cmra.v 58.3 KB
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From iris.algebra Require Export ofe.
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Set Default Proof Using "Type".
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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.

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(* The inclusion quantifies over [A], not [option A].  This means we do not get
   reflexivity.  However, if we used [option A], the following would no longer
   hold:
     x ≼ y ↔ x.1 ≼ y.1 ∧ x.2 ≼ y.2
*)
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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 (x : A) : NonExpansive (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;
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  mixin_cmra_pcore_mono x y cx :
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    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 
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     z1 z2, x  z1  z2  z1 {n} y1  z2 {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;
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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_ofe_mixin : OfeMixin 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' _ {_ _ _ _ _ _} _ _ _.
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(* Given [m : CMRAMixin A], the notation [CMRAT A m] provides a smart
constructor, which uses [ofe_mixin_of A] to infer the canonical OFE mixin of
the type [A], so that it does not have to be given manually. *)
Notation CMRAT A m := (CMRAT' A (ofe_mixin_of A%type) m A) (only parsing).

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Arguments cmra_car : simpl never.
Arguments cmra_equiv : simpl never.
Arguments cmra_dist : 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.
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Arguments cmra_ofe_mixin : simpl never.
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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_ofeC (A : cmraT) : ofeT := OfeT A (cmra_ofe_mixin A).
Canonical Structure cmra_ofeC.
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Definition cmra_mixin_of' A {Ac : cmraT} (f : Ac  A) : CMRAMixin Ac := cmra_mixin Ac.
Notation cmra_mixin_of A :=
  ltac:(let H := eval hnf in (cmra_mixin_of' A id) in exact H) (only parsing).

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(** Lifting properties from the mixin *)
Section cmra_mixin.
  Context {A : cmraT}.
  Implicit Types x y : A.
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  Global Instance cmra_op_ne (x : A) : NonExpansive (op x).
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  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.
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  Lemma cmra_pcore_mono x y cx :
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    x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy.
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  Proof. apply (mixin_cmra_pcore_mono _ (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 
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     z1 z2, x  z1  z2  z1 {n} y1  z2 {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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Hint Mode Persistent + ! : typeclass_instances.
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Instance: Params (@Persistent) 1.
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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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Hint Mode Exclusive + ! : typeclass_instances.
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Instance: Params (@Exclusive) 1.
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(** * Cancelable elements. *)
Class Cancelable {A : cmraT} (x : A) :=
  cancelableN n y z : {n}(x  y)  x  y {n} x  z  y {n} z.
Arguments cancelableN {_} _ {_} _ _ _ _.
Hint Mode Cancelable + ! : typeclass_instances.
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Instance: Params (@Cancelable) 1.
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(** * Identity-free elements. *)
Class IdFree {A : cmraT} (x : A) :=
  id_free0_r y : {0}x  x  y {0} x  False.
Arguments id_free0_r {_} _ {_} _ _.
Hint Mode IdFree + ! : typeclass_instances.
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Instance: Params (@IdFree) 1.
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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_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;
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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;
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  ucmra_ofe_mixin : OfeMixin ucmra_car;
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  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' _ {_ _ _ _ _ _ _} _ _ _ _.
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Notation UCMRAT A m :=
  (UCMRAT' A (ofe_mixin_of A%type) (cmra_mixin_of A%type) m A) (only parsing).
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Arguments ucmra_car : simpl never.
Arguments ucmra_equiv : simpl never.
Arguments ucmra_dist : 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.
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Arguments ucmra_ofe_mixin : simpl never.
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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_ofeC (A : ucmraT) : ofeT := OfeT A (ucmra_ofe_mixin A).
Canonical Structure ucmra_ofeC.
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Coercion ucmra_cmraR (A : ucmraT) : cmraT :=
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  CMRAT' A (ucmra_ofe_mixin A) (ucmra_cmra_mixin A) A.
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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.
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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 :> NonExpansive f;
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  cmra_monotone_validN n x : {n} x  {n} f x;
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  cmra_monotone x y : x  y  f x  f y
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}.
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Arguments cmra_monotone_validN {_ _} _ {_} _ _ _.
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Arguments cmra_monotone {_ _} _ {_} _ _ _.
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(* Not all intended homomorphisms preserve validity, in particular it does not
hold for the [ownM] and [own] connectives. *)
Class CMRAHomomorphism {A B : cmraT} (f : A  B) := {
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  cmra_homomorphism_ne :> NonExpansive f;
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  cmra_homomorphism x y : f (x  y)  f x  f y
}.
Arguments cmra_homomorphism {_ _} _ _ _ _.

