Require Export modures.cofe. 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. Class ValidN (A : Type) := validN : nat → A → Prop. Instance: Params (@validN) 3. Notation "✓{ n }" := (validN n) (at level 1, format "✓{ n }"). 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. 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. Hint Extern 0 (?x ≼{_} ?y) => reflexivity. Record CMRAMixin A `{Dist A, Equiv A, Unit A, Op A, ValidN A, Minus A} := { (* setoids *) 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; mixin_cmra_validN_ne n : Proper (dist n ==> impl) (✓{n}); mixin_cmra_minus_ne n : Proper (dist n ==> dist n ==> dist n) minus; (* valid *) mixin_cmra_validN_0 x : ✓{0} x; mixin_cmra_validN_S n x : ✓{S n} x → ✓{n} x; (* monoid *) 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; 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; mixin_cmra_op_minus n x y : x ≼{n} y → x ⋅ y ⩪ x ={n}= y }. 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 }. (** 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; cmra_cofe_mixin : CofeMixin cmra_car; cmra_mixin : CMRAMixin cmra_car; cmra_extend_mixin : CMRAExtendMixin cmra_car }. Arguments CMRAT {_ _ _ _ _ _ _ _} _ _ _. 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. Add Printing Constructor cmraT. Existing Instances cmra_unit cmra_op cmra_validN cmra_minus. Coercion cmra_cofeC (A : cmraT) : cofeT := CofeT (cmra_cofe_mixin A). Canonical Structure cmra_cofeC. (** 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. Global Instance cmra_validN_ne n : Proper (dist n ==> impl) (@validN A _ n). Proof. apply (mixin_cmra_validN_ne _ (cmra_mixin A)). Qed. 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. 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. 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. 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 ∅ }. (** * Morphisms *) 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) }. (** * Frame preserving updates *) Definition cmra_updateP {A : cmraT} (x : A) (P : A → Prop) := ∀ z n, ✓{S n} (x ⋅ z) → ∃ y, P y ∧ ✓{S n} (y ⋅ z). Instance: Params (@cmra_updateP) 3. Infix "⇝:" := cmra_updateP (at level 70). Definition cmra_update {A : cmraT} (x y : A) := ∀ z n, ✓{S n} (x ⋅ z) → ✓{S n} (y ⋅ z). Infix "⇝" := cmra_update (at level 70). Instance: Params (@cmra_update) 3. (** * Properties **) Section cmra. Context {A : cmraT}. Implicit Types x y z : A. Implicit Types xs ys zs : list A. (** ** 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. (** ** 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 *) 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. 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. 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. Proof. intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using cmra_included_l. by rewrite Hx1 Hx2. Qed. (** ** Minus *) Lemma cmra_op_minus' x y : x ≼ y → x ⋅ y ⩪ x ≡ y. Proof. rewrite cmra_included_includedN equiv_dist; eauto using cmra_op_minus. Qed. (** ** Timeless *) Lemma cmra_timeless_included_l x y : Timeless x → ✓{1} y → x ≼{1} y → x ≼ y. Proof. intros ?? [x' ?]. destruct (cmra_extend_op 1 y x x') as ([z z']&Hy&Hz&Hz'); auto; simpl in *. by exists z'; rewrite Hy (timeless x z). 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. Lemma cmra_op_timeless x1 x2 : ✓ (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2). Proof. intros ??? z Hz. destruct (cmra_extend_op 1 z x1 x2) as ([y1 y2]&Hz'&?