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From algebra Require Export cofe.
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Class Unit (A : Type) := unit : A  A.
Instance: Params (@unit) 2.

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 Div (A : Type) := div : A  A  A.
Instance: Params (@div) 2.
Infix "÷" := div : C_scope.
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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
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    `{Dist A, Equiv A, Unit A, Op A, Valid A, ValidN A, Div A} := {
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  (* setoids *)
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  mixin_cmra_op_ne n (x : A) : Proper (dist n ==> dist n) (op x);
  mixin_cmra_unit_ne n : Proper (dist n ==> dist n) unit;
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  mixin_cmra_validN_ne n : Proper (dist n ==> impl) (validN n);
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  mixin_cmra_div_ne n : Proper (dist n ==> dist n ==> dist n) div;
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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_unit_l x : unit x  x  x;
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  mixin_cmra_unit_idemp x : unit (unit x)  unit x;
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  mixin_cmra_unit_preserving x y : x  y  unit x  unit y;
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  mixin_cmra_validN_op_l n x y : {n} (x  y)  {n} x;
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  mixin_cmra_op_div x y : x  y  x  y ÷ x  y;
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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 *)
Structure cmraT := CMRAT {
  cmra_car :> Type;
  cmra_equiv : Equiv cmra_car;
  cmra_dist : Dist cmra_car;
  cmra_compl : Compl cmra_car;
  cmra_unit : Unit cmra_car;
  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_div : Div 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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}.
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Arguments CMRAT {_ _ _ _ _ _ _ _ _} _ _.
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Arguments cmra_car : simpl never.
Arguments cmra_equiv : simpl never.
Arguments cmra_dist : simpl never.
Arguments cmra_compl : simpl never.
Arguments cmra_unit : simpl never.
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_div : simpl never.
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Arguments cmra_cofe_mixin : simpl never.
Arguments cmra_mixin : simpl never.
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Add Printing Constructor cmraT.
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Existing Instances cmra_unit cmra_op cmra_valid cmra_validN cmra_div.
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Coercion cmra_cofeC (A : cmraT) : cofeT := CofeT (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.
  Global Instance cmra_unit_ne n : Proper (dist n ==> dist n) (@unit A _).
  Proof. apply (mixin_cmra_unit_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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  Global Instance cmra_div_ne n :
    Proper (dist n ==> dist n ==> dist n) (@div A _).
  Proof. apply (mixin_cmra_div_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_unit_l x : unit x  x  x.
  Proof. apply (mixin_cmra_unit_l _ (cmra_mixin A)). Qed.
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  Lemma cmra_unit_idemp x : unit (unit x)  unit x.
  Proof. apply (mixin_cmra_unit_idemp _ (cmra_mixin A)). Qed.
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  Lemma cmra_unit_preserving x y : x  y  unit x  unit y.
  Proof. apply (mixin_cmra_unit_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_op_div x y : x  y  x  y ÷ x  y.
  Proof. apply (mixin_cmra_op_div _ (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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(** * CMRAs with a global identity element *)
(** We use the notation  because for most instances (maps, sets, etc) the
`empty' element is the global identity. *)
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Class CMRAIdentity (A : cmraT) `{Empty A} := {
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  cmra_empty_valid :  ;
  cmra_empty_left_id :> LeftId ()  ();
  cmra_empty_timeless :> Timeless 
}.
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Instance cmra_identity_inhabited `{CMRAIdentity A} : Inhabited A := populate .
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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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(** * Local updates *)
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(** The idea is that lemams taking this class will usually have L explicit,
    and leave Lv implicit - it will be inferred by the typeclass machinery. *)
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Class LocalUpdate {A : cmraT} (Lv : A  Prop) (L : A  A) := {
  local_update_ne n :> Proper (dist n ==> dist n) L;
  local_updateN n x y : Lv x  {n} (x  y)  L (x  y) {n} L x  y
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}.
Arguments local_updateN {_ _} _ {_} _ _ _ _ _.

