Commit 2717af5d by Robbert Krebbers

### Clean up algebra/option.

`Improve names, simplify definition of dist.`
parent 908be24e
 ... ... @@ -4,40 +4,50 @@ From iris.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. Inductive option_dist' (n : nat) : relation (option A) := | Some_dist x y : x ≡{n}≡ y → option_dist' n (Some x) (Some y) | None_dist : option_dist' n None None. Instance option_dist : Dist (option A) := option_dist'. Lemma dist_option_Forall2 n mx my : mx ≡{n}≡ my ↔ option_Forall2 (dist n) mx my. Proof. split; destruct 1; constructor; auto. Qed. Program Definition option_chain (c : chain (option A)) (x : A) : chain A := {| chain_car n := from_option x (c n) |}. Next Obligation. intros c x n i ?; simpl. by destruct (chain_cauchy c n i). Qed. Instance option_compl : Compl (option A) := λ c, match c 0 with Some x => Some (compl (option_chain c x)) | None => 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. intros Hxy; destruct (Hxy 0); 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. - destruct 1; constructor; by apply dist_S. - intros n c; rewrite /compl /option_compl. feed inversion (chain_cauchy c 0 n); first auto with lia; constructor. rewrite (conv_compl n (option_chain c _)) /=. destruct (c n); naive_solver. 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. Proof. destruct 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. Proof. destruct 1; split; eauto. Qed. Global Instance Some_dist_inj : Inj (dist n) (dist n) (@Some A). Proof. by inversion_clear 1. Qed. Global Instance from_option_ne n : Proper (dist n ==> dist n ==> dist n) (@from_option A). Proof. by destruct 2. Qed. Global Instance None_timeless : Timeless (@None A). Proof. inversion_clear 1; constructor. Qed. Global Instance Some_timeless x : Timeless x → Timeless (Some x). ... ... @@ -125,16 +135,16 @@ Global Instance option_persistent (mx : option A) : Proof. intros. destruct mx. apply _. apply cmra_unit_persistent. Qed. (** Internalized properties *) Lemma option_equivI {M} (x y : option A) : (x ≡ y) ⊣⊢ (match x, y with | Some a, Some b => a ≡ b | None, None => True | _, _ => False end : uPred M). Lemma option_equivI {M} (mx my : option A) : (mx ≡ my) ⊣⊢ (match mx, my with | Some x, Some y => x ≡ y | None, None => True | _, _ => False end : uPred M). Proof. uPred.unseal. do 2 split. by destruct 1. by destruct x, y; try constructor. uPred.unseal. do 2 split. by destruct 1. by destruct mx, my; try constructor. Qed. Lemma option_validI {M} (x : option A) : (✓ x) ⊣⊢ (match x with Some a => ✓ a | None => True end : uPred M). Proof. uPred.unseal. by destruct x. Qed. Lemma option_validI {M} (mx : option A) : (✓ mx) ⊣⊢ (match mx with Some x => ✓ x | None => True end : uPred M). Proof. uPred.unseal. by destruct mx. Qed. (** Updates *) Lemma option_updateP (P : A → Prop) (Q : option A → Prop) x : ... ... @@ -146,7 +156,7 @@ Proof. 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. x ~~>: P → Some x ~~>: λ my, default False my P. Proof. eauto using option_updateP. Qed. Lemma option_update x y : x ~~> y → Some x ~~> Some y. Proof. ... ...
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