&?); auto; simpl in *. { by rewrite -?Hz. } by rewrite Hz' (timeless x1 y1) // (timeless x2 y2). Qed. (** ** 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. (** ** Updates *) Global Instance cmra_update_preorder : PreOrder (@cmra_update A). Proof. split. by intros x y. intros x y y' ?? z ?; naive_solver. Qed. Lemma cmra_update_updateP x y : x ⇝ y ↔ x ⇝: (y =). Proof. split. * by intros Hx z ?; exists y; split; [done|apply (Hx z)]. * by intros Hx z n ?; destruct (Hx z n) as (?&<-&?). Qed. Lemma ra_updateP_id (P : A → Prop) x : P x → x ⇝: P. Proof. by intros ? z n ?; exists x. Qed. Lemma ra_updateP_compose (P Q : A → Prop) x : x ⇝: P → (∀ y, P y → y ⇝: Q) → x ⇝: Q. Proof. intros Hx Hy z n ?. destruct (Hx z n) as (y&?&?); auto. by apply (Hy y). Qed. Lemma ra_updateP_weaken (P Q : A → Prop) x : x ⇝: P → (∀ y, P y → Q y) → x ⇝: Q. Proof. eauto using ra_updateP_compose, ra_updateP_id. Qed. End cmra. Hint Extern 0 (_ ≼{0} _) => apply cmra_includedN_0. (** * Properties about monotone functions *) Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A). Proof. by split. Qed. Instance cmra_monotone_compose {A B C : cmraT} (f : A → B) (g : B → C) : CMRAMonotone f → CMRAMonotone g → CMRAMonotone (g ∘ f). Proof. split. * move=> n x y Hxy /=. by apply includedN_preserving, includedN_preserving. * move=> n x Hx /=. by apply validN_preserving, validN_preserving. Qed. 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 }. Section discrete. Context {A : cofeT} `{∀ x : A, Timeless x}. Context `{Unit A, Op A, Valid A, Minus A} (ra : RA A). Instance discrete_validN : ValidN A := λ n x, match n with 0 => True | S n => ✓ x end. Definition discrete_cmra_mixin : CMRAMixin A. Proof. 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. Qed. Definition discrete_extend_mixin : CMRAExtendMixin A. Proof. 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. Qed. Definition discreteRA : cmraT := CMRAT (cofe_mixin A) discrete_cmra_mixin discrete_extend_mixin. Lemma discrete_updateP (x : discreteRA) (P : A → Prop) : (∀ z, ✓ (x ⋅ z) → ∃ y, P y ∧ ✓ (y ⋅ z)) → x ⇝: P. Proof. intros Hvalid z n; apply Hvalid. Qed. Lemma discrete_update (x y : discreteRA) : (∀ z, ✓ (x ⋅ z) → ✓ (y ⋅ z)) → x ⇝ y. Proof. intros Hvalid z n; apply Hvalid. Qed. End discrete. (** ** 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. (** ** Product *) 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. * by split. * by intros n x [??]; split; apply cmra_validN_S. * split; simpl; apply (associative _). * split; simpl; apply (commutative _). * split; simpl; apply cmra_unit_l. * split; simpl; apply cmra_unit_idempotent. * intros n x y; rewrite !prod_includedN. by intros [??]; split; apply cmra_unit_preservingN. * intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l. * 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. 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. Lemma prod_update x y : x.1 ⇝ y.1 → x.2 ⇝ y.2 → x ⇝ y. Proof. intros ?? z n [??]; split; simpl in *; auto. Qed. Lemma prod_updateP (P : A * B → Prop) P1 P2 x : x.1 ⇝: P1 → x.2 ⇝: P2 → (∀ a b, P1 a → P2 b → P (a,b)) → x ⇝: P. 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. 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). Proof. split. * intros n x y; rewrite !prod_includedN; intros [??]; simpl. by split; apply includedN_preserving. * by intros n x [??]; split; simpl; apply validN_preserving. Qed. Definition prodRA_map {A A' B B' : cmraT} (f : A -n> A') (g : B -n> B') : prodRA A B -n> prodRA A' B' := CofeMor (prod_map f g : prodRA A B → prodRA A' B'). Instance prodRA_map_ne {A A' B B'} n : Proper (dist n==> dist n==> dist n) (@prodRA_map A A' B B') := prodC_map_ne n.