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(** * Frame preserving updates *)
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Definition cmra_updateP {A : cmraT} (x : A) (P : A  Prop) :=  n z,
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  {n} (x  z)   y, P y  {n} (y  z).
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Instance: Params (@cmra_updateP) 1.
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Infix "~~>:" := cmra_updateP (at level 70).
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Definition cmra_update {A : cmraT} (x y : A) :=  n z,
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  {n} (x  z)  {n} (y  z).
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Infix "~~>" := cmra_update (at level 70).
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Instance: Params (@cmra_update) 1.
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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 *)
Global Instance cmra_unit_proper : Proper (() ==> ()) (@unit 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.
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  by rewrite Hy (comm _ x1) Hx (comm _ y2).
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Qed.
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.
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Global Instance cmra_div_proper : Proper (() ==> () ==> ()) (@div A _).
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Proof. apply (ne_proper_2 _). 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_update_proper :
  Proper (() ==> () ==> iff) (@cmra_update A).
Proof.
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  intros x1 x2 Hx y1 y2 Hy; split=>? n z; [rewrite -Hx -Hy|rewrite Hx Hy]; auto.
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Qed.
Global Instance cmra_updateP_proper :
  Proper (() ==> pointwise_relation _ iff ==> iff) (@cmra_updateP A).
Proof.
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  intros x1 x2 Hx P1 P2 HP; split=>Hup n z;
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    [rewrite -Hx; setoid_rewrite <-HP|rewrite Hx; setoid_rewrite HP]; auto.
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.

(** ** Units *)
Lemma cmra_unit_r x : x  unit x  x.
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Proof. by rewrite (comm _ x) cmra_unit_l. Qed.
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Lemma cmra_unit_unit x : unit x  unit x  unit x.
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Proof. by rewrite -{2}(cmra_unit_idemp x) cmra_unit_r. Qed.
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Lemma cmra_unit_validN n x : {n} x  {n} unit x.
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Proof. rewrite -{1}(cmra_unit_l x); apply cmra_validN_op_l. Qed.
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Lemma cmra_unit_valid x :  x   unit x.
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Proof. rewrite -{1}(cmra_unit_l x); apply cmra_valid_op_l. Qed.

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(** ** Div *)
Lemma cmra_op_div' n x y : x {n} y  x  y ÷ x {n} y.
Proof. intros [z ->]. by rewrite cmra_op_div; last exists z. Qed.
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(** ** Order *)
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Lemma cmra_included_includedN x y : x  y   n, x {n} y.
Proof.
  split; [by intros [z Hz] n; exists z; rewrite Hz|].
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  intros Hxy; exists (y ÷ x); apply equiv_dist=> n.
  by rewrite cmra_op_div'.
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Qed.
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Global Instance cmra_includedN_preorder n : PreOrder (@includedN A _ _ n).
Proof.
  split.
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  - by intros x; exists (unit x); rewrite cmra_unit_r.
  - intros x y z [z1 Hy] [z2 Hz]; exists (z1  z2).
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    by rewrite assoc -Hy -Hz.
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Qed.
Global Instance cmra_included_preorder: PreOrder (@included A _ _).
Proof.
  split; red; intros until 0; rewrite !cmra_included_includedN; first done.
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  intros; etrans; eauto.
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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. rewrite cmra_included_includedN; eauto using cmra_validN_includedN. 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_unit_preservingN n x y : x {n} y  unit x {n} unit y.
Proof.
  intros [z ->].
  apply cmra_included_includedN, cmra_unit_preserving, cmra_included_l.
Qed.
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Lemma cmra_included_unit x : unit x  x.
Proof. by exists x; rewrite cmra_unit_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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(** ** 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.
  split; first by rewrite cmra_included_includedN.
  intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l.
Qed.
Lemma cmra_discrete_updateP `{CMRADiscrete A} (x : A) (P : A  Prop) :
  ( z,  (x  z)   y, P y   (y  z))  x ~~>: P.
Proof. intros ? n. by setoid_rewrite <-cmra_discrete_valid_iff. Qed.
Lemma cmra_discrete_update `{CMRADiscrete A} (x y : A) :
  ( z,  (x  z)   (y  z))  x ~~> y.
Proof. intros ? n. by setoid_rewrite <-cmra_discrete_valid_iff. Qed.

