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From iris.algebra Require Export excl local_updates.
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From iris.base_logic Require Import base_logic.
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From iris.proofmode Require Import classes.
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Set Default Proof Using "Type".
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Record auth (A : Type) := Auth { authoritative : excl' A; auth_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 auth_own {_} _.
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Notation "◯ a" := (Auth None a) (at level 20).
Notation "● a" := (Auth (Excl' a) ) (at level 20).
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(* COFE *)
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Section cofe.
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Context {A : ofeT}.
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Implicit Types a : excl' A.
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Implicit Types b : A.
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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  auth_own x  auth_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  auth_own x {n} auth_own y.
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Global Instance Auth_ne : NonExpansive2 (@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: NonExpansive (@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 : NonExpansive (@auth_own A).
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Proof. by destruct 1. Qed.
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Global Instance own_proper : Proper (() ==> ()) (@auth_own A).
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Proof. by destruct 1. Qed.
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Definition auth_ofe_mixin : OfeMixin (auth A).
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Proof. by apply (iso_ofe_mixin (λ x, (authoritative x, auth_own x))). Qed.
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Canonical Structure authC := OfeT (auth A) auth_ofe_mixin.

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Global Instance auth_cofe `{Cofe A} : Cofe authC.
Proof.
  apply (iso_cofe (λ y : _ * _, Auth (y.1) (y.2))
    (λ x, (authoritative x, auth_own x))); by repeat intro.
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Qed.
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Global Instance Auth_timeless a b :
  Timeless a  Timeless b  Timeless (Auth a b).
Proof. by intros ?? [??] [??]; split; apply: timeless. Qed.
Global Instance auth_discrete : Discrete A  Discrete authC.
Proof. intros ? [??]; apply _. Qed.
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Global Instance auth_leibniz : LeibnizEquiv A  LeibnizEquiv (auth A).
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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.
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Context {A : ucmraT}.
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Implicit Types a b : A.
Implicit Types x y : auth A.
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Instance auth_valid : Valid (auth A) := λ x,
  match authoritative x with
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  | Excl' a => ( n, auth_own x {n} a)   a
  | None =>  auth_own x
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  | ExclBot' => False
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  end.
Global Arguments auth_valid !_ /.
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Instance auth_validN : ValidN (auth A) := λ n x,
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  match authoritative x with
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  | Excl' a => auth_own x {n} a  {n} a
  | None => {n} auth_own x
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  | ExclBot' => False
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  end.
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Global Arguments auth_validN _ !_ /.
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Instance auth_pcore : PCore (auth A) := λ x,
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  Some (Auth (core (authoritative x)) (core (auth_own x))).
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Instance auth_op : Op (auth A) := λ x y,
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  Auth (authoritative x  authoritative y) (auth_own x  auth_own y).
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Definition auth_valid_eq :
  valid = λ x, match authoritative x with
               | Excl' a => ( n, auth_own x {n} a)   a
               | None =>  auth_own x
               | ExclBot' => False
               end := eq_refl _.
Definition auth_validN_eq :
  validN = λ n x, match authoritative x with
                  | Excl' a => auth_own x {n} a  {n} a
                  | None => {n} auth_own x
                  | ExclBot' => False
                  end := eq_refl _.

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Lemma auth_included (x y : auth A) :
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  x  y  authoritative x  authoritative y  auth_own x  auth_own y.
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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 : {n} x  {n} authoritative x.
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Proof. by destruct x as [[[]|]]. Qed.
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Lemma auth_own_validN n x : {n} x  {n} auth_own x.
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Proof.
  rewrite auth_validN_eq.
  destruct x as [[[]|]]; naive_solver eauto using cmra_validN_includedN.
Qed.
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Lemma auth_valid_discrete `{CMRADiscrete A} x :
   x  match authoritative x with
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        | Excl' a => auth_own x  a   a
        | None =>  auth_own x
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        | ExclBot' => False
        end.
Proof.
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  rewrite auth_valid_eq. destruct x as [[[?|]|] ?]; simpl; try done.
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  setoid_rewrite <-cmra_discrete_included_iff; naive_solver eauto using 0.
Qed.
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Lemma auth_validN_2 n a b : {n} ( a   b)  b {n} a  {n} a.
Proof. by rewrite auth_validN_eq /= left_id. Qed.
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Lemma auth_valid_discrete_2 `{CMRADiscrete A} a b :  ( a   b)  b  a   a.
Proof. by rewrite auth_valid_discrete /= left_id. Qed.
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Lemma authoritative_valid  x :  x   authoritative x.
Proof. by destruct x as [[[]|]]. Qed.
Lemma auth_own_valid `{CMRADiscrete A} x :  x   auth_own x.
Proof.
  rewrite auth_valid_discrete.
  destruct x as [[[]|]]; naive_solver eauto using cmra_valid_included.
Qed.

