From iris.algebra Require Export cmra. From iris.algebra Require Import list. From iris.base_logic Require Import base_logic. Local Arguments validN _ _ _ !_ /. Local Arguments valid _ _ !_ /. Local Arguments op _ _ _ !_ /. Local Arguments pcore _ _ !_ /. Record agree (A : Type) : Type := Agree { agree_car : A; agree_with : list A; }. Arguments Agree {_} _ _. Arguments agree_car {_} _. Arguments agree_with {_} _. (* Some theory about set-inclusion on lists and lists of which all elements are equal. TODO: Move this elsewhere. *) Definition list_setincl `(R : relation A) (al bl : list A) := ∀ a, a ∈ al → ∃ b, b ∈ bl ∧ R a b. Definition list_setequiv `(R : relation A) (al bl : list A) := list_setincl R al bl ∧ list_setincl R bl al. (* list_agrees is carefully written such that, when applied to a singleton, it is convertible to True. This makes working with agreement much more pleasant. *) Definition list_agrees `(R : relation A) (al : list A) := match al with | [] => True | [a] => True | a :: al => ∀ b, b ∈ al → R a b end. Lemma list_agrees_alt `(R : relation A) `{Equivalence _ R} al : list_agrees R al ↔ (∀ a b, a ∈ al → b ∈ al → R a b). Proof. destruct al as [|a [|b al]]. - split; last done. intros _ ? ? []%elem_of_nil. - split; last done. intros _ ? ? ->%elem_of_list_singleton ->%elem_of_list_singleton. done. - simpl. split. + intros Hl a' b' [->|Ha']%elem_of_cons. * intros [->|Hb']%elem_of_cons; first done. auto. * intros [->|Hb']%elem_of_cons; first by (symmetry; auto). trans a; last by auto. symmetry. auto. + intros Hl b' Hb'. apply Hl; set_solver. Qed. Section list_theory. Context `(R: relation A) `{Equivalence A R}. Collection Hyps := Type H. Local Set Default Proof Using "Hyps". Global Instance: PreOrder (list_setincl R). Proof. split. - intros al a Ha. set_solver. - intros al bl cl Hab Hbc a Ha. destruct (Hab _ Ha) as (b & Hb & Rab). destruct (Hbc _ Hb) as (c & Hc & Rbc). exists c. split; first done. by trans b. Qed. Global Instance: Equivalence (list_setequiv R). Proof. split. - by split. - intros ?? [??]. split; auto. - intros ??? [??] [??]. split; etrans; done. Qed. Global Instance list_setincl_subrel `(R' : relation A) : subrelation R R' → subrelation (list_setincl R) (list_setincl R'). Proof using. intros HRR' al bl Hab. intros a Ha. destruct (Hab _ Ha) as (b & Hb & HR). exists b. split; first done. exact: HRR'. Qed. Global Instance list_setequiv_subrel `(R' : relation A) : subrelation R R' → subrelation (list_setequiv R) (list_setequiv R'). Proof using. intros HRR' ?? [??]. split; exact: list_setincl_subrel. Qed. Global Instance list_setincl_perm : subrelation (≡ₚ) (list_setincl R). Proof. intros al bl Hab a Ha. exists a. split; last done. by rewrite -Hab. Qed. Global Instance list_setincl_app l : Proper (list_setincl R ==> list_setincl R) (app l). Proof. intros al bl Hab a [Ha|Ha]%elem_of_app. - exists a. split; last done. apply elem_of_app. by left. - destruct (Hab _ Ha) as (b & Hb & HR). exists b. split; last done. apply elem_of_app. by right. Qed. Global Instance list_setequiv_app l : Proper (list_setequiv R ==> list_setequiv R) (app l). Proof. intros al bl [??]