diff --git a/modures/agree.v b/modures/agree.v index 3020bf8c83d863cca8759ee0c829cb533c49b170..8aa273ee8e85bd6c6a0b4b2248c1d97693f34b93 100644 --- a/modures/agree.v +++ b/modures/agree.v @@ -12,27 +12,27 @@ Arguments agree_car {_} _ _. Arguments agree_is_valid {_} _ _. Section agree. -Context `{Cofe A}. +Context {A : cofeT}. -Global Instance agree_validN : ValidN (agree A) := λ n x, +Instance agree_validN : ValidN (agree A) := λ n x, agree_is_valid x n ∧ ∀ n', n' ≤ n → x n' ={n'}= x n. Lemma agree_valid_le (x : agree A) n n' : agree_is_valid x n → n' ≤ n → agree_is_valid x n'. Proof. induction 2; eauto using agree_valid_S. Qed. -Global Instance agree_valid : Valid (agree A) := λ x, ∀ n, ✓{n} x. -Global Instance agree_equiv : Equiv (agree A) := λ x y, +Instance agree_valid : Valid (agree A) := λ x, ∀ n, ✓{n} x. +Instance agree_equiv : Equiv (agree A) := λ x y, (∀ n, agree_is_valid x n ↔ agree_is_valid y n) ∧ (∀ n, agree_is_valid x n → x n ={n}= y n). -Global Instance agree_dist : Dist (agree A) := λ n x y, +Instance agree_dist : Dist (agree A) := λ n x y, (∀ n', n' ≤ n → agree_is_valid x n' ↔ agree_is_valid y n') ∧ (∀ n', n' ≤ n → agree_is_valid x n' → x n' ={n'}= y n'). -Global Program Instance agree_compl : Compl (agree A) := λ c, +Program Instance agree_compl : Compl (agree A) := λ c, {| agree_car n := c n n; agree_is_valid n := agree_is_valid (c n) n |}. Next Obligation. intros; apply agree_valid_0. Qed. Next Obligation. intros c n ?; apply (chain_cauchy c n (S n)), agree_valid_S; auto. Qed. -Instance agree_cofe : Cofe (agree A). +Definition agree_cofe_mixin : CofeMixin (agree A). Proof. split. * intros x y; split. @@ -49,14 +49,15 @@ Proof. by split; intros; apply agree_valid_0. * by intros c n; split; intros; apply (chain_cauchy c). Qed. +Canonical Structure agreeC := CofeT agree_cofe_mixin. -Global Program Instance agree_op : Op (agree A) := λ x y, +Program Instance agree_op : Op (agree A) := λ x y, {| agree_car := x; agree_is_valid n := agree_is_valid x n ∧ agree_is_valid y n ∧ x ={n}= y |}. Next Obligation. by intros; simpl; split_ands; try apply agree_valid_0. Qed. Next Obligation. naive_solver eauto using agree_valid_S, dist_S. Qed. -Global Instance agree_unit : Unit (agree A) := id. -Global Instance agree_minus : Minus (agree A) := λ x y, x. +Instance agree_unit : Unit (agree A) := id. +Instance agree_minus : Minus (agree A) := λ x y, x. Instance: Commutative (≡) (@op (agree A) _). Proof. intros x y; split; [naive_solver|by intros n (?&?&Hxy); apply Hxy]. Qed. Definition agree_idempotent (x : agree A) : x ⋅ x ≡ x. @@ -70,7 +71,7 @@ Proof. * etransitivity; [apply Hxy|symmetry; apply Hy, Hy']; eauto using agree_valid_le. Qed. -Instance: Proper (dist n ==> dist n ==> dist n) op. +Instance: Proper (dist n ==> dist n ==> dist n) (@op (agree A) _). Proof. by intros n x1 x2 Hx y1 y2 Hy; rewrite Hy !(commutative _ _ y2) Hx. Qed. Instance: Proper ((≡) ==> (≡) ==> (≡)) op := ne_proper_2 _. Instance: Associative (≡) (@op (agree A) _). @@ -84,7 +85,7 @@ Proof. split; [|by intros ?; exists y]. by intros [z Hz]; rewrite Hz (associative _) agree_idempotent. Qed. -Global Instance agree_cmra : CMRA (agree A). +Definition agree_cmra_mixin : CMRAMixin (agree A). Proof. split; try (apply _ || done). * intros n x y Hxy [? Hx]; split; [by apply Hxy|intros n' ?]. @@ -103,12 +104,15 @@ Qed. Lemma agree_op_inv (x y1 y2 : agree A) n : ✓{n} x → x ={n}= y1 ⋅ y2 → y1 ={n}= y2. Proof. by intros [??] Hxy; apply Hxy. Qed. -Global Instance agree_extend : CMRAExtend (agree A). +Definition agree_cmra_extend_mixin : CMRAExtendMixin (agree A). Proof. intros n x y1 y2 ? Hx; exists (x,x); simpl; split. * by rewrite agree_idempotent. * by rewrite Hx (agree_op_inv x y1 y2) // agree_idempotent. Qed. +Canonical Structure agreeRA : cmraT := + CMRAT agree_cofe_mixin agree_cmra_mixin agree_cmra_extend_mixin. + Program Definition to_agree (x : A) : agree A := {| agree_car n := x; agree_is_valid n := True |}. Solve Obligations with done. @@ -125,12 +129,20 @@ Proof. Qed. End agree. +Arguments agreeC : clear implicits. +Arguments agreeRA : clear implicits. + Program Definition agree_map {A B} (f : A → B) (x : agree A) : agree B := {| agree_car n := f (x n); agree_is_valid := agree_is_valid x |}. Solve Obligations with auto using agree_valid_0, agree_valid_S. +Lemma agree_map_id {A} (x : agree A) : agree_map id x = x. +Proof. by destruct x. Qed. +Lemma agree_map_compose {A B C} (f : A → B) (g : B → C) + (x : agree A) : agree_map (g ∘ f) x = agree_map g (agree_map f x). +Proof. done. Qed. Section agree_map. - Context `{Cofe A, Cofe B} (f : A → B) `{Hf: ∀ n, Proper (dist n ==> dist n) f}. + Context {A B : cofeT} (f : A → B) `{Hf: ∀ n, Proper (dist n ==> dist n) f}. Global Instance agree_map_ne n : Proper (dist n ==> dist n) (agree_map f). Proof. by intros x1 x2 Hx; split; simpl; intros; [apply Hx|apply Hf, Hx]. Qed. Global Instance agree_map_proper : @@ -147,13 +159,7 @@ Section agree_map. try apply Hxy; try apply Hf; eauto using @agree_valid_le. Qed. End agree_map. -Lemma agree_map_id {A} (x : agree A) : agree_map id x = x. -Proof. by destruct x. Qed. -Lemma agree_map_compose {A B C} (f : A → B) (g : B → C) (x : agree A) : - agree_map (g ∘ f) x = agree_map g (agree_map f x). -Proof. done. Qed. -Canonical Structure agreeRA (A : cofeT) : cmraT := CMRAT (agree A). Definition agreeRA_map {A B} (f : A -n> B) : agreeRA A -n> agreeRA B := CofeMor (agree_map f : agreeRA A → agreeRA B). Instance agreeRA_map_ne A B n : Proper (dist n ==> dist n) (@agreeRA_map A B). diff --git a/modures/auth.v b/modures/auth.v index 44f63688500c3fe76a07f9f7c17e61ef7a01c984..4585be08b3e85e4b40acba74a0edda278056ddd6 100644 --- a/modures/auth.v +++ b/modures/auth.v @@ -10,22 +10,24 @@ Notation "◯ x" := (Auth ExclUnit x) (at level 20). Notation "● x" := (Auth (Excl x) ∅) (at level 20). (* COFE *) -Instance auth_equiv `{Equiv A} : Equiv (auth A) := λ x y, +Section cofe. +Context {A : cofeT}. + +Instance auth_equiv : Equiv (auth A) := λ x y, authoritative x ≡ authoritative y ∧ own x ≡ own y. -Instance auth_dist `{Dist A} : Dist (auth A) := λ n x y, +Instance auth_dist : Dist (auth A) := λ n x y, authoritative x ={n}= authoritative y ∧ own x ={n}= own y. -Instance Auth_ne `{Dist A} : Proper (dist n ==> dist n ==> dist n) (@Auth A). +Global Instance Auth_ne : Proper (dist n ==> dist n ==> dist n) (@Auth A). Proof. by split. Qed. -Instance authoritative_ne `{Dist A} : - Proper (dist n ==> dist n) (@authoritative A). +Global Instance authoritative_ne: Proper (dist n ==> dist n) (@authoritative A). Proof. by destruct 1. Qed. -Instance own_ne `{Dist A} : Proper (dist n ==> dist n) (@own A). +Global Instance own_ne : Proper (dist n ==> dist n) (@own A). Proof. by destruct 1. Qed. -Instance auth_compl `{Cofe A} : Compl (auth A) := λ c, +Instance auth_compl : Compl (auth A) := λ c, Auth (compl (chain_map authoritative c)) (compl (chain_map own c)). -Local Instance auth_cofe `{Cofe A} : Cofe (auth A). +Definition auth_cofe_mixin : CofeMixin (auth A). Proof. split. * intros x y; unfold dist, auth_dist, equiv, auth_equiv. @@ -39,53 +41,59 @@ Proof. * intros c n; split. apply (conv_compl (chain_map authoritative c) n). apply (conv_compl (chain_map own c) n). Qed. -Instance Auth_timeless `{Dist A, Equiv A} (x : excl A) (y : A) : +Canonical Structure authC := CofeT auth_cofe_mixin. +Instance Auth_timeless (x : excl A) (y : A) : Timeless x → Timeless y → Timeless (Auth x y). -Proof. by intros ?? [??] [??]; split; apply (timeless _). Qed. +Proof. by intros ?? [??] [??]; split; simpl in *; apply (timeless _). Qed. +End cofe. + +Arguments authC : clear implicits. (* CMRA *) -Instance auth_empty `{Empty A} : Empty (auth A) := Auth ∅ ∅. -Instance auth_valid `{Equiv A, Valid A, Op A} : Valid (auth A) := λ x, +Section cmra. +Context {A : cmraT}. + +Global Instance auth_empty `{Empty A} : Empty (auth A) := Auth ∅ ∅. +Instance auth_valid : Valid (auth A) := λ x, match authoritative x with | Excl a => own x ≼ a ∧ ✓ a | ExclUnit => ✓ (own x) | ExclBot => False end. -Arguments auth_valid _ _ _ _ !_ /. -Instance auth_validN `{Dist A, ValidN A, Op A} : ValidN (auth A) := λ n x, +Global Arguments auth_valid !_ /. +Instance auth_validN : ValidN (auth A) := λ n x, match authoritative x with | Excl a => own x ≼{n} a ∧ ✓{n} a | ExclUnit => ✓{n} (own x) | ExclBot => n = 0 end. -Arguments auth_validN _ _ _ _ _ !_ /. -Instance auth_unit `{Unit A} : Unit (auth A) := λ x, +Global Arguments auth_validN _ !