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Require Export algebra.excl.
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Local Arguments validN _ _ _ !_ /.
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Record auth (A : Type) : Type := Auth { authoritative : excl A ; own : A }.
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Add Printing Constructor auth.
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Arguments Auth {_} _ _.
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Arguments authoritative {_} _.
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Arguments own {_} _.
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Notation "◯ a" := (Auth ExclUnit a) (at level 20).
Notation "● a" := (Auth (Excl a) ) (at level 20).
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(* COFE *)
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Section cofe.
Context {A : cofeT}.
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Implicit Types a b : A.
Implicit Types x y : auth A.
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Instance auth_equiv : Equiv (auth A) := λ x y,
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  authoritative x  authoritative y  own x  own y.
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Instance auth_dist : Dist (auth A) := λ n x y,
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  authoritative x ={n}= authoritative y  own x ={n}= own y.

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Global Instance Auth_ne : Proper (dist n ==> dist n ==> dist n) (@Auth A).
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Proof. by split. Qed.
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Global Instance Auth_proper : Proper (() ==> () ==> ()) (@Auth A).
Proof. by split. Qed.
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Global Instance authoritative_ne: Proper (dist n ==> dist n) (@authoritative A).
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Proof. by destruct 1. Qed.
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Global Instance authoritative_proper : Proper (() ==> ()) (@authoritative A).
Proof. by destruct 1. Qed.
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Global Instance own_ne : Proper (dist n ==> dist n) (@own A).
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Proof. by destruct 1. Qed.
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Global Instance own_proper : Proper (() ==> ()) (@own A).
Proof. by destruct 1. Qed.
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Instance auth_compl : Compl (auth A) := λ c,
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  Auth (compl (chain_map authoritative c)) (compl (chain_map own c)).
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Definition auth_cofe_mixin : CofeMixin (auth A).
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Proof.
  split.
  * intros x y; unfold dist, auth_dist, equiv, auth_equiv.
    rewrite !equiv_dist; naive_solver.
  * intros n; split.
    + by intros ?; split.
    + by intros ?? [??]; split; symmetry.
    + intros ??? [??] [??]; split; etransitivity; eauto.
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  * by intros ? [??] [??] [??]; split; apply dist_S.
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  * by split.
  * intros c n; split. apply (conv_compl (chain_map authoritative c) n).
    apply (conv_compl (chain_map own c) n).
Qed.
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Canonical Structure authC := CofeT auth_cofe_mixin.
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Instance Auth_timeless (ea : excl A) (b : A) :
  Timeless ea  Timeless b  Timeless (Auth ea b).
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Proof. by intros ?? [??] [??]; split; simpl in *; apply (timeless _). Qed.
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Global Instance auth_leibniz : LeibnizEquiv A  LeibnizEquiv (auth A).
Proof. by intros ? [??] [??] [??]; f_equal'; apply leibniz_equiv. Qed.
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End cofe.

Arguments authC : clear implicits.
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(* CMRA *)
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Section cmra.
Context {A : cmraT}.
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Implicit Types a b : A.
Implicit Types x y : auth A.
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Global Instance auth_empty `{Empty A} : Empty (auth A) := Auth  .
Instance auth_validN : ValidN (auth A) := λ n x,
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  match authoritative x with
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  | Excl a => own x {n} a  {n} a
  | ExclUnit => {n} (own x)
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  | ExclBot => n = 0
  end.
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Global Arguments auth_validN _ !_ /.
Instance auth_unit : Unit (auth A) := λ x,
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  Auth (unit (authoritative x)) (unit (own x)).
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Instance auth_op : Op (auth A) := λ x y,
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  Auth (authoritative x  authoritative y) (own x  own y).
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Instance auth_minus : Minus (auth A) := λ x y,
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  Auth (authoritative x  authoritative y) (own x  own y).
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Lemma auth_included (x y : auth A) :
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  x  y  authoritative x  authoritative y  own x  own y.
Proof.
  split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|].
  intros [[z1 Hz1] [z2 Hz2]]; exists (Auth z1 z2); split; auto.
Qed.
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Lemma auth_includedN n (x y : auth A) :
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  x {n} y  authoritative x {n} authoritative y  own x {n} own y.
Proof.
  split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|].
  intros [[z1 Hz1] [z2 Hz2]]; exists (Auth z1 z2); split; auto.
Qed.
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Lemma authoritative_validN n (x : auth A) : {n} x  {n} (authoritative x).
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Proof. by destruct x as [[]]. Qed.
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Lemma own_validN n (x : auth A) : {n} x  {n} (own x).
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Proof. destruct x as [[]]; naive_solver eauto using cmra_validN_includedN. Qed.
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Definition auth_cmra_mixin : CMRAMixin (auth A).
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Proof.
  split.
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  * by intros n x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
  * by intros n y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
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  * intros n [x a] [y b] [Hx Ha]; simpl in *;
      destruct Hx as [[][]| | |]; intros ?; cofe_subst; auto.
  * by intros n x1 x2 [Hx Hx'] y1 y2 [Hy Hy'];
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      split; simpl; rewrite ?Hy ?Hy' ?Hx ?Hx'.
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  * by intros [[] ?]; simpl.
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  * intros n [[] ?] ?; naive_solver eauto using cmra_includedN_S, cmra_validN_S.
  * by split; simpl; rewrite associative.
  * by split; simpl; rewrite commutative.
  * by split; simpl; rewrite ?cmra_unit_l.
  * by split; simpl; rewrite ?cmra_unit_idempotent.
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  * intros n ??; rewrite! auth_includedN; intros [??].
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    by split; simpl; apply cmra_unit_preservingN.
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  * assert ( n (a b1 b2 : A), b1  b2 {n} a  b1 {n} a).
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    { intros n a b1 b2 <-; apply cmra_includedN_l. }
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   intros n [[a1| |] b1] [[a2| |] b2];
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     naive_solver eauto using cmra_validN_op_l, cmra_validN_includedN.
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  * by intros n ??; rewrite auth_includedN;
      intros [??]; split; simpl; apply cmra_op_minus.
Qed.
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Definition auth_cmra_extend_mixin : CMRAExtendMixin (auth A).
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Proof.
  intros n x y1 y2 ? [??]; simpl in *.
  destruct (cmra_extend_op n (authoritative x) (authoritative y1)
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    (authoritative y2)) as (ea&?&?&?); auto using authoritative_validN.
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  destruct (cmra_extend_op n (own x) (own y1) (own y2))
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    as (b&?&?&?); auto using own_validN.
  by exists (Auth (ea.1) (b.1), Auth (ea.2) (b.2)).
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Qed.
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Canonical Structure authRA : cmraT :=
  CMRAT auth_cofe_mixin auth_cmra_mixin auth_cmra_extend_mixin.
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(** The notations ◯ and ● only work for CMRAs with an empty element. So, in
what follows, we assume we have an empty element. *)
Context `{Empty A, !CMRAIdentity A}.