. split; apply list_setincl_app; done. Qed. Global Instance: subrelation (≡ₚ) (flip (list_setincl R)). Proof. intros ???. apply list_setincl_perm. done. Qed. Global Instance list_agrees_setincl : Proper (flip (list_setincl R) ==> impl) (list_agrees R). Proof. move=> al bl /= Hab /list_agrees_alt Hal. apply (list_agrees_alt _) => a b Ha Hb. destruct (Hab _ Ha) as (a' & Ha' & HRa). destruct (Hab _ Hb) as (b' & Hb' & HRb). trans a'; first done. etrans; last done. eapply Hal; done. Qed. Global Instance list_agrees_setequiv : Proper (list_setequiv R ==> iff) (list_agrees R). Proof. intros ?? [??]. split; by apply: list_agrees_setincl. Qed. Lemma list_setincl_contains al bl : (∀ x, x ∈ al → x ∈ bl) → list_setincl R al bl. Proof. intros Hin a Ha. exists a. split; last done. naive_solver. Qed. Lemma list_setequiv_equiv al bl : (∀ x, x ∈ al ↔ x ∈ bl) → list_setequiv R al bl. Proof. intros Hin. split; apply list_setincl_contains; naive_solver. Qed. Lemma list_agrees_contains al bl : (∀ x, x ∈ bl → x ∈ al) → list_agrees R al → list_agrees R bl. Proof. intros ?. by eapply (list_agrees_setincl _),list_setincl_contains. Qed. Lemma list_agrees_equiv al bl : (∀ x, x ∈ bl ↔ x ∈ al) → list_agrees R al ↔ list_agrees R bl. Proof. intros ?. by eapply (list_agrees_setequiv _), list_setequiv_equiv. Qed. Lemma list_setincl_singleton a b : R a b → list_setincl R [a] [b]. Proof. intros HR c ->%elem_of_list_singleton. exists b. split; last done. apply elem_of_list_singleton. done. Qed. Lemma list_setincl_singleton_rev a b : list_setincl R [a] [b] → R a b. Proof using. intros Hl. destruct (Hl a) as (? & ->%elem_of_list_singleton & HR); last done. by apply elem_of_list_singleton. Qed. Lemma list_setequiv_singleton a b : R a b → list_setequiv R [a] [b]. Proof. intros ?. split; by apply list_setincl_singleton. Qed. Lemma list_agrees_iff_setincl al a : a ∈ al → list_agrees R al ↔ list_setincl R al [a]. Proof. intros Hin. split. - move=>/list_agrees_alt Hl b Hb. exists a. split; first set_solver+. exact: Hl. - intros Hl. apply (list_agrees_alt _)=> b c Hb Hc. destruct (Hl _ Hb) as (? & ->%elem_of_list_singleton & ?). destruct (Hl _ Hc) as (? & ->%elem_of_list_singleton & ?). by trans a. Qed. Lemma list_setincl_singleton_in al a : a ∈ al → list_setincl R [a] al. Proof. intros Hin b ->%elem_of_list_singleton. exists a. split; done. Qed. Global Instance list_setincl_ext : subrelation (Forall2 R) (list_setincl R). Proof. move=>al bl. induction 1. - intros ? []%elem_of_nil. - intros a [->|Ha]%elem_of_cons. + eexists. split; first constructor. done. + destruct (IHForall2 _ Ha) as (b & ? & ?). exists b. split; first by constructor. done. Qed. Global Instance list_setequiv_ext : subrelation (Forall2 R) (list_setequiv R). Proof. move=>al bl ?. split; apply list_setincl_ext; done. Qed. Lemma list_agrees_subrel `(R' : relation A) `{Equivalence _ R'} : subrelation R R' → ∀ l, list_agrees R l → list_agrees R' l. Proof. move=> HR l /list_agrees_alt Hl. apply (list_agrees_alt _)=> a b Ha Hb. by apply HR, Hl. Qed. Section fmap. Context `(R' : relation B) (f : A → B) {Hf: Proper (R ==> R') f}. Collection Hyps := Type Hf. Local Set Default Proof Using "Hyps". Global Instance list_setincl_fmap : Proper (list_setincl R ==> list_setincl R') (fmap f). Proof using Hf. intros al bl Hab a' (a & -> & Ha)%elem_of_list_fmap. destruct (Hab _ Ha) as (b & Hb & HR). exists (f b). split; first eapply elem_of_list_fmap; eauto. Qed. Global Instance list_setequiv_fmap : Proper (list_setequiv R ==> list_setequiv R') (fmap f). Proof using Hf. intros ?? [??]