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(** ** RAs with an empty element *)
Section identity.
  Context `{Empty A, !CMRAIdentity A}.
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  Lemma cmra_empty_validN n : {n} .
  Proof. apply cmra_valid_validN, cmra_empty_valid. Qed.
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  Lemma cmra_empty_leastN n x :  {n} x.
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  Proof. by exists x; rewrite left_id. Qed.
  Lemma cmra_empty_least x :   x.
  Proof. by exists x; rewrite left_id. Qed.
  Global Instance cmra_empty_right_id : RightId ()  ().
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  Proof. by intros x; rewrite (comm op) left_id. Qed.
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  Lemma cmra_unit_empty : unit   .
  Proof. by rewrite -{2}(cmra_unit_l ) right_id. Qed.
End identity.
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(** ** Local updates *)
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Global Instance local_update_proper Lv (L : A  A) :
  LocalUpdate Lv L  Proper (() ==> ()) L.
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Proof. intros; apply (ne_proper _). Qed.

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Lemma local_update L `{!LocalUpdate Lv L} x y :
  Lv x   (x  y)  L (x  y)  L x  y.
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Proof.
  by rewrite cmra_valid_validN equiv_dist=>?? n; apply (local_updateN L).
Qed.
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Global Instance local_update_op x : LocalUpdate (λ _, True) (op x).
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Proof. split. apply _. by intros n y1 y2 _ _; rewrite assoc. Qed.
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Global Instance local_update_id : LocalUpdate (λ _, True) (@id A).
Proof. split; auto with typeclass_instances. Qed.

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(** ** Updates *)
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Global Instance cmra_update_preorder : PreOrder (@cmra_update A).
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Proof. split. by intros x y. intros x y y' ?? z ?; naive_solver. Qed.
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Lemma cmra_update_updateP x y : x ~~> y  x ~~>: (y =).
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Proof.
  split.
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  - by intros Hx z ?; exists y; split; [done|apply (Hx z)].
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  - by intros Hx n z ?; destruct (Hx n z) as (?&<-&?).
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Qed.
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Lemma cmra_updateP_id (P : A  Prop) x : P x  x ~~>: P.
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Proof. by intros ? n z ?; exists x. Qed.
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Lemma cmra_updateP_compose (P Q : A  Prop) x :
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  x ~~>: P  ( y, P y  y ~~>: Q)  x ~~>: Q.
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Proof.
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  intros Hx Hy n z ?. destruct (Hx n z) as (y&?&?); auto. by apply (Hy y).
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Qed.
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Lemma cmra_updateP_compose_l (Q : A  Prop) x y : x ~~> y  y ~~>: Q  x ~~>: Q.
Proof.
  rewrite cmra_update_updateP.
  intros; apply cmra_updateP_compose with (y =); intros; subst; auto.
Qed.
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Lemma cmra_updateP_weaken (P Q : A  Prop) x : x ~~>: P  ( y, P y  Q y)  x ~~>: Q.
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Proof. eauto using cmra_updateP_compose, cmra_updateP_id. Qed.
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Lemma cmra_updateP_op (P1 P2 Q : A  Prop) x1 x2 :
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  x1 ~~>: P1  x2 ~~>: P2  ( y1 y2, P1 y1  P2 y2  Q (y1  y2))  x1  x2 ~~>: Q.
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Proof.
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  intros Hx1 Hx2 Hy n z ?.
  destruct (Hx1 n (x2  z)) as (y1&?&?); first by rewrite assoc.
  destruct (Hx2 n (y1  z)) as (y2&?&?);
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    first by rewrite assoc (comm _ x2) -assoc.
  exists (y1  y2); split; last rewrite (comm _ y1) -assoc; auto.
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Qed.
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Lemma cmra_updateP_op' (P1 P2 : A  Prop) x1 x2 :
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  x1 ~~>: P1  x2 ~~>: P2  x1  x2 ~~>: λ y,  y1 y2, y = y1  y2  P1 y1  P2 y2.
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Proof. eauto 10 using cmra_updateP_op. Qed.
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Lemma cmra_update_op x1 x2 y1 y2 : x1 ~~> y1  x2 ~~> y2  x1  x2 ~~> y1  y2.
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Proof.
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  rewrite !cmra_update_updateP; eauto using cmra_updateP_op with congruence.
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Qed.
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Lemma cmra_update_id x : x ~~> x.
Proof. intro. auto. Qed.
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Section identity_updates.
  Context `{Empty A, !CMRAIdentity A}.
  Lemma cmra_update_empty x : x ~~> .
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  Proof. intros n z; rewrite left_id; apply cmra_validN_op_r. Qed.
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  Lemma cmra_update_empty_alt y :  ~~> y   x, x ~~> y.
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  Proof. split; [intros; trans |]; auto using cmra_update_empty. Qed.
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End identity_updates.
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End cmra.

