From algebra Require Export cmra. From algebra Require Import upred. (* COFE *) Section cofe. Context {A : cofeT}. Inductive option_dist : Dist (option A) := | Some_dist n x y : x ≡{n}≡ y → Some x ≡{n}≡ Some y | None_dist n : None ≡{n}≡ None. Existing Instance option_dist. Program Definition option_chain (c : chain (option A)) (x : A) (H : c 0 = Some x) : chain A := {| chain_car n := from_option x (c n) |}. Next Obligation. intros c x ? n i ?; simpl. destruct (c 0) eqn:?; simplify_eq/=. by feed inversion (chain_cauchy c n i). Qed. Instance option_compl : Compl (option A) := λ c, match Some_dec (c 0) with | inleft (exist x H) => Some (compl (option_chain c x H)) | inright _ => None end. Definition option_cofe_mixin : CofeMixin (option A). Proof. split. - intros mx my; 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 [x|]; constructor. + by destruct 1; constructor. + destruct 1; inversion_clear 1; constructor; etrans; eauto. - by inversion_clear 1; constructor; apply dist_S. - intros n c; unfold compl, option_compl. destruct (Some_dec (c 0)) as [[x Hx]|]. { assert (is_Some (c n)) as [y Hy]. { feed inversion (chain_cauchy c 0 n); eauto with lia congruence. } rewrite Hy; constructor. by rewrite (conv_compl n (option_chain c x Hx)) /= Hy. } feed inversion (chain_cauchy c 0 n); eauto with lia congruence. constructor. Qed. Canonical Structure optionC := CofeT option_cofe_mixin. Global Instance option_discrete : Discrete A → Discrete optionC. Proof. inversion_clear 2; constructor; by apply (timeless _). Qed. Global Instance Some_ne : Proper (dist n ==> dist n) (@Some A). Proof. by constructor. Qed. Global Instance is_Some_ne n : Proper (dist n ==> iff) (@is_Some A). Proof. inversion_clear 1; split; eauto. Qed. Global Instance Some_dist_inj : Inj (dist n) (dist n) (@Some A). Proof. by inversion_clear 1. Qed. Global Instance None_timeless : Timeless (@None A). Proof. inversion_clear 1; constructor. Qed. Global Instance Some_timeless x : Timeless x → Timeless (Some x). Proof. by intros ?; inversion_clear 1; constructor; apply timeless. Qed. End cofe. Arguments optionC : clear implicits. (* CMRA *) Section cmra. Context {A : cmraT}. Instance option_valid : Valid (option A) := λ mx, match mx with Some x => ✓ x | None => True end. Instance option_validN : ValidN (option A) := λ n mx, match mx with Some x => ✓{n} x | None => True end. Global Instance option_empty : Empty (option A) := None. Instance option_core : Core (option A) := fmap core. Instance option_op : Op (option A) := union_with (λ x y, Some (x ⋅ y)). Instance option_div : Div (option A) := difference_with (λ x y, Some (x ÷ y)). Definition Some_valid a : ✓ Some a ↔ ✓ a := reflexivity _. Definition Some_op a b : Some (a ⋅ b) = Some a ⋅ Some b := eq_refl. Lemma option_included (mx my : option A) : mx ≼ my ↔ mx = None ∨ ∃ x y, mx = Some x ∧ my = Some y ∧ x ≼ y. Proof. split. - intros [mz Hmz]. destruct mx as [x|]; [right|by left]. destruct my as [y|]; [exists x, y|destruct mz; inversion_clear Hmz]. destruct mz as [z|]; inversion_clear Hmz; split_and?; auto; setoid_subst; eauto using cmra_included_l. - intros [->|(x&y&->&->&z&Hz)]; try (by exists my; destruct my; constructor). by exists (Some z); constructor. Qed. Definition option_cmra_mixin : CMRAMixin (option A). Proof. split. - by intros n [x|]; destruct 1; constructor; cofe_subst. - by destruct 1; constructor; cofe_subst. - by destruct 1; rewrite /validN /option_validN //=; cofe_subst. - by destruct 1; inversion_clear 1; constructor; cofe_subst. - intros [x|]; [apply cmra_valid_validN|done]. - intros n [x|]; unfold validN, option_validN; eauto using cmra_validN_S. - intros [x|] [y|] [z|]; constructor; rewrite ?assoc; auto. - intros [x|] [y|]; constructor; rewrite 1?comm; auto. - by intros [x|]; constructor; rewrite cmra_core_l. - by intros [x|]; constructor; rewrite cmra_core_idemp. - intros mx my; rewrite !option_included ;intros [->|(x&y&->&->&?)]; auto. right; exists (core x), (core y); eauto using cmra_core_preserving. - intros n [x|] [y|]; rewrite /validN /option_validN /=; eauto using cmra_validN_op_l. - intros mx my; rewrite option_included. intros [->|(x&y&->&->&?)]; [by destruct my|]. by constructor; apply cmra_op_div. - intros n mx my1 my2. destruct mx as [x|], my1 as [y1|], my2 as [y2|]; intros Hx Hx'; try (by exfalso; inversion Hx'; auto). + destruct (cmra_extend n x y1 y2) as ([z1 z2]&?&?