Class UCMRAHomomorphism {A B : ucmraT} (f : A  B) := {
  ucmra_homomorphism :> CMRAHomomorphism f;
  ucmra_homomorphism_unit : f   
}.
Arguments ucmra_homomorphism_unit {_ _} _ _.

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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' : NonExpansive (@pcore A _).
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Proof.
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  intros n x y Hxy. destruct (pcore x) as [cx|] eqn:?.
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  { 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.
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Global Instance cmra_op_ne' : NonExpansive2 (@op A _).
Proof. intros n x1 x2 Hx y1 y2 Hy. by rewrite Hy (comm _ x1) Hx (comm _ y2). Qed.
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Global Instance cmra_op_proper' : Proper (() ==> () ==> ()) (@op A _).
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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 : NonExpansive2 (@opM A).
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Proof. destruct 2; by ofe_subst. Qed.
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Global Instance cmra_opM_proper : Proper (() ==> () ==> ()) (@opM A).
Proof. destruct 2; by setoid_subst. Qed.
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Global Instance Persistent_proper : Proper (() ==> iff) (@Persistent A).
Proof. solve_proper. Qed.
Global Instance Exclusive_proper : Proper (() ==> iff) (@Exclusive A).
Proof. intros x y Hxy. rewrite /Exclusive. by setoid_rewrite Hxy. Qed.
Global Instance Cancelable_proper : Proper (() ==> iff) (@Cancelable A).
Proof. intros x y Hxy. rewrite /Cancelable. by setoid_rewrite Hxy. Qed.
Global Instance IdFree_proper : Proper (() ==> iff) (@IdFree A).
Proof. intros x y Hxy. rewrite /IdFree. by setoid_rewrite Hxy. 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.
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Proof. intros. rewrite comm. by apply cmra_pcore_l. Qed.
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Lemma cmra_pcore_r' x cx : pcore x  Some cx  x  cx  x.
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Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_r. Qed.
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Lemma cmra_pcore_idemp' x cx : pcore x  Some cx  pcore cx  Some cx.
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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.
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Proof. intros. eapply (exclusive0_l x y), cmra_validN_le; eauto with lia. Qed.
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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.
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Proof. destruct my as [y|]. move=> /(exclusiveN_l _ x) []. done. Qed.
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Lemma exclusive_includedN n x `{!Exclusive x} y : x {n} y  {n} y  False.
Proof. intros [? ->]. by apply exclusiveN_l. Qed.
Lemma exclusive_included x `{!Exclusive x} y : x  y   y  False.
Proof. intros [? ->]. by apply exclusive_l. 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_valid_included x y :  y  x  y   x.
Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_valid_op_l. 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 ?]; ofe_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_mono' x y cx :
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  x  y  pcore x  Some cx   cy, pcore y = Some cy  cx  cy.
Proof.
  intros ? (cx'&?&Hcx)%equiv_Some_inv_r'.
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  destruct (cmra_pcore_mono x y cx') as (cy&->&?); auto.
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  exists cy; by rewrite Hcx.
Qed.
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Lemma cmra_pcore_monoN' n x y cx :
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  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'.
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  destruct (cmra_pcore_mono x (x  z) cx')
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    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_monoN_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_mono_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_monoN_r n x y z : x {n} y  x  z {n} y  z.
Proof. by intros; rewrite -!(comm _ z); apply cmra_monoN_l. Qed.
Lemma cmra_mono_r x y z : x  y  x  z  y  z.
Proof. by intros; rewrite -!(comm _ z); apply cmra_mono_l. Qed.
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Lemma cmra_monoN n x1 x2 y1 y2 : x1 {n} y1  x2 {n} y2  x1  x2 {n} y1  y2.
Proof. intros; etrans; eauto using cmra_monoN_l, cmra_monoN_r. Qed.
Lemma cmra_mono x1 x2 y1 y2 : x1  y1  x2  y2  x1  x2  y1  y2.
Proof. intros; etrans; eauto using cmra_mono_l, cmra_mono_r. Qed.
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Global Instance cmra_monoN' n :
  Proper (includedN n ==> includedN n ==> includedN n) (@op A _).
Proof. intros x1 x2 Hx y1 y2 Hy. by apply cmra_monoN. Qed.
Global Instance cmra_mono' :
  Proper (included ==> included ==> included) (@op A _).
Proof. intros x1 x2 Hx y1 y2 Hy. by apply cmra_mono. 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.
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  Local Set Default Proof Using "Type*".
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  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.
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  Lemma cmra_core_mono x y : x  y  core x  core y.
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  Proof.
    intros; destruct (cmra_total x) as [cx Hcx].
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    destruct (cmra_pcore_mono x y cx) as (cy&Hcy&?); auto.
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    by rewrite /core /= Hcx Hcy.
  Qed.

