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From iris.algebra Require Export cofe.
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Class PCore (A : Type) := pcore : A  option A.
Instance: Params (@pcore) 2.
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Class Op (A : Type) := op : A  A  A.
Instance: Params (@op) 2.
Infix "⋅" := op (at level 50, left associativity) : C_scope.
Notation "(⋅)" := op (only parsing) : C_scope.

Definition included `{Equiv A, Op A} (x y : A) :=  z, y  x  z.
Infix "≼" := included (at level 70) : C_scope.
Notation "(≼)" := included (only parsing) : C_scope.
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Hint Extern 0 (_  _) => reflexivity.
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Instance: Params (@included) 3.

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Class ValidN (A : Type) := validN : nat  A  Prop.
Instance: Params (@validN) 3.
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Notation "✓{ n } x" := (validN n x)
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  (at level 20, n at next level, format "✓{ n }  x").
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Class Valid (A : Type) := valid : A  Prop.
Instance: Params (@valid) 2.
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Notation "✓ x" := (valid x) (at level 20) : C_scope.
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Definition includedN `{Dist A, Op A} (n : nat) (x y : A) :=  z, y {n} x  z.
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Notation "x ≼{ n } y" := (includedN n x y)
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  (at level 70, n at next level, format "x  ≼{ n }  y") : C_scope.
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Instance: Params (@includedN) 4.
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Hint Extern 0 (_ {_} _) => reflexivity.
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Record CMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A} := {
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  (* setoids *)
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  mixin_cmra_op_ne n (x : A) : Proper (dist n ==> dist n) (op x);
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  mixin_cmra_pcore_ne n x y cx :
    x {n} y  pcore x = Some cx   cy, pcore y = Some cy  cx {n} cy;
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  mixin_cmra_validN_ne n : Proper (dist n ==> impl) (validN n);
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  (* valid *)
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  mixin_cmra_valid_validN x :  x   n, {n} x;
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  mixin_cmra_validN_S n x : {S n} x  {n} x;
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  (* monoid *)
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  mixin_cmra_assoc : Assoc () ();
  mixin_cmra_comm : Comm () ();
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  mixin_cmra_pcore_l x cx : pcore x = Some cx  cx  x  x;
  mixin_cmra_pcore_idemp x cx : pcore x = Some cx  pcore cx  Some cx;
  mixin_cmra_pcore_preserving x y cx :
    x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy;
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  mixin_cmra_validN_op_l n x y : {n} (x  y)  {n} x;
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  mixin_cmra_extend n x y1 y2 :
    {n} x  x {n} y1  y2 
    { z | x  z.1  z.2  z.1 {n} y1  z.2 {n} y2 }
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}.
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(** Bundeled version *)
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Structure cmraT := CMRAT' {
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  cmra_car :> Type;
  cmra_equiv : Equiv cmra_car;
  cmra_dist : Dist cmra_car;
  cmra_compl : Compl cmra_car;
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  cmra_pcore : PCore cmra_car;
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  cmra_op : Op cmra_car;
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  cmra_valid : Valid cmra_car;
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  cmra_validN : ValidN cmra_car;
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  cmra_cofe_mixin : CofeMixin cmra_car;
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  cmra_mixin : CMRAMixin cmra_car;
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  _ : Type
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}.
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Arguments CMRAT' _ {_ _ _ _ _ _ _} _ _ _.
Notation CMRAT A m m' := (CMRAT' A m m' A).
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Arguments cmra_car : simpl never.
Arguments cmra_equiv : simpl never.
Arguments cmra_dist : simpl never.
Arguments cmra_compl : simpl never.
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Arguments cmra_pcore : simpl never.
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Arguments cmra_op : simpl never.
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Arguments cmra_valid : simpl never.
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Arguments cmra_validN : simpl never.
Arguments cmra_cofe_mixin : simpl never.
Arguments cmra_mixin : simpl never.
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Add Printing Constructor cmraT.
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Hint Extern 0 (PCore _) => eapply (@cmra_pcore _) : typeclass_instances.
Hint Extern 0 (Op _) => eapply (@cmra_op _) : typeclass_instances.
Hint Extern 0 (Valid _) => eapply (@cmra_valid _) : typeclass_instances.
Hint Extern 0 (ValidN _) => eapply (@cmra_validN _) : typeclass_instances.
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Coercion cmra_cofeC (A : cmraT) : cofeT := CofeT A (cmra_cofe_mixin A).
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Canonical Structure cmra_cofeC.

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(** Lifting properties from the mixin *)
Section cmra_mixin.
  Context {A : cmraT}.
  Implicit Types x y : A.
  Global Instance cmra_op_ne n (x : A) : Proper (dist n ==> dist n) (op x).
  Proof. apply (mixin_cmra_op_ne _ (cmra_mixin A)). Qed.
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  Lemma cmra_pcore_ne n x y cx :
    x {n} y  pcore x = Some cx   cy, pcore y = Some cy  cx {n} cy.
  Proof. apply (mixin_cmra_pcore_ne _ (cmra_mixin A)). Qed.
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  Global Instance cmra_validN_ne n : Proper (dist n ==> impl) (@validN A _ n).
  Proof. apply (mixin_cmra_validN_ne _ (cmra_mixin A)). Qed.
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  Lemma cmra_valid_validN x :  x   n, {n} x.
  Proof. apply (mixin_cmra_valid_validN _ (cmra_mixin A)). Qed.
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  Lemma cmra_validN_S n x : {S n} x  {n} x.
  Proof. apply (mixin_cmra_validN_S _ (cmra_mixin A)). Qed.
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  Global Instance cmra_assoc : Assoc () (@op A _).
  Proof. apply (mixin_cmra_assoc _ (cmra_mixin A)). Qed.
  Global Instance cmra_comm : Comm () (@op A _).
  Proof. apply (mixin_cmra_comm _ (cmra_mixin A)). Qed.
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  Lemma cmra_pcore_l x cx : pcore x = Some cx  cx  x  x.
  Proof. apply (mixin_cmra_pcore_l _ (cmra_mixin A)). Qed.
  Lemma cmra_pcore_idemp x cx : pcore x = Some cx  pcore cx  Some cx.
  Proof. apply (mixin_cmra_pcore_idemp _ (cmra_mixin A)). Qed.
  Lemma cmra_pcore_preserving x y cx :
    x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy.
  Proof. apply (mixin_cmra_pcore_preserving _ (cmra_mixin A)). Qed.
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  Lemma cmra_validN_op_l n x y : {n} (x  y)  {n} x.
  Proof. apply (mixin_cmra_validN_op_l _ (cmra_mixin A)). Qed.
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  Lemma cmra_extend n x y1 y2 :
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    {n} x  x {n} y1  y2 
    { z | x  z.1  z.2  z.1 {n} y1  z.2 {n} y2 }.
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  Proof. apply (mixin_cmra_extend _ (cmra_mixin A)). Qed.
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End cmra_mixin.

