# Library iris.algebra.deprecated

From iris.algebra Require Import ofe cmra.
Set Default Proof Using "Type".

Notation cofeT := ofeT (only parsing).

Module dec_agree.
Local Arguments validN _ _ _ !_ /.
Local Arguments valid _ _ !_ /.
Local Arguments op _ _ _ !_ /.
Local Arguments pcore _ _ !_ /.

Inductive dec_agree (A : Type) : Type :=
| DecAgree : A dec_agree A
| DecAgreeBot : dec_agree A.
Arguments DecAgree {_} _.
Arguments DecAgreeBot {_}.
Instance maybe_DecAgree {A} : Maybe (@DecAgree A) := λ x,
match x with DecAgree aSome a | _None end.

Section dec_agree.
Context `{EqDecision A}.
Implicit Types a b : A.
Implicit Types x y : dec_agree A.

Instance dec_agree_valid : Valid (dec_agree A) := λ x,
if x is DecAgree _ then True else False.
Canonical Structure dec_agreeO : ofeT := leibnizO (dec_agree A).

Instance dec_agree_op : Op (dec_agree A) := λ x y,
match x, y with
| DecAgree a, DecAgree bif decide (a = b) then DecAgree a else DecAgreeBot
| _, _DecAgreeBot
end.
Instance dec_agree_pcore : PCore (dec_agree A) := Some.

Definition dec_agree_ra_mixin : RAMixin (dec_agree A).
Proof.
apply ra_total_mixin; apply _ || eauto.
- intros [?|] [?|] [?|]; by repeat (simplify_eq/= || case_match).
- intros [?|] [?|]; by repeat (simplify_eq/= || case_match).
- intros [?|]; by repeat (simplify_eq/= || case_match).
- by intros [?|] [?|] ?.
Qed.

Canonical Structure dec_agreeR : cmraT :=
discreteR (dec_agree A) dec_agree_ra_mixin.

Global Instance dec_agree_cmra_discrete : CmraDiscrete dec_agreeR.
Proof. apply discrete_cmra_discrete. Qed.
Global Instance dec_agree_cmra_total : CmraTotal dec_agreeR.
Proof. intros x. by x. Qed.

Global Instance dec_agree_core_id (x : dec_agreeR) : CoreId x.
Proof. by constructor. Qed.

Lemma dec_agree_ne a b : a b DecAgree a DecAgree b = DecAgreeBot.
Proof. intros. by rewrite /= decide_False. Qed.

Lemma dec_agree_idemp (x : dec_agree A) : x x = x.
Proof. destruct x; by rewrite /= ?decide_True. Qed.

Lemma dec_agree_op_inv (x1 x2 : dec_agree A) : (x1 x2) x1 = x2.
Proof. destruct x1, x2; by repeat (simplify_eq/= || case_match). Qed.

Lemma DecAgree_included a b : DecAgree a DecAgree b a = b.
Proof.
split. intros [[c|] [=]%leibniz_equiv]. by simplify_option_eq. by intros →.
Qed.
End dec_agree.

Arguments dec_agreeO : clear implicits.
Arguments dec_agreeR _ {_}.
End dec_agree.