cmra.v 26.2 KB
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Require Export algebra.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.
Hint Extern 0 (?x  ?y) => reflexivity.
Instance: Params (@included) 3.

Class Minus (A : Type) := minus : A  A  A.
Instance: Params (@minus) 2.
Infix "⩪" := minus (at level 40) : C_scope.
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Class ValidN (A : Type) := validN : nat  A  Prop.
Instance: Params (@validN) 3.
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Notation "✓{ n }" := (validN n) (at level 1, format "✓{ n }").
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Class Valid (A : Type) := valid : A  Prop.
Instance: Params (@valid) 2.
Notation "✓" := valid (at level 1).
Instance validN_valid `{ValidN A} : Valid A := λ x,  n, {n} x.

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Definition includedN `{Dist A, Op A} (n : nat) (x y : A) :=  z, y ={n}= x  z.
Notation "x ≼{ n } y" := (includedN n x y)
  (at level 70, format "x  ≼{ n }  y") : C_scope.
Instance: Params (@includedN) 4.
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Hint Extern 0 (?x {_} ?y) => reflexivity.
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Record CMRAMixin A `{Dist A, Equiv A, Unit A, Op A, ValidN A, Minus 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) ({n});
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  mixin_cmra_minus_ne n : Proper (dist n ==> dist n ==> dist n) minus;
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  (* valid *)
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  mixin_cmra_validN_0 x : {0} x;
  mixin_cmra_validN_S n x : {S n} x  {n} x;
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  (* monoid *)
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  mixin_cmra_associative : Associative () ();
  mixin_cmra_commutative : Commutative () ();
  mixin_cmra_unit_l x : unit x  x  x;
  mixin_cmra_unit_idempotent x : unit (unit x)  unit x;
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  mixin_cmra_unit_preservingN n x y : x {n} y  unit x {n} unit y;
  mixin_cmra_validN_op_l n x y : {n} (x  y)  {n} x;
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  mixin_cmra_op_minus n x y : x {n} y  x  y  x ={n}= y
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}.
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Definition CMRAExtendMixin A `{Equiv A, Dist A, Op A, ValidN A} :=  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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(** 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;
  cmra_validN : ValidN cmra_car;
  cmra_minus : Minus cmra_car;
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  cmra_cofe_mixin : CofeMixin cmra_car;
  cmra_mixin : CMRAMixin cmra_car;
  cmra_extend_mixin : CMRAExtendMixin 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.
Arguments cmra_validN : simpl never.
Arguments cmra_minus : simpl never.
Arguments cmra_cofe_mixin : simpl never.
Arguments cmra_mixin : simpl never.
Arguments cmra_extend_mixin : simpl never.
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Add Printing Constructor cmraT.
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Existing Instances cmra_unit cmra_op cmra_validN cmra_minus.
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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_minus_ne n :
    Proper (dist n ==> dist n ==> dist n) (@minus A _).
  Proof. apply (mixin_cmra_minus_ne _ (cmra_mixin A)). Qed.
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  Lemma cmra_validN_0 x : {0} x.
  Proof. apply (mixin_cmra_validN_0 _ (cmra_mixin A)). Qed.
  Lemma cmra_validN_S n x : {S n} x  {n} x.
  Proof. apply (mixin_cmra_validN_S _ (cmra_mixin A)). Qed.
  Global Instance cmra_associative : Associative () (@op A _).
  Proof. apply (mixin_cmra_associative _ (cmra_mixin A)). Qed.
  Global Instance cmra_commutative : Commutative () (@op A _).
  Proof. apply (mixin_cmra_commutative _ (cmra_mixin A)). Qed.
  Lemma cmra_unit_l x : unit x  x  x.
  Proof. apply (mixin_cmra_unit_l _ (cmra_mixin A)). Qed.
  Lemma cmra_unit_idempotent x : unit (unit x)  unit x.
  Proof. apply (mixin_cmra_unit_idempotent _ (cmra_mixin A)). Qed.
  Lemma cmra_unit_preservingN n x y : x {n} y  unit x {n} unit y.
  Proof. apply (mixin_cmra_unit_preservingN _ (cmra_mixin A)). Qed.
  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_minus n x y : x {n} y  x  y  x ={n}= y.
  Proof. apply (mixin_cmra_op_minus _ (cmra_mixin A)). Qed.
  Lemma cmra_extend_op n x y1 y2 :
    {n} x  x ={n}= y1  y2 
    { z | x  z.1  z.2  z.1 ={n}= y1  z.2 ={n}= y2 }.
  Proof. apply (cmra_extend_mixin A). Qed.
End cmra_mixin.