. split; apply list_setincl_fmap; done. Qed. Lemma list_agrees_fmap `{Equivalence _ R'} al : list_agrees R al → list_agrees R' (f <\$> al). Proof using Type*. move=> /list_agrees_alt Hl. apply (list_agrees_alt R') => a' b'. intros (a & -> & Ha)%elem_of_list_fmap (b & -> & Hb)%elem_of_list_fmap. apply Hf. exact: Hl. Qed. End fmap. End list_theory. Section agree. Local Set Default Proof Using "Type". Context {A : ofeT}. Definition agree_list (x : agree A) := agree_car x :: agree_with x. Instance agree_validN : ValidN (agree A) := λ n x, list_agrees (dist n) (agree_list x). Instance agree_valid : Valid (agree A) := λ x, list_agrees (equiv) (agree_list x). Instance agree_dist : Dist (agree A) := λ n x y, list_setequiv (dist n) (agree_list x) (agree_list y). Instance agree_equiv : Equiv (agree A) := λ x y, ∀ n, list_setequiv (dist n) (agree_list x) (agree_list y). Definition agree_dist_incl n (x y : agree A) := list_setincl (dist n) (agree_list x) (agree_list y). Definition agree_ofe_mixin : OfeMixin (agree A). Proof. split. - intros x y; split; intros Hxy; done. - split; rewrite /dist /agree_dist; intros ? *. + reflexivity. + by symmetry. + intros. etrans; eassumption. - intros ???. apply list_setequiv_subrel=>??. apply dist_S. Qed. Canonical Structure agreeC := OfeT (agree A) agree_ofe_mixin. Program Instance agree_op : Op (agree A) := λ x y, {| agree_car := agree_car x; agree_with := agree_with x ++ agree_car y :: agree_with y |}. Instance agree_pcore : PCore (agree A) := Some. Instance: Comm (≡) (@op (agree A) _). Proof. intros x y n. apply: list_setequiv_equiv. set_solver. Qed. Lemma agree_idemp (x : agree A) : x ⋅ x ≡ x. Proof. intros n. apply: list_setequiv_equiv. set_solver. Qed. Instance: ∀ n : nat, Proper (dist n ==> impl) (@validN (agree A) _ n). Proof. intros n x y. rewrite /dist /validN /agree_dist /agree_validN. by intros ->. Qed. Instance: ∀ n : nat, Proper (equiv ==> iff) (@validN (agree A) _ n). Proof. intros n ???. assert (x ≡{n}≡ y) as Hxy by by apply equiv_dist. split; rewrite Hxy; done. Qed. Instance: ∀ x : agree A, NonExpansive (op x). Proof. intros x n y1 y2. rewrite /dist /agree_dist /agree_list /=. rewrite !app_comm_cons. apply: list_setequiv_app. Qed. Instance: NonExpansive2 (@op (agree A) _). Proof. by intros n x1 x2 Hx y1 y2 Hy; rewrite Hy !(comm _ _ y2) Hx. Qed. Instance: Proper ((≡) ==> (≡) ==> (≡)) op := ne_proper_2 _. Instance: Assoc (≡) (@op (agree A) _). Proof. intros x y z n. apply: list_setequiv_equiv. set_solver. Qed. Lemma agree_included (x y : agree A) : x ≼ y ↔ y ≡ x ⋅ y. Proof. split; [|by intros ?; exists y]. by intros [z Hz]; rewrite Hz assoc agree_idemp. Qed. Lemma agree_op_inv_inclN n x1 x2 : ✓{n} (x1 ⋅ x2) → agree_dist_incl n x1 x2. Proof. rewrite /validN /= => /list_agrees_alt Hv a /elem_of_cons Ha. exists (agree_car x2). split; first by constructor. eapply Hv. - simpl. destruct Ha as [->|Ha]; set_solver. - simpl. set_solver+. Qed. Lemma agree_op_invN n (x1 x2 : agree A) : ✓{n} (x1 ⋅ x2) → x1 ≡{n}≡ x2. Proof. intros Hxy. split; apply agree_op_inv_inclN; first done. by rewrite comm. Qed. Lemma