_ /. +Instance auth_unit : Unit (auth A) := λ x, Auth (unit (authoritative x)) (unit (own x)). -Instance auth_op `{Op A} : Op (auth A) := λ x y, +Instance auth_op : Op (auth A) := λ x y, Auth (authoritative x ⋅ authoritative y) (own x ⋅ own y). -Instance auth_minus `{Minus A} : Minus (auth A) := λ x y, +Instance auth_minus : Minus (auth A) := λ x y, Auth (authoritative x ⩪ authoritative y) (own x ⩪ own y). -Lemma auth_included `{Equiv A, Op A} (x y : auth A) : +Lemma auth_included (x y : auth A) : x ≼ y ↔ authoritative x ≼ authoritative y ∧ own x ≼ own y. Proof. split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|]. intros [[z1 Hz1] [z2 Hz2]]; exists (Auth z1 z2); split; auto. Qed. -Lemma auth_includedN `{Dist A, Op A} n (x y : auth A) : +Lemma auth_includedN n (x y : auth A) : x ≼{n} y ↔ authoritative x ≼{n} authoritative y ∧ own x ≼{n} own y. Proof. split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|]. intros [[z1 Hz1] [z2 Hz2]]; exists (Auth z1 z2); split; auto. Qed. -Lemma authoritative_validN `{CMRA A} n (x : auth A) : - ✓{n} x → ✓{n} (authoritative x). +Lemma authoritative_validN n (x : auth A) : ✓{n} x → ✓{n} (authoritative x). Proof. by destruct x as [[]]. Qed. -Lemma own_validN `{CMRA A} n (x : auth A) : ✓{n} x → ✓{n} (own x). +Lemma own_validN n (x : auth A) : ✓{n} x → ✓{n} (own x). Proof. destruct x as [[]]; naive_solver eauto using cmra_valid_includedN. Qed. -Instance auth_cmra `{CMRA A} : CMRA (auth A). + +Definition auth_cmra_mixin : CMRAMixin (auth A). Proof. split. - * apply _. * by intros n x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'. * by intros n y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'. * intros n [x a] [y b] [Hx Ha]; simpl in *; @@ -103,14 +111,14 @@ Proof. * by split; simpl; rewrite ?(ra_unit_idempotent _). * intros n ??; rewrite! auth_includedN; intros [??]. by split; simpl; apply cmra_unit_preserving. - * assert (∀ n a b1 b2, b1 ⋅ b2 ≼{n} a → b1 ≼{n} a). + * assert (∀ n (a b1 b2 : A), b1 ⋅ b2 ≼{n} a → b1 ≼{n} a). { intros n a b1 b2 <-; apply cmra_included_l. } intros n [[a1| |] b1] [[a2| |] b2]; naive_solver eauto using cmra_valid_op_l, cmra_valid_includedN. * by intros n ??; rewrite auth_includedN; intros [??]; split; simpl; apply cmra_op_minus. Qed. -Instance auth_cmra_extend `{CMRA A, !CMRAExtend A} : CMRAExtend (auth A). +Definition auth_cmra_extend_mixin : CMRAExtendMixin (auth A). Proof. intros n x y1 y2 ? [??]; simpl in *. destruct (cmra_extend_op n (authoritative x) (authoritative y1) @@ -119,39 +127,49 @@ Proof. as (z2&?&?&?); auto using own_validN. by exists (Auth (z1.1) (z2.1), Auth (z1.2) (z2.2)). Qed. -Instance auth_ra_empty `{CMRA A, Empty A, !RAIdentity A} : RAIdentity (auth A). +Canonical Structure authRA : cmraT := + CMRAT auth_cofe_mixin auth_cmra_mixin auth_cmra_extend_mixin. +Instance auth_ra_empty `{Empty A} : RAIdentity A → RAIdentity (auth A). Proof. - split; [apply (ra_empty_valid (A:=A))|]. + split; simpl; [apply ra_empty_valid|]. by intros x; constructor; simpl; rewrite (left_id _ _). Qed. -Instance auth_frag_valid_timeless `{CMRA A} (x : A) : +Global Instance auth_frag_valid_timeless (x : A) : ValidTimeless x → ValidTimeless (◯ x). Proof. by intros ??; apply (valid_timeless x). Qed. -Instance auth_valid_timeless `{CMRA A, Empty A, !RAIdentity A} (x : A) : +Global Instance auth_valid_timeless `{Empty A, !RAIdentity A} (x : A) : ValidTimeless x → ValidTimeless (● x). Proof. by intros ? [??]; split; [apply ra_empty_least|apply (valid_timeless x)]. Qed. -Lemma auth_frag_op `{CMRA A} a b : ◯ (a ⋅ b) ≡ ◯ a ⋅ ◯ b. +Lemma auth_frag_op (a b : A) : ◯ (a ⋅ b) ≡ ◯ a ⋅ ◯ b. Proof. done. Qed. +Lemma auth_includedN' n (x y : authC A) : + x ≼{n} y ↔ authoritative x ≼{n} authoritative y ∧ own x ≼{n} own y. +Proof. + split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|]. + intros [[z1 Hz1] [z2 Hz2]]; exists (Auth z1 z2); split; auto. +Qed. +End cmra. + +Arguments authRA : clear implicits. (* Functor *) -Definition authRA (A : cmraT) : cmraT := CMRAT (auth A). Instance auth_fmap : FMap auth := λ A B f x, Auth (f <\$> authoritative x) (f (own x)). -Instance auth_fmap_cmra_ne `{Dist A, Dist B} n : +Instance auth_fmap_cmra_ne {A B : cmraT} n : Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@fmap auth _ A B). Proof. intros f g Hf [??] [??] [??]; split; [by apply excl_fmap_cmra_ne|by apply Hf]. Qed. -Instance auth_fmap_cmra_monotone `{CMRA A, CMRA B} (f : A → B) : +Instance auth_fmap_cmra_monotone {A B : cmraT} (f : A → B) : (∀ n, Proper (dist n ==> dist n) f) → CMRAMonotone f → CMRAMonotone (fmap f : auth A → auth B). Proof. split. - * by intros n [x a] [y b]; rewrite !auth_includedN; simpl; - intros [??]; split; apply includedN_preserving. - * intros n [[a| |] b]; + * by intros n [x a] [y b]; rewrite !auth_includedN /=; + intros [??]; split; simpl; apply: includedN_preserving. + * intros n [[a| |] b]; rewrite /= /cmra_validN; naive_solver eauto using @includedN_preserving, @validN_preserving. Qed. Definition authRA_map {A B : cmraT} (f : A -n> B) : authRA A -n> authRA B := diff --git a/modures/cmra.v b/modures/cmra.v index d974534afd84b53fe927ce579ef87da602dc7141..c31f970922625c0cbdb5db833635062d88e488ee 100644 --- a/modures/cmra.v +++ b/modures/cmra.v @@ -10,37 +10,29 @@ Notation "x ≼{ n } y" := (includedN n x y) 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} := { +Record CMRAMixin A + `{Dist A, Equiv 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; + 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_valid_ne n : Proper (dist n ==> impl) (✓{n}); + mixin_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; + mixin_cmra_valid_0 x : ✓{0} x; + mixin_cmra_valid_S n x : ✓{S n} x → ✓{n} x; + mixin_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 + 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_preserving n x y : x ≼{n} y → unit x ≼{n} unit y; + mixin_cmra_valid_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 }. -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. +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 { @@ -53,32 +45,73 @@ Structure cmraT := CMRAT { 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 + 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_valid _ _ : simpl never. -Arguments cmra_validN _ _ _ : simpl never. -Arguments cmra_minus _ _ _ : simpl never. -Arguments cmra_cmra _ : simpl never. +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_cofe_mixin : simpl never. +Arguments cmra_mixin : simpl never. +Arguments cmra_extend_mixin : 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. +Existing Instances cmra_unit cmra_op cmra_valid 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_valid_ne n : Proper (dist n ==> impl) (@validN A _ n). + Proof. apply (mixin_cmra_valid_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_valid_0 x : ✓{0} x. + Proof. apply (mixin_cmra_valid_0 _ (cmra_mixin A)). Qed. + Lemma cmra_valid_S n x : ✓{S n} x → ✓{n} x. + Proof. apply (mixin_cmra_valid_S _ (cmra_mixin A)). Qed. + Lemma cmra_valid_validN x : ✓ x ↔ ∀ n, ✓{n} x. + Proof. apply (mixin_cmra_valid_validN _ (cmra_mixin A)). Qed. + Lemma cmra_unit_preserving n x y : x ≼{n} y → unit x ≼{n} unit y. + Proof. apply (mixin_cmra_unit_preserving _ (cmra_mixin A)). Qed. + Lemma cmra_valid_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x. + Proof. apply (mixin_cmra_valid_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_valid_0. + +(** 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) +}. + (** Updates *) -Definition cmra_updateP `{Op A, ValidN A} (x : A) (P : A → Prop) := ∀ z n, +Definition cmra_updateP {A : cmraT} (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, +Definition cmra_update {A : cmraT} (x y : A) := ∀ z n, ✓{n} (x ⋅ z) → ✓{n} (y ⋅ z). Infix "⇝" := cmra_update (at level 70). Instance: Params (@cmra_update) 3. @@ -86,13 +119,13 @@ Instance: Params (@cmra_update) 3. (** Timeless validity *) (* Not sure whether this is useful, see the rule [uPred_valid_elim_timeless] in logic.v *) -Class ValidTimeless `{Valid A, ValidN A} (x : A) := +Class ValidTimeless {A : cmraT} (x : A) := valid_timeless : validN 1 x → valid x. -Arguments valid_timeless {_ _ _} _ {_} _. +Arguments valid_timeless {_} _ {_} _. (** Properties **) Section cmra. -Context `{cmra : CMRA A}. +Context {A : cmraT}. Implicit Types x y z : A. Lemma cmra_included_includedN x y : x ≼ y ↔ ∀ n, x ≼{n} y. @@ -102,17 +135,17 @@ Proof. symmetry; apply cmra_op_minus, Hxy. Qed. -Global Instance cmra_valid_ne' : Proper (dist n ==> iff) (✓{n}) | 1. +Global Instance cmra_valid_ne' : Proper (dist n ==> iff) (✓{n} : A → _) | 1. Proof. by split; apply cmra_valid_ne. Qed. -Global Instance cmra_valid_proper : Proper ((≡) ==> iff) (✓{n}) | 1. +Global Instance cmra_valid_proper : Proper ((≡) ==> iff) (✓{n} : A → _) | 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; + split; try by (destruct (@cmra_mixin A); 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. + 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. @@ -122,17 +155,16 @@ 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) (⋅). +Global Instance ra_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. Lemma cmra_unit_valid x n : ✓{n} x → ✓{n} (unit x). -Proof. rewrite -{1}(cmra_unit_l x); apply cmra_valid_op_l. Qed. +Proof. rewrite -{1}(ra_unit_l x); 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. +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 *. @@ -140,7 +172,7 @@ Proof. 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 : +Lemma cmra_op_timeless x1 x2 : ✓ (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2). Proof. intros ??? z Hz. @@ -151,13 +183,13 @@ Qed. (** * Included *) Global Instance cmra_included_ne n : - Proper (dist n ==> dist n ==> iff) (includedN n) | 1. + Proper (dist n ==> dist n ==> iff) (includedN n : relation A) | 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. + Proper ((≡) ==> (≡) ==> iff) (includedN n : relation A) | 1. Proof. intros n x x' Hx y y' Hy; unfold includedN. by setoid_rewrite Hx; setoid_rewrite Hy. @@ -171,7 +203,7 @@ 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). +Global Instance cmra_included_ord n : PreOrder (includedN n : relation A). Proof. split. * by intros x; exists (unit x); rewrite ra_unit_r. @@ -179,9 +211,9 @@ Proof. 