x # T xl ~~>> T y #' T yl. Proof. destruct H; eauto using cmra_updateP_weaken. Qed. End cmra_transport. (** * ucmra Transport *) Section ucmra_transport. Context {A B : ucmraT} (H : A = B). Notation T := (ucmra_transport H). Lemma ucmra_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 ucmra_transport_updateP' (P : A → Prop) x : x ~~>: P → T x ~~>: λ y, ∃ y', y = ucmra_transport H y' ∧ P y'. Proof. eauto using ucmra_transport_updateP. Qed. Lemma ucmra_transport_update x y : x ~~> y → T x ~~> T y. Proof. destruct H; eauto using cmra_updateP_weaken. Qed. Lemma ucmra_transport_step_updateP (P : A → A → Prop) (Q : B → B → Prop) x xl : x # xl ~~>>: P → (∀ y yl, P y yl → Q (T y) (T yl)) → T x # T xl ~~>>: Q. Proof. destruct H. eauto using cmra_step_updateP_weaken. Qed. Lemma ucmra_transport_step_updateP' (P : A → A → Prop) (Q : B → B → Prop) x xl : x # xl ~~>>: P → T x # T xl ~~>>: (λ y yl, ∃ y' yl', y = ucmra_transport H y' ∧ yl = ucmra_transport H yl' ∧ P y' yl'). Proof. intros. eapply ucmra_transport_step_updateP; eauto. Qed. Lemma ucmra_transport_step_update x xl y yl : x # xl ~~>> y #' yl → T x # T xl ~~>> T y #' T yl. Proof. destruct H; eauto using cmra_updateP_weaken. Qed. End ucmra_transport. (** * 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. Proof. rewrite !cmra_update_updateP; eauto using option_updateP with subst. Qed. End option.