&?); auto. { by inversion_clear Hx'. } by exists (Some z1, Some z2); repeat constructor. + by exists (Some x,None); inversion Hx'; repeat constructor. + by exists (None,Some x); inversion Hx'; repeat constructor. + exists (None,None); repeat constructor. Qed. Canonical Structure optionR := CMRAT option_cofe_mixin option_cmra_mixin. Global Instance option_cmra_identity : CMRAIdentity optionR. Proof. split. done. by intros []. by inversion_clear 1. Qed. Global Instance option_cmra_discrete : CMRADiscrete A → CMRADiscrete optionR. Proof. split; [apply _|]. by intros [x|]; [apply (cmra_discrete_valid x)|]. Qed. (** Misc *) Global Instance Some_cmra_monotone : CMRAMonotone Some. Proof. split; [apply _|done|intros x y [z ->]; by exists (Some z)]. Qed. Lemma op_is_Some mx my : is_Some (mx ⋅ my) ↔ is_Some mx ∨ is_Some my. Proof. destruct mx, my; rewrite /op /option_op /= -!not_eq_None_Some; naive_solver. Qed. Lemma option_op_positive_dist_l n mx my : mx ⋅ my ≡{n}≡ None → mx ≡{n}≡ None. Proof. by destruct mx, my; inversion_clear 1. Qed. Lemma option_op_positive_dist_r n mx my : mx ⋅ my ≡{n}≡ None → my ≡{n}≡ None. Proof. by destruct mx, my; inversion_clear 1. Qed. (** Internalized properties *) Lemma option_equivI {M} (x y : option A) : (x ≡ y)%I ≡ (match x, y with | Some a, Some b => a ≡ b | None, None => True | _, _ => False end : uPred M)%I. Proof. uPred.unseal. do 2 split. by destruct 1. by destruct x, y; try constructor. Qed. Lemma option_validI {M} (x : option A) : (✓ x)%I ≡ (match x with Some a => ✓ a | None => True end : uPred M)%I. Proof. uPred.unseal. by destruct x. Qed. (** Updates *) 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 n [y|] ?. { destruct (Hx n y) as (y'&?&?); auto. exists (Some y'); auto. } destruct (Hx n (core x)) as (y'&?&?); rewrite ?cmra_core_r; auto. by exists (Some y'); split; [auto|apply cmra_validN_op_l with (core x)]. Qed. Lemma option_updateP' (P : A → Prop) x : x ~~>: P → Some x ~~>: λ y, default False y P. 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 congruence. Qed. Lemma option_update_None `{Empty A, !CMRAIdentity A} : ∅ ~~> Some ∅. Proof. intros n [x|] ?; rewrite /op /cmra_op /validN /cmra_validN /= ?left_id; auto using cmra_empty_validN. Qed. End cmra. Arguments optionR : clear implicits. (** Functor *) Instance option_fmap_ne {A B : cofeT} (f : A → B) n: Proper (dist n ==> dist n) f → Proper (dist n==>dist n) (fmap (M:=option) f). Proof. by intros Hf; destruct 1; constructor; apply Hf. Qed. Instance option_fmap_cmra_monotone {A B : cmraT} (f: A → B) `{!CMRAMonotone f} : CMRAMonotone (fmap f : option A → option B). Proof. split; first apply _. - intros n [x|] ?; rewrite /cmra_validN //=. by apply (validN_preserving f). - intros mx my; rewrite !option_included. intros [->|(x&y&->&->&?)]; simpl; eauto 10 using @included_preserving. Qed. Definition optionC_map {A B} (f : A -n> B) : optionC A -n> optionC B := CofeMor (fmap f : optionC A → optionC B). Instance optionC_map_ne A B n : Proper (dist n ==> dist n) (@optionC_map A B). Proof. by intros f f' Hf []; constructor; apply Hf. Qed. Program Definition optionCF (F : cFunctor) : cFunctor := {| cFunctor_car A B := optionC (cFunctor_car F A B); cFunctor_map A1 A2 B1 B2 fg := optionC_map (cFunctor_map F fg) |}. Next Obligation. by intros F A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, cFunctor_ne. Qed. Next Obligation. intros F A B x. rewrite /= -{2}(option_fmap_id x). apply option_fmap_setoid_ext=>y; apply cFunctor_id. Qed. Next Obligation. intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -option_fmap_compose. apply option_fmap_setoid_ext=>y; apply cFunctor_compose. Qed. Instance optionCF_contractive F : cFunctorContractive F → cFunctorContractive (optionCF F). Proof. by intros ? A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, cFunctor_contractive. Qed. Program Definition optionRF (F : rFunctor) : rFunctor := {| rFunctor_car A B := optionR (rFunctor_car F A B); rFunctor_map A1 A2 B1 B2 fg := optionC_map (rFunctor_map F fg) |}. Next Obligation. by intros F A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, rFunctor_ne. Qed. Next Obligation. intros F A B x. rewrite /= -{2}(option_fmap_id x). apply option_fmap_setoid_ext=>y; apply rFunctor_id. Qed. Next Obligation. intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -option_fmap_compose. apply option_fmap_setoid_ext=>y; apply rFunctor_compose. Qed. Instance optionRF_contractive F : rFunctorContractive F → rFunctorContractive (optionRF F). Proof. by intros ? A1 A2 B1 B2 n f g Hfg; apply optionC_map_ne, rFunctor_contractive. Qed.