Require Export modures.ra modures.cofe. Class ValidN (A : Type) := validN : nat → A → Prop. Instance: Params (@validN) 3. Notation "✓{ n }" := (validN n) (at level 1, format "✓{ n }"). 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 ≼{_} ?x) => reflexivity. Class CMRA A `{Equiv A, Compl A, Unit A, Op A, Valid A, ValidN A, Minus A} := { (* setoids *) cmra_cofe :> Cofe A; cmra_op_ne n x :> Proper (dist n ==> dist n) (op x); cmra_unit_ne n :> Proper (dist n ==> dist n) unit; cmra_valid_ne n :> Proper (dist n ==> impl) (✓{n}); cmra_minus_ne n :> Proper (dist n ==> dist n ==> dist n) minus; (* valid *) cmra_valid_0 x : ✓{0} x; cmra_valid_S n x : ✓{S n} x → ✓{n} x; cmra_valid_validN x : ✓ x ↔ ∀ n, ✓{n} x; (* monoid *) cmra_associative : Associative (≡) (⋅); cmra_commutative : Commutative (≡) (⋅); cmra_unit_l x : unit x ⋅ x ≡ x; cmra_unit_idempotent x : unit (unit x) ≡ unit x; cmra_unit_preserving n x y : x ≼{n} y → unit x ≼{n} unit y; cmra_valid_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x; cmra_op_minus n x y : x ≼{n} y → x ⋅ y ⩪ x ={n}= y }. Class CMRAExtend A `{Equiv A, Dist A, Op A, ValidN A} := cmra_extend_op n x y1 y2 : ✓{n} x → x ={n}= y1 ⋅ y2 → { z | x ≡ z.1 ⋅ z.2 ∧ z ={n}= (y1,y2) }. Class CMRAMonotone `{Dist A, Op A, ValidN A, Dist B, Op B, ValidN B} (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) }. Hint Extern 0 (✓{0} _) => apply cmra_valid_0. (** 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_valid : Valid cmra_car; cmra_validN : ValidN cmra_car; cmra_minus : Minus cmra_car; cmra_cmra : CMRA cmra_car; cmra_extend : CMRAExtend 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_valid _ _ : simpl never. Arguments cmra_validN _ _ _ : simpl never. Arguments cmra_minus _ _ _ : simpl never. Arguments cmra_cmra _ : simpl never. Add Printing Constructor cmraT. Existing Instances cmra_equiv cmra_dist cmra_compl cmra_unit cmra_op cmra_valid cmra_validN cmra_minus cmra_cmra cmra_extend. Coercion cmra_cofeC (A : cmraT) : cofeT := CofeT A. Canonical Structure cmra_cofeC. (** Updates *) Definition cmra_updateP `{Op A, ValidN A} (x : A) (P : A → Prop) := ∀ z n, ✓{n} (x ⋅ z) → ∃ y, P y ∧ ✓{n} (y ⋅ z). Instance: Params (@cmra_updateP) 3. Infix "⇝:" := cmra_updateP (at level 70). Definition cmra_update `{Op A, ValidN A} (x y : A) := ∀ z n, ✓{n} (x ⋅ z) → ✓{n} (y ⋅ z). Infix "⇝" := cmra_update (at level 70). Instance: Params (@cmra_update) 3. (** Timeless elements *) Class ValidTimeless `{Valid A, ValidN A} (x : A) := valid_timeless : validN 1 x → valid x. Arguments valid_timeless {_ _ _} _ {_} _. (** Properties **) Section cmra. Context `{cmra : CMRA A}. Implicit Types x y z : A. 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_valid_ne' : Proper (dist n ==> iff) (✓{n}) | 1. Proof. by split; apply cmra_valid_ne. Qed. Global Instance cmra_valid_proper : Proper ((≡) ==> iff) (✓{n}) | 1. Proof. by intros n x1 x2 Hx; apply cmra_valid_ne', equiv_dist. Qed. Global Instance cmra_ra : RA A. Proof. split; try by (destruct cmra; eauto using ne_proper, ne_proper_2 with typeclass_instances). * by intros x1 x2 Hx; rewrite !cmra_valid_validN; intros ? n; rewrite <-Hx. * intros x y; rewrite !cmra_included_includedN. eauto using cmra_unit_preserving. * intros x y; rewrite !cmra_valid_validN; intros ? n. by apply cmra_valid_op_l with y. * intros x y [z Hz]; apply equiv_dist; intros n. by apply cmra_op_minus; exists z; rewrite