From iris.algebra Require Export cmra. From iris.base_logic Require Import base_logic. Local Arguments validN _ _ _ !_ /. Local Arguments valid _ _ !_ /. Inductive excl (A : Type) := | Excl : A → excl A | ExclBot : excl A. Arguments Excl {_} _. Arguments ExclBot {_}. Notation excl' A := (option (excl A)). Notation Excl' x := (Some (Excl x)). Notation ExclBot' := (Some ExclBot). Instance maybe_Excl {A} : Maybe (@Excl A) := λ x, match x with Excl a => Some a | _ => None end. Section excl. Context {A : cofeT}. Implicit Types a b : A. Implicit Types x y : excl A. (* Cofe *) Inductive excl_equiv : Equiv (excl A) := | Excl_equiv a b : a ≡ b → Excl a ≡ Excl b | ExclBot_equiv : ExclBot ≡ ExclBot. Existing Instance excl_equiv. Inductive excl_dist : Dist (excl A) := | Excl_dist a b n : a ≡{n}≡ b → Excl a ≡{n}≡ Excl b | ExclBot_dist n : ExclBot ≡{n}≡ ExclBot. Existing Instance excl_dist. Global Instance Excl_ne n : Proper (dist n ==> dist n) (@Excl A). Proof. by constructor. Qed. Global Instance Excl_proper : Proper ((≡) ==> (≡)) (@Excl A). Proof. by constructor. Qed. Global Instance Excl_inj : Inj (≡) (≡) (@Excl A). Proof. by inversion_clear 1. Qed. Global Instance Excl_dist_inj n : Inj (dist n) (dist n) (@Excl A). Proof. by inversion_clear 1. Qed. Program Definition excl_chain (c : chain (excl A)) (a : A) : chain A := {| chain_car n := match c n return _ with Excl y => y | _ => a end |}. Next Obligation. intros c a n i ?; simpl. by destruct (chain_cauchy c n i). Qed. Instance excl_compl : Compl (excl A) := λ c, match c 0 with Excl a => Excl (compl (excl_chain c a)) | x => x end. Definition excl_cofe_mixin : CofeMixin (excl A). Proof. split. - intros x y; split; [by destruct 1; constructor; apply equiv_dist|]. intros Hxy; feed inversion (Hxy 1); subst; constructor; apply equiv_dist. by intros n; feed inversion (Hxy n). - intros n; split. + by intros []; constructor. + by destruct 1; constructor. + destruct 1; inversion_clear 1; constructor; etrans; eauto. - by inversion_clear 1; constructor; apply dist_S. - intros n c; rewrite /compl /excl_compl. feed inversion (chain_cauchy c 0 n); first auto with lia; constructor. rewrite (conv_compl n (excl_chain c _)) /=. destruct (c n); naive_solver. Qed. Canonical Structure exclC : cofeT := CofeT (excl A) excl_cofe_mixin. Global Instance excl_discrete : Discrete A → Discrete exclC. Proof. by inversion_clear 2; constructor; apply (timeless _). Qed. Global Instance excl_leibniz : LeibnizEquiv A → LeibnizEquiv (excl A). Proof. by destruct 2; f_equal; apply leibniz_equiv. Qed. Global Instance Excl_timeless a : Timeless a → Timeless (Excl a). Proof. by inversion_clear 2; constructor; apply (timeless _). Qed. Global Instance ExclBot_timeless : Timeless (@ExclBot A). Proof. by inversion_clear 1; constructor. Qed. (* CMRA *) Instance excl_valid : Valid (excl A) := λ x, match x with Excl _ => True | ExclBot => False end. Instance excl_validN : ValidN (excl A) := λ n x, match x with Excl _ => True | ExclBot => False end. Instance excl_pcore : PCore (excl A) := λ _, None. Instance excl_op : Op (excl A) := λ x y, ExclBot. Lemma excl_cmra_mixin : CMRAMixin (excl A). Proof. split; try discriminate. - by intros n []; destruct 1; constructor. - by destruct 1; intros ?. - intros x; split. done. by move=> /(_ 0). - intros n [?