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Hint Extern 0 ({0} _) => apply cmra_validN_0.

(** * CMRAs with a global identity element *)
(** We use the notation ∅ because for most instances (maps, sets, etc) the
`empty' element is the global identity. *)
Class CMRAIdentity (A : cmraT) `{Empty A} : Prop := {
  cmra_empty_valid :  ;
  cmra_empty_left_id :> LeftId ()  ();
  cmra_empty_timeless :> Timeless 
}.
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(** * Morphisms *)
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Class CMRAMonotone {A B : cmraT} (f : A  B) := {
  includedN_preserving n x y : x {n} y  f x {n} f y;
  validN_preserving n x : {n} x  {n} (f x)
}.

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(** * Frame preserving updates *)
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Definition cmra_updateP {A : cmraT} (x : A) (P : A  Prop) :=  z n,
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  {S n} (x  z)   y, P y  {S 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) :=  z n,
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  {S n} (x  z)  {S 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.
  by rewrite Hy (commutative _ x1) Hx (commutative _ y2).
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.
Global Instance cmra_minus_proper : Proper (() ==> () ==> ()) (@minus A _).
Proof. apply (ne_proper_2 _). Qed.

Global Instance cmra_valid_proper : Proper (() ==> iff) (@valid A _).
Proof. by intros x y Hxy; split; intros ? n; [rewrite -Hxy|rewrite Hxy]. Qed.
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.
  intros x1 x2 Hx y1 y2 Hy; split=>? z n; [rewrite -Hx -Hy|rewrite Hx Hy]; auto.
Qed.
Global Instance cmra_updateP_proper :
  Proper (() ==> pointwise_relation _ iff ==> iff) (@cmra_updateP A).
Proof.
  intros x1 x2 Hx P1 P2 HP; split=>Hup z n;
    [rewrite -Hx; setoid_rewrite <-HP|rewrite Hx; setoid_rewrite HP]; auto.
Qed.
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(** ** Validity *)
Lemma cmra_valid_validN x :  x   n, {n} x.
Proof. done. Qed.
Lemma cmra_validN_le x n n' : {n} x  n'  n  {n'} x.
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.
Lemma cmra_validN_op_r x y n : {n} (x  y)  {n} y.
Proof. rewrite (commutative _ x); apply cmra_validN_op_l. Qed.
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.
Proof. by rewrite (commutative _ x) cmra_unit_l. Qed.
Lemma cmra_unit_unit x : unit x  unit x  unit x.
Proof. by rewrite -{2}(cmra_unit_idempotent x) cmra_unit_r. Qed.
Lemma cmra_unit_validN x n : {n} x  {n} (unit x).
Proof. rewrite -{1}(cmra_unit_l x); apply cmra_validN_op_l. Qed.
Lemma cmra_unit_valid x :  x   (unit x).
Proof. rewrite -{1}(cmra_unit_l x); apply cmra_valid_op_l. Qed.

(** ** 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|].
  intros Hxy; exists (y  x); apply equiv_dist; intros n.
  symmetry; apply cmra_op_minus, Hxy.
Qed.
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Global Instance cmra_includedN_preorder n : PreOrder (@includedN A _ _ n).
Proof.
  split.
  * by intros x; exists (unit x); rewrite cmra_unit_r.
  * intros x y z [z1 Hy] [z2 Hz]; exists (z1  z2).
    by rewrite (associative _) -Hy -Hz.
Qed.
Global Instance cmra_included_preorder: PreOrder (@included A _ _).
Proof.
  split; red; intros until 0; rewrite !cmra_included_includedN; first done.
  intros; etransitivity; eauto.
Qed.
Lemma cmra_validN_includedN x y n : {n} y  x {n} y  {n} x.
Proof. intros Hyv [z ?]; cofe_subst y; eauto using cmra_validN_op_l. Qed.
Lemma cmra_validN_included x y n : {n} y  x  y  {n} x.
Proof. rewrite cmra_included_includedN; eauto using cmra_validN_includedN. Qed.