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]; revert Hyv; rewrite Hy; apply cmra_valid_op_l. Qed. +Proof. intros Hyv [z ?]; cofe_subst y; 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. +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. @@ -190,7 +222,7 @@ Proof. Qed. (** * Properties of [(⇝)] relation *) -Global Instance cmra_update_preorder : PreOrder cmra_update. +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. @@ -200,9 +232,14 @@ Proof. Qed. End cmra. -Instance cmra_monotone_id `{CMRA A} : CMRAMonotone (@id A). +Hint Resolve cmra_ra. +Hint Extern 0 (_ ≼{0} _) => apply cmra_included_0. +(* Also via [cmra_cofe; cofe_equivalence] *) +Hint Cut [!*; ra_equivalence; cmra_ra] : typeclass_instances. + +Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A). Proof. by split. Qed. -Instance cmra_monotone_ra_monotone `{CMRA A, CMRA B} (f : A → B) : +Instance cmra_monotone_ra_monotone {A B : cmraT} (f : A → B) : CMRAMonotone f → RAMonotone f. Proof. split. @@ -212,18 +249,13 @@ Proof. 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. + Definition discrete_cmra_mixin : CMRAMixin A. Proof. split; try by (destruct ra; auto). * by intros [|n]; try apply ra_op_proper. @@ -236,91 +268,93 @@ Section discrete. * 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. + Definition discrete_extend_mixin : CMRAExtendMixin A. Proof. - intros [|n] x y1 y2 ??; [|by exists (y1,y2)]. + intros [|n] x y1 y2 ??; [|by exists (y1,y2); rewrite /dist /discrete_dist]. by exists (x,unit x); simpl; rewrite ra_unit_r. Qed. - Global Instance discrete_timeless (x : A) : ValidTimeless x. + Definition discreteRA : cmraT := + CMRAT discrete_cofe_mixin discrete_cmra_mixin discrete_extend_mixin. + Global Instance discrete_timeless (x : A) : ValidTimeless (x : discreteRA). 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. + (∀ z, ✓ (x ⋅ z) → ∃ y, P y ∧ ✓ (y ⋅ z)) → (x : discreteRA) ⇝: 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. + Lemma discrete_update (x y : A) : + (∀ z, ✓ (x ⋅ z) → ✓ (y ⋅ z)) → (x : discreteRA) ⇝ 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 `{RAIdentity A, RAIdentity B} : RAIdentity (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). +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_valid : Valid (A * B) := λ x, ✓ (x.1) ∧ ✓ (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. + * 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. + Global Instance prod_ra_empty `{Empty A, Empty B} : + RAIdentity A → RAIdentity B → RAIdentity (A * B). + Proof. + repeat split; simpl; repeat (apply ra_empty_valid || apply (left_id _ _)). + 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. + Instance pair_timeless (x : A) (y : B) : + ValidTimeless x → ValidTimeless y → ValidTimeless (x,y). + Proof. by intros ?? [??]; split; apply (valid_timeless _). 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 x1 x2; rewrite !prod_includedN; intros [??]; simpl. + * intros n x y; rewrite !prod_includedN; intros [??]; simpl. by split; apply includedN_preserving. * by intros n x [??]; split; simpl; apply validN_preserving. Qed. diff --git a/modures/cofe.v b/modures/cofe.v index bca6b342113729ea2cc88ba4bf69f7b709fb7a83..b508eb45fb94b62ded4c4ee3f41365d81eb08073 100644 --- a/modures/cofe.v +++ b/modures/cofe.v @@ -29,14 +29,13 @@ Arguments chain_car {_ _} _ _. Arguments chain_cauchy {_ _} _ _ _ _. Class Compl A `{Dist A} := compl : chain A → A. -Class Cofe A `{Equiv A, Compl A} := { - equiv_dist x y : x ≡ y ↔ ∀ n, x ={n}= y; - dist_equivalence n :> Equivalence (dist n); - dist_S n x y : x ={S n}= y → x ={n}= y; - dist_0 x y : x ={0}= y; - conv_compl (c : chain A) n : compl c ={n}= c n +Record CofeMixin A `{Equiv A, Compl A} := { + mixin_equiv_dist x y : x ≡ y ↔ ∀ n, x ={n}= y; + mixin_dist_equivalence n : Equivalence (dist n); + mixin_dist_S n x y : x ={S n}= y → x ={n}= y; + mixin_dist_0 x y : x ={0}= y; + mixin_conv_compl (c : chain A) n : compl c ={n}= c n }. -Hint Extern 0 (_ ={0}= _) => apply dist_0. Class Contractive `{Dist A, Dist B} (f : A -> B) := contractive n : Proper (dist n ==> dist (S n)) f. @@ -46,20 +45,39 @@ Structure cofeT := CofeT { cofe_equiv : Equiv cofe_car; cofe_dist : Dist cofe_car; cofe_compl : Compl cofe_car; - cofe_cofe : Cofe cofe_car + cofe_mixin : CofeMixin cofe_car }. -Arguments CofeT _ {_ _ _ _}. +Arguments CofeT {_ _ _ _} _. Add Printing Constructor cofeT. -Existing Instances cofe_equiv cofe_dist cofe_compl cofe_cofe. -Arguments cofe_car _ : simpl never. -Arguments cofe_equiv _ _ _ : simpl never. -Arguments cofe_dist _ _ _ _ : simpl never. -Arguments cofe_compl _ _ : simpl never. -Arguments cofe_cofe _ : simpl never. +Existing Instances cofe_equiv cofe_dist cofe_compl. +Arguments cofe_car : simpl never. +Arguments cofe_equiv : simpl never. +Arguments cofe_dist : simpl never. +Arguments cofe_compl : simpl never. +Arguments cofe_mixin : simpl never. + +(** Lifting properties from the mixin *) +Section cofe_mixin. + Context {A : cofeT}. + Implicit Types x y : A. + Lemma equiv_dist x y : x ≡ y ↔ ∀ n, x ={n}= y. + Proof. apply (mixin_equiv_dist _ (cofe_mixin A)). Qed. + Global Instance dist_equivalence n : Equivalence (@dist A _ n). + Proof. apply (mixin_dist_equivalence _ (cofe_mixin A)). Qed. + Lemma dist_S n x y : x ={S n}= y → x ={n}= y. + Proof. apply (mixin_dist_S _ (cofe_mixin A)). Qed. + Lemma dist_0 x y : x ={0}= y. + Proof. apply (mixin_dist_0 _ (cofe_mixin A)). Qed. + Lemma conv_compl (c : chain A) n : compl c ={n}= c n. + Proof. apply (mixin_conv_compl _ (cofe_mixin A)). Qed. +End cofe_mixin. + +Hint Extern 0 (_ ={0}= _) => apply dist_0. (** General properties *) Section cofe. - Context `{Cofe A}. + Context {A : cofeT}. + Implicit Types x y : A. Global Instance cofe_equivalence : Equivalence ((≡) : relation A). Proof. split. @@ -67,24 +85,24 @@ Section cofe. * by intros x y; rewrite !equiv_dist. * by intros x y z; rewrite !equiv_dist; intros; transitivity y. Qed. - Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (dist n). + Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n). Proof. intros x1 x2 ? y1 y2 ?; split; intros. * by transitivity x1; [|transitivity y1]. * by transitivity x2; [|transitivity y2]. Qed. - Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (dist n). + Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (@dist A _ n). Proof. by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n). Qed. Global Instance dist_proper_2 n x : Proper ((≡) ==> iff) (dist n x). Proof. by apply dist_proper. Qed. - Lemma dist_le x y n n' : x ={n}= y → n' ≤ n → x ={n'}= y. + Lemma dist_le (x y : A) n n' : x ={n}= y → n' ≤ n → x ={n'}= y. Proof. induction 2; eauto using dist_S. Qed. - Instance ne_proper `{Cofe B} (f : A → B) + Instance ne_proper {B : cofeT} (f : A → B) `{!∀ n, Proper (dist n ==> dist n) f} : Proper ((≡) ==> (≡)) f | 100. Proof. by intros x1 x2; rewrite !equiv_dist; intros Hx n; rewrite (Hx n). Qed. - Instance ne_proper_2 `{Cofe B, Cofe C} (f : A → B → C) + Instance ne_proper_2 {B C : cofeT} (f : A → B → C) `{!∀ n, Proper (dist n ==> dist n ==> dist n) f} : Proper ((≡) ==> (≡) ==> (≡)) f | 100. Proof. @@ -95,10 +113,10 @@ Section cofe. Proof. intros. by rewrite (conv_compl c1 n) (conv_compl c2 n). Qed. Lemma compl_ext (c1 c2 : chain A) : (∀ i, c1 i ≡ c2 i) → compl c1 ≡ compl c2. Proof. setoid_rewrite equiv_dist; naive_solver eauto using compl_ne. Qed. - Global Instance contractive_ne `{Cofe B} (f : A → B) `{!Contractive f} n : + Global Instance contractive_ne {B : cofeT} (f : A → B) `{!Contractive f} n : Proper (dist n ==> dist n) f | 100. Proof. by intros x1 x2 ?; apply dist_S, contractive. Qed. - Global Instance contractive_proper `{Cofe B} (f : A → B) `{!Contractive f} : + Global Instance contractive_proper {B : cofeT} (f : A → B) `{!Contractive f} : Proper ((≡) ==> (≡)) f | 100 := _. End cofe. @@ -106,24 +124,24 @@ End cofe. Program Definition chain_map `{Dist A, Dist B} (f : A → B) `{!