agree_valid_includedN n (x y : agree A) : ✓{n} y → x ≼{n} y → x ≡{n}≡ y. Proof. move=> Hval [z Hy]; move: Hval; rewrite Hy. by move=> /agree_op_invN->; rewrite agree_idemp. Qed. Definition agree_cmra_mixin : CMRAMixin (agree A). Proof. apply cmra_total_mixin; try apply _ || by eauto. - move=>x. split. + move=>/list_agrees_alt Hx n. apply (list_agrees_alt _)=> a b Ha Hb. apply equiv_dist, Hx; done. + intros Hx. apply (list_agrees_alt _)=> a b Ha Hb. apply equiv_dist=>n. eapply (list_agrees_alt _); first (by apply Hx); done. - intros n x. apply (list_agrees_subrel _ _)=>??. apply dist_S. - intros x. apply agree_idemp. - intros ??? Hl. apply: list_agrees_contains Hl. set_solver. - intros n x y1 y2 Hval Hx; exists x, x; simpl; split. + by rewrite agree_idemp. + by move: Hval; rewrite Hx; move=> /agree_op_invN->; rewrite agree_idemp. Qed. Canonical Structure agreeR : cmraT := CMRAT (agree A) agree_ofe_mixin agree_cmra_mixin. Global Instance agree_total : CMRATotal agreeR. Proof. rewrite /CMRATotal; eauto. Qed. Global Instance agree_persistent (x : agree A) : Persistent x. Proof. by constructor. Qed. Lemma agree_op_inv (x1 x2 : agree A) : ✓ (x1 ⋅ x2) → x1 ≡ x2. Proof. intros ?. apply equiv_dist=>n. by apply agree_op_invN, cmra_valid_validN. Qed. Global Instance agree_discrete : Discrete A → CMRADiscrete agreeR. Proof. intros HD. split. - intros x y Hxy n. eapply list_setequiv_subrel; last exact Hxy. clear -HD. intros x y ?. apply equiv_dist, HD. done. - rewrite /valid /cmra_valid /agree_valid /validN /cmra_validN /agree_validN /=. move=> x. apply (list_agrees_subrel _ _). clear -HD. intros x y. apply HD. Qed. Definition to_agree (x : A) : agree A := {| agree_car := x; agree_with := [] |}. Global Instance to_agree_ne : NonExpansive to_agree. Proof. intros x1 x2 Hx; rewrite /= /dist /agree_dist /=. exact: list_setequiv_singleton. Qed. Global Instance to_agree_proper : Proper ((≡) ==> (≡)) to_agree := ne_proper _. Global Instance to_agree_injN n : Inj (dist n) (dist n) (to_agree). Proof. intros a b [Hxy%list_setincl_singleton_rev _]. done. Qed. Global Instance to_agree_inj : Inj (≡) (≡) (to_agree). Proof. intros a b ?. apply equiv_dist=>n. apply to_agree_injN. by apply equiv_dist. Qed. Lemma to_agree_uninjN n (x : agree A) : ✓{n} x → ∃ y : A, to_agree y ≡{n}≡ x. Proof. intros Hl. exists (agree_car x). rewrite /dist /agree_dist /=. split. - apply: list_setincl_singleton_in. set_solver+. - apply (list_agrees_iff_setincl _); first set_solver+. done. Qed. Lemma to_agree_uninj (x : agree A) : ✓ x → ∃ y : A, to_agree y ≡ x. Proof. intros Hl. exists (agree_car x). rewrite /dist /agree_dist /=. split. - apply: list_setincl_singleton_in. set_solver+. - apply (list_agrees_iff_setincl _); first set_solver+. eapply list_agrees_subrel; last exact: Hl; [apply _..