From algebra Require Export cmra. From algebra Require Import functor upred. Local Arguments validN _ _ _ !_ /. Local Arguments valid _ _ !_ /. Local Arguments op _ _ _ !_ /. Local Arguments unit _ _ !_ /. (* This is isomorphic to optiob, but has a very different RA structure. *) Inductive dec_agree (A : Type) : Type := | DecAgree : A → dec_agree A | DecAgreeBot : dec_agree A. Arguments DecAgree {_} _. Arguments DecAgreeBot {_}. Section dec_agree. Context {A : Type} `{∀ x y : A, Decision (x = y)}. Instance dec_agree_valid : Valid (dec_agree A) := λ x, if x is DecAgree _ then True else False. Instance dec_agree_equiv : Equiv (dec_agree A) := equivL. Canonical Structure dec_agreeC : cofeT := leibnizC (dec_agree A). Instance dec_agree_op : Op (dec_agree A) := λ x y, match x, y with | DecAgree a, DecAgree b => if decide (a = b) then DecAgree a else DecAgreeBot | _, _ => DecAgreeBot end. Instance dec_agree_unit : Unit (dec_agree A) := id. Instance dec_agree_minus : Minus (dec_agree A) := λ x y, x. Definition dec_agree_ra : RA (dec_agree A). Proof. split. - apply _. - apply _. - apply _. - apply _. - intros [?|] [?|] [?|]; simpl; repeat (case_match; simpl); subst; congruence. - intros [?|] [?|]; simpl; repeat (case_match; simpl); try subst; congruence. - intros [?|]; simpl; repeat (case_match; simpl); try subst; congruence. - intros [?|]; simpl; repeat (case_match; simpl); try subst; congruence. - intros [?|] [?|] ?; simpl; done. - intros [?|] [?|] ?; simpl; done. - intros [?|] [?|] [[?|]]; simpl; repeat (case_match; simpl); subst; try congruence; []. case=>EQ. destruct EQ. done. Qed. Canonical Structure dec_agreeRA : cmraT := discreteRA dec_agree_ra. (* Some properties of this CMRA *) Lemma dec_agree_idemp (x : dec_agree A) : x ⋅ x ≡ x. Proof. destruct x as [x|]; simpl; repeat (case_match; simpl); try subst; congruence. Qed. Lemma dec_agree_op_inv (x1 x2 : dec_agree A) : ✓ (x1 ⋅ x2) → x1 ≡ x2. Proof. destruct x1 as [x1|], x2 as [x2|]; simpl;repeat (case_match; simpl); by subst. Qed. Lemma dec_agree_equivI {M} a b : (DecAgree a ≡ DecAgree b)%I ≡ (a = b : uPred M)%I. Proof. split. by case. by destruct 1. Qed. Lemma dec_agree_validI {M} (x y : dec_agreeRA) : ✓ (x ⋅ y) ⊑ (x = y : uPred M). Proof. intros r n _ ?. by apply: dec_agree_op_inv. Qed. End dec_agree.