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Definition opM {A : cmraT} (x : A) (my : option A) :=
  match my with Some y => x  y | None => x end.
Infix "⋅?" := opM (at level 50, left associativity) : C_scope.

(** * Persistent elements *)
Class Persistent {A : cmraT} (x : A) := persistent : pcore x  Some x.
Arguments persistent {_} _ {_}.

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(** * Exclusive elements (i.e., elements that cannot have a frame). *)
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Class Exclusive {A : cmraT} (x : A) := exclusive0_l y : {0} (x  y)  False.
Arguments exclusive0_l {_} _ {_} _ _.
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(** * CMRAs whose core is total *)
(** The function [core] may return a dummy when used on CMRAs without total
core. *)
Class CMRATotal (A : cmraT) := cmra_total (x : A) : is_Some (pcore x).

Class Core (A : Type) := core : A  A.
Instance: Params (@core) 2.

Instance core' `{PCore A} : Core A := λ x, from_option id x (pcore x).
Arguments core' _ _ _ /.

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(** * CMRAs with a unit element *)
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(** We use the notation ∅ because for most instances (maps, sets, etc) the
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`empty' element is the unit. *)
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Record UCMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, Empty A} := {
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  mixin_ucmra_unit_valid :  ;
  mixin_ucmra_unit_left_id : LeftId ()  ();
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  mixin_ucmra_unit_timeless : Timeless ;
  mixin_ucmra_pcore_unit : pcore   Some 
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}.
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Structure ucmraT := UCMRAT' {
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  ucmra_car :> Type;
  ucmra_equiv : Equiv ucmra_car;
  ucmra_dist : Dist ucmra_car;
  ucmra_compl : Compl ucmra_car;
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  ucmra_pcore : PCore ucmra_car;
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  ucmra_op : Op ucmra_car;
  ucmra_valid : Valid ucmra_car;
  ucmra_validN : ValidN ucmra_car;
  ucmra_empty : Empty ucmra_car;
  ucmra_cofe_mixin : CofeMixin ucmra_car;
  ucmra_cmra_mixin : CMRAMixin ucmra_car;
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  ucmra_mixin : UCMRAMixin ucmra_car;
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  _ : Type;
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}.
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Arguments UCMRAT' _ {_ _ _ _ _ _ _ _} _ _ _ _.
Notation UCMRAT A m m' m'' := (UCMRAT' A m m' m'' A).
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Arguments ucmra_car : simpl never.
Arguments ucmra_equiv : simpl never.
Arguments ucmra_dist : simpl never.
Arguments ucmra_compl : simpl never.
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Arguments ucmra_pcore : simpl never.
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Arguments ucmra_op : simpl never.
Arguments ucmra_valid : simpl never.
Arguments ucmra_validN : simpl never.
Arguments ucmra_cofe_mixin : simpl never.
Arguments ucmra_cmra_mixin : simpl never.
Arguments ucmra_mixin : simpl never.
Add Printing Constructor ucmraT.
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Hint Extern 0 (Empty _) => eapply (@ucmra_empty _) : typeclass_instances.
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Coercion ucmra_cofeC (A : ucmraT) : cofeT := CofeT A (ucmra_cofe_mixin A).
Canonical Structure ucmra_cofeC.
Coercion ucmra_cmraR (A : ucmraT) : cmraT :=
  CMRAT A (ucmra_cofe_mixin A) (ucmra_cmra_mixin A).
Canonical Structure ucmra_cmraR.

(** Lifting properties from the mixin *)
Section ucmra_mixin.
  Context {A : ucmraT}.
  Implicit Types x y : A.
  Lemma ucmra_unit_valid :  ( : A).
  Proof. apply (mixin_ucmra_unit_valid _ (ucmra_mixin A)). Qed.
  Global Instance ucmra_unit_left_id : LeftId ()  (@op A _).
  Proof. apply (mixin_ucmra_unit_left_id _ (ucmra_mixin A)). Qed.
  Global Instance ucmra_unit_timeless : Timeless ( : A).
  Proof. apply (mixin_ucmra_unit_timeless _ (ucmra_mixin A)). Qed.
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  Lemma ucmra_pcore_unit : pcore (:A)  Some .
  Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin A)). Qed.
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End ucmra_mixin.
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(** * Discrete CMRAs *)
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Class CMRADiscrete (A : cmraT) := {
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  cmra_discrete :> Discrete A;
  cmra_discrete_valid (x : A) : {0} x   x
}.