∀ n, Proper (dist n ==> dist n) f} (c : chain A) : chain B := {| chain_car n := f (c n) |}. -Next Obligation. by intros A ? B ? f Hf c n i ?; apply Hf, chain_cauchy. Qed. +Next Obligation. by intros ? A ? B f Hf c n i ?; apply Hf, chain_cauchy. Qed. (** Timeless elements *) -Class Timeless `{Dist A, Equiv A} (x : A) := timeless y : x ={1}= y → x ≡ y. -Arguments timeless {_ _ _} _ {_} _ _. +Class Timeless {A : cofeT} (x : A) := timeless y : x ={1}= y → x ≡ y. +Arguments timeless {_} _ {_} _ _. (** Fixpoint *) -Program Definition fixpoint_chain `{Cofe A, Inhabited A} (f : A → A) +Program Definition fixpoint_chain {A : cofeT} `{Inhabited A} (f : A → A) `{!Contractive f} : chain A := {| chain_car i := Nat.iter i f inhabitant |}. Next Obligation. - intros A ???? f ? x n; induction n as [|n IH]; intros i ?; [done|]. + intros A ? f ? n; induction n as [|n IH]; intros i ?; first done. destruct i as [|i]; simpl; first lia; apply contractive, IH; auto with lia. Qed. -Program Definition fixpoint `{Cofe A, Inhabited A} (f : A → A) +Program Definition fixpoint {A : cofeT} `{Inhabited A} (f : A → A) `{!Contractive f} : A := compl (fixpoint_chain f). Section fixpoint. - Context `{Cofe A, Inhabited A} (f : A → A) `{!Contractive f}. + Context {A : cofeT} `{Inhabited A} (f : A → A) `{!Contractive f}. Lemma fixpoint_unfold : fixpoint f ≡ f (fixpoint f). Proof. apply equiv_dist; intros n; unfold fixpoint. @@ -153,46 +171,51 @@ Arguments CofeMor {_ _} _ {_}. Add Printing Constructor cofeMor. Existing Instance cofe_mor_ne. -Instance cofe_mor_proper `(f : cofeMor A B) : Proper ((≡) ==> (≡)) f := _. -Instance cofe_mor_equiv {A B : cofeT} : Equiv (cofeMor A B) := λ f g, - ∀ x, f x ≡ g x. -Instance cofe_mor_dist (A B : cofeT) : Dist (cofeMor A B) := λ n f g, - ∀ x, f x ={n}= g x. -Program Definition fun_chain `(c : chain (cofeMor A B)) (x : A) : chain B := - {| chain_car n := c n x |}. -Next Obligation. intros A B c x n i ?. by apply (chain_cauchy c). Qed. -Program Instance cofe_mor_compl (A B : cofeT) : Compl (cofeMor A B) := λ c, - {| cofe_mor_car x := compl (fun_chain c x) |}. -Next Obligation. - intros A B c n x y Hxy. - rewrite (conv_compl (fun_chain c x) n) (conv_compl (fun_chain c y) n) /= Hxy. - apply (chain_cauchy c); lia. -Qed. -Instance cofe_mor_cofe (A B : cofeT) : Cofe (cofeMor A B). -Proof. - split. - * intros X Y; split; [intros HXY n k; apply equiv_dist, HXY|]. - intros HXY k; apply equiv_dist; intros n; apply HXY. - * intros n; split. - + by intros f x. - + by intros f g ? x. - + by intros f g h ?? x; transitivity (g x). - * by intros n f g ? x; apply dist_S. - * by intros f g x. - * intros c n x; simpl. - rewrite (conv_compl (fun_chain c x) n); apply (chain_cauchy c); lia. -Qed. -Instance cofe_mor_car_ne A B n : - Proper (dist n ==> dist n ==> dist n) (@cofe_mor_car A B). -Proof. intros f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed. -Instance cofe_mor_car_proper A B : - Proper ((≡) ==> (≡) ==> (≡)) (@cofe_mor_car A B) := ne_proper_2 _. -Lemma cofe_mor_ext {A B} (f g : cofeMor A B) : f ≡ g ↔ ∀ x, f x ≡ g x. -Proof. done. Qed. -Canonical Structure cofe_mor (A B : cofeT) : cofeT := CofeT (cofeMor A B). +Section cofe_mor. + Context {A B : cofeT}. + Global Instance cofe_mor_proper (f : cofeMor A B) : Proper ((≡) ==> (≡)) f. + Proof. apply ne_proper, cofe_mor_ne. Qed. + Instance cofe_mor_equiv : Equiv (cofeMor A B) := λ f g, ∀ x, f x ≡ g x. + Instance cofe_mor_dist : Dist (cofeMor A B) := λ n f g, ∀ x, f x ={n}= g x. + Program Definition fun_chain `(c : chain (cofeMor A B)) (x : A) : chain B := + {| chain_car n := c n x |}. + Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed. + Program Instance cofe_mor_compl : Compl (cofeMor A B) := λ c, + {| cofe_mor_car x := compl (fun_chain c x) |}. + Next Obligation. + intros c n x y Hx. + rewrite (conv_compl (fun_chain c x) n) (conv_compl (fun_chain c y) n) /= Hx. + apply (chain_cauchy c); lia. + Qed. + Definition cofe_mor_cofe_mixin : CofeMixin (cofeMor A B). + Proof. + split. + * intros X Y; split; [intros HXY n k; apply equiv_dist, HXY|]. + intros HXY k; apply equiv_dist; intros n; apply HXY. + * intros n; split. + + by intros f x. + + by intros f g ? x. + + by intros f g h ?? x; transitivity (g x). + * by intros n f g ? x; apply dist_S. + * by intros f g x. + * intros c n x; simpl. + rewrite (conv_compl (fun_chain c x) n); apply (chain_cauchy c); lia. + Qed. + Canonical Structure cofe_mor : cofeT := CofeT cofe_mor_cofe_mixin. + + Global Instance cofe_mor_car_ne n : + Proper (dist n ==> dist n ==> dist n) (@cofe_mor_car A B). + Proof. intros f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed. + Global Instance cofe_mor_car_proper : + Proper ((≡) ==> (≡) ==> (≡)) (@cofe_mor_car A B) := ne_proper_2 _. + Lemma cofe_mor_ext (f g : cofeMor A B) : f ≡ g ↔ ∀ x, f x ≡ g x. + Proof. done. Qed. +End cofe_mor. + +Arguments cofe_mor : clear implicits. Infix "-n>" := cofe_mor (at level 45, right associativity). -Instance cofe_more_inhabited (A B : cofeT) - `{Inhabited B} : Inhabited (A -n> B) := populate (CofeMor (λ _, inhabitant)). +Instance cofe_more_inhabited {A B : cofeT} `{Inhabited B} : + Inhabited (A -n> B) := populate (CofeMor (λ _, inhabitant)). (** Identity and composition *) Definition cid {A} : A -n> A := CofeMor id. @@ -205,46 +228,47 @@ Lemma ccompose_ne {A B C} (f1 f2 : B -n> C) (g1 g2 : A -n> B) n : f1 ={n}= f2 → g1 ={n}= g2 → f1 ◎ g1 ={n}= f2 ◎ g2. Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed. -(** Pre-composition as a functor *) -Local Instance ccompose_l_ne' {A B C} (f : B -n> A) n : - Proper (dist n ==> dist n) (λ g : A -n> C, g ◎ f). -Proof. by intros g1 g2 ?; apply ccompose_ne. Qed. -Definition ccompose_l {A B C} (f : B -n> A) : (A -n> C) -n> (B -n> C) := - CofeMor (λ g : A -n> C, g ◎ f). -Instance ccompose_l_ne {A B C} : Proper (dist n ==> dist n) (@ccompose_l A B C). -Proof. by intros n f1 f2 Hf g x; apply ccompose_ne. Qed. - (** unit *) -Instance unit_dist : Dist unit := λ _ _ _, True. -Instance unit_compl : Compl unit := λ _, (). -Instance unit_cofe : Cofe unit. -Proof. by repeat split; try exists 0. Qed. +Section unit. + Instance unit_dist : Dist unit := λ _ _ _, True. + Instance unit_compl : Compl unit := λ _, (). + Definition unit_cofe_mixin : CofeMixin unit. + Proof. by repeat split; try exists 0. Qed. + Canonical Structure unitC : cofeT := CofeT unit_cofe_mixin. +End unit. (** Product *) -Instance prod_dist `{Dist A, Dist B} : Dist (A * B) := λ n, - prod_relation (dist n) (dist n). -Instance pair_ne `{Dist A, Dist B} : - Proper (dist n ==> dist n ==> dist n) (@pair A B) := _. -Instance fst_ne `{Dist A, Dist B} : Proper (dist n ==> dist n) (@fst A B) := _. -Instance snd_ne `{Dist A, Dist B} : Proper (dist n ==> dist n) (@snd A B) := _. -Instance prod_compl `{Compl A, Compl B} : Compl (A * B) := λ c, - (compl (chain_map fst c), compl (chain_map snd c)). -Instance prod_cofe `{Cofe A, Cofe B} : Cofe (A * B). -Proof. - split. - * intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation. - rewrite !equiv_dist; naive_solver. - * apply _. - * by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. - * by split. - * intros c n; split. apply (conv_compl (chain_map fst c) n). - apply (conv_compl (chain_map snd c) n). -Qed. -Instance pair_timeless `{Dist A, Equiv A, Dist B, Equiv B} (x : A) (y : B) : - Timeless x → Timeless y → Timeless (x,y). -Proof. by intros ?? [x' y'] [??]; split; apply (timeless _). Qed. -Canonical Structure prodC (A B : cofeT) : cofeT := CofeT (A * B). -Instance prod_map_ne `{Dist A, Dist A', Dist B, Dist B'} n : +Section product. + Context {A B : cofeT}. + + Instance prod_dist : Dist (A * B) := λ n, prod_relation (dist n) (dist n). + Global Instance pair_ne : + Proper (dist n ==> dist n ==> dist n) (@pair A B) := _. + Global Instance fst_ne : Proper (dist n ==> dist n) (@fst A B) := _. + Global Instance snd_ne : Proper (dist n ==> dist n) (@snd A B) := _. + Instance prod_compl : Compl (A * B) := λ c, + (compl (chain_map fst c), compl (chain_map snd c)). + Definition prod_cofe_mixin : CofeMixin (A * B). + Proof. + split. + * intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation. + rewrite !equiv_dist; naive_solver. + * apply _. + * by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. + * by split. + * intros c n; split. apply (conv_compl (chain_map fst c) n). + apply (conv_compl (chain_map snd c) n). + Qed. + Canonical Structure prodC : cofeT := CofeT prod_cofe_mixin. + Global Instance pair_timeless (x : A) (y : B) : + Timeless x → Timeless y → Timeless (x,y). + Proof. by intros ?? [x' y'] [??]; split; apply (timeless _). Qed. +End product. + +Arguments prodC : clear implicits. +Typeclasses Opaque prod_dist. + +Instance prod_map_ne {A A' B B' : cofeT} n : Proper ((dist n ==> dist n) ==> (dist n ==> dist n) ==> dist n ==> dist n) (@prod_map A A' B B'). Proof. by intros f f' Hf g g' Hg ?? [??]; split; [apply Hf|apply Hg]. Qed. @@ -254,15 +278,13 @@ Instance prodC_map_ne {A A' B B'} n : Proper (dist n ==> dist n ==> dist n) (@prodC_map A A' B B'). Proof. intros f f' Hf g g' Hg [??]; split; [apply Hf|apply Hg]. Qed. -Typeclasses Opaque prod_dist. - (** Discrete cofe *) Section discrete_cofe. Context `{Equiv A, @Equivalence A (≡)}. Instance discrete_dist : Dist A := λ n x y, match n with 0 => True | S n => x ≡ y end. Instance discrete_compl : Compl A := λ c, c 1. - Instance discrete_cofe : Cofe A. + Definition discrete_cofe_mixin : CofeMixin A. Proof. split. * intros x y; split; [by intros ? []|intros Hn; apply (Hn 1)]. @@ -271,9 +293,9 @@ Section discrete_cofe. * done. * intros c [|n]; [done|apply (chain_cauchy c 1 (S n)); lia]. Qed. - Global Instance discrete_timeless (x : A) : Timeless x. + Definition discreteC : cofeT := CofeT discrete_cofe_mixin. + Global Instance discrete_timeless (x : A) : Timeless (x : discreteC). Proof. by intros y. Qed. - Definition discreteC : cofeT := CofeT A. End discrete_cofe. Arguments discreteC _ {_ _}. @@ -288,16 +310,15 @@ Arguments Later {_} _. Arguments later_car {_} _. Section later. - Instance later_equiv `{Equiv A} : Equiv (later A) := λ x y, - later_car x ≡ later_car y. - Instance later_dist `{Dist A} : Dist (later A) := λ n x y, + Context {A : cofeT}. + Instance later_equiv : Equiv (later A) := λ x y, later_car x ≡ later_car y. + Instance later_dist : Dist (later A) := λ n x y, match n with 0 => True | S n => later_car x ={n}= later_car y end. - Program Definition later_chain `{Dist A} (c : chain (later A)) : chain A := + Program Definition later_chain (c : chain (later A)) : chain A := {| chain_car n := later_car (c (S n)) |}. - Next Obligation. intros A ? c n i ?; apply (chain_cauchy c (S n)); lia. Qed. - Instance later_compl `{Compl A} : Compl (later