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  Global Instance cmra_core_ne : NonExpansive (@core A _).
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  Proof.
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    intros n x y Hxy. destruct (cmra_total x) as [cx Hcx].
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    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.
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  Lemma cmra_core_monoN n x y : x {n} y  core x {n} core y.
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  Proof.
    intros [z ->].
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    apply cmra_included_includedN, cmra_core_mono, cmra_included_l.
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  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 x y : Timeless y  x {0} y  x  y.
Proof. intros ? [x' ?]. exists x'. by apply (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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(** Cancelable elements  *)
Global Instance cancelable_proper : Proper (equiv ==> iff) (@Cancelable A).
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Proof. unfold Cancelable. intros x x' EQ. by setoid_rewrite EQ. Qed.
Lemma cancelable x `{!Cancelable x} y z : (x  y)  x  y  x  z  y  z.
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Proof. rewrite !equiv_dist cmra_valid_validN. intros. by apply (cancelableN x). Qed.
Lemma discrete_cancelable x `{CMRADiscrete A}:
  ( y z, (x  y)  x  y  x  z  y  z)  Cancelable x.
Proof. intros ????. rewrite -!timeless_iff -cmra_discrete_valid_iff. auto. Qed.
Global Instance cancelable_op x y :
  Cancelable x  Cancelable y  Cancelable (x  y).
Proof.
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  intros ?? n z z' ??. apply (cancelableN y), (cancelableN x).
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  - eapply cmra_validN_op_r. by rewrite assoc.
  - by rewrite assoc.
  - by rewrite !assoc.
Qed.
Global Instance exclusive_cancelable (x : A) : Exclusive x  Cancelable x.
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Proof. intros ? n z z' []%(exclusiveN_l _ x). Qed.
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(** Id-free elements  *)
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Global Instance id_free_ne n : Proper (dist n ==> iff) (@IdFree A).
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Proof.
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  intros x x' EQ%(dist_le _ 0); last lia. rewrite /IdFree.
  split=> y ?; (rewrite -EQ || rewrite EQ); eauto.
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Qed.
Global Instance id_free_proper : Proper (equiv ==> iff) (@IdFree A).
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Proof. by move=> P Q /equiv_dist /(_ 0)=> ->. Qed.
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Lemma id_freeN_r n n' x `{!IdFree x} y : {n}x  x  y {n'} x  False.
Proof. eauto using cmra_validN_le, dist_le with lia. Qed.
Lemma id_freeN_l n n' x `{!IdFree x} y : {n}x  y  x {n'} x  False.
Proof. rewrite comm. eauto using id_freeN_r. Qed.
Lemma id_free_r x `{!IdFree x} y : x  x  y  x  False.
Proof. move=> /cmra_valid_validN ? /equiv_dist. eauto. Qed.
Lemma id_free_l x `{!IdFree x} y : x  y  x  x  False.
Proof. rewrite comm. eauto using id_free_r. Qed.
Lemma discrete_id_free x `{CMRADiscrete A}:
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  ( y,  x  x  y  x  False)  IdFree x.
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Proof. repeat intro. eauto using cmra_discrete_valid, cmra_discrete, timeless. Qed.
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Global Instance id_free_op_r x y : IdFree y  Cancelable x  IdFree (x  y).
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Proof.
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  intros ?? z ? Hid%symmetry. revert Hid. rewrite -assoc=>/(cancelableN x) ?.
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  eapply (id_free0_r _); [by eapply cmra_validN_op_r |symmetry; eauto].
Qed.
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Global Instance id_free_op_l x y : IdFree x  Cancelable y  IdFree (x  y).
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Proof. intros. rewrite comm. apply _. Qed.
Global Instance exclusive_id_free x : Exclusive x  IdFree x.
Proof. intros ? z ? Hid. apply (exclusiveN_l 0 x z). by rewrite Hid. 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.

  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.
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    intros x. destruct (cmra_pcore_mono'  x ) as (cx&->&?);
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      eauto using ucmra_unit_least, (persistent (:A)).
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  Qed.
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  Global Instance empty_cancelable : Cancelable (:A).
  Proof. intros ???. by rewrite !left_id. Qed.
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End ucmra.
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Hint Immediate cmra_unit_total.

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(** * Properties about CMRAs with Leibniz equality *)
Section cmra_leibniz.
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  Local Set Default Proof Using "Type*".
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  Context {A : cmraT} `{!LeibnizEquiv A}.
  Implicit Types x y : A.