Hz. Qed. Lemma cmra_valid_op_r x y n : ✓{n} (x ⋅ y) → ✓{n} y. Proof. rewrite (commutative _ x); apply cmra_valid_op_l. Qed. Lemma cmra_valid_le x n n' : ✓{n} x → n' ≤ n → ✓{n'} x. Proof. induction 2; eauto using cmra_valid_S. Qed. Global Instance ra_op_ne n : Proper (dist n ==> dist n ==> dist n) (⋅). Proof. intros x1 x2 Hx y1 y2 Hy. by rewrite Hy, (commutative _ x1), Hx, (commutative _ y2). Qed. Lemma cmra_unit_valid x n : ✓{n} x → ✓{n} (unit x). Proof. rewrite <-(cmra_unit_l x) at 1; apply cmra_valid_op_l. Qed. (* Timeless *) Lemma cmra_timeless_included_l `{!CMRAExtend A} 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) by done. 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 `{!CMRAExtend A} 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; apply cmra_valid_validN. } by rewrite Hz', (timeless x1 y1), (timeless x2 y2). Qed. (** * Included *) Global Instance cmra_included_ne n : Proper (dist n ==> dist n ==> iff) (includedN n) | 1. Proof. intros x x' Hx y y' Hy; unfold includedN. by setoid_rewrite Hx; setoid_rewrite Hy. Qed. Global Instance cmra_included_proper : Proper ((≡) ==> (≡) ==> iff) (includedN n) | 1. Proof. intros n x x' Hx y y' Hy; unfold includedN. by setoid_rewrite Hx; setoid_rewrite Hy. Qed. Lemma cmra_included_0 x y : x ≼{0} y. Proof. by exists (unit x). Qed. Lemma cmra_included_S x y n : x ≼{S n} y → x ≼{n} y. Proof. by intros [z Hz]; exists z; apply dist_S. Qed. Lemma cmra_included_l n x y : x ≼{n} x ⋅ y. Proof. by exists y. Qed. Lemma cmra_included_r n x y : y ≼{n} x ⋅ y. Proof. rewrite (commutative op); apply cmra_included_l. Qed. Global Instance cmra_included_ord n : PreOrder (includedN n). Proof. split. * by intros x; exists (unit x); rewrite ra_unit_r. * intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite (associative _), <-Hy, <-Hz. Qed. Lemma cmra_valid_includedN x y n : ✓{n} y → x ≼{n} y → ✓{n} x. Proof. intros Hyv [z Hy]; rewrite Hy in Hyv; eauto using cmra_valid_op_l. Qed. Lemma cmra_valid_included x y n : ✓{n} y → x ≼ y → ✓{n} x. Proof. rewrite cmra_included_includedN; eauto using cmra_valid_includedN. 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 ra_included_l. by rewrite Hx1, Hx2. Qed. (** * Properties of [(⇝)] relation *) Global Instance cmra_update_preorder : PreOrder cmra_update. 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. End cmra. Instance cmra_monotone_id `{CMRA A} : CMRAMonotone (@id A). Proof. by split. Qed. Instance cmra_monotone_ra_monotone `{CMRA A, CMRA B} (f : A → B) : CMRAMonotone f → RAMonotone f. Proof. split. * intros x y; rewrite !cmra_included_includedN. by intros ? n; apply includedN_preserving. * intros x; rewrite !cmra_valid_validN. by intros ? n; apply validN_preserving. Qed. Hint Extern 0 (_ ≼{0} _) => apply cmra_included_0. (* Also via [cmra_cofe; cofe_equivalence] *) Hint Cut [!*; ra_equivalence; cmra_ra] : typeclass_instances. (** Discrete *) Section discrete. Context `{ra : RA A}. Existing Instances discrete_dist discrete_compl. Let discrete_cofe' : Cofe A := discrete_cofe. Instance discrete_validN : ValidN A := λ n x, match n with 0 => True | S n => ✓ x end. Instance discrete_cmra : CMRA A. Proof. split; try by (destruct ra; auto). * by intros [|n]; try apply ra_op_proper. * by intros [|n]; try apply ra_unit_proper. * by intros [|n]; try apply ra_valid_proper. * by intros [|n]; try apply ra_minus_proper. * by intros [|n]. * intros x; split; intros