A) := λ c, - Later (compl (later_chain c)). - Instance later_cofe `{Cofe A} : Cofe (later A). + Next Obligation. intros c n i ?; apply (chain_cauchy c (S n)); lia. Qed. + Instance later_compl : Compl (later A) := λ c, Later (compl (later_chain c)). + Definition later_cofe_mixin : CofeMixin (later A). Proof. split. * intros x y; unfold equiv, later_equiv; rewrite !equiv_dist. @@ -310,23 +331,25 @@ Section later. * done. * intros c [|n]; [done|by apply (conv_compl (later_chain c) n)]. Qed. - Canonical Structure laterC (A : cofeT) : cofeT := CofeT (later A). - - Global Instance Later_contractive `{Dist A} : Contractive (@Later A). + Canonical Structure laterC : cofeT := CofeT later_cofe_mixin. + Global Instance Later_contractive : Contractive (@Later A). Proof. by intros n ??. Qed. - Definition later_map {A B} (f : A → B) (x : later A) : later B := - Later (f (later_car x)). - Global Instance later_map_ne `{Cofe A, Cofe B} (f : A → B) n : - Proper (dist (pred n) ==> dist (pred n)) f → - Proper (dist n ==> dist n) (later_map f) | 0. - Proof. destruct n as [|n]; intros Hf [x] [y] ?; do 2 red; simpl; auto. Qed. - Lemma later_fmap_id {A} (x : later A) : later_map id x = x. - Proof. by destruct x. Qed. - Lemma later_fmap_compose {A B C} (f : A → B) (g : B → C) (x : later A) : - later_map (g ∘ f) x = later_map g (later_map f x). - Proof. by destruct x. Qed. - Definition laterC_map {A B} (f : A -n> B) : laterC A -n> laterC B := - CofeMor (later_map f). - Instance laterC_map_contractive (A B : cofeT) : Contractive (@laterC_map A B). - Proof. intros n f g Hf n'; apply Hf. Qed. End later. + +Arguments laterC : clear implicits. + +Definition later_map {A B} (f : A → B) (x : later A) : later B := + Later (f (later_car x)). +Instance later_map_ne {A B : cofeT} (f : A → B) n : + Proper (dist (pred n) ==> dist (pred n)) f → + Proper (dist n ==> dist n) (later_map f) | 0. +Proof. destruct n as [|n]; intros Hf [x] [y] ?; do 2 red; simpl; auto. Qed. +Lemma later_map_id {A} (x : later A) : later_map id x = x. +Proof. by destruct x. Qed. +Lemma later_map_compose {A B C} (f : A → B) (g : B → C) (x : later A) : + later_map (g ∘ f) x = later_map g (later_map f x). +Proof. by destruct x. Qed. +Definition laterC_map {A B} (f : A -n> B) : laterC A -n> laterC B := + CofeMor (later_map f). +Instance laterC_map_contractive (A B : cofeT) : Contractive (@laterC_map A B). +Proof. intros n f g Hf n'; apply Hf. Qed. diff --git a/modures/cofe_solver.v b/modures/cofe_solver.v index d9dd6166ec057b6231dc391090a85ce8569734ba..984246d6b68b5b1b7e992b68042925b770a83f30 100644 --- a/modures/cofe_solver.v +++ b/modures/cofe_solver.v @@ -2,7 +2,7 @@ Require Export modures.cofe. Section solver. Context (F : cofeT → cofeT → cofeT). -Context `{Finhab : Inhabited (F (CofeT unit) (CofeT unit))}. +Context `{Finhab : Inhabited (F unitC unitC)}. Context (map : ∀ {A1 A2 B1 B2 : cofeT}, ((A2 -n> A1) * (B1 -n> B2)) → (F A1 B1 -n> F A2 B2)). Arguments map {_ _ _ _} _. @@ -20,11 +20,11 @@ Lemma map_ext {A1 A2 B1 B2 : cofeT} Proof. by rewrite -!cofe_mor_ext; intros -> -> ->. Qed. Fixpoint A (k : nat) : cofeT := - match k with 0 => CofeT unit | S k => F (A k) (A k) end. + match k with 0 => unitC | S k => F (A k) (A k) end. Fixpoint f {k} : A k -n> A (S k) := match k with 0 => CofeMor (λ _, inhabitant) | S k => map (g,f) end with g {k} : A (S k) -n> A k := - match k with 0 => CofeMor (λ _, () : CofeT ()) | S k => map (f,g) end. + match k with 0 => CofeMor (λ _, () : unitC) | S k => map (f,g) end. Definition f_S k (x : A (S k)) : f x = map (g,f) x := eq_refl. Definition g_S k (x : A (S (S k))) : g x = map (f,g) x := eq_refl. Lemma gf {k} (x : A k) : g (f x) ≡ x. @@ -58,7 +58,7 @@ Next Obligation. rewrite (conv_compl (tower_chain c k) n). by rewrite (conv_compl (tower_chain c (S k)) n) /= (g_tower (c n) k). Qed. -Instance tower_cofe : Cofe tower. +Definition tower_cofe_mixin : CofeMixin tower. Proof. split. * intros X Y; split; [by intros HXY n k; apply equiv_dist|]. @@ -73,7 +73,7 @@ Proof. * intros c n k; rewrite /= (conv_compl (tower_chain c k) n). apply (chain_cauchy c); lia. Qed. -Definition T : cofeT := CofeT tower. +Definition T : cofeT := CofeT tower_cofe_mixin. Fixpoint ff {k} (i : nat) : A k -n> A (i + k) := match i with 0 => cid | S i => f ◎ ff i end. diff --git a/modures/excl.v b/modures/excl.v index 1014ece16e034d961b3671a701390819ea802aa9..4e84b333b458b4455940386129cd60891b5d9bd4 100644 --- a/modures/excl.v +++ b/modures/excl.v @@ -12,8 +12,11 @@ Arguments ExclBot {_}. Instance maybe_Excl {A} : Maybe (@Excl A) := λ x, match x with Excl a => Some a | _ => None end. +Section excl. +Context {A : cofeT}. + (* Cofe *) -Inductive excl_equiv `{Equiv A} : Equiv (excl A) := +Inductive excl_equiv : Equiv (excl A) := | Excl_equiv (x y : A) : x ≡ y → Excl x ≡ Excl y | ExclUnit_equiv : ExclUnit ≡ ExclUnit | ExclBot_equiv : ExclBot ≡ ExclBot. @@ -24,21 +27,21 @@ Inductive excl_dist `{Dist A} : Dist (excl A) := | ExclUnit_dist n : ExclUnit ={n}= ExclUnit | ExclBot_dist n : ExclBot ={n}= ExclBot. Existing Instance excl_dist. -Program Definition excl_chain `{Cofe A} +Program Definition excl_chain (c : chain (excl A)) (x : A) (H : maybe Excl (c 1) = Some x) : chain A := {| chain_car n := match c n return _ with Excl y => y | _ => x end |}. Next Obligation. - intros A ???? c x ? n i ?; simpl; destruct (c 1) eqn:?; simplify_equality'. + intros c x ? n i ?; simpl; destruct (c 1) eqn:?; simplify_equality'. destruct (decide (i = 0)) as [->|]. { by replace n with 0 by lia. } feed inversion (chain_cauchy c 1 i); auto with lia congruence. feed inversion (chain_cauchy c n i); simpl; auto with lia congruence. Qed. -Instance excl_compl `{Cofe A} : Compl (excl A) := λ c, +Instance excl_compl : Compl (excl A) := λ c, match Some_dec (maybe Excl (c 1)) with | inleft (exist x H) => Excl (compl (excl_chain c x H)) | inright _ => c 1 end. -Local Instance excl_cofe `{Cofe A} : Cofe (excl A). +Definition excl_cofe_mixin : CofeMixin (excl A). Proof. split. * intros mx my; split; [by destruct 1; constructor; apply equiv_dist|]. @@ -61,39 +64,40 @@ Proof. feed inversion (chain_cauchy c 1 n); auto with lia; constructor. destruct (c 1); simplify_equality'. Qed. -Instance Excl_timeless `{Equiv A, Dist A} (x:A) :Timeless x → Timeless (Excl x). +Canonical Structure exclC : cofeT := CofeT excl_cofe_mixin. + +Global Instance Excl_timeless (x : A) : Timeless x → Timeless (Excl x). Proof. by inversion_clear 2; constructor; apply (timeless _). Qed. -Instance ExclUnit_timeless `{Equiv A, Dist A} : Timeless (@ExclUnit A). +Global Instance ExclUnit_timeless : Timeless (@ExclUnit A). Proof. by inversion_clear 1; constructor. Qed. -Instance ExclBot_timeless `{Equiv A, Dist A} : Timeless (@ExclBot A). +Global Instance ExclBot_timeless : Timeless (@ExclBot A). Proof. by inversion_clear 1; constructor. Qed. -Instance excl_timeless `{Equiv A, Dist A} : +Global Instance excl_timeless : (∀ x : A, Timeless x) → ∀ x : excl A, Timeless x. Proof. intros ? []; apply _. Qed. (* CMRA *) -Instance excl_valid {A} : Valid (excl A) := λ x, +Instance excl_valid : Valid (excl A) := λ x, match x with Excl _ | ExclUnit => True | ExclBot => False end. -Instance excl_validN {A} : ValidN (excl A) := λ n x, +Instance excl_validN : ValidN (excl A) := λ n x, match x with Excl _ | ExclUnit => True | ExclBot => n = 0 end. -Instance excl_empty {A} : Empty (excl A) := ExclUnit. -Instance excl_unit {A} : Unit (excl A) := λ _, ∅. -Instance excl_op {A} : Op (excl A) := λ x y, +Global Instance excl_empty : Empty (excl A) := ExclUnit. +Instance excl_unit : Unit (excl A) := λ _, ∅. +Instance excl_op : Op (excl A) := λ x y, match x, y with | Excl x, ExclUnit | ExclUnit, Excl x => Excl x | ExclUnit, ExclUnit => ExclUnit | _, _=> ExclBot end. -Instance excl_minus {A} : Minus (excl A) := λ x y, +Instance excl_minus : Minus (excl A) := λ x y, match x, y with | _, ExclUnit => x | Excl _, Excl _ => ExclUnit | _, _ => ExclBot end. -Instance excl_cmra `{Cofe A} : CMRA (excl A). +Definition excl_cmra_mixin : CMRAMixin (excl A). Proof. split. - * apply _. * by intros n []; destruct 1; constructor. * constructor. * by destruct 1 as [? []| | |]; intros ?. @@ -110,9 +114,9 @@ Proof. * by intros n [?| |] [?| |]. * by intros n [?| |] [?| |] [[?| |] Hz]; inversion_clear Hz; constructor. Qed. -Instance excl_empty_ra `{Cofe A} : RAIdentity (excl A). +Global Instance excl_empty_ra : RAIdentity (excl A). Proof. split. done. by intros []. Qed. -Instance excl_extend `{Cofe A} : CMRAExtend (excl A). +Definition excl_cmra_extend_mixin : CMRAExtendMixin (excl A). Proof. intros [|n] x y1 y2 ? Hx; [by exists (x,∅); destruct x|]. by exists match y1, y2 with @@ -121,23 +125,28 @@ Proof. | ExclUnit, _ => (ExclUnit, x) | _, ExclUnit => (x, ExclUnit) end; destruct y1, y2; inversion_clear Hx; repeat constructor. Qed. -Instance excl_valid_timeless {A} (x : excl A) : ValidTimeless x. +Canonical Structure exclRA : cmraT := + CMRAT excl_cofe_mixin excl_cmra_mixin excl_cmra_extend_mixin. +Global Instance excl_valid_timeless (x : excl A) : ValidTimeless x. Proof. by destruct x; intros ?. Qed. (* Updates *) -Lemma excl_update {A} (x : A) y : ✓ y → Excl x ⇝ y. +Lemma excl_update (x : A) y : ✓ y → Excl x ⇝ y. Proof. by destruct y; intros ? [?