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Lemma auth_cmra_mixin : CMRAMixin (auth A).
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Proof.
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  apply cmra_total_mixin.
  - eauto.
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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 *. rewrite !auth_validN_eq.
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    destruct Hx as [?? Hx|]; first destruct Hx; intros ?; ofe_subst; auto.
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  - intros [[[?|]|] ?]; rewrite /= ?auth_valid_eq
      ?auth_validN_eq /= ?cmra_included_includedN ?cmra_valid_validN;
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      naive_solver eauto using O.
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  - intros n [[[]|] ?]; rewrite !auth_validN_eq /=;
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      naive_solver eauto using cmra_includedN_S, cmra_validN_S.
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  - by split; simpl; rewrite assoc.
  - by split; simpl; rewrite comm.
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  - by split; simpl; rewrite ?cmra_core_l.
  - by split; simpl; rewrite ?cmra_core_idemp.
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  - intros ??; rewrite! auth_included; intros [??].
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    by split; simpl; apply cmra_core_mono.
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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]; rewrite auth_validN_eq;
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     naive_solver eauto using cmra_validN_op_l, cmra_validN_includedN.
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  - intros n x y1 y2 ? [??]; simpl in *.
    destruct (cmra_extend n (authoritative x) (authoritative y1)
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      (authoritative y2)) as (ea1&ea2&?&?&?); auto using authoritative_validN.
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    destruct (cmra_extend n (auth_own x) (auth_own y1) (auth_own y2))
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      as (b1&b2&?&?&?); auto using auth_own_validN.
    by exists (Auth ea1 b1), (Auth ea2 b2).
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Qed.
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Canonical Structure authR := CMRAT (auth A) auth_cmra_mixin.
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Global Instance auth_cmra_discrete : CMRADiscrete A  CMRADiscrete authR.
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Proof.
  split; first apply _.
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  intros [[[?|]|] ?]; rewrite auth_valid_eq auth_validN_eq /=; auto.
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  - setoid_rewrite <-cmra_discrete_included_iff.
    rewrite -cmra_discrete_valid_iff. tauto.
  - by rewrite -cmra_discrete_valid_iff.
Qed.
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Instance auth_empty : Empty (auth A) := Auth  .
Lemma auth_ucmra_mixin : UCMRAMixin (auth A).
Proof.
  split; simpl.
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  - rewrite auth_valid_eq /=. apply ucmra_unit_valid.
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  - by intros x; constructor; rewrite /= left_id.
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  - do 2 constructor; simpl; apply (persistent_core _).
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Qed.
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Canonical Structure authUR := UCMRAT (auth A) auth_ucmra_mixin.
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Global Instance auth_frag_persistent a : Persistent a  Persistent ( a).
Proof. do 2 constructor; simpl; auto. by apply persistent_core. Qed.

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(** Internalized properties *)
Lemma auth_equivI {M} (x y : auth A) :
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  x  y  (authoritative x  authoritative y  auth_own x  auth_own y : uPred M).
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Proof. by uPred.unseal. Qed.
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Lemma auth_validI {M} (x : auth A) :
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   x  (match authoritative x with
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          | Excl' a => ( b, a  auth_own x  b)   a
          | None =>  auth_own x
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          | ExclBot' => False
          end : uPred M).
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Proof. uPred.unseal. by destruct x as [[[]|]]. 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_frag_mono a b : a  b   a   b.
Proof. intros [c ->]. rewrite auth_frag_op. apply cmra_included_l. Qed.
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Global Instance auth_frag_sep_homomorphism : MonoidHomomorphism op op (Auth None).
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Proof. done. Qed.

Lemma auth_both_op a b : Auth (Excl' a) b   a   b.
Proof. by rewrite /op /auth_op /= left_id. Qed.
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Lemma auth_auth_valid a :  a   ( a).
Proof. intros; split; simpl; auto using ucmra_unit_leastN. Qed.
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Lemma auth_update a b a' b' :
  (a,b) ~l~> (a',b')   a   b ~~>  a'   b'.
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Proof.
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  intros Hup; apply cmra_total_update.
  move=> n [[[?|]|] bf1] // [[bf2 Ha] ?]; do 2 red; simpl in *.
  move: Ha; rewrite !left_id -assoc=> Ha.
  destruct (Hup n (Some (bf1  bf2))); auto.
  split; last done. exists bf2. by rewrite -assoc.
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Qed.
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Lemma auth_update_alloc a a' b' : (a,) ~l~> (a',b')   a ~~>  a'   b'.
Proof. intros. rewrite -(right_id _ _ ( a)). by apply auth_update. Qed.
Lemma auth_update_dealloc a b a' : (a,b) ~l~> (a',)   a   b ~~>  a'.
Proof. intros. rewrite -(right_id _ _ ( a')). by apply auth_update. Qed.
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Lemma auth_local_update (a b0 b1 a' b0' b1': A) :
  (b0, b1) ~l~> (b0', b1')  b0'  a'   a' 
  ( a   b0,  a   b1) ~l~> ( a'   b0',  a'   b1').
Proof.
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  rewrite !local_update_unital=> Hup ? ? n /=.
  move=> [[[ac|]|] bc] /auth_validN_2 [Le Val] [] /=;
    inversion_clear 1 as [?? Ha|]; inversion_clear Ha. (* need setoid_discriminate! *)
  rewrite !left_id=> ?.
  destruct (Hup n bc) as [Hval' Heq]; eauto using cmra_validN_includedN.
  rewrite -!auth_both_op auth_validN_eq /=.
  split_and!; [by apply cmra_included_includedN|by apply cmra_valid_validN|done].
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Qed.
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End cmra.