Lemma cmra_includedN_0 x y : x {0} y.
Proof. by exists (unit x). Qed.
Lemma cmra_includedN_S x y n : x {S n} y  x {n} y.
Proof. by intros [z Hz]; exists z; apply dist_S. Qed.
Lemma cmra_includedN_le x y n n' : x {n} y  n'  n  x {n'} y.
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.
Proof. rewrite (commutative op); apply cmra_includedN_l. Qed.
Lemma cmra_included_r x y : y  x  y.
Proof. rewrite (commutative op); apply cmra_included_l. Qed.
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Lemma cmra_unit_preserving x y : x  y  unit x  unit y.
Proof. rewrite !cmra_included_includedN; eauto using cmra_unit_preservingN. Qed.
Lemma cmra_included_unit x : unit x  x.
Proof. by exists x; rewrite cmra_unit_l. Qed.
Lemma cmra_preserving_l x y z : x  y  z  x  z  y.
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (associative op). Qed.
Lemma cmra_preserving_r x y z : x  y  x  z  y  z.
Proof. by intros; rewrite -!(commutative _ z); apply cmra_preserving_l. Qed.

Lemma cmra_included_dist_l x1 x2 x1' n :
  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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(** ** Minus *)
Lemma cmra_op_minus' x y : x  y  x  y  x  y.
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Proof.
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  rewrite cmra_included_includedN equiv_dist; eauto using cmra_op_minus.
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Qed.
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(** ** Timeless *)
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Lemma cmra_timeless_included_l x y : Timeless x  {1} y  x {1} y  x  y.
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Proof.
  intros ?? [x' ?].
  destruct (cmra_extend_op 1 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.
Lemma cmra_timeless_included_r n x y : Timeless y  x {1} y  x {n} y.
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_op 1 z x1 x2) as ([y1 y2]&Hz'&?&?); auto; simpl in *.
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  { by rewrite -?Hz. }
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  by rewrite Hz' (timeless x1 y1) // (timeless x2 y2).
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Qed.
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(** ** RAs with an empty element *)
Section identity.
  Context `{Empty A, !CMRAIdentity A}.
  Lemma cmra_empty_leastN  n x :  {n} x.
  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 ()  ().
  Proof. by intros x; rewrite (commutative op) left_id. Qed.
  Lemma cmra_unit_empty : unit   .
  Proof. by rewrite -{2}(cmra_unit_l ) right_id. Qed.
End identity.
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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.
  * by intros Hx z ?; exists y; split; [done|apply (Hx z)].
  * by intros Hx z n ?; destruct (Hx z n) as (?&<-&?).
Qed.
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Lemma cmra_updateP_id (P : A  Prop) x : P x  x ~~>: P.
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Proof. by intros ? z n ?; 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.
  intros Hx Hy z n ?. destruct (Hx z n) as (y&?&?); auto. by apply (Hy y).
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.
  intros Hx1 Hx2 Hy z n ?.
  destruct (Hx1 (x2  z) n) as (y1&?&?); first by rewrite associative.
  destruct (Hx2 (y1  z) n) as (y2&?&?);
    first by rewrite associative (commutative _ x2) -associative.
  exists (y1  y2); split; last rewrite (commutative _ y1) -associative; auto.
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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End cmra.

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Hint Extern 0 (_ {0} _) => apply cmra_includedN_0.
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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. by split. 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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  * move=> n x y Hxy /=. by apply includedN_preserving, includedN_preserving.
  * move=> n x Hx /=. by apply validN_preserving, validN_preserving.
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Qed.
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Section cmra_monotone.
  Context {A B : cmraT} (f : A  B) `{!CMRAMonotone f}.
  Lemma included_preserving x y : x  y  f x  f y.
  Proof.
    rewrite !cmra_included_includedN; eauto using includedN_preserving.
  Qed.
  Lemma valid_preserving x :  x   (f x).
  Proof. rewrite !cmra_valid_validN; eauto using validN_preserving. Qed.
End cmra_monotone.