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(** * Morphisms *)
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Class CMRAMonotone {A B : cmraT} (f : A  B) := {
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  cmra_monotone_ne n :> Proper (dist n ==> dist n) f;
  validN_preserving n x : {n} x  {n} f x;
  included_preserving x y : x  y  f x  f y
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}.
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Arguments validN_preserving {_ _} _ {_} _ _ _.
Arguments included_preserving {_ _} _ {_} _ _ _.
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(** * Properties **)
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Section cmra.
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Context {A : cmraT}.
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Implicit Types x y z : A.
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Implicit Types xs ys zs : list A.
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(** ** Setoids *)
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Global Instance cmra_pcore_ne' n : Proper (dist n ==> dist n) (@pcore A _).
Proof.
  intros x y Hxy. destruct (pcore x) as [cx|] eqn:?.
  { destruct (cmra_pcore_ne n x y cx) as (cy&->&->); auto. }
  destruct (pcore y) as [cy|] eqn:?; auto.
  destruct (cmra_pcore_ne n y x cy) as (cx&?&->); simplify_eq/=; auto.
Qed.
Lemma cmra_pcore_proper x y cx :
  x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy.
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Proof.
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  intros. destruct (cmra_pcore_ne 0 x y cx) as (cy&?&?); auto.
  exists cy; split; [done|apply equiv_dist=> n].
  destruct (cmra_pcore_ne n x y cx) as (cy'&?&?); naive_solver.
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Qed.
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Global Instance cmra_pcore_proper' : Proper (() ==> ()) (@pcore A _).
Proof. apply (ne_proper _). Qed.
Global Instance cmra_op_ne' n : Proper (dist n ==> dist n ==> dist n) (@op A _).
Proof. intros x1 x2 Hx y1 y2 Hy. by rewrite Hy (comm _ x1) Hx (comm _ y2). Qed.
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Global Instance ra_op_proper' : Proper (() ==> () ==> ()) (@op A _).
Proof. apply (ne_proper_2 _). Qed.
Global Instance cmra_validN_ne' : Proper (dist n ==> iff) (@validN A _ n) | 1.
Proof. by split; apply cmra_validN_ne. Qed.
Global Instance cmra_validN_proper : Proper (() ==> iff) (@validN A _ n) | 1.
Proof. by intros n x1 x2 Hx; apply cmra_validN_ne', equiv_dist. Qed.

Global Instance cmra_valid_proper : Proper (() ==> iff) (@valid A _).
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Proof.
  intros x y Hxy; rewrite !cmra_valid_validN.
  by split=> ? n; [rewrite -Hxy|rewrite Hxy].
Qed.
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Global Instance cmra_includedN_ne n :
  Proper (dist n ==> dist n ==> iff) (@includedN A _ _ n) | 1.
Proof.
  intros x x' Hx y y' Hy.
  by split; intros [z ?]; exists z; [rewrite -Hx -Hy|rewrite Hx Hy].
Qed.
Global Instance cmra_includedN_proper n :
  Proper (() ==> () ==> iff) (@includedN A _ _ n) | 1.
Proof.
  intros x x' Hx y y' Hy; revert Hx Hy; rewrite !equiv_dist=> Hx Hy.
  by rewrite (Hx n) (Hy n).
Qed.
Global Instance cmra_included_proper :
  Proper (() ==> () ==> iff) (@included A _ _) | 1.
Proof.
  intros x x' Hx y y' Hy.
  by split; intros [z ?]; exists z; [rewrite -Hx -Hy|rewrite Hx Hy].
Qed.
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Global Instance cmra_opM_ne n : Proper (dist n ==> dist n ==> dist n) (@opM A).
Proof. destruct 2; by cofe_subst. Qed.
Global Instance cmra_opM_proper : Proper (() ==> () ==> ()) (@opM A).
Proof. destruct 2; by setoid_subst. Qed.
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(** ** Op *)
Lemma cmra_opM_assoc x y mz : (x  y) ? mz  x  (y ? mz).
Proof. destruct mz; by rewrite /= -?assoc. Qed.

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(** ** Validity *)
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Lemma cmra_validN_le n n' x : {n} x  n'  n  {n'} x.
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Proof. induction 2; eauto using cmra_validN_S. Qed.
Lemma cmra_valid_op_l x y :  (x  y)   x.
Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_l. Qed.
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Lemma cmra_validN_op_r n x y : {n} (x  y)  {n} y.
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Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed.
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Lemma cmra_valid_op_r x y :  (x  y)   y.
Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_r. Qed.

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(** ** Core *)
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Lemma cmra_pcore_l' x cx : pcore x  Some cx  cx  x  x.
Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_l. Qed.
Lemma cmra_pcore_r x cx : pcore x = Some cx  x  cx  x.
Proof. intros. rewrite comm. by apply cmra_pcore_l. Qed. 
Lemma cmra_pcore_r' x cx : pcore x  Some cx  x  cx  x.
Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_r. Qed. 
Lemma cmra_pcore_idemp' x cx : pcore x  Some cx  pcore cx  Some cx.
Proof. intros (cx'&?&->)%equiv_Some_inv_r'. eauto using cmra_pcore_idemp. Qed. 
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Lemma cmra_pcore_dup x cx : pcore x = Some cx  cx  cx  cx.
Proof. intros; symmetry; eauto using cmra_pcore_r', cmra_pcore_idemp. Qed.
Lemma cmra_pcore_dup' x cx : pcore x  Some cx  cx  cx  cx.
Proof. intros; symmetry; eauto using cmra_pcore_r', cmra_pcore_idemp'. Qed.
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Lemma cmra_pcore_validN n x cx : {n} x  pcore x = Some cx  {n} cx.
Proof.
  intros Hvx Hx%cmra_pcore_l. move: Hvx; rewrite -Hx. apply cmra_validN_op_l.
Qed.
Lemma cmra_pcore_valid x cx :  x  pcore x = Some cx   cx.
Proof.
  intros Hv Hx%cmra_pcore_l. move: Hv; rewrite -Hx. apply cmra_valid_op_l.
Qed.
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(** ** Persistent elements *)
Lemma persistent_dup x `{!Persistent x} : x  x  x.
Proof. by apply cmra_pcore_dup' with x. Qed.