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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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(** * 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.
  Lemma cmra_transport_unit x : T (unit x) = unit (T x).
  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.
  Lemma cmra_transport_updateP (P : A  Prop) (Q : B  Prop) x :
    x ~~>: P  ( y, P y  Q (T y))  T x ~~>: Q.
  Proof. destruct H; eauto using cmra_updateP_weaken. Qed.
  Lemma cmra_transport_updateP' (P : A  Prop) x :
    x ~~>: P  T x ~~>: λ y,  y', y = cmra_transport H y'  P y'.
  Proof. eauto using cmra_transport_updateP. Qed.
End cmra_transport.

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(** * Instances *)
(** ** Discrete CMRA *)
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Class RA A `{Equiv A, Unit A, Op A, Valid A, Div A} := {
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  (* setoids *)
  ra_op_ne (x : A) : Proper (() ==> ()) (op x);
  ra_unit_ne :> Proper (() ==> ()) unit;
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  ra_validN_ne :> Proper (() ==> impl) valid;
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  ra_div_ne :> Proper (() ==> () ==> ()) div;
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  (* monoid *)
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  ra_assoc :> Assoc () ();
  ra_comm :> Comm () ();
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  ra_unit_l x : unit x  x  x;
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  ra_unit_idemp x : unit (unit x)  unit x;
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  ra_unit_preserving x y : x  y  unit x  unit y;
  ra_valid_op_l x y :  (x  y)   x;
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  ra_op_div x y : x  y  x  y ÷ x  y
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}.

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Section discrete.
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  Context {A : cofeT} `{Discrete A}.
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  Context `{Unit A, Op A, Valid A, Div A} (ra : RA A).
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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; split; unfold Proper, respectful, includedN;
      try setoid_rewrite <-(timeless_iff _ _); try done.
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    - intros x; split; first done. by move=> /(_ 0).
    - intros n x y1 y2 ??; exists (y1,y2); split_and?; auto.
      apply (timeless _), dist_le with n; auto with lia.
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  Qed.
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  Definition discreteR : cmraT := CMRAT (cofe_mixin A) discrete_cmra_mixin.
  Global Instance discrete_cmra_discrete : CMRADiscrete discreteR.
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  Proof. split. change (Discrete A); apply _. by intros x ?. Qed.
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End discrete.