(** * Instances *)
(** ** Discrete CMRA *)
Class RA A `{Equiv A, Unit A, Op A, Valid A, Minus A} := {
  (* setoids *)
  ra_op_ne (x : A) : Proper (() ==> ()) (op x);
  ra_unit_ne :> Proper (() ==> ()) unit;
  ra_validN_ne :> Proper (() ==> impl) ;
  ra_minus_ne :> Proper (() ==> () ==> ()) minus;
  (* monoid *)
  ra_associative :> Associative () ();
  ra_commutative :> Commutative () ();
  ra_unit_l x : unit x  x  x;
  ra_unit_idempotent x : unit (unit x)  unit x;
  ra_unit_preserving x y : x  y  unit x  unit y;
  ra_valid_op_l x y :  (x  y)   x;
  ra_op_minus x y : x  y  x  y  x  y
}.

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Section discrete.
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  Context {A : cofeT} `{ x : A, Timeless x}.
  Context `{Unit A, Op A, Valid A, Minus A} (ra : RA A).

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  Instance discrete_validN : ValidN A := λ n x,
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    match n with 0 => True | S n =>  x end.
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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;
      repeat match goal with
      | |-  n : nat, _ => intros [|?]
      end; try setoid_rewrite <-(timeless_S _ _ _ _); try done.
    by intros x y ?; exists x.
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  Qed.
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  Definition discrete_extend_mixin : CMRAExtendMixin A.
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  Proof.
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    intros [|n] x y1 y2 ??.
    * by exists (unit x, x); rewrite /= ra_unit_l.
    * exists (y1,y2); split_ands; auto.
      apply (timeless _), dist_le with (S n); auto with lia.
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  Qed.
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  Definition discreteRA : cmraT :=
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    CMRAT (cofe_mixin A) discrete_cmra_mixin discrete_extend_mixin.
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  Lemma discrete_updateP (x : discreteRA) (P : A  Prop) :
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    ( z,  (x  z)   y, P y   (y  z))  x ~~>: P.
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  Proof. intros Hvalid z n; apply Hvalid. Qed.
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  Lemma discrete_update (x y : discreteRA) :
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    ( z,  (x  z)   (y  z))  x ~~> y.
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  Proof. intros Hvalid z n; apply Hvalid. 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, ().
  Instance unit_minus : Minus () := λ x y, ().
  Global Instance unit_empty : Empty () := ().
  Definition unit_ra : RA ().
  Proof. by split. Qed.
  Canonical Structure unitRA : cmraT :=
    Eval cbv [unitC discreteRA cofe_car] in discreteRA unit_ra.
  Global Instance unit_cmra_identity : CMRAIdentity unitRA.
  Proof. by split; intros []. Qed.
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)).
  Instance prod_validN : ValidN (A * B) := λ n x, {n} (x.1)  {n} (x.2).
  Instance prod_minus : Minus (A * B) := λ x y, (x.1  y.1, x.2  y.2).
  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 _.
    * 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];
        split; rewrite /= ?Hx1 ?Hx2 ?Hy1 ?Hy2.
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    * by split.
    * by intros n x [??]; split; apply cmra_validN_S.
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    * split; simpl; apply (associative _).
    * split; simpl; apply (commutative _).
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    * split; simpl; apply cmra_unit_l.
    * split; simpl; apply cmra_unit_idempotent.
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    * intros n x y; rewrite !prod_includedN.
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      by intros [??]; split; apply cmra_unit_preservingN.
    * intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l.
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    * intros x y n; rewrite prod_includedN; intros [??].
      by split; apply cmra_op_minus.
  Qed.
  Definition prod_cmra_extend_mixin : CMRAExtendMixin (A * B).
  Proof.
    intros n x y1 y2 [??] [??]; simpl in *.
    destruct (cmra_extend_op n (x.1) (y1.1) (y2.1)) as (z1&?&?&?); auto.
    destruct (cmra_extend_op n (x.2) (y1.2) (y2.2)) as (z2&?&?&?); auto.
    by exists ((z1.1,z2.1),(z1.2,z2.2)).
  Qed.
  Canonical Structure prodRA : cmraT :=
    CMRAT prod_cofe_mixin prod_cmra_mixin prod_cmra_extend_mixin.
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  Global Instance prod_cmra_identity `{Empty A, Empty B} :
    CMRAIdentity A  CMRAIdentity B  CMRAIdentity prodRA.
  Proof.
    split.
    * split; apply cmra_empty_valid.
    * by split; rewrite /=left_id.
    * by intros ? [??]; split; apply (timeless _).
  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 ?? z n [??]; 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.
    intros Hx1 Hx2 HP z n [??]; simpl in *.
    destruct (Hx1 (z.1) n) as (a&?&?), (Hx2 (z.2) n) as (b&?&?); auto.
    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.
Arguments prodRA : clear implicits.