| |]. Qed. +End excl. + +Arguments exclC : clear implicits. +Arguments exclRA : clear implicits. (* Functor *) -Definition exclRA (A : cofeT) : cmraT := CMRAT (excl A). Instance excl_fmap : FMap excl := λ A B f x, match x with | Excl a => Excl (f a) | ExclUnit => ExclUnit | ExclBot => ExclBot end. -Instance excl_fmap_cmra_ne `{Dist A, Dist B} n : +Instance excl_fmap_cmra_ne {A B : cofeT} n : Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@fmap excl _ A B). Proof. by intros f f' Hf; destruct 1; constructor; apply Hf. Qed. -Instance excl_fmap_cmra_monotone `{Cofe A, Cofe B} : +Instance excl_fmap_cmra_monotone {A B : cofeT} : (∀ n, Proper (dist n ==> dist n) f) → CMRAMonotone (fmap f : excl A → excl B). Proof. split. @@ -145,7 +154,7 @@ Proof. by destruct x, z; constructor. * by intros n [a| |]. Qed. -Definition exclRA_map {A B : cofeT} (f : A -n> B) : exclRA A -n> exclRA B := +Definition exclRA_map {A B} (f : A -n> B) : exclRA A -n> exclRA B := CofeMor (fmap f : exclRA A → exclRA B). Lemma exclRA_map_ne A B n : Proper (dist n ==> dist n) (@exclRA_map A B). Proof. by intros f f' Hf []; constructor; apply Hf. Qed. diff --git a/modures/fin_maps.v b/modures/fin_maps.v index 9c25aa3320fd3775a3e96b6d9398963d5f3f35c2..a5331a804684f3b0d5e695c21013d52c2b701f3c 100644 --- a/modures/fin_maps.v +++ b/modures/fin_maps.v @@ -1,18 +1,17 @@ -Require Export modures.cmra prelude.gmap. -Require Import modures.option. +Require Export modures.cmra prelude.gmap modures.option. -Section map. -Context `{Countable K}. +Section cofe. +Context `{Countable K} {A : cofeT}. (* COFE *) -Global Instance map_dist `{Dist A} : Dist (gmap K A) := λ n m1 m2, +Instance map_dist : Dist (gmap K A) := λ n m1 m2, ∀ i, m1 !! i ={n}= m2 !! i. -Program Definition map_chain `{Dist A} (c : chain (gmap K A)) +Program Definition map_chain (c : chain (gmap K A)) (k : K) : chain (option A) := {| chain_car n := c n !! k |}. -Next Obligation. by intros A ? c k n i ?; apply (chain_cauchy c). Qed. -Global Instance map_compl `{Cofe A} : Compl (gmap K A) := λ c, +Next Obligation. by intros c k n i ?; apply (chain_cauchy c). Qed. +Instance map_compl : Compl (gmap K A) := λ c, map_imap (λ i _, compl (map_chain c i)) (c 1). -Global Instance map_cofe `{Cofe A} : Cofe (gmap K A). +Definition map_cofe_mixin : CofeMixin (gmap K A). Proof. split. * intros m1 m2; split. @@ -29,30 +28,32 @@ Proof. feed inversion (λ H, chain_cauchy c 1 n H k); simpl; auto with lia. by rewrite conv_compl; simpl; apply reflexive_eq. Qed. -Global Instance lookup_ne `{Dist A} n k : +Canonical Structure mapC : cofeT := CofeT map_cofe_mixin. + +Global Instance lookup_ne n k : Proper (dist n ==> dist n) (lookup k : gmap K A → option A). Proof. by intros m1 m2. Qed. -Global Instance insert_ne `{Dist A} (i : K) n : +Global Instance insert_ne (i : K) n : Proper (dist n ==> dist n ==> dist n) (insert (M:=gmap K A) i). Proof. intros x y ? m m' ? j; destruct (decide (i = j)); simplify_map_equality; [by constructor|by apply lookup_ne]. Qed. -Global Instance singleton_ne `{Cofe A} (i : K) n : +Global Instance singleton_ne (i : K) n : Proper (dist n ==> dist n) (singletonM i : A → gmap K A). Proof. by intros ???; apply insert_ne. Qed. -Global Instance delete_ne `{Dist A} (i : K) n : +Global Instance delete_ne (i : K) n : Proper (dist n ==> dist n) (delete (M:=gmap K A) i). Proof. intros m m' ? j; destruct (decide (i = j)); simplify_map_equality; [by constructor|by apply lookup_ne]. Qed. -Global Instance map_empty_timeless `{Dist A, Equiv A} : Timeless (∅ : gmap K A). +Global Instance map_empty_timeless : Timeless (∅ : gmap K A). Proof. intros m Hm i; specialize (Hm i); rewrite lookup_empty in Hm |- *. inversion_clear Hm; constructor. Qed. -Global Instance map_lookup_timeless `{Cofe A} (m : gmap K A) i : +Global Instance map_lookup_timeless (m : gmap K A) i : Timeless m → Timeless (m !! i). Proof. intros ? [x|] Hx; [|by symmetry; apply (timeless _)]. @@ -60,51 +61,41 @@ Proof. by (by symmetry in Hx; inversion Hx; cofe_subst; rewrite insert_id). by rewrite (timeless m (<[i:=x]>m)) // lookup_insert. Qed. -Global Instance map_ra_insert_timeless `{Cofe A} (m : gmap K A) i x : +Global Instance map_ra_insert_timeless (m : gmap K A) i x : Timeless x → Timeless m → Timeless (<[i:=x]>m). Proof. intros ?? m' Hm j; destruct (decide (i = j)); simplify_map_equality. { by apply (timeless _); rewrite -Hm lookup_insert. } by apply (timeless _); rewrite -Hm lookup_insert_ne. Qed. -Global Instance map_ra_singleton_timeless `{Cofe A} (i : K) (x : A) : +Global Instance map_ra_singleton_timeless (i : K) (x : A) : Timeless x → Timeless ({[ i ↦ x ]} : gmap K A) := _. -Global Instance map_fmap_ne `{Dist A, Dist B} (f : A → B) n : - Proper (dist n ==> dist n) f → - Proper (dist n ==> dist n) (fmap (M:=gmap K) f). -Proof. by intros ? m m' Hm k; rewrite !lookup_fmap; apply option_fmap_ne. Qed. -Canonical Structure mapC (A : cofeT) : cofeT := CofeT (gmap K A). -Definition mapC_map {A B} (f: A -n> B) : mapC A -n> mapC B := - CofeMor (fmap f : mapC A → mapC B). -Global Instance mapC_map_ne {A B} n : - Proper (dist n ==> dist n) (@mapC_map A B). -Proof. - intros f g Hf m k; rewrite /= !lookup_fmap. - destruct (_ !! k) eqn:?; simpl; constructor; apply Hf. -Qed. +End cofe. +Arguments mapC _ {_ _} _. (* CMRA *) -Global Instance map_op `{Op A} : Op (gmap K A) := merge op. -Global Instance map_unit `{Unit A} : Unit (gmap K A) := fmap unit. -Global Instance map_valid `{Valid A} : Valid (gmap K A) := λ m, ∀ i, ✓ (m !! i). -Global Instance map_validN `{ValidN A} : ValidN (gmap K A) := λ n m, - ∀ i, ✓{n} (m!!i). -Global Instance map_minus `{Minus A} : Minus (gmap K A) := merge minus. -Lemma lookup_op `{Op A} m1 m2 i : (m1 ⋅ m2) !! i = m1 !! i ⋅ m2 !! i. +Section cmra. +Context `{Countable K} {A : cmraT}. + +Instance map_op : Op (gmap K A) := merge op. +Instance map_unit : Unit (gmap K A) := fmap unit. +Instance map_valid : Valid (gmap K A) := λ m, ∀ i, ✓ (m !! i). +Instance map_validN : ValidN (gmap K A) := λ n m, ∀ i, ✓{n} (m!!i). +Instance map_minus : Minus (gmap K A) := merge minus. +Lemma lookup_op m1 m2 i : (m1 ⋅ m2) !! i = m1 !! i ⋅ m2 !! i. Proof. by apply lookup_merge. Qed. -Lemma lookup_minus `{Minus A} m1 m2 i : (m1 ⩪ m2) !! i = m1 !! i ⩪ m2 !! i. +Lemma lookup_minus m1 m2 i : (m1 ⩪ m2) !! i = m1 !! i ⩪ m2 !! i. Proof. by apply lookup_merge. Qed. -Lemma lookup_unit `{Unit A} m i : unit m !! i = unit (m !! i). +Lemma lookup_unit m i : unit m !! i = unit (m !! i). Proof. by apply lookup_fmap. Qed. -Lemma map_included_spec `{CMRA A} (m1 m2 : gmap K A) : - m1 ≼ m2 ↔ ∀ i, m1 !! i ≼ m2 !! i. +Lemma map_included_spec (m1 m2 : gmap K A) : m1 ≼ m2 ↔ ∀ i, m1 !! i ≼ m2 !! i. Proof. split. * by intros [m Hm]; intros i; exists (m !! i); rewrite -lookup_op Hm. * intros Hm; exists (m2 ⩪ m1); intros i. by rewrite lookup_op lookup_minus ra_op_minus. Qed. -Lemma map_includedN_spec `{CMRA A} (m1 m2 : gmap K A) n : +Lemma map_includedN_spec (m1 m2 : gmap K A) n : m1 ≼{n} m2 ↔ ∀ i, m1 !! i ≼{n} m2 !! i. Proof. split. @@ -112,10 +103,9 @@ Proof. * intros Hm; exists (m2 ⩪ m1); intros i. by rewrite lookup_op lookup_minus cmra_op_minus. Qed. -Global Instance map_cmra `{CMRA A} : CMRA (gmap K A). +Definition map_cmra_mixin : CMRAMixin (gmap K A). Proof. split. - * apply _. * by intros n m1 m2 m3 Hm i; rewrite !lookup_op (Hm i). * by intros n m1 m2 Hm i; rewrite !lookup_unit (Hm i). * by intros n m1 m2 Hm ? i; rewrite -(Hm i). @@ -135,13 +125,13 @@ Proof. * intros x y n; rewrite map_includedN_spec; intros ? i. by rewrite lookup_op lookup_minus cmra_op_minus. Qed. -Global Instance map_ra_empty `{RA A} : RAIdentity (gmap K A). +Global Instance map_ra_empty : RAIdentity (gmap K A). Proof. split. * by intros ?; rewrite lookup_empty. - * by intros m i; rewrite /= lookup_op lookup_empty; destruct (m !! i). + * by intros m i; rewrite /= lookup_op lookup_empty (left_id_L None _). Qed. -Global Instance map_cmra_extend `{CMRA A, !CMRAExtend A} : CMRAExtend (gmap K A). +Definition map_cmra_extend_mixin : CMRAExtendMixin (gmap K A). Proof. intros n m m1 m2 Hm Hm12. assert (∀ i, m !! i ={n}= m1 !! i ⋅ m2 !! i) as Hm12' @@ -150,20 +140,21 @@ Proof. set (f_proj i := proj1_sig (f i)). exists (map_imap (λ i _, (f_proj i).1) m, map_imap (λ i _, (f_proj i).2) m); repeat split; intros i; rewrite /= ?lookup_op !lookup_imap. - * destruct (m !! i) as [x|] eqn:Hx; simpl; [|constructor]. + * destruct (m !! i) as [x|] eqn:Hx; rewrite !Hx /=; [|constructor]. rewrite -Hx; apply (proj2_sig (f i)). - * destruct (m !! i) as [x|] eqn:Hx; simpl; [apply (proj2_sig (f i))|]. + * destruct (m !! i) as [x|] eqn:Hx; rewrite /=; [apply (proj2_sig (f i))|]. pose proof (Hm12' i) as Hm12''; rewrite Hx in Hm12''. - by destruct (m1 !! i), (m2 !! i); inversion_clear Hm12''. + by symmetry; apply option_op_positive_dist_l with (m2 !! i). * destruct (m !! i) as [x|] eqn:Hx; simpl; [apply (proj2_sig (f i))|]. pose proof (Hm12' i) as Hm12''; rewrite Hx in Hm12''. - by destruct (m1 !! i), (m2 !! i); inversion_clear Hm12''. + by symmetry; apply option_op_positive_dist_r with (m1 !! i). Qed. -Global Instance map_empty_valid_timeless `{Valid A, ValidN A} : - ValidTimeless (∅ : gmap K A). +Canonical Structure mapRA : cmraT := + CMRAT map_cofe_mixin map_cmra_mixin map_cmra_extend_mixin. + +Global Instance map_empty_valid_timeless : ValidTimeless (∅ : gmap K A). Proof. by intros ??; rewrite lookup_empty. Qed. -Global Instance map_ra_insert_valid_timeless `{Valid A, ValidN