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(** ** CMRA for the unit type *)
Section unit.
  Instance unit_valid : Valid () := λ x, True.
  Instance unit_unit : Unit () := λ x, x.
  Instance unit_op : Op () := λ x y, ().
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  Instance unit_div : Div () := λ x y, ().
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  Global Instance unit_empty : Empty () := ().
  Definition unit_ra : RA ().
  Proof. by split. Qed.
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  Canonical Structure unitR : cmraT :=
    Eval cbv [unitC discreteR cofe_car] in discreteR unit_ra.
  Global Instance unit_cmra_identity : CMRAIdentity unitR.
  Global Instance unit_cmra_discrete : CMRADiscrete unitR.
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  Proof. by apply discrete_cmra_discrete. Qed.
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End unit.
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(** ** Product *)
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Section prod.
  Context {A B : cmraT}.
  Instance prod_op : Op (A * B) := λ x y, (x.1  y.1, x.2  y.2).
  Global Instance prod_empty `{Empty A, Empty B} : Empty (A * B) := (, ).
  Instance prod_unit : Unit (A * B) := λ x, (unit (x.1), unit (x.2)).
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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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  Instance prod_div : Div (A * B) := λ x y, (x.1 ÷ y.1, x.2 ÷ y.2).
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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.
  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.
    - by intros n y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
    - by intros n y1 y2 [Hy1 Hy2] [??]; split; rewrite /= -?Hy1 -?Hy2.
    - by intros n x1 x2 [Hx1 Hx2] y1 y2 [Hy1 Hy2];
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        split; rewrite /= ?Hx1 ?Hx2 ?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.
    - by split; rewrite /= cmra_unit_l.
    - by split; rewrite /= cmra_unit_idemp.
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    - intros x y; rewrite !prod_included.
      by intros [??]; split; apply cmra_unit_preserving.
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    - intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l.
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    - intros x y; rewrite prod_included; intros [??].
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      by split; apply cmra_op_div.
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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 : cmraT := CMRAT prod_cofe_mixin prod_cmra_mixin.
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  Global Instance prod_cmra_identity `{Empty A, Empty B} :
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    CMRAIdentity A  CMRAIdentity B  CMRAIdentity prodR.
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  Proof.
    split.
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    - split; apply cmra_empty_valid.
    - by split; rewrite /=left_id.
    - by intros ? [??]; split; apply (timeless _).
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  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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  Lemma prod_update x y : x.1 ~~> y.1  x.2 ~~> y.2  x ~~> y.
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  Proof. intros ?? n z [??]; split; simpl in *; auto. Qed.
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  Lemma prod_updateP P1 P2 (Q : A * B  Prop)  x :
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    x.1 ~~>: P1  x.2 ~~>: P2  ( a b, P1 a  P2 b  Q (a,b))  x ~~>: Q.
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  Proof.
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    intros Hx1 Hx2 HP n z [??]; simpl in *.
    destruct (Hx1 n (z.1)) as (a&?&?), (Hx2 n (z.2)) as (b&?&?); auto.
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    exists (a,b); repeat split; auto.
  Qed.
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  Lemma prod_updateP' P1 P2 x :
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    x.1 ~~>: P1  x.2 ~~>: P2  x ~~>: λ y, P1 (y.1)  P2 (y.2).
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  Proof. eauto using prod_updateP. Qed.
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End prod.
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Arguments prodR : clear implicits.
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Instance prod_map_cmra_monotone {A A' B B' : cmraT} (f : A  A') (g : B  B') :
  CMRAMonotone f  CMRAMonotone g  CMRAMonotone (prod_map f g).
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Proof.
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  split; first apply _.
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  - by intros n x [??]; split; simpl; apply validN_preserving.
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  - intros x y; rewrite !prod_included=> -[??] /=.
    by split; apply included_preserving.
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Qed.
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(** Functors *)
Structure rFunctor := RFunctor {
  rFunctor_car : cofeT  cofeT -> cmraT;
  rFunctor_map {A1 A2 B1 B2} :
    ((A2 -n> A1) * (B1 -n> B2))  rFunctor_car A1 B1 -n> rFunctor_car A2 B2;
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  rFunctor_ne A1 A2 B1 B2 n :
    Proper (dist n ==> dist n) (@rFunctor_map A1 A2 B1 B2);
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  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) 
}.
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Existing Instances rFunctor_ne rFunctor_mono.
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Instance: Params (@rFunctor_map) 5.

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

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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.

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Instance constRF_contractive B : rFunctorContractive (constRF B).
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Proof. rewrite /rFunctorContractive; apply _. Qed.
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Program Definition prodRF (F1 F2 : rFunctor) : rFunctor := {|
  rFunctor_car A B := prodR (rFunctor_car F1 A B) (rFunctor_car F2 A B);
  rFunctor_map A1 A2 B1 B2 fg :=
    prodC_map (rFunctor_map F1 fg) (rFunctor_map F2 fg)
|}.
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Next Obligation.
  intros F1 F2 A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply rFunctor_ne.
Qed.
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Next Obligation. by intros F1 F2 A B [??]; rewrite /= !rFunctor_id. Qed.
Next Obligation.
  intros F1 F2 A1 A2 A3 B1 B2 B3 f g f' g' [??]; simpl.
  by rewrite !rFunctor_compose.
Qed.
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Instance prodRF_contractive F1 F2 :
  rFunctorContractive F1  rFunctorContractive F2 
  rFunctorContractive (prodRF F1 F2).
Proof.
  intros ?? A1 A2 B1 B2 n ???;
    by apply prodC_map_ne; apply rFunctor_contractive.
Qed.