dec_agree.v 1.88 KB
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From algebra Require Export cmra.
Local Arguments validN _ _ _ !_ /.
Local Arguments valid _ _  !_ /.
Local Arguments op _ _ _ !_ /.
Local Arguments unit _ _ !_ /.

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(* This is isomorphic to option, but has a very different RA structure. *)
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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 _.
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  - intros [?|] [?|] [?|]; by repeat (simplify_eq/= || case_match).
  - intros [?|] [?|]; by repeat (simplify_eq/= || case_match).
  - intros [?|]; by repeat (simplify_eq/= || case_match).
  - intros [?|]; by repeat (simplify_eq/= || case_match).
  - by intros [?|] [?|] ?.
  - by intros [?|] [?|] ?.
  - intros [?|] [?|] [[?|]]; fold_leibniz;
      intros; by repeat (simplify_eq/= || case_match).
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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.
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Proof. destruct x; by repeat (simplify_eq/= || case_match). Qed.
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Lemma dec_agree_op_inv (x1 x2 : dec_agree A) :  (x1  x2)  x1  x2.
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Proof. destruct x1, x2; by repeat (simplify_eq/= || case_match). Qed.
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End dec_agree.
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Arguments dec_agreeRA _ {_}.