  Global Instance cmra_assoc_L : Assoc (=) (@op A _).
  Proof. intros x y z. unfold_leibniz. by rewrite assoc. Qed.
  Global Instance cmra_comm_L : Comm (=) (@op A _).
  Proof. intros x y. unfold_leibniz. by rewrite comm. Qed.

  Lemma cmra_pcore_l_L x cx : pcore x = Some cx  cx  x = x.
  Proof. unfold_leibniz. apply cmra_pcore_l'. Qed.
  Lemma cmra_pcore_idemp_L x cx : pcore x = Some cx  pcore cx = Some cx.
  Proof. unfold_leibniz. apply cmra_pcore_idemp'. Qed.

  Lemma cmra_opM_assoc_L x y mz : (x  y) ? mz = x  (y ? mz).
  Proof. unfold_leibniz. apply cmra_opM_assoc. Qed.

  (** ** Core *)
  Lemma cmra_pcore_r_L x cx : pcore x = Some cx  x  cx = x.
  Proof. unfold_leibniz. apply cmra_pcore_r'. Qed.
  Lemma cmra_pcore_dup_L x cx : pcore x = Some cx  cx = cx  cx.
  Proof. unfold_leibniz. apply cmra_pcore_dup'. Qed.

  (** ** Persistent elements *)
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  Lemma persistent_dup_L x `{!Persistent x} : x = x  x.
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  Proof. unfold_leibniz. by apply persistent_dup. Qed.

  (** ** Total core *)
  Section total_core.
    Context `{CMRATotal A}.

    Lemma cmra_core_r_L x : x  core x = x.
    Proof. unfold_leibniz. apply cmra_core_r. Qed.
    Lemma cmra_core_l_L x : core x  x = x.
    Proof. unfold_leibniz. apply cmra_core_l. Qed.
    Lemma cmra_core_idemp_L x : core (core x) = core x.
    Proof. unfold_leibniz. apply cmra_core_idemp. Qed.
    Lemma cmra_core_dup_L x : core x = core x  core x.
    Proof. unfold_leibniz. apply cmra_core_dup. Qed.
    Lemma persistent_total_L x : Persistent x  core x = x.
    Proof. unfold_leibniz. apply persistent_total. Qed.
    Lemma persistent_core_L x `{!Persistent x} : core x = x.
    Proof. by apply persistent_total_L. Qed.
  End total_core.
End cmra_leibniz.

Section ucmra_leibniz.
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  Local Set Default Proof Using "Type*".
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  Context {A : ucmraT} `{!LeibnizEquiv A}.
  Implicit Types x y z : A.

  Global Instance ucmra_unit_left_id_L : LeftId (=)  (@op A _).
  Proof. intros x. unfold_leibniz. by rewrite left_id. Qed.
  Global Instance ucmra_unit_right_id_L : RightId (=)  (@op A _).
  Proof. intros x. unfold_leibniz. by rewrite right_id. Qed.
End ucmra_leibniz.

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(** * 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)).
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  Context (op_ne :  (x : A), NonExpansive (op x)).
  Context (core_ne : NonExpansive (@core A _)).
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  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).
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  Context (core_mono :  x y : A, x  y  core x  core y).
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  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 
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     z1 z2, x  z1  z2  z1 {n} y1  z2 {n} y2).
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  Lemma cmra_total_mixin : CMRAMixin A.
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  Proof using Type*.
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    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.
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    - intros x y cx Hxy%core_mono Hx. move: Hxy.
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      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 cmra_monotone_validN, cmra_monotone_validN.
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  - move=> x y Hxy /=. by apply cmra_monotone, cmra_monotone.
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Qed.
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Section cmra_monotone.
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  Local Set Default Proof Using "Type*".
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  Context {A B : cmraT} (f : A  B) `{!CMRAMonotone f}.
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  Global Instance cmra_monotone_proper : Proper (() ==> ()) f := ne_proper _.
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  Lemma cmra_monotoneN 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, (cmra_monotone f), cmra_included_l.
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  Qed.
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  Lemma cmra_monotone_valid x :  x   f x.
  Proof. rewrite !cmra_valid_validN; eauto using cmra_monotone_validN. Qed.
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End cmra_monotone.

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Instance cmra_homomorphism_id {A : cmraT} : CMRAHomomorphism (@id A).
Proof. repeat split; by try apply _. Qed.
Instance cmra_homomorphism_compose {A B C : cmraT} (f : A  B) (g : B  C) :
  CMRAHomomorphism f  CMRAHomomorphism g  CMRAHomomorphism (g  f).
Proof.
  split.
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  - apply _.
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  - move=> x y /=. rewrite -(cmra_homomorphism g).
    by apply (ne_proper _), cmra_homomorphism.
Qed.

Instance cmra_homomorphism_proper {A B : cmraT} (f