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 _ _ !_ /. (** Define an agreement construction such that Agree A is discrete when A is discrete. Notice that this construction is NOT complete. The fullowing is due to Aleš: Proposition: Ag(T) is not necessarily complete. Proof. Let T be the set of binary streams (infinite sequences) with the usual ultrametric, measuring how far they agree. Let Aₙ be the set of all binary strings of length n. Thus for Aₙ to be a subset of T we have them continue as a stream of zeroes. Now Aₙ is a finite non-empty subset of T. Moreover {Aₙ} is a Cauchy sequence in the defined (Hausdorff) metric. However the limit (if it were to exist as an element of Ag(T)) would have to be the set of all binary streams, which is not exactly finite. Thus Ag(T) is not necessarily complete. *) Record agree (A : Type) : Type := Agree { agree_car : list A; agree_not_nil : bool_decide (agree_car = []) = false }. Arguments Agree {_} _ _. Arguments agree_car {_} _. Arguments agree_not_nil {_} _. Local Coercion agree_car : agree >-> list. Lemma elem_of_agree {A} (x : agree A) : ∃ a, a ∈ agree_car x. Proof. destruct x as [[|a ?] ?]; set_solver+. Qed. Lemma agree_eq {A} (x y : agree A) : agree_car x = agree_car y → x = y. Proof. destruct x as [a ?], y as [b ?]; simpl. intros ->; f_equal. apply (proof_irrel _). Qed. Section agree. Local Set Default Proof Using "Type". Context {A : ofeT}. Implicit Types a b : A. Implicit Types x y : agree A. (* OFE *) Instance agree_dist : Dist (agree A) := λ n x y, (∀ a, a ∈ agree_car x → ∃ b, b ∈ agree_car y ∧ a ≡{n}≡ b) ∧ (∀ b, b ∈ agree_car y → ∃ a, a ∈ agree_car x ∧ a ≡{n}≡ b). Instance agree_equiv : Equiv (agree A) := λ x y, ∀ n, x ≡{n}≡ y. Definition agree_ofe_mixin : OfeMixin (agree A). Proof. split. - done. - intros n; split; rewrite /dist /agree_dist. + intros x; split; eauto. + intros x y [??]. naive_solver eauto. + intros x y z [H1 H1'] [H2 H2']; split. * intros a ?. destruct (H1 a) as (b&?&?); auto. destruct (H2 b) as (c&?&?); eauto. by exists c; split; last etrans. * intros a ?. destruct (H2' a) as (b&?&?); auto. destruct (H1' b) as (c&?&?); eauto. by exists c; split; last etrans. - intros n x y [??]; split; naive_solver eauto using dist_S. Qed. Canonical Structure agreeC := OfeT (agree A) agree_ofe_mixin. (* CMRA *) (* agree_validN is carefully written such that, when applied to a singleton, it is convertible to True. This makes working with agreement much more pleasant. *) Instance agree_validN : ValidN (agree A) := λ n x, match agree_car x with | [a] => True | _ => ∀ a b, a ∈ agree_car x → b ∈ agree_car x → a ≡{n}≡ b end. Instance agree_valid : Valid (agree A) := λ x, ∀ n, ✓{n} x. Program Instance agree_op : Op (agree A) := λ x y, Agree (agree_car x ++ agree_car y) _. Next Obligation. by intros [[|??]] y. Qed. Instance agree_pcore : PCore (agree A) := Some. Lemma agree_validN_def n x : ✓{n} x ↔ ∀ a b, a ∈ agree_car x → b ∈ agree_car x → a ≡{n}≡ b. Proof. rewrite /validN /agree_validN. destruct (agree_car _) as [|? [|??]]; auto. setoid_rewrite elem_of_list_singleton; naive_solver. Qed. Instance agree_comm : Comm (≡) (@op (agree A) _). Proof. intros x y n; split=> a /=; setoid_rewrite elem_of_app; naive_solver. Qed. Instance agree_assoc : Assoc (≡) (@op (agree A) _). Proof. intros x y z n; split=> a /=; repeat setoid_rewrite elem_of_app; naive_solver. Qed. Lemma agree_idemp (x : agree A) : x ⋅ x ≡ x. Proof. intros n; split=> a /=; setoid_rewrite elem_of_app; naive_solver. Qed. Instance agree_validN_ne n : Proper (dist n ==> impl) (@validN (agree A) _ n). Proof. intros x y [H H']; rewrite /impl !agree_validN_def; intros Hv a b Ha Hb. destruct (H' a) as (a'&?&<-); auto. destruct (H' b) as (b'&?&<-); auto. Qed. Instance agree_validN_proper n : Proper (equiv ==> iff) (@validN (agree A) _ n). Proof. move=> x y /equiv_dist H. by split; rewrite (H n). Qed. Instance agree_op_ne' x : NonExpansive (op x). Proof. intros n y1 y2 [H H']; split=> a /=; setoid_rewrite elem_of_app; naive_solver. Qed. Instance agree_op_ne : NonExpansive2 (@op (agree A) _). Proof. by intros n x1 x2 Hx y1 y2 Hy; rewrite Hy !(comm _ _ y2) Hx. Qed. Instance agree_op_proper : Proper ((≡) ==> (≡) ==> (≡)) op := ne_proper_2 _. 