A} - (m : gmap K A) i x: +Global Instance map_ra_insert_valid_timeless (m : gmap K A) i x: ValidTimeless x → ValidTimeless m → m !! i = None → ValidTimeless (<[i:=x]>m). Proof. @@ -173,48 +164,40 @@ Proof. intros j; destruct (decide (i = j)); simplify_map_equality;[by rewrite Hi|]. by specialize (Hm j); simplify_map_equality. Qed. -Global Instance map_ra_singleton_valid_timeless `{Valid A, ValidN A} (i : K) x : - ValidTimeless x → ValidTimeless ({[ i ↦ x ]} : gmap K A). +Global Instance map_ra_singleton_valid_timeless (i : K) x : + ValidTimeless x → ValidTimeless {[ i ↦ x ]}. Proof. intros ?; apply (map_ra_insert_valid_timeless _ _ _ _ _). by rewrite lookup_empty. Qed. -Lemma map_insert_valid `{ValidN A} (m : gmap K A) n i x : - ✓{n} x → ✓{n} m → ✓{n} (<[i:=x]>m). +End cmra. +Arguments mapRA _ {_ _} _. + +Section properties. +Context `{Countable K} {A: cmraT}. + +Lemma map_insert_valid (m :gmap K A) n i x : ✓{n} x → ✓{n} m → ✓{n} (<[i:=x]>m). Proof. by intros ?? j; destruct (decide (i = j)); simplify_map_equality. Qed. -Lemma map_insert_op `{Op A} (m1 m2 : gmap K A) i x : +Lemma map_insert_op (m1 m2 : gmap K A) i x : m2 !! i = None → <[i:=x]>(m1 ⋅ m2) = <[i:=x]>m1 ⋅ m2. -Proof. by intros Hi; apply (insert_merge_l _); rewrite Hi. Qed. -Lemma map_singleton_op `{Op A} (i : K) (x y : A) : - ({[ i ↦ x ⋅ y ]} : gmap K A) = {[ i ↦ x ]} ⋅ {[ i ↦ y ]}. -Proof. by apply symmetry, merge_singleton. Qed. - -Canonical Structure mapRA (A : cmraT) : cmraT := CMRAT (gmap K A). -Global Instance map_fmap_cmra_monotone `{CMRA A, CMRA B} (f : A → B) - `{!CMRAMonotone f} : CMRAMonotone (fmap f : gmap K A → gmap K B). -Proof. - split. - * intros m1 m2 n; rewrite !map_includedN_spec; intros Hm i. - by rewrite !lookup_fmap; apply includedN_preserving. - * by intros n m ? i; rewrite lookup_fmap; apply validN_preserving. -Qed. -Definition mapRA_map {A B : cmraT} (f : A -n> B) : mapRA A -n> mapRA B := - CofeMor (fmap f : mapRA A → mapRA B). -Global Instance mapRA_map_ne {A B} n : - Proper (dist n ==> dist n) (@mapRA_map A B) := mapC_map_ne n. -Global Instance mapRA_map_monotone {A B : cmraT} (f : A -n> B) - `{!CMRAMonotone f} : CMRAMonotone (mapRA_map f) := _. +Proof. by intros Hi; apply (insert_merge_l _ m1 m2); rewrite Hi. Qed. +Lemma map_unit_singleton (i : K) (x : A) : + unit ({[ i ↦ x ]} : gmap K A) = {[ i ↦ unit x ]}. +Proof. apply map_fmap_singleton. Qed. +Lemma map_op_singleton (i : K) (x y : A) : + {[ i ↦ x ]} ⋅ {[ i ↦ y ]} = ({[ i ↦ x ⋅ y ]} : gmap K A). +Proof. by apply (merge_singleton _ _ _ x y). Qed. Context `{Fresh K (gset K), !FreshSpec K (gset K)}. -Lemma map_dom_op `{Op A} (m1 m2 : gmap K A) : +Lemma map_dom_op (m1 m2 : gmap K A) : dom (gset K) (m1 ⋅ m2) ≡ dom _ m1 ∪ dom _ m2. Proof. apply elem_of_equiv; intros i; rewrite elem_of_union !elem_of_dom. unfold is_Some; setoid_rewrite lookup_op. destruct (m1 !! i), (m2 !! i); naive_solver. Qed. -Lemma map_update_alloc `{CMRA A} (m : gmap K A) x : +Lemma map_update_alloc (m : gmap K A) x : ✓ x → m ⇝: λ m', ∃ i, m' = <[i:=x]>m ∧ m !! i = None. Proof. intros ? mf n Hm. set (i := fresh (dom (gset K) (m ⋅ mf))). @@ -225,6 +208,31 @@ Proof. * rewrite -map_insert_op; last by apply not_elem_of_dom. by apply map_insert_valid; [apply cmra_valid_validN|]. Qed. -End map. -Arguments mapC _ {_ _} _. -Arguments mapRA _ {_ _} _. +End properties. + +Instance map_fmap_ne `{Countable K} {A B : cofeT} (f : A → B) n : + Proper (dist n ==> dist n) f → Proper (dist n ==>dist n) (fmap (M:=gmap K) f). +Proof. by intros ? m m' Hm k; rewrite !lookup_fmap; apply option_fmap_ne. Qed. +Definition mapC_map `{Countable K} {A B} (f: A -n> B) : mapC K A -n> mapC K B := + CofeMor (fmap f : mapC K A → mapC K B). +Instance mapC_map_ne `{Countable K} {A B} n : + Proper (dist n ==> dist n) (@mapC_map K _ _ A B). +Proof. + intros f g Hf m k; rewrite /= !lookup_fmap. + destruct (_ !! k) eqn:?; simpl; constructor; apply Hf. +Qed. + +Instance map_fmap_cmra_monotone `{Countable K} {A B : cmraT} (f : A → B) + `{!CMRAMonotone f} : CMRAMonotone (fmap f : gmap K A → gmap K B). +Proof. + split. + * intros m1 m2 n; rewrite !map_includedN_spec; intros Hm i. + by rewrite !lookup_fmap; apply: includedN_preserving. + * by intros n m ? i; rewrite lookup_fmap; apply validN_preserving. +Qed. +Definition mapRA_map `{Countable K} {A B : cmraT} (f : A -n> B) : + mapRA K A -n> mapRA K B := CofeMor (fmap f : mapRA K A → mapRA K B). +Instance mapRA_map_ne `{Countable K} {A B} n : + Proper (dist n ==> dist n) (@mapRA_map K _ _ A B) := mapC_map_ne n. +Instance mapRA_map_monotone `{Countable K} {A B : cmraT} (f : A -n> B) + `{!CMRAMonotone f} : CMRAMonotone (mapRA_map f) := _. diff --git a/modures/logic.v b/modures/logic.v index c717b2b87505e6c09527e6c0e150011df064c7b1..791a9366fd5138eecca0ae68926020fbfdebb7e8 100644 --- a/modures/logic.v +++ b/modures/logic.v @@ -15,36 +15,41 @@ Hint Resolve uPred_0. Add Printing Constructor uPred. Instance: Params (@uPred_holds) 3. -Instance uPred_equiv (M : cmraT) : Equiv (uPred M) := λ P Q, ∀ x n, - ✓{n} x → P n x ↔ Q n x. -Instance uPred_dist (M : cmraT) : Dist (uPred M) := λ n P Q, ∀ x n', - n' ≤ n → ✓{n'} x → P n' x ↔ Q n' x. -Program Instance uPred_compl (M : cmraT) : Compl (uPred M) := λ c, - {| uPred_holds n x := c n n x |}. -Next Obligation. by intros M c x y n ??; simpl in *; apply uPred_ne with x. Qed. -Next Obligation. by intros M c x; simpl. Qed. -Next Obligation. - intros M c x1 x2 n1 n2 ????; simpl in *. - apply (chain_cauchy c n2 n1); eauto using uPred_weaken. -Qed. -Instance uPred_cofe (M : cmraT) : Cofe (uPred M). -Proof. - split. - * intros P Q; split; [by intros HPQ n x i ??; apply HPQ|]. - intros HPQ x n ?; apply HPQ with n; auto. - * intros n; split. - + by intros P x i. - + by intros P Q HPQ x i ??; symmetry; apply HPQ. - + by intros P Q Q' HP HQ x i ??; transitivity (Q i x); [apply HP|apply HQ]. - * intros n P Q HPQ x i ??; apply HPQ; auto. - * intros P Q x i; rewrite Nat.le_0_r; intros ->; split; intros; apply uPred_0. - * by intros c n x i ??; apply (chain_cauchy c i n). -Qed. +Section cofe. + Context {M : cmraT}. + Instance uPred_equiv : Equiv (uPred M) := λ P Q, ∀ x n, + ✓{n} x → P n x ↔ Q n x. + Instance uPred_dist : Dist (uPred M) := λ n P Q, ∀ x n', + n' ≤ n → ✓{n'} x → P n' x ↔ Q n' x. + Program Instance uPred_compl : Compl (uPred M) := λ c, + {| uPred_holds n x := c n n x |}. + Next Obligation. by intros c x y n ??; simpl in *; apply uPred_ne with x. Qed. + Next Obligation. by intros c x; simpl. Qed. + Next Obligation. + intros c x1 x2 n1 n2 ????; simpl in *. + apply (chain_cauchy c n2 n1); eauto using uPred_weaken. + Qed. + Definition uPred_cofe_mixin : CofeMixin (uPred M). + Proof. + split. + * intros P Q; split; [by intros HPQ n x i ??; apply HPQ|]. + intros HPQ x n ?; apply HPQ with n; auto. + * intros n; split. + + by intros P x i. + + by intros P Q HPQ x i ??; symmetry; apply HPQ. + + by intros P Q Q' HP HQ x i ??; transitivity (Q i x); [apply HP|apply HQ]. + * intros n P Q HPQ x i ??; apply HPQ; auto. + * intros P Q x i; rewrite Nat.le_0_r; intros ->; split; intros; apply uPred_0. + * by intros c n x i ??; apply (chain_cauchy c i n). + Qed. + Canonical Structure uPredC : cofeT := CofeT uPred_cofe_mixin. +End cofe. +Arguments uPredC : clear implicits. + Instance uPred_holds_ne {M} (P : uPred M) n : Proper (dist n ==> iff) (P n). Proof. intros x1 x2 Hx; split; eauto using uPred_ne. Qed. Instance uPred_holds_proper {M} (P : uPred M) n : Proper ((≡) ==> iff) (P n). Proof. by intros x1 x2 Hx; apply uPred_holds_ne, equiv_dist. Qed. -Canonical Structure uPredC (M : cmraT) : cofeT := CofeT (uPred M). (** functor *) Program Definition uPred_map {M1 M2 : cmraT} (f : M2 → M1) @@ -125,7 +130,7 @@ Next Obligation. by intros M P Q x; exists x, x. Qed. Next Obligation. intros M P Q x y n1 n2 (x1&x2&Hx&?&?) Hxy ??. assert (∃ x2', y ={n2}= x1 ⋅ x2' ∧ x2 ≼ x2') as (x2'&Hy&?). - { destruct Hxy as [z Hy]; exists (x2 ⋅ z); split; eauto using ra_included_l. + { destruct Hxy as [z Hy]; exists (x2 ⋅ z); split; eauto using @ra_included_l. apply dist_le with n1; auto. by rewrite (associative op) -Hx Hy. } exists x1, x2'; split_ands; [done| |]. * cofe_subst y; apply uPred_weaken with x1 n1; eauto using cmra_valid_op_l. @@ -144,7 +149,7 @@ Next Obligation. intros M P Q x1 x2 [|n]; auto with lia. Qed. Next Obligation. intros M P Q x1 x2 n1 n2 HPQ ??? x3 n3 ???; simpl in *. apply uPred_weaken with (x1 ⋅ x3) n3; - eauto using cmra_valid_included, ra_preserving_r. + eauto using cmra_valid_included, @ra_preserving_r. Qed. Program Definition uPred_later {M} (P : uPred M) : uPred M := @@ -160,7 +165,7 @@ Next Obligation. by intros M P x1 x2 n ? Hx; simpl in *; rewrite <-Hx. Qed. Next Obligation. by intros; simpl. Qed. Next Obligation. intros M P x1 x2 n1 n2 ????; eapply uPred_weaken with (unit x1) n1; - auto using ra_unit_preserving, cmra_unit_valid. + eauto using @ra_unit_preserving, cmra_unit_valid. Qed. Program Definition uPred_own {M : cmraT} (a : M) : uPred M := @@ -480,7 +485,7 @@ Qed. Global Instance True_sep : LeftId (≡) True%I (@uPred_sep M). Proof. intros P x n Hvalid; split. - * intros (x1&x2&?&_&?); cofe_subst; eauto using uPred_weaken, ra_included_r. + * intros (x1&x2&?