Require Export algebra.cmra. Require Import algebra.functor. (* COFE *) Section cofe. Context {A : cofeT}. Inductive option_dist : Dist (option A) := | option_0_dist (x y : option A) : x ={0}= y | 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 1 = Some x) : chain A := {| chain_car n := from_option x (c n) |}. Next Obligation. intros c x ? n i ?; simpl; destruct (decide (i = 0)) as [->|]. { by replace n with 0 by lia. } feed inversion (chain_cauchy c 1 i); auto with lia congruence. feed inversion (chain_cauchy c n i); simpl; auto with lia congruence. Qed. Instance option_compl : Compl (option A) := λ c, match Some_dec (c 1) 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; etransitivity; eauto. * by inversion_clear 1; constructor; apply dist_S. * constructor. * intros c n; unfold compl, option_compl. destruct (decide (n = 0)) as [->|]; [constructor|]. destruct (Some_dec (c 1)) as [[x Hx]|]. { assert (is_Some (c n)) as [y Hy]. { feed inversion (chain_cauchy c 1 n); try congruence; eauto with lia. } rewrite Hy; constructor. by rewrite (conv_compl (option_chain c x Hx) n); simpl; rewrite Hy. } feed inversion (chain_cauchy c 1 n); auto with lia congruence; constructor. Qed. Canonical Structure optionC := CofeT option_cofe_mixin. Global Instance Some_ne : Proper (dist n ==> dist n) (@Some A). Proof. by constructor. Qed. Global Instance is_Some_ne n : Proper (dist (S n) ==> iff) (@is_Some A). Proof. inversion_clear 1; split; eauto. Qed. Global Instance Some_dist_inj : Injective (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_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_unit : Unit (option A) := fmap unit. Instance option_op : Op (option A) := union_with (λ x y, Some (x ⋅ y)). Instance option_minus : Minus (option A) := difference_with (λ x y, Some (x ⩪ y)). Lemma option_includedN n (mx my : option A) : mx ≼{n} my ↔ n = 0 ∨ mx = None ∨ ∃ x y, mx = Some x ∧ my = Some y ∧ x ≼{n} y. Proof. split. * intros [mz Hmz]; destruct n as [|n]; [by left|right]. 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_ands; auto; cofe_subst; eauto using cmra_includedN_l. * intros [->|[->|(x&y&->&->&z&Hz)]]; try (by exists my; destruct my; constructor). by exists (Some z); constructor. Qed. Lemma None_includedN n (mx : option A) : None ≼{n} mx. Proof. rewrite option_includedN; auto. Qed. Lemma Some_Some_includedN n (x y : A) : x ≼{n} y → Some x ≼{n} Some y. Proof. rewrite option_includedN; eauto 10. Qed. Definition option_cmra_mixin : CMRAMixin (option A). Proof. split. * by intros n [x|]; destruct 1; constructor; repeat apply (_ : Proper (dist _ ==> _ ==> _) _). * by destruct 1; constructor; apply (_ : Proper (dist n ==> _) _). * destruct 1 as [[?|] [?|]| |]; unfold validN, option_validN; simpl; intros ?; auto using cmra_validN_0; eapply (_ : Proper (dist _ ==> impl) (✓{_})); eauto. * by destruct 1; inversion_clear 1; constructor; repeat apply (_ : Proper (dist _ ==> _ ==> _) _). * intros [x|]; unfold validN, option_validN; auto using cmra_validN_0. * intros n [x|]; unfold validN, option_validN; eauto using cmra_validN_S. * intros [x|] [y|] [z|]; constructor; rewrite ?associative; auto. * intros [x|] [y|]; constructor; rewrite 1?commutative; auto. * by intros [x|]; constructor; rewrite cmra_unit_l. * by intros [x|]; constructor; rewrite cmra_unit_idempotent. * intros n mx my; rewrite !option_includedN;intros [|[->|(x&y&->&->&?)]];auto. do 2 right; exists (unit x), (unit y); eauto using cmra_unit_preservingN. * intros n [x|] [y|]; rewrite /validN /option_validN /=; eauto using cmra_validN_op_l. * intros n mx my; rewrite option_includedN. intros [->|[->|(x&y&->&->&?)]]; [done|by destruct my|]. by constructor; apply cmra_op_minus. Qed. Definition option_cmra_extend_mixin : CMRAExtendMixin (option A). Proof. intros n mx my1 my2; destruct (decide (n = 0)) as [->|]. { by exists (mx, None); repeat constructor; destruct mx; constructor. } destruct mx as [x|], my1 as [y1|], my2 as [y2|]; intros Hx Hx'; try (by exfalso; inversion Hx'; auto). * destruct (cmra_extend_op 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 optionRA := CMRAT option_cofe_mixin option_cmra_mixin option_cmra_extend_mixin. Global Instance option_cmra_identity : CMRAIdentity optionRA. Proof. split. done. by intros []. by inversion_clear 1. 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. 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 [y|] n ?. { destruct (Hx y n) as (y'&?&?); auto. exists (Some y'); auto. } destruct (Hx (unit x) n) as (y'&?&?); rewrite ?cmra_unit_r; auto. by exists (Some y'); split; [auto|apply cmra_validN_op_l with (unit 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. End cmra. Arguments optionRA : 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. * intros n mx my; rewrite !option_includedN. intros [->|[->|(x&y&->&->&?)]]; simpl; eauto 10 using @includedN_preserving. * by intros n [x|] ?; rewrite /cmra_validN /=; try apply validN_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 optionF (Σ : iFunctor) : iFunctor := {| ifunctor_car := optionRA ∘ Σ; ifunctor_map A B := optionC_map ∘ ifunctor_map Σ |}. Next Obligation. by intros Σ A B n f g Hfg; apply optionC_map_ne, ifunctor_map_ne. Qed. Next Obligation. intros Σ A x. rewrite /= -{2}(option_fmap_id x). apply option_fmap_setoid_ext=>y; apply ifunctor_map_id. Qed. Next Obligation. intros Σ A B C f g x. rewrite /= -option_fmap_compose. apply option_fmap_setoid_ext=>y; apply ifunctor_map_compose. Qed.