Global Instance auth_cmra_identity : CMRAIdentity authRA.
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Proof.
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  split; simpl.
  * by apply (@cmra_empty_valid A _).
  * by intros x; constructor; rewrite /= left_id.
  * apply Auth_timeless; apply _.
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Qed.
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Lemma auth_frag_op a b :  (a  b)   a   b.
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Proof. done. Qed.
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Lemma auth_update a a' b b' :
  ( n af, {n} a  a ={n}= a'  af  b ={n}= b'  af  {n} b) 
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   a   a' ~~>  b   b'.
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Proof.
  move=> Hab [[] bf1] n // =>-[[bf2 Ha] ?]; do 2 red; simpl in *.
  destruct (Hab (S n) (bf1  bf2)) as [Ha' ?]; auto.
  { by rewrite Ha left_id associative. }
  split; [by rewrite Ha' left_id associative; apply cmra_includedN_l|done].
Qed.
Lemma auth_update_op_l a a' b :
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   (b  a)   a   a' ~~>  (b  a)   (b  a').
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Proof.
  intros; apply auth_update.
  by intros n af ? Ha; split; [by rewrite Ha associative|].
Qed.
Lemma auth_update_op_r a a' b :
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   (a  b)   a   a' ~~>  (a  b)   (a'  b).
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Proof. rewrite -!(commutative _ b); apply auth_update_op_l. Qed.
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End cmra.

Arguments authRA : clear implicits.
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(* Functor *)
Instance auth_fmap : FMap auth := λ A B f x,
  Auth (f <$> authoritative x) (f (own x)).
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Arguments auth_fmap _ _ _ !_ /.
Lemma auth_fmap_id {A} (x : auth A) : id <$> x = x.
Proof. by destruct x; rewrite /= excl_fmap_id. Qed.
Lemma excl_fmap_compose {A B C} (f : A  B) (g : B  C) (x : auth A) :
  g  f <$> x = g <$> f <$> x.
Proof. by destruct x; rewrite /= excl_fmap_compose. Qed.
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Instance auth_fmap_cmra_ne {A B : cmraT} n :
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  Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@fmap auth _ A B).
Proof.
  intros f g Hf [??] [??] [??]; split; [by apply excl_fmap_cmra_ne|by apply Hf].
Qed.
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Instance auth_fmap_cmra_monotone {A B : cmraT} (f : A  B) :
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  ( n, Proper (dist n ==> dist n) f)  CMRAMonotone f 
  CMRAMonotone (fmap f : auth A  auth B).
Proof.
  split.
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  * by intros n [x a] [y b]; rewrite !auth_includedN /=;
      intros [??]; split; simpl; apply: includedN_preserving.
  * intros n [[a| |] b]; rewrite /= /cmra_validN;
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      naive_solver eauto using @includedN_preserving, @validN_preserving.
Qed.
Definition authRA_map {A B : cmraT} (f : A -n> B) : authRA A -n> authRA B :=
  CofeMor (fmap f : authRA A  authRA B).
Lemma authRA_map_ne A B n : Proper (dist n ==> dist n) (@authRA_map A B).
Proof. intros f f' Hf [[a| |] b]; repeat constructor; apply Hf. Qed.