|]. intros ???. by apply equiv_dist. Qed. Lemma to_agree_included (a b : A) : to_agree a ≼ to_agree b ↔ a ≡ b. Proof. split. - intros (x & Heq). apply equiv_dist=>n. destruct (Heq n) as [_ Hincl]. (* TODO: This could become a generic lemma about list_setincl. *) destruct (Hincl a) as (? & ->%elem_of_list_singleton & ?); first set_solver+. done. - intros Hab. rewrite Hab. eexists. symmetry. eapply agree_idemp. Qed. Lemma to_agree_comp_valid (a b : A) : ✓ (to_agree a ⋅ to_agree b) ↔ a ≡ b. Proof. split. - (* TODO: can this be derived from other stuff? Otherwise, should probably become sth. generic about list_agrees. *) intros Hv. apply Hv; simpl; set_solver. - intros ->. rewrite agree_idemp. done. Qed. (** Internalized properties *) Lemma agree_equivI {M} a b : to_agree a ≡ to_agree b ⊣⊢ (a ≡ b : uPred M). Proof. uPred.unseal. do 2 split. - intros Hx. exact: to_agree_injN. - intros Hx. exact: to_agree_ne. Qed. Lemma agree_validI {M} x y : ✓ (x ⋅ y) ⊢ (x ≡ y : uPred M). Proof. uPred.unseal; split=> r n _ ?; by apply: agree_op_invN. Qed. End agree. Arguments agreeC : clear implicits. Arguments agreeR : clear implicits. Program Definition agree_map {A B} (f : A → B) (x : agree A) : agree B := {| agree_car := f (agree_car x); agree_with := f <\$> (agree_with x) |}. Lemma agree_map_id {A} (x : agree A) : agree_map id x = x. Proof. rewrite /agree_map /= list_fmap_id. by destruct x. Qed. Lemma agree_map_compose {A B C} (f : A → B) (g : B → C) (x : agree A) : agree_map (g ∘ f) x = agree_map g (agree_map f x). Proof. rewrite /agree_map /= list_fmap_compose. done. Qed. Section agree_map. Context {A B : ofeT} (f : A → B) `{Hf: NonExpansive f}. Collection Hyps := Type Hf. Instance agree_map_ne : NonExpansive (agree_map f). Proof using Hyps. intros n x y Hxy. change (list_setequiv (dist n)(f <\$> (agree_list x))(f <\$> (agree_list y))). eapply list_setequiv_fmap; last exact Hxy. apply _. Qed. Instance agree_map_proper : Proper ((≡) ==> (≡)) (agree_map f) := ne_proper _. Lemma agree_map_ext (g : A → B) x : (∀ x, f x ≡ g x) → agree_map f x ≡ agree_map g x. Proof. intros Hfg n. apply: list_setequiv_ext. change (f <\$> (agree_list x) ≡{n}≡ g <\$> (agree_list x)). apply list_fmap_ext_ne=>y. by apply equiv_dist. Qed. Global Instance agree_map_monotone : CMRAMonotone (agree_map f). Proof using Hyps. split; first apply _. - intros n x. rewrite /cmra_validN /validN /= /agree_validN /= => ?. change (list_agrees (dist n) (f <\$> agree_list x)). eapply (list_agrees_fmap _ _ _); done. - intros x y; rewrite !agree_included=> ->. rewrite /equiv /agree_equiv /agree_map /agree_op /agree_list /=. rewrite !fmap_app=>n. apply: list_setequiv_equiv. set_solver+. Qed. End agree_map. Definition agreeC_map {A B} (f : A -n> B) : agreeC A -n> agreeC B := CofeMor (agree_map f : agreeC A → agreeC B). Instance agreeC_map_ne A B : NonExpansive (@agreeC_map A B). Proof. intros n f g Hfg x. apply: list_setequiv_ext. change (f <\$> (agree_list x) ≡{n}≡ g <\$> (agree_list x)). apply list_fmap_ext_ne. done. Qed. Program Definition agreeRF (F : cFunctor) : rFunctor := {| rFunctor_car A B := agreeR (cFunctor_car F A B); rFunctor_map A1 A2 B1 B2 fg := agreeC_map (cFunctor_map F fg) |}. Next Obligation. intros ? A1 A2 B1 B2 n ???; simpl. by apply agreeC_map_ne, cFunctor_ne. Qed. Next Obligation. intros F A B x; simpl. rewrite -{2}(agree_map_id x). apply agree_map_ext=>y. by rewrite cFunctor_id. Qed. Next Obligation. intros F A1 A2 A3 B1 B2 B3 f g f' g' x; simpl. rewrite -agree_map_compose. apply agree_map_ext=>y; apply cFunctor_compose. Qed. Instance agreeRF_contractive F : cFunctorContractive F → rFunctorContractive (agreeRF F). Proof. intros ? A1 A2 B1 B2 n ???; simpl. by apply agreeC_map_ne, cFunctor_contractive. Qed.