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_invN n (x1 x2 : agree A) : ✓{n} (x1 ⋅ x2) → x1 ≡{n}≡ x2. Proof. rewrite agree_validN_def /=. setoid_rewrite elem_of_app=> Hv; split=> a Ha. - destruct (elem_of_agree x2); naive_solver. - destruct (elem_of_agree x1); naive_solver. Qed. Definition agree_cmra_mixin : CmraMixin (agree A). Proof. apply cmra_total_mixin; try apply _ || by eauto. - intros n x; rewrite !agree_validN_def; eauto using dist_S. - intros x. apply agree_idemp. - intros n x y; rewrite !agree_validN_def /=. setoid_rewrite elem_of_app; naive_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_cmra_mixin. Global Instance agree_cmra_total : CmraTotal agreeR. Proof. rewrite /CmraTotal; eauto. Qed. Global Instance agree_core_id (x : agree A) : CoreId x. Proof. by constructor. Qed. Global Instance agree_cmra_discrete : OfeDiscrete A → CmraDiscrete agreeR. Proof. intros HD. split. - intros x y [H H'] n; split=> a; setoid_rewrite <-(discrete_iff_0 _ _); auto. - intros x; rewrite agree_validN_def=> Hv n. apply agree_validN_def=> a b ??. apply discrete_iff_0; auto. Qed. Program Definition to_agree (a : A) : agree A := {| agree_car := [a]; agree_not_nil := eq_refl |}. Global Instance to_agree_ne : NonExpansive to_agree. Proof. intros n a1 a2 Hx; split=> b /=; setoid_rewrite elem_of_list_singleton; naive_solver. 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. move=> a b [_] /=. setoid_rewrite elem_of_list_singleton. naive_solver. Qed. Global Instance to_agree_inj : Inj (≡) (≡) (to_agree). Proof. intros a b ?. apply equiv_dist=>n. by apply to_agree_injN, equiv_dist. Qed. Lemma to_agree_uninjN n (x : agree A) : ✓{n} x → ∃ a : A, to_agree a ≡{n}≡ x. Proof. rewrite agree_validN_def=> Hv. destruct (elem_of_agree x) as [a ?]. exists a; split=> b /=; setoid_rewrite elem_of_list_singleton; naive_solver. Qed. Lemma to_agree_uninj (x : agree A) : ✓ x → ∃ y : A, to_agree y ≡ x. Proof. rewrite /valid /agree_valid; setoid_rewrite agree_validN_def. destruct (elem_of_agree x) as [a ?]. exists a; split=> b /=; setoid_rewrite elem_of_list_singleton; naive_solver. Qed. Lemma agree_valid_includedN n x y : ✓{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. Lemma to_agree_included a b : to_agree a ≼ to_agree b ↔ a ≡ b. Proof. split; last by intros ->. intros (x & Heq). apply equiv_dist=>n. destruct (Heq n) as [_ Hincl]. by destruct (Hincl a) as (? & ->%elem_of_list_singleton & ?); first set_solver+. Qed. Global Instance agree_cancelable x : Cancelable x. Proof. intros n y z Hv Heq. destruct (to_agree_uninjN n x) as [x' EQx]; first by eapply cmra_validN_op_l. destruct (to_agree_uninjN n y) as [y' EQy]; first by eapply cmra_validN_op_r. destruct (to_agree_uninjN n z) as [z' EQz]. { eapply (cmra_validN_op_r n x z). by rewrite -Heq. } assert (Hx'y' : x' ≡{n}≡ y'). { apply (inj to_agree), agree_op_invN. by rewrite EQx EQy. } assert (Hx'z' : x' ≡{n}≡ z'). { apply (inj to_agree), agree_op_invN. by rewrite EQx EQz -Heq. } by rewrite -EQy -EQz -Hx'y' -Hx'z'. Qed. Lemma agree_op_inv x y : ✓ (x ⋅ y) → x ≡ y. Proof. intros ?. apply equiv_dist=>n. by apply agree_op_invN, cmra_valid_validN. Qed. Lemma agree_op_inv' a b : ✓ (to_agree a ⋅ to_agree b) → a ≡ b. Proof. by intros ?%agree_op_inv%(inj _). Qed. Lemma agree_op_invL' `{!LeibnizEquiv A} a b : ✓ (to_agree a ⋅ to_agree b) → a = b. Proof. by intros ?%agree_op_inv'%leibniz_equiv. 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. Instance: Params (@to_agree) 1. 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 |}. Next Obligation. by intros A B f [[|??] ?]. Qed. Lemma agree_map_id {A} (x : agree A) : agree_map id x = x. Proof. apply agree_eq. by rewrite /= list_fmap_id. 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. apply agree_eq. by rewrite /= list_fmap_compose. Qed. Section agree_map. Context {A B : ofeT} (f : A → B) `{Hf: NonExpansive f}. Instance agree_map_ne : NonExpansive (agree_map f). Proof. intros n x y [H H']; split=> b /=; setoid_rewrite elem_of_list_fmap. - intros (a&->&?). destruct (H a) as (a'&?&?); auto. naive_solver. - intros (a&->&?). destruct (H' a) as (a'&?&?); auto. naive_solver. Qed. Instance agree_map_proper : Proper ((≡) ==> (≡)) (agree_map f) := ne_proper _. Lemma agree_map_ext (g : A → B) x : (∀ a, f a ≡ g a) → agree_map f x ≡ agree_map g x. Proof using Hf. intros Hfg n; split=> b /=; setoid_rewrite elem_of_list_fmap. - intros (a&->&?). exists (g a). rewrite Hfg; eauto. - intros (a&->&?). exists (f a). rewrite -Hfg; eauto. Qed. Global Instance agree_map_morphism : CmraMorphism (agree_map f). Proof using Hf. split; first apply _. - intros n x. rewrite !agree_validN_def=> Hv b b' /=. intros (a&->&?)%elem_of_list_fmap (a'&->&?)%elem_of_list_fmap. apply Hf; eauto. - done. - intros x y n; split=> b /=; rewrite !fmap_app; setoid_rewrite elem_of_app; eauto. 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; split=> b /=; setoid_rewrite elem_of_list_fmap; naive_solver. 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.