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.
  split.
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  * intros n x y; rewrite !prod_includedN; intros [??]; simpl.
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    by split; apply includedN_preserving.
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  * by intros n x [??]; split; simpl; apply validN_preserving.
Qed.
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(** ** Indexed product *)
Section iprod_cmra.
  Context {A} {B : A  cmraT}.
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  Implicit Types f g : iprod B.
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  Instance iprod_op : Op (iprod B) := λ f g x, f x  g x.
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  Definition iprod_lookup_op f g x : (f  g) x = f x  g x := eq_refl.
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  Instance iprod_unit : Unit (iprod B) := λ f x, unit (f x).
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  Definition iprod_lookup_unit f x : (unit f) x = unit (f x) := eq_refl.
  Global Instance iprod_empty `{ x, Empty (B x)} : Empty (iprod B) := λ x, .
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  Instance iprod_validN : ValidN (iprod B) := λ n f,  x, {n} (f x).
  Instance iprod_minus : Minus (iprod B) := λ f g x, f x  g x.
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  Definition iprod_lookup_minus f g x : (f  g) x = f x  g x := eq_refl.
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  Lemma iprod_includedN_spec (f g : iprod B) n : f {n} g   x, f x {n} g x.
  Proof.
    split.
    * by intros [h Hh] x; exists (h x); rewrite /op /iprod_op (Hh x).
    * intros Hh; exists (g  f)=> x; specialize (Hh x).
      by rewrite /op /iprod_op /minus /iprod_minus cmra_op_minus.
  Qed.
  Definition iprod_cmra_mixin : CMRAMixin (iprod B).
  Proof.
    split.
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    * by intros n f1 f2 f3 Hf x; rewrite iprod_lookup_op (Hf x).
    * by intros n f1 f2 Hf x; rewrite iprod_lookup_unit (Hf x).
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    * by intros n f1 f2 Hf ? x; rewrite -(Hf x).
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    * by intros n f f' Hf g g' Hg i; rewrite iprod_lookup_minus (Hf i) (Hg i).
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    * by intros f x.
    * intros n f Hf x; apply cmra_validN_S, Hf.
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    * by intros f1 f2 f3 x; rewrite iprod_lookup_op associative.
    * by intros f1 f2 x; rewrite iprod_lookup_op commutative.
    * by intros f x; rewrite iprod_lookup_op iprod_lookup_unit cmra_unit_l.
    * by intros f x; rewrite iprod_lookup_unit cmra_unit_idempotent.
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    * intros n f1 f2; rewrite !iprod_includedN_spec=> Hf x.
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      by rewrite iprod_lookup_unit; apply cmra_unit_preservingN, Hf.
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    * intros n f1 f2 Hf x; apply cmra_validN_op_l with (f2 x), Hf.
    * intros n f1 f2; rewrite iprod_includedN_spec=> Hf x.
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      by rewrite iprod_lookup_op iprod_lookup_minus cmra_op_minus; try apply Hf.
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  Qed.
  Definition iprod_cmra_extend_mixin : CMRAExtendMixin (iprod B).
  Proof.
    intros n f f1 f2 Hf Hf12.
    set (g x := cmra_extend_op n (f x) (f1 x) (f2 x) (Hf x) (Hf12 x)).
    exists ((λ x, (proj1_sig (g x)).1), (λ x, (proj1_sig (g x)).2)).
    split_ands; intros x; apply (proj2_sig (g x)).
  Qed.
  Canonical Structure iprodRA : cmraT :=
    CMRAT iprod_cofe_mixin iprod_cmra_mixin iprod_cmra_extend_mixin.
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  Global Instance iprod_cmra_identity `{ x, Empty (B x)} :
    ( x, CMRAIdentity (B x))  CMRAIdentity iprodRA.
  Proof.
    intros ?; split.
    * intros n x; apply cmra_empty_valid.
    * by intros f x; rewrite iprod_lookup_op left_id.
    * by intros f Hf x; apply (timeless _).
  Qed.