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(** ** Exclusive elements *)
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Lemma exclusiveN_l n x `{!Exclusive x} y : {n} (x  y)  False.
Proof. intros ?%cmra_validN_le%exclusive0_l; auto with arith. Qed.
Lemma exclusiveN_r n x `{!Exclusive x} y : {n} (y  x)  False.
Proof. rewrite comm. by apply exclusiveN_l. Qed.
Lemma exclusive_l x `{!Exclusive x} y :  (x  y)  False.
Proof. by move /cmra_valid_validN /(_ 0) /exclusive0_l. Qed.
Lemma exclusive_r x `{!Exclusive x} y :  (y  x)  False.
Proof. rewrite comm. by apply exclusive_l. Qed.
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Lemma exclusiveN_opM n x `{!Exclusive x} my : {n} (x ? my)  my = None.
Proof. destruct my. move=> /(exclusiveN_l _ x) []. done. Qed.
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(** ** Order *)
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Lemma cmra_included_includedN n x y : x  y  x {n} y.
Proof. intros [z ->]. by exists z. Qed.
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Global Instance cmra_includedN_trans n : Transitive (@includedN A _ _ n).
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Proof.
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  intros x y z [z1 Hy] [z2 Hz]; exists (z1  z2). by rewrite assoc -Hy -Hz.
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Qed.
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Global Instance cmra_included_trans: Transitive (@included A _ _).
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Proof.
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  intros x y z [z1 Hy] [z2 Hz]; exists (z1  z2). by rewrite assoc -Hy -Hz.
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Qed.
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Lemma cmra_validN_includedN n x y : {n} y  x {n} y  {n} x.
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Proof. intros Hyv [z ?]; cofe_subst y; eauto using cmra_validN_op_l. Qed.
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Lemma cmra_validN_included n x y : {n} y  x  y  {n} x.
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Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_validN_op_l. Qed.
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Lemma cmra_includedN_S n x y : x {S n} y  x {n} y.
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Proof. by intros [z Hz]; exists z; apply dist_S. Qed.
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Lemma cmra_includedN_le n n' x y : x {n} y  n'  n  x {n'} y.
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Proof. induction 2; auto using cmra_includedN_S. Qed.

Lemma cmra_includedN_l n x y : x {n} x  y.
Proof. by exists y. Qed.
Lemma cmra_included_l x y : x  x  y.
Proof. by exists y. Qed.
Lemma cmra_includedN_r n x y : y {n} x  y.
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Proof. rewrite (comm op); apply cmra_includedN_l. Qed.
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Lemma cmra_included_r x y : y  x  y.
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Proof. rewrite (comm op); apply cmra_included_l. Qed.
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Lemma cmra_pcore_preserving' x y cx :
  x  y  pcore x  Some cx   cy, pcore y = Some cy  cx  cy.
Proof.
  intros ? (cx'&?&Hcx)%equiv_Some_inv_r'.
  destruct (cmra_pcore_preserving x y cx') as (cy&->&?); auto.
  exists cy; by rewrite Hcx.
Qed.
Lemma cmra_pcore_preservingN' n x y cx :
  x {n} y  pcore x {n} Some cx   cy, pcore y = Some cy  cx {n} cy.
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Proof.
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  intros [z Hy] (cx'&?&Hcx)%dist_Some_inv_r'.
  destruct (cmra_pcore_preserving x (x  z) cx')
    as (cy&Hxy&?); auto using cmra_included_l.
  assert (pcore y {n} Some cy) as (cy'&?&Hcy')%dist_Some_inv_r'.
  { by rewrite Hy Hxy. }
  exists cy'; split; first done.
  rewrite Hcx -Hcy'; auto using cmra_included_includedN.
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Qed.
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Lemma cmra_included_pcore x cx : pcore x = Some cx  cx  x.
Proof. exists x. by rewrite cmra_pcore_l. Qed.
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Lemma cmra_preservingN_l n x y z : x {n} y  z  x {n} z  y.
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Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed.
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Lemma cmra_preserving_l x y z : x  y  z  x  z  y.
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Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed.
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Lemma cmra_preservingN_r n x y z : x {n} y  x  z {n} y  z.
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Proof. by intros; rewrite -!(comm _ z); apply cmra_preservingN_l. Qed.
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Lemma cmra_preserving_r x y z : x  y  x  z  y  z.
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Proof. by intros; rewrite -!(comm _ z); apply cmra_preserving_l. Qed.
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Lemma cmra_included_dist_l n x1 x2 x1' :
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  x1  x2  x1' {n} x1   x2', x1'  x2'  x2' {n} x2.
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Proof.
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  intros [z Hx2] Hx1; exists (x1'  z); split; auto using cmra_included_l.
  by rewrite Hx1 Hx2.
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Qed.
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(** ** Total core *)
Section total_core.
  Context `{CMRATotal A}.

  Lemma cmra_core_l x : core x  x  x.
  Proof.
    destruct (cmra_total x) as [cx Hcx]. by rewrite /core /= Hcx cmra_pcore_l.
  Qed.
  Lemma cmra_core_idemp x : core (core x)  core x.
  Proof.
    destruct (cmra_total x) as [cx Hcx]. by rewrite /core /= Hcx cmra_pcore_idemp.
  Qed.
  Lemma cmra_core_preserving x y : x  y  core x  core y.
  Proof.
    intros; destruct (cmra_total x) as [cx Hcx].
    destruct (cmra_pcore_preserving x y cx) as (cy&Hcy&?); auto.
    by rewrite /core /= Hcx Hcy.
  Qed.

  Global Instance cmra_core_ne n : Proper (dist n ==> dist n) (@core A _).
  Proof.
    intros x y Hxy. destruct (cmra_total x) as [cx Hcx].
    by rewrite /core /= -Hxy Hcx.
  Qed.
  Global Instance cmra_core_proper : Proper (() ==> ()) (@core A _).
  Proof. apply (ne_proper _). Qed.

  Lemma cmra_core_r x : x  core x  x.
  Proof. by rewrite (comm _ x) cmra_core_l. Qed.
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  Lemma cmra_core_dup x : core x  core x  core x.
  Proof. by rewrite -{3}(cmra_core_idemp x) cmra_core_r. Qed.
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  Lemma cmra_core_validN n x : {n} x  {n} core x.
  Proof. rewrite -{1}(cmra_core_l x); apply cmra_validN_op_l. Qed.
  Lemma cmra_core_valid x :  x   core x.
  Proof. rewrite -{1}(cmra_core_l x); apply cmra_valid_op_l. Qed.