&_&?); cofe_subst; eauto using uPred_weaken, @ra_included_r. * by destruct n as [|n]; [|intros ?; exists (unit x), x; rewrite ra_unit_l]. Qed. Global Instance sep_commutative : Commutative (≡) (@uPred_sep M). @@ -555,11 +560,11 @@ Proof. by apply forall_intro; intros a; rewrite ->forall_elim. Qed. (* Later *) Lemma later_mono P Q : P ⊆ Q → (▷ P)%I ⊆ (▷ Q)%I. -Proof. intros HP x [|n] ??; [done|apply HP; auto using cmra_valid_S]. Qed. +Proof. intros HP x [|n] ??; [done|apply HP; eauto using cmra_valid_S]. Qed. Lemma later_intro P : P ⊆ (▷ P)%I. Proof. intros x [|n] ??; simpl in *; [done|]. - apply uPred_weaken with x (S n); auto using cmra_valid_S. + apply uPred_weaken with x (S n); eauto using cmra_valid_S. Qed. Lemma lub P : (▷ P → P)%I ⊆ P. Proof. @@ -582,10 +587,10 @@ Proof. intros x n ?; split. * destruct n as [|n]; simpl; [by exists x, x|intros (x1&x2&Hx&?&?)]. destruct (cmra_extend_op n x x1 x2) - as ([y1 y2]&Hx'&Hy1&Hy2); auto using cmra_valid_S; simpl in *. + as ([y1 y2]&Hx'&Hy1&Hy2); eauto using cmra_valid_S; simpl in *. exists y1, y2; split; [by rewrite Hx'|by rewrite Hy1 Hy2]. * destruct n as [|n]; simpl; [done|intros (x1&x2&Hx&?&?)]. - exists x1, x2; eauto using (dist_S (A:=M)). + exists x1, x2; eauto using dist_S. Qed. (* Later derived *) @@ -604,7 +609,8 @@ Lemma always_const (P : Prop) : (□ uPred_const P : uPred M)%I ≡ uPred_const Proof. done. Qed. Lemma always_elim P : (□ P)%I ⊆ P. Proof. - intros x n ??; apply uPred_weaken with (unit x) n;auto using ra_included_unit. + intros x n ??; apply uPred_weaken with (unit x) n; + eauto using @ra_included_unit. Qed. Lemma always_intro P Q : (□ P)%I ⊆ Q → (□ P)%I ⊆ (□ Q)%I. Proof. diff --git a/modures/option.v b/modures/option.v index aec24387ae421e6826a2729e93c6bd58b3dd5303..2df111ca30ce3a8e2f2061c30c771963601ce534 100644 --- a/modures/option.v +++ b/modures/option.v @@ -1,25 +1,27 @@ Require Export modures.cmra. (* COFE *) -Inductive option_dist `{Dist A} : Dist (option A) := +Section cofe. +Context {A : cofeT}. +Inductive option_dist : Dist (option A) := | option_0_dist (x y : option A) : x ={0}= y | Some_dist n x y : x ={n}= y → Some x ={n}= Some y | None_dist n : None ={n}= None. Existing Instance option_dist. -Program Definition option_chain `{Cofe A} +Program Definition option_chain (c : chain (option A)) (x : A) (H : c 1 = Some x) : chain A := {| chain_car n := from_option x (c n) |}. Next Obligation. - intros A ???? c x ? n i ?; simpl; destruct (decide (i = 0)) as [->|]. + intros c x ? n i ?; simpl; destruct (decide (i = 0)) as [->|]. { by replace n with 0 by lia. } feed inversion (chain_cauchy c 1 i); auto with lia congruence. feed inversion (chain_cauchy c n i); simpl; auto with lia congruence. Qed. -Instance option_compl `{Cofe A} : Compl (option A) := λ c, +Instance option_compl : Compl (option A) := λ c, match Some_dec (c 1) with | inleft (exist x H) => Some (compl (option_chain c x H)) | inright _ => None end. -Instance option_cofe `{Cofe A} : Cofe (option A). +Definition option_cofe_mixin : CofeMixin (option A). Proof. split. * intros mx my; split; [by destruct 1; constructor; apply equiv_dist|]. @@ -40,28 +42,36 @@ Proof. by rewrite (conv_compl (option_chain c x Hx) n); simpl; rewrite Hy. } feed inversion (chain_cauchy c 1 n); auto with lia congruence; constructor. Qed. -Instance Some_ne `{Dist A} : Proper (dist n ==> dist n) Some. +Canonical Structure optionC := CofeT option_cofe_mixin. +Global Instance Some_ne : Proper (dist n ==> dist n) (@Some A). Proof. by constructor. Qed. -Instance None_timeless `{Dist A, Equiv A} : Timeless (@None A). +Global Instance None_timeless : Timeless (@None A). Proof. inversion_clear 1; constructor. Qed. -Instance Some_timeless `{Dist A, Equiv A} x : Timeless x → Timeless (Some x). +Global Instance Some_timeless x : Timeless x → Timeless (Some x). Proof. by intros ?; inversion_clear 1; constructor; apply timeless. Qed. -Instance option_fmap_ne `{Dist A, Dist B} (f : A → B) n: +End cofe. + +Arguments optionC : clear implicits. + +Instance option_fmap_ne {A B : cofeT} (f : A → B) n: Proper (dist n ==> dist n) f → Proper (dist n==>dist n) (fmap (M:=option) f). Proof. by intros Hf; destruct 1; constructor; apply Hf. Qed. -Instance is_Some_ne `{Dist A} : Proper (dist (S n) ==> iff) (@is_Some A). +Instance is_Some_ne `{A : cofeT} : Proper (dist (S n) ==> iff) (@is_Some A). Proof. intros n; inversion_clear 1; split; eauto. Qed. (* CMRA *) -Instance option_valid `{Valid A} : Valid (option A) := λ mx, +Section cmra. +Context {A : cmraT}. + +Instance option_valid : Valid (option A) := λ mx, match mx with Some x => ✓ x | None => True end. -Instance option_validN `{ValidN A} : ValidN (option A) := λ n mx, +Instance option_validN : ValidN (option A) := λ n mx, match mx with Some x => ✓{n} x | None => True end. -Instance option_unit `{Unit A} : Unit (option A) := fmap unit. -Instance option_op `{Op A} : Op (option A) := union_with (λ x y, Some (x ⋅ y)). -Instance option_minus `{Minus A} : Minus (option A) := +Instance option_unit : Unit (option A) := fmap unit. +Instance option_op : Op (option A) := union_with (λ x y, Some (x ⋅ y)). +Instance option_minus : Minus (option A) := difference_with (λ x y, Some (x ⩪ y)). -Lemma option_included `{CMRA A} mx my : +Lemma option_included mx my : mx ≼ my ↔ mx = None ∨ ∃ x y, mx = Some x ∧ my = Some y ∧ x ≼ y. Proof. split. @@ -72,7 +82,7 @@ Proof. * intros [->|(x&y&->&->&z&Hz)]; try (by exists my; destruct my; constructor). by exists (Some z); constructor. Qed. -Lemma option_includedN `{CMRA A} n mx my : +Lemma option_includedN n (mx my : option A) : mx ≼{n} my ↔ n = 0 ∨ mx = None ∨ ∃ x y, mx = Some x ∧ my = Some y ∧ x ≼{n} y. Proof. split. @@ -80,15 +90,14 @@ Proof. destruct mx as [x|]; [right|by left]. destruct my as [y|]; [exists x, y|destruct mz; inversion_clear Hmz]. destruct mz as [z|]; inversion_clear Hmz; split_ands; auto; - cofe_subst; auto using cmra_included_l. + cofe_subst; eauto using @cmra_included_l. * intros [->|[->|(x&y&->&->&z&Hz)]]; try (by exists my; destruct my; constructor). by exists (Some z); constructor. Qed. -Instance option_cmra `{CMRA A} : CMRA (option A). +Definition option_cmra_mixin : CMRAMixin (option A). Proof. split. - * apply _. * by intros n [x|]; destruct 1; constructor; repeat apply (_ : Proper (dist _ ==> _ ==> _) _). * by destruct 1; constructor; apply (_ : Proper (dist n ==> _) _). @@ -98,22 +107,22 @@ Proof. * by destruct 1; inversion_clear 1; constructor; repeat apply (_ : Proper (dist _ ==> _ ==> _) _). * intros [x|]; unfold validN, option_validN; auto using cmra_valid_0. - * intros n [x|]; unfold validN, option_validN; auto using cmra_valid_S. + * intros n [x|]; unfold validN, option_validN; eauto using @cmra_valid_S. * by intros [x|]; unfold valid, validN, option_validN, option_valid; - auto using cmra_valid_validN. + eauto using cmra_valid_validN. * intros [x|] [y|] [z|]; constructor; rewrite ?(associative _); auto. * intros [x|] [y|]; constructor; rewrite 1?(commutative _); auto. - * by intros [x|]; constructor; rewrite cmra_unit_l. - * by intros [x|]; constructor; rewrite cmra_unit_idempotent. + * by intros [x|]; constructor; rewrite ra_unit_l. + * by intros [x|]; constructor; rewrite ra_unit_idempotent. * intros n mx my; rewrite !option_includedN;intros [|[->|(x&y&->&->&?)]];auto. - do 2 right; exists (unit x), (unit y); eauto using cmra_unit_preserving. + do 2 right; exists (unit x), (unit y); eauto using @cmra_unit_preserving. * intros n [x|] [y|]; unfold validN, option_validN; simpl; - eauto using cmra_valid_op_l. + eauto using @cmra_valid_op_l. * intros n mx my; rewrite option_includedN. intros [->|[->|(x&y&->&->&?)]]; [done|by destruct my|]. by constructor; apply cmra_op_minus. Qed. -Instance option_cmra_extend `{CMRA A, !CMRAExtend A} : CMRAExtend (option A). +Definition option_cmra_extend_mixin : CMRAExtendMixin (option A). Proof. intros n mx my1 my2; destruct (decide (n = 0)) as [->|]. { by exists (mx, None); repeat constructor; destruct mx; constructor. } @@ -126,17 +135,28 @@ Proof. * by exists (None,Some x); inversion Hx'; repeat constructor. * exists (None,None); repeat constructor. Qed. -Instance None_valid_timeless `{Valid A, ValidN A} : ValidTimeless (@None A). +Canonical Structure optionRA := + CMRAT option_cofe_mixin option_cmra_mixin option_cmra_extend_mixin. + +Lemma option_op_positive_dist_l n x y : x ⋅ y ={n}= None → x ={n}= None. +Proof. by destruct x, y; inversion_clear 1. Qed. +Lemma option_op_positive_dist_r n x y : x ⋅ y ={n}= None → y ={n}= None. +Proof. by destruct x, y; inversion_clear 1. Qed. + +Global Instance None_valid_timeless : ValidTimeless (@None A). Proof. done. Qed. -Instance Some_valid_timeless `{Valid A, ValidN A} x : +Global Instance Some_valid_timeless x : ValidTimeless x → ValidTimeless (Some x). Proof. by intros ? y; apply (valid_timeless x). Qed. -Instance option_fmap_cmra_monotone `{CMRA A, CMRA B} (f : A → B) - `{!CMRAMonotone f} : CMRAMonotone (fmap f : option A → option B). +End cmra. + +Arguments optionRA : clear implicits. + +Instance option_fmap_cmra_monotone {A B : cmraT} (f: A → B) `{!CMRAMonotone f} : + CMRAMonotone (fmap f : option A → option B). Proof. split. * intros n mx my; rewrite !option_includedN. intros [->|[->|(x&y&->&->&?)]]; simpl; eauto 10 using @includedN_preserving. - * by intros n [x|] ?; - unfold validN, option_validN; simpl; try apply validN_preserving. -Qed. \ No newline at end of file + * by intros n [x|] ?; rewrite /cmra_validN /=; try apply validN_preserving. +Qed.