  Context `{ x x' : A, Decision (x = x')}.
  Lemma iprod_insert_updateP x (P : B x  Prop) (Q : iprod B  Prop) g y1 :
    y1 ~~>: P  ( y2, P y2  Q (iprod_insert x y2 g)) 
    iprod_insert x y1 g ~~>: Q.
  Proof.
    intros Hy1 HP gf n Hg. destruct (Hy1 (gf x) n) as (y2&?&?).
    { move: (Hg x). by rewrite iprod_lookup_op iprod_lookup_insert. }
    exists (iprod_insert x y2 g); split; [auto|].
    intros x'; destruct (decide (x' = x)) as [->|];
      rewrite iprod_lookup_op ?iprod_lookup_insert //.
    move: (Hg x'). by rewrite iprod_lookup_op !iprod_lookup_insert_ne.
  Qed.
  Lemma iprod_insert_updateP' x (P : B x  Prop) g y1 :
    y1 ~~>: P 
    iprod_insert x y1 g ~~>: λ g',  y2, g' = iprod_insert x y2 g  P y2.
  Proof. eauto using iprod_insert_updateP. Qed.
  Lemma iprod_insert_update g x y1 y2 :
    y1 ~~> y2  iprod_insert x y1 g ~~> iprod_insert x y2 g.
  Proof.
    rewrite !cmra_update_updateP;
      eauto using iprod_insert_updateP with congruence.
  Qed.

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  Context `{ x, Empty (B x)} `{ x, CMRAIdentity (B x)}.
  Lemma iprod_op_singleton (x : A) (y1 y2 : B x) :
    iprod_singleton x y1  iprod_singleton x y2  iprod_singleton x (y1  y2).
  Proof.
    intros x'; destruct (decide (x' = x)) as [->|].
    * by rewrite iprod_lookup_op !iprod_lookup_singleton.
    * by rewrite iprod_lookup_op !iprod_lookup_singleton_ne // left_id.
  Qed.
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  Lemma iprod_singleton_updateP x (P : B x  Prop) (Q : iprod B  Prop) y1 :
    y1 ~~>: P  ( y2, P y2  Q (iprod_singleton x y2)) 
    iprod_singleton x y1 ~~>: Q.
  Proof. rewrite /iprod_singleton; eauto using iprod_insert_updateP. Qed.
  Lemma iprod_singleton_updateP' x (P : B x  Prop) y1 :
    y1 ~~>: P 
    iprod_singleton x y1 ~~>: λ g',  y2, g' = iprod_singleton x y2  P y2.
  Proof. eauto using iprod_singleton_updateP. Qed.
  Lemma iprod_singleton_update x y1 y2 :
    y1 ~~> y2  iprod_singleton x y1 ~~> iprod_singleton x y2.
  Proof. by intros; apply iprod_insert_update. Qed.
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End iprod_cmra.

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Arguments iprodRA {_} _.
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Instance iprod_map_cmra_monotone {A} {B1 B2: A  cmraT} (f :  x, B1 x  B2 x) :
  ( x, CMRAMonotone (f x))  CMRAMonotone (iprod_map f).
Proof.
  split.
  * intros n g1 g2; rewrite !iprod_includedN_spec=> Hf x.
    rewrite /iprod_map; apply includedN_preserving, Hf.
  * intros n g Hg x; rewrite /iprod_map; apply validN_preserving, Hg.
Qed.