  Lemma persistent_total x : Persistent x  core x  x.
  Proof.
    split; [intros; by rewrite /core /= (persistent x)|].
    rewrite /Persistent /core /=.
    destruct (cmra_total x) as [? ->]. by constructor.
  Qed.
  Lemma persistent_core x `{!Persistent x} : core x  x.
  Proof. by apply persistent_total. Qed.

  Global Instance cmra_core_persistent x : Persistent (core x).
  Proof.
    destruct (cmra_total x) as [cx Hcx].
    rewrite /Persistent /core /= Hcx /=. eauto using cmra_pcore_idemp.
  Qed.

  Lemma cmra_included_core x : core x  x.
  Proof. by exists x; rewrite cmra_core_l. Qed.
  Global Instance cmra_includedN_preorder n : PreOrder (@includedN A _ _ n).
  Proof.
    split; [|apply _]. by intros x; exists (core x); rewrite cmra_core_r.
  Qed.
  Global Instance cmra_included_preorder : PreOrder (@included A _ _).
  Proof.
    split; [|apply _]. by intros x; exists (core x); rewrite cmra_core_r.
  Qed.
  Lemma cmra_core_preservingN n x y : x {n} y  core x {n} core y.
  Proof.
    intros [z ->].
    apply cmra_included_includedN, cmra_core_preserving, cmra_included_l.
  Qed.
End total_core.

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(** ** Timeless *)
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Lemma cmra_timeless_included_l x y : Timeless x  {0} y  x {0} y  x  y.
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Proof.
  intros ?? [x' ?].
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  destruct (cmra_extend 0 y x x') as ([z z']&Hy&Hz&Hz'); auto; simpl in *.
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  by exists z'; rewrite Hy (timeless x z).
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Qed.
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Lemma cmra_timeless_included_r n x y : Timeless y  x {0} y  x {n} y.
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Proof. intros ? [x' ?]. exists x'. by apply equiv_dist, (timeless y). Qed.
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Lemma cmra_op_timeless x1 x2 :
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   (x1  x2)  Timeless x1  Timeless x2  Timeless (x1  x2).
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Proof.
  intros ??? z Hz.
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  destruct (cmra_extend 0 z x1 x2) as ([y1 y2]&Hz'&?&?); auto; simpl in *.
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  { rewrite -?Hz. by apply cmra_valid_validN. }
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  by rewrite Hz' (timeless x1 y1) // (timeless x2 y2).
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Qed.
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(** ** Discrete *)
Lemma cmra_discrete_valid_iff `{CMRADiscrete A} n x :  x  {n} x.
Proof.
  split; first by rewrite cmra_valid_validN.
  eauto using cmra_discrete_valid, cmra_validN_le with lia.
Qed.
Lemma cmra_discrete_included_iff `{Discrete A} n x y : x  y  x {n} y.
Proof.
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  split; first by apply cmra_included_includedN.
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  intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l.
Qed.
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End cmra.

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(** * Properties about CMRAs with a unit element **)
Section ucmra.
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  Context {A : ucmraT}.
  Implicit Types x y z : A.

  Global Instance ucmra_unit_inhabited : Inhabited A := populate .

  Lemma ucmra_unit_validN n : {n} (:A).
  Proof. apply cmra_valid_validN, ucmra_unit_valid. Qed.
  Lemma ucmra_unit_leastN n x :  {n} x.
  Proof. by exists x; rewrite left_id. Qed.
  Lemma ucmra_unit_least x :   x.
  Proof. by exists x; rewrite left_id. Qed.
  Global Instance ucmra_unit_right_id : RightId ()  (@op A _).
  Proof. by intros x; rewrite (comm op) left_id. Qed.
  Global Instance ucmra_unit_persistent : Persistent (:A).
  Proof. apply ucmra_pcore_unit. Qed.

  Global Instance cmra_unit_total : CMRATotal A.
  Proof.
    intros x. destruct (cmra_pcore_preserving'  x ) as (cx&->&?);
      eauto using ucmra_unit_least, (persistent ).
  Qed.
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End ucmra.
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Hint Immediate cmra_unit_total.

(** * Constructing a CMRA with total core *)
Section cmra_total.
  Context A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A}.
  Context (total :  x, is_Some (pcore x)).
  Context (op_ne :  n (x : A), Proper (dist n ==> dist n) (op x)).
  Context (core_ne :  n, Proper (dist n ==> dist n) (@core A _)).
  Context (validN_ne :  n, Proper (dist n ==> impl) (@validN A _ n)).
  Context (valid_validN :  (x : A),  x   n, {n} x).
  Context (validN_S :  n (x : A), {S n} x  {n} x).
  Context (op_assoc : Assoc () (@op A _)).
  Context (op_comm : Comm () (@op A _)).
  Context (core_l :  x : A, core x  x  x).
  Context (core_idemp :  x : A, core (core x)  core x).
  Context (core_preserving :  x y : A, x  y  core x  core y).
  Context (validN_op_l :  n (x y : A), {n} (x  y)  {n} x).
  Context (extend :  n (x y1 y2 : A),
    {n} x  x {n} y1  y2 
    { z | x  z.1  z.2  z.1 {n} y1  z.2 {n} y2 }).
  Lemma cmra_total_mixin : CMRAMixin A.
  Proof.
    split; auto.
    - intros n x y ? Hcx%core_ne Hx; move: Hcx. rewrite /core /= Hx /=.
      case (total y)=> [cy ->]; eauto.
    - intros x cx Hcx. move: (core_l x). by rewrite /core /= Hcx.
    - intros x cx Hcx. move: (core_idemp x). rewrite /core /= Hcx /=.
      case (total cx)=>[ccx ->]; by constructor.
    - intros x y cx Hxy%core_preserving Hx. move: Hxy.
      rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto.
  Qed.
End cmra_total.
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(** * Properties about monotone functions *)
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Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A).
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Proof. repeat split; by try apply _. Qed.
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Instance cmra_monotone_compose {A B C : cmraT} (f : A  B) (g : B  C) :
  CMRAMonotone f  CMRAMonotone g  CMRAMonotone (g  f).
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Proof.
  split.
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  - apply _. 
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  - move=> n x Hx /=. by apply validN_preserving, validN_preserving.
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  - move=> x y Hxy /=. by apply included_preserving, included_preserving.
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Qed.
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Section cmra_monotone.
  Context {A B : cmraT} (f : A  B) `{!CMRAMonotone f}.
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  Global Instance cmra_monotone_proper : Proper (() ==> ()) f := ne_proper _.
  Lemma includedN_preserving n x y : x {n} y  f x {n} f y.
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  Proof.
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    intros [z ->].
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    apply cmra_included_includedN, (included_preserving f), cmra_included_l.
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  Qed.
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  Lemma valid_preserving x :  x   f x.
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  Proof. rewrite !cmra_valid_validN; eauto using validN_preserving. Qed.
End cmra_monotone.

