dec_agree.v 2.33 KB
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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.
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(* 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.
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Proof. do 2 split. by case. by destruct 1. Qed.
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Lemma dec_agree_validI {M} (x y : dec_agreeRA) :  (x  y)  (x = y : uPred M).
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Proof. split=> r n _ ?. by apply: dec_agree_op_inv. Qed.
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End dec_agree.