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Arguments authR : clear implicits.
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Arguments authUR : clear implicits.
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(* Proof mode class instances *)
Instance from_op_auth_frag {A : ucmraT} (a b1 b2 : A) :
  FromOp a b1 b2  FromOp ( a) ( b1) ( b2).
Proof. done. Qed.
Instance into_op_auth_frag {A : ucmraT} (a b1 b2 : A) :
  IntoOp a b1 b2  IntoOp ( a) ( b1) ( b2).
Proof. done. Qed.

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(* Functor *)
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Definition auth_map {A B} (f : A  B) (x : auth A) : auth B :=
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  Auth (excl_map f <$> authoritative x) (f (auth_own x)).
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Lemma auth_map_id {A} (x : auth A) : auth_map id x = x.
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Proof. by destruct x as [[[]|]]. Qed.
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Lemma auth_map_compose {A B C} (f : A  B) (g : B  C) (x : auth A) :
  auth_map (g  f) x = auth_map g (auth_map f x).
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Proof. by destruct x as [[[]|]]. Qed.
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Lemma auth_map_ext {A B : ofeT} (f g : A  B) x :
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  ( x, f x  g x)  auth_map f x  auth_map g x.
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Proof.
  constructor; simpl; auto.
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  apply option_fmap_equiv_ext=> a; by apply excl_map_ext.
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Qed.
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Instance auth_map_ne {A B : ofeT} n :
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  Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@auth_map A B).
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Proof.
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  intros f g Hf [??] [??] [??]; split; simpl in *; [|by apply Hf].
  apply option_fmap_ne; [|done]=> x y ?; by apply excl_map_ne.
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Qed.
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Instance auth_map_cmra_morphism {A B : ucmraT} (f : A  B) :
  CMRAMorphism f  CMRAMorphism (auth_map f).
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Proof.
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  split; try apply _.
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  - intros n [[[a|]|] b]; rewrite !auth_validN_eq; try
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      naive_solver eauto using cmra_morphism_monotoneN, cmra_morphism_validN.
  - intros [??]. apply Some_proper. by f_equiv; rewrite /= cmra_morphism_core.
  - intros [[?|]?] [[?|]?]; try apply Auth_proper=>//=; by rewrite cmra_morphism_op.
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Qed.
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Definition authC_map {A B} (f : A -n> B) : authC A -n> authC B :=
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  CofeMor (auth_map f).
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Lemma authC_map_ne A B : NonExpansive (@authC_map A B).
Proof. intros n f f' Hf [[[a|]|] b]; repeat constructor; apply Hf. Qed.
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Program Definition authRF (F : urFunctor) : rFunctor := {|
  rFunctor_car A B := authR (urFunctor_car F A B);
  rFunctor_map A1 A2 B1 B2 fg := authC_map (urFunctor_map F fg)
|}.
Next Obligation.
  by intros F A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_ne.
Qed.
Next Obligation.
  intros F A B x. rewrite /= -{2}(auth_map_id x).
  apply auth_map_ext=>y; apply urFunctor_id.
Qed.
Next Obligation.
  intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -auth_map_compose.
  apply auth_map_ext=>y; apply urFunctor_compose.
Qed.

Instance authRF_contractive F :
  urFunctorContractive F  rFunctorContractive (authRF F).
Proof.
  by intros ? A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_contractive.
Qed.

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Program Definition authURF (F : urFunctor) : urFunctor := {|
  urFunctor_car A B := authUR (urFunctor_car F A B);
  urFunctor_map A1 A2 B1 B2 fg := authC_map (urFunctor_map F fg)
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|}.
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Next Obligation.
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  by intros F A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_ne.
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Qed.
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Next Obligation.
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  intros F A B x. rewrite /= -{2}(auth_map_id x).
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  apply auth_map_ext=>y; apply urFunctor_id.
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Qed.
Next Obligation.
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  intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -auth_map_compose.
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  apply auth_map_ext=>y; apply urFunctor_compose.
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Qed.
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Instance authURF_contractive F :
  urFunctorContractive F  urFunctorContractive (authURF F).
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Proof.
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  by intros ? A1 A2 B1 B2 n f g Hfg; apply authC_map_ne, urFunctor_contractive.
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Qed.