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(** Functors *)
Structure rFunctor := RFunctor {
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  rFunctor_car : cofeT  cofeT  cmraT;
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  rFunctor_map {A1 A2 B1 B2} :
    ((A2 -n> A1) * (B1 -n> B2))  rFunctor_car A1 B1 -n> rFunctor_car A2 B2;
  rFunctor_ne A1 A2 B1 B2 n :
    Proper (dist n ==> dist n) (@rFunctor_map A1 A2 B1 B2);
  rFunctor_id {A B} (x : rFunctor_car A B) : rFunctor_map (cid,cid) x  x;
  rFunctor_compose {A1 A2 A3 B1 B2 B3}
      (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x :
    rFunctor_map (fg, g'f') x  rFunctor_map (g,g') (rFunctor_map (f,f') x);
  rFunctor_mono {A1 A2 B1 B2} (fg : (A2 -n> A1) * (B1 -n> B2)) :
    CMRAMonotone (rFunctor_map fg) 
}.
Existing Instances rFunctor_ne rFunctor_mono.
Instance: Params (@rFunctor_map) 5.

Class rFunctorContractive (F : rFunctor) :=
  rFunctor_contractive A1 A2 B1 B2 :> Contractive (@rFunctor_map F A1 A2 B1 B2).

Definition rFunctor_diag (F: rFunctor) (A: cofeT) : cmraT := rFunctor_car F A A.
Coercion rFunctor_diag : rFunctor >-> Funclass.

Program Definition constRF (B : cmraT) : rFunctor :=
  {| rFunctor_car A1 A2 := B; rFunctor_map A1 A2 B1 B2 f := cid |}.
Solve Obligations with done.

Instance constRF_contractive B : rFunctorContractive (constRF B).
Proof. rewrite /rFunctorContractive; apply _. Qed.

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Structure urFunctor := URFunctor {
  urFunctor_car : cofeT  cofeT  ucmraT;
  urFunctor_map {A1 A2 B1 B2} :
    ((A2 -n> A1) * (B1 -n> B2))  urFunctor_car A1 B1 -n> urFunctor_car A2 B2;
  urFunctor_ne A1 A2 B1 B2 n :
    Proper (dist n ==> dist n) (@urFunctor_map A1 A2 B1 B2);
  urFunctor_id {A B} (x : urFunctor_car A B) : urFunctor_map (cid,cid) x  x;
  urFunctor_compose {A1 A2 A3 B1 B2 B3}
      (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x :
    urFunctor_map (fg, g'f') x  urFunctor_map (g,g') (urFunctor_map (f,f') x);
  urFunctor_mono {A1 A2 B1 B2} (fg : (A2 -n> A1) * (B1 -n> B2)) :
    CMRAMonotone (urFunctor_map fg) 
}.
Existing Instances urFunctor_ne urFunctor_mono.
Instance: Params (@urFunctor_map) 5.

Class urFunctorContractive (F : urFunctor) :=
  urFunctor_contractive A1 A2 B1 B2 :> Contractive (@urFunctor_map F A1 A2 B1 B2).

Definition urFunctor_diag (F: urFunctor) (A: cofeT) : ucmraT := urFunctor_car F A A.
Coercion urFunctor_diag : urFunctor >-> Funclass.

Program Definition constURF (B : ucmraT) : urFunctor :=
  {| urFunctor_car A1 A2 := B; urFunctor_map A1 A2 B1 B2 f := cid |}.
Solve Obligations with done.

Instance constURF_contractive B : urFunctorContractive (constURF B).
Proof. rewrite /urFunctorContractive; apply _. Qed.

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(** * Transporting a CMRA equality *)
Definition cmra_transport {A B : cmraT} (H : A = B) (x : A) : B :=
  eq_rect A id x _ H.

Section cmra_transport.
  Context {A B : cmraT} (H : A = B).
  Notation T := (cmra_transport H).
  Global Instance cmra_transport_ne n : Proper (dist n ==> dist n) T.
  Proof. by intros ???; destruct H. Qed.
  Global Instance cmra_transport_proper : Proper (() ==> ()) T.
  Proof. by intros ???; destruct H. Qed.
  Lemma cmra_transport_op x y : T (x  y) = T x  T y.
  Proof. by destruct H. Qed.
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  Lemma cmra_transport_core x : T (core x) = core (T x).
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  Proof. by destruct H. Qed.
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  Lemma cmra_transport_validN n x : {n} T x  {n} x.
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  Proof. by destruct H. Qed.
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  Lemma cmra_transport_valid x :  T x   x.
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  Proof. by destruct H. Qed.
  Global Instance cmra_transport_timeless x : Timeless x  Timeless (T x).
  Proof. by destruct H. Qed.
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  Global Instance cmra_transport_persistent x : Persistent x  Persistent (T x).
  Proof. by destruct H. Qed.
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End cmra_transport.