|]; simpl; auto with lia. - by intros [?|] [?|] [?|]; constructor. - by intros [?|] [?|]; constructor. - by intros n [?|] [?|]. - intros n x [?|] [?|] ?; inversion_clear 1; eauto. Qed. Canonical Structure exclR := CMRAT (excl A) excl_cofe_mixin excl_cmra_mixin. Global Instance excl_cmra_discrete : Discrete A → CMRADiscrete exclR. Proof. split. apply _. by intros []. Qed. (** Internalized properties *) Lemma excl_equivI {M} (x y : excl A) : x ≡ y ⊣⊢ (match x, y with | Excl a, Excl b => a ≡ b | ExclBot, ExclBot => True | _, _ => False end : uPred M). Proof. uPred.unseal. do 2 split. by destruct 1. by destruct x, y; try constructor. Qed. Lemma excl_validI {M} (x : excl A) : ✓ x ⊣⊢ (if x is ExclBot then False else True : uPred M). Proof. uPred.unseal. by destruct x. Qed. (** Exclusive *) Global Instance excl_exclusive x : Exclusive x. Proof. by destruct x; intros n []. Qed. (** Option excl *) Lemma excl_validN_inv_l n mx a : ✓{n} (Excl' a ⋅ mx) → mx = None. Proof. by destruct mx. Qed. Lemma excl_validN_inv_r n mx a : ✓{n} (mx ⋅ Excl' a) → mx = None. Proof. by destruct mx. Qed. Lemma Excl_includedN n a b : Excl' a ≼{n} Excl' b → a ≡{n}≡ b. Proof. by intros [[c|] Hb%(inj Some)]; inversion_clear Hb. Qed. Lemma Excl_included a b : Excl' a ≼ Excl' b → a ≡ b. Proof. by intros [[c|] Hb%(inj Some)]; inversion_clear Hb. Qed. End excl. Arguments exclC : clear implicits. Arguments exclR : clear implicits. (* Functor *) Definition excl_map {A B} (f : A → B) (x : excl A) : excl B := match x with Excl a => Excl (f a) | ExclBot => ExclBot end. Lemma excl_map_id {A} (x : excl A) : excl_map id x = x. Proof. by destruct x. Qed. Lemma excl_map_compose {A B C} (f : A → B) (g : B → C) (x : excl A) : excl_map (g ∘ f) x = excl_map g (excl_map f x). Proof. by destruct x. Qed. Lemma excl_map_ext {A B : cofeT} (f g : A → B) x : (∀ x, f x ≡ g x) → excl_map f x ≡ excl_map g x. Proof. by destruct x; constructor. Qed. Instance excl_map_ne {A B : cofeT} n : Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@excl_map A B). Proof. by intros f f' Hf; destruct 1; constructor; apply Hf. Qed. Instance excl_map_cmra_monotone {A B : cofeT} (f : A → B) : (∀ n, Proper (dist n ==> dist n) f) → CMRAMonotone (excl_map f). Proof. split; try apply _. - by intros n [a|]. - intros x y [z Hy]; exists (excl_map f z); apply equiv_dist=> n. move: Hy=> /equiv_dist /(_ n) ->; by destruct x, z. Qed. Definition exclC_map {A B} (f : A -n> B) : exclC A -n> exclC B := CofeMor (excl_map f). Instance exclC_map_ne A B n : Proper (dist n ==> dist n) (@exclC_map A B). Proof. by intros f f' Hf []; constructor; apply Hf. Qed. Program Definition exclRF (F : cFunctor) : rFunctor := {| rFunctor_car A B := (exclR (cFunctor_car F A B)); rFunctor_map A1 A2 B1 B2 fg := exclC_map (cFunctor_map F fg) |}. Next Obligation. intros F A1 A2 B1 B2 n x1 x2 ??. by apply exclC_map_ne, cFunctor_ne. Qed. Next Obligation. intros F A B x; simpl. rewrite -{2}(excl_map_id x). apply excl_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 -excl_map_compose. apply excl_map_ext=>y; apply cFunctor_compose. Qed. Instance exclRF_contractive F : cFunctorContractive F → rFunctorContractive (exclRF F). Proof. intros A1 A2 B1 B2 n x1 x2 ??. by apply exclC_map_ne, cFunctor_contractive. Qed.