Hvalid. by intros [|n]. apply (Hvalid 1). * by intros [|n]; try apply ra_unit_preserving. * by intros [|n]; try apply ra_valid_op_l. * by intros [|n] x y; try apply ra_op_minus. Qed. Instance discrete_extend : CMRAExtend A. Proof. intros [|n] x y1 y2 ??; [|by exists (y1,y2)]. by exists (x,unit x); simpl; rewrite ra_unit_r. Qed. Global Instance discrete_timeless (x : A) : ValidTimeless x. Proof. by intros ?. Qed. Definition discreteRA : cmraT := CMRAT A. Lemma discrete_updateP (x : A) (P : A → Prop) `{!Inhabited (sig P)} : (∀ z, ✓ (x ⋅ z) → ∃ y, P y ∧ ✓ (y ⋅ z)) → x ⇝: P. Proof. intros Hvalid z [|n]; [|apply Hvalid]. by destruct (_ : Inhabited (sig P)) as [[y ?]]; exists y. Qed. Lemma discrete_update (x y : A) : (∀ z, ✓ (x ⋅ z) → ✓ (y ⋅ z)) → x ⇝ y. Proof. intros Hvalid z [|n]; [done|apply Hvalid]. Qed. End discrete. Arguments discreteRA _ {_ _ _ _ _ _}. (** Product *) Instance prod_op `{Op A, Op B} : Op (A * B) := λ x y, (x.1 ⋅ y.1, x.2 ⋅ y.2). Instance prod_empty `{Empty A, Empty B} : Empty (A * B) := (∅, ∅). Instance prod_unit `{Unit A, Unit B} : Unit (A * B) := λ x, (unit (x.1), unit (x.2)). Instance prod_valid `{Valid A, Valid B} : Valid (A * B) := λ x, ✓ (x.1) ∧ ✓ (x.2). Instance prod_validN `{ValidN A, ValidN B} : ValidN (A * B) := λ n x, ✓{n} (x.1) ∧ ✓{n} (x.2). Instance prod_minus `{Minus A, Minus B} : Minus (A * B) := λ x y, (x.1 ⩪ y.1, x.2 ⩪ y.2). Lemma prod_included `{Equiv A, Equiv B, Op A, Op B} (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 `{Dist A, Dist B, Op A, Op B} (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. Instance prod_cmra `{CMRA A, CMRA B} : CMRA (A * B). Proof. split; try apply _. * by intros n x y1 y2 [Hy1 Hy2]; split; simpl in *; rewrite ?Hy1, ?Hy2. * by intros n y1 y2 [Hy1 Hy2]; split; simpl in *; rewrite ?Hy1, ?Hy2. * by intros n y1 y2 [Hy1 Hy2] [??]; split; simpl in *; rewrite <-?Hy1, <-?Hy2. * by intros n x1 x2 [Hx1 Hx2] y1 y2 [Hy1 Hy2]; split; simpl in *; rewrite ?Hx1, ?Hx2, ?Hy1, ?Hy2. * split; apply cmra_valid_0. * by intros n x [??]; split; apply cmra_valid_S. * intros x; split; [by intros [??] n; split; apply cmra_valid_validN|]. intros Hvalid; split; apply cmra_valid_validN; intros n; apply Hvalid. * split; simpl; apply (associative _). * split; simpl; apply (commutative _). * split; simpl; apply ra_unit_l. * split; simpl; apply ra_unit_idempotent. * intros n x y; rewrite !prod_includedN. by intros [??]; split; apply cmra_unit_preserving. * intros n x y [??]; split; simpl in *; eapply cmra_valid_op_l; eauto. * intros x y n; rewrite prod_includedN; intros [??]. by split; apply cmra_op_minus. Qed. Instance prod_ra_empty `{RAEmpty A, RAEmpty B} : RAEmpty (A * B). Proof. repeat split; simpl; repeat apply ra_empty_valid; repeat apply (left_id _ _). Qed. Instance prod_cmra_extend `{CMRAExtend A, CMRAExtend B} : CMRAExtend (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. Instance pair_timeless `{Valid A, ValidN A, Valid B, ValidN B} (x : A) (y : B) : ValidTimeless x → ValidTimeless y → ValidTimeless (x,y). Proof. by intros ?? [??]; split; apply (valid_timeless _). Qed. Canonical Structure prodRA (A B : cmraT) : cmraT := CMRAT (A * B). Instance prod_map_cmra_monotone `{CMRA A, CMRA A', CMRA B, CMRA B'} (f : A → A') (g : B → B') `{!CMRAMonotone f, !CMRAMonotone g} : CMRAMonotone (prod_map f g). Proof. split. * intros n x1 x2; 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.