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(** * Instances *)
(** ** Discrete CMRA *)
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Record RAMixin A `{Equiv A, PCore A, Op A, Valid A} := {
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  (* setoids *)
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  ra_op_proper (x : A) : Proper (() ==> ()) (op x);
  ra_core_proper x y cx :
    x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy;
  ra_validN_proper : Proper (() ==> impl) valid;
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  (* monoid *)
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  ra_assoc : Assoc () ();
  ra_comm : Comm () ();
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  ra_pcore_l x cx : pcore x = Some cx  cx  x  x;
  ra_pcore_idemp x cx : pcore x = Some cx  pcore cx  Some cx;
  ra_pcore_preserving x y cx :
    x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy;
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  ra_valid_op_l x y :  (x  y)   x
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}.

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Section discrete.
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  Context `{Equiv A, PCore A, Op A, Valid A, @Equivalence A ()}.
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  Context (ra_mix : RAMixin A).
  Existing Instances discrete_dist discrete_compl.
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  Instance discrete_validN : ValidN A := λ n x,  x.
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  Definition discrete_cmra_mixin : CMRAMixin A.
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  Proof.
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    destruct ra_mix; split; try done.
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    - intros x; split; first done. by move=> /(_ 0).
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    - intros n x y1 y2 ??; by exists (y1,y2).
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  Qed.
End discrete.

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Notation discreteR A ra_mix :=
  (CMRAT A discrete_cofe_mixin (discrete_cmra_mixin ra_mix)).
Notation discreteLeibnizR A ra_mix :=
  (CMRAT A (@discrete_cofe_mixin _ equivL _) (discrete_cmra_mixin ra_mix)).

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Global Instance discrete_cmra_discrete `{Equiv A, PCore A, Op A, Valid A,
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  @Equivalence A ()} (ra_mix : RAMixin A) : CMRADiscrete (discreteR A ra_mix).
Proof. split. apply _. done. Qed.

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Section ra_total.
  Context A `{Equiv A, PCore A, Op A, Valid A}.
  Context (total :  x, is_Some (pcore x)).
  Context (op_proper :  (x : A), Proper (() ==> ()) (op x)).
  Context (core_proper: Proper (() ==> ()) (@core A _)).
  Context (valid_proper : Proper (() ==> impl) (@valid A _)).
  Context (op_assoc : Assoc () (@op A _)).
  Context (op_comm : Comm () (@op A _)).
  Context (core_l :  x : A, core x  x  x).
  Context (core_idemp :  x : A, core (core x)  core x).
  Context (core_preserving :  x y : A, x  y  core x  core y).
  Context (valid_op_l :  x y : A,  (x  y)   x).
  Lemma ra_total_mixin : RAMixin A.
  Proof.
    split; auto.
    - intros x y ? Hcx%core_proper Hx; move: Hcx. rewrite /core /= Hx /=.
      case (total y)=> [cy ->]; eauto.
    - intros x cx Hcx. move: (core_l x). by rewrite /core /= Hcx.
    - intros x cx Hcx. move: (core_idemp x). rewrite /core /= Hcx /=.
      case (total cx)=>[ccx ->]; by constructor.
    - intros x y cx Hxy%core_preserving Hx. move: Hxy.
      rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto.
  Qed.
End ra_total.

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(** ** CMRA for the unit type *)
Section unit.
  Instance unit_valid : Valid () := λ x, True.
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  Instance unit_validN : ValidN () := λ n x, True.
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  Instance unit_pcore : PCore () := λ x, Some x.
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  Instance unit_op : Op () := λ x y, ().
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  Lemma unit_cmra_mixin : CMRAMixin ().
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  Proof. apply discrete_cmra_mixin, ra_total_mixin; by eauto. Qed.
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  Canonical Structure unitR : cmraT := CMRAT () unit_cofe_mixin unit_cmra_mixin.
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  Instance unit_empty : Empty () := ().
  Lemma unit_ucmra_mixin : UCMRAMixin ().
  Proof. done. Qed.
  Canonical Structure unitUR : ucmraT :=
    UCMRAT () unit_cofe_mixin unit_cmra_mixin unit_ucmra_mixin.

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  Global Instance unit_cmra_discrete : CMRADiscrete unitR.
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  Proof. done. Qed.
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  Global Instance unit_persistent (x : ()) : Persistent x.
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  Proof. by constructor. Qed.
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End unit.
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(** ** Natural numbers *)
Section nat.
  Instance nat_valid : Valid nat := λ x, True.
  Instance nat_validN : ValidN nat := λ n x, True.
  Instance nat_pcore : PCore nat := λ x, Some 0.
  Instance nat_op : Op nat := plus.
  Lemma nat_included (x y : nat) : x  y  x  y.
  Proof.
    split.
    - intros [z ->]; unfold op, nat_op; lia.
    - exists (y - x). by apply le_plus_minus.
  Qed.
  Lemma nat_cmra_mixin : CMRAMixin nat.
  Proof.
    apply discrete_cmra_mixin, ra_total_mixin; try by eauto.
    - solve_proper.
    - intros x y z. apply Nat.add_assoc.
    - intros x y. apply Nat.add_comm.
    - by exists 0.
  Qed.
  Canonical Structure natR : cmraT :=
    CMRAT nat (@discrete_cofe_mixin _ equivL _) nat_cmra_mixin.

  Instance nat_empty : Empty nat := 0.
  Lemma nat_ucmra_mixin : UCMRAMixin nat.
  Proof. split; apply _ || done. Qed.
  Canonical Structure natUR : ucmraT :=
    UCMRAT nat (@discrete_cofe_mixin _ equivL _) nat_cmra_mixin nat_ucmra_mixin.

  Global Instance nat_cmra_discrete : CMRADiscrete natR.
  Proof. constructor; apply _ || done. Qed.
  Global Instance nat_persistent (x : ()) : Persistent x.
  Proof. by constructor. Qed.
End nat.

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(** ** Product *)
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Section prod.
  Context {A B : cmraT}.
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  Local Arguments pcore _ _ !_ /.
  Local Arguments cmra_pcore _ !_/.

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  Instance prod_op : Op (A * B) := λ x y, (x.1  y.1, x.2  y.2).
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  Instance prod_pcore : PCore (A * B) := λ x,
    c1  pcore (x.1); c2  pcore (x.2); Some (c1, c2).
  Arguments prod_pcore !_ /.
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  Instance prod_valid : Valid (A * B) := λ x,  x.1   x.2.
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  Instance prod_validN : ValidN (A * B) := λ n x, {n} x.1  {n} x.2.
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  Lemma prod_pcore_Some (x cx : A * B) :
    pcore x = Some cx  pcore (x.1) = Some (cx.1)  pcore (x.2) = Some (cx.2).
  Proof. destruct x, cx; by intuition simplify_option_eq. Qed.
  Lemma prod_pcore_Some' (x cx : A * B) :
    pcore x  Some cx  pcore (x.1)  Some (cx.1)  pcore (x.2)  Some (cx.2).
  Proof.
    split; [by intros (cx'&[-> ->]%prod_pcore_Some&->)%equiv_Some_inv_r'|].
    rewrite {3}/pcore /prod_pcore. (* TODO: use setoid rewrite *)
    intros [Hx1 Hx2]; inversion_clear Hx1; simpl; inversion_clear Hx2.
    by constructor.
  Qed.

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  Lemma prod_included (x y : A * B) : x  y  x.1  y.1  x.2  y.2.
  Proof.
    split; [intros [z Hz]; split; [exists (z.1)|exists (z.2)]; apply Hz|].
    intros [[z1 Hz1] [z2 Hz2]]; exists (z1,z2); split; auto.
  Qed.
  Lemma prod_includedN (x y : A * B) n : x {n} y  x.1 {n} y.1  x.2 {n} y.2.
  Proof.
    split; [intros [z Hz]; split; [exists (z.1)|exists (z.2)]; apply Hz|].
    intros [[z1 Hz1] [z2 Hz2]]; exists (z1,z2); split; auto.
  Qed.
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  Definition prod_cmra_mixin : CMRAMixin (A * B).
  Proof.
    split; try apply _.
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    - by intros n x y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
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    - intros n x y cx; setoid_rewrite prod_pcore_Some=> -[??] [??].
      destruct (cmra_pcore_ne n (x.1) (y.1) (cx.1)) as (z1&->&?); auto.
      destruct (cmra_pcore_ne n (x.2) (y.2) (cx.2)) as (z2&->&?); auto.
      exists (z1,z2); repeat constructor; auto.
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    - by intros n y1 y2 [Hy1 Hy2] [??]; split; rewrite /= -?Hy1 -?Hy2.
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    - intros x; split.
      + intros [??] n; split; by apply cmra_valid_validN.
      + intros Hxy; split; apply cmra_valid_validN=> n; apply Hxy.
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    - by intros n x [??]; split; apply cmra_validN_S.
    - by split; rewrite /= assoc.
    - by split; rewrite /= comm.
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    - intros x y [??]%prod_pcore_Some;
        constructor; simpl; eauto using cmra_pcore_l.
    - intros x y; rewrite prod_pcore_Some prod_pcore_Some'.
      naive_solver eauto using cmra_pcore_idemp.
    - intros x y cx; rewrite prod_included prod_pcore_Some=> -[??] [??].
      destruct (cmra_pcore_preserving (x.1) (y.1) (cx.1)) as (z1&?&?); auto.
      destruct (cmra_pcore_preserving (x.2) (y.2) (cx.2)) as (z2&?&?); auto.
      exists (z1,z2). by rewrite prod_included prod_pcore_Some.
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    - intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l.
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    - intros n x y1 y2 [??] [??]; simpl in *.
      destruct (cmra_extend n (x.1) (y1.1) (y2.1)) as (z1&?&?&?); auto.
      destruct (cmra_extend n (x.2) (y1.2) (y2.2)) as (z2&?&?&?); auto.
      by exists ((z1.1,z2.1),(z1.2,z2.2)).
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  Qed.
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  Canonical Structure prodR :=
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    CMRAT (A * B) prod_cofe_mixin prod_cmra_mixin.
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  Lemma pair_op (a a' : A) (b b' : B) : (a, b)  (a', b') = (a  a', b  b').
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  Proof. done. Qed.

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  Global Instance prod_cmra_total : CMRATotal A  CMRATotal B  CMRATotal prodR.
  Proof.
    intros H1 H2 [a b]. destruct (H1 a) as [ca ?], (H2 b) as [cb ?].
    exists (ca,cb); by simplify_option_eq.
  Qed.

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  Global Instance prod_cmra_discrete :
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    CMRADiscrete A  CMRADiscrete B  CMRADiscrete prodR.
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  Proof. split. apply _. by intros ? []; split; apply cmra_discrete_valid. Qed.

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  Global Instance pair_persistent x y :
    Persistent x  Persistent y  Persistent (x,y).
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  Proof. by rewrite /Persistent prod_pcore_Some'. Qed.
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  Global Instance pair_exclusive_l x y : Exclusive x  Exclusive (x,y).
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  Proof. by intros ?[][?%exclusive0_l]. Qed.
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  Global Instance pair_exclusive_r x y : Exclusive y  Exclusive (x,y).
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  Proof. by intros ?[][??%exclusive0_l]. Qed.
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End prod.
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Arguments prodR : clear implicits.
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Section prod_unit.
  Context {A B : ucmraT}.

  Instance prod_empty `{Empty A, Empty B} : Empty (A * B) := (, ).
  Lemma prod_ucmra_mixin : UCMRAMixin (A * B).
  Proof.
    split.
    - split; apply ucmra_unit_valid.
    - by split; rewrite /=left_id.
    - by intros ? [??]; split; apply (timeless _).
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    - rewrite prod_pcore_Some'; split; apply (persistent _).
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  Qed.
  Canonical Structure prodUR :=
    UCMRAT (A * B) prod_cofe_mixin prod_cmra_mixin prod_ucmra_mixin.

  Lemma pair_split (x : A) (y : B) : (x, y)  (x, )  (, y).
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  Proof. by rewrite pair_op left_id right_id. Qed.
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End prod_unit.

Arguments prodUR : clear implicits.