Commit 5696cb01 by Robbert Krebbers

### More CMRAs

```* Framepreserving updates are now on CMRAs rather than RAs
* Excl and auth are now CMRAs
* Show that excl and auth are functors
* STS is now an CMRA```
parent 780f6b82
 Require Export iris.excl. Local Arguments disjoint _ _ !_ !_ /. Local Arguments valid _ _ !_ /. Local Arguments validN _ _ _ !_ /. Record auth (A : Type) : Type := Auth { authoritative : excl A ; own : A }. Arguments Auth {_} _ _. ... ... @@ -8,11 +9,54 @@ Arguments own {_} _. Notation "∘ x" := (Auth ExclUnit x) (at level 20). Notation "∙ x" := (Auth (Excl x) ∅) (at level 20). Instance auth_empty `{Empty A} : Empty (auth A) := Auth ∅ ∅. Instance auth_valid `{Equiv A, Valid A, Op A} : Valid (auth A) := λ x, valid (authoritative x) ∧ excl_above (own x) (authoritative x). (* COFE *) Instance auth_equiv `{Equiv A} : 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, authoritative x ={n}= authoritative y ∧ own x ={n}= own y. Instance Auth_ne `{Dist A} : 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). Proof. by destruct 1. Qed. Instance own_ne `{Dist A} : Proper (dist n ==> dist n) (@own A). Proof. by destruct 1. Qed. Instance auth_compl `{Cofe A} : 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). Proof. split. * intros x y; unfold dist, auth_dist, equiv, auth_equiv. rewrite !equiv_dist; naive_solver. * intros n; split. + by intros ?; split. + by intros ?? [??]; split; symmetry. + intros ??? [??] [??]; split; etransitivity; eauto. * by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. * by split. * intros c n; split. apply (conv_compl (chain_map authoritative c) n). apply (conv_compl (chain_map own c) n). Qed. (* CMRA *) Instance auth_empty `{Empty A} : Empty (auth A) := Auth ∅ ∅. Instance auth_valid `{Equiv A, Valid A, Op A} : Valid (auth A) := λ x, match authoritative x with | Excl a => own x ≼ a ∧ valid a | ExclUnit => valid (own x) | ExclBot => False end. Arguments auth_valid _ _ _ _ !_ /. Instance auth_validN `{Dist A, ValidN A, Op A} : ValidN (auth A) := λ n x, match authoritative x with | Excl a => own x ≼{n} a ∧ validN n a | ExclUnit => validN n (own x) | ExclBot => n = 0 end. Arguments auth_validN _ _ _ _ _ !_ /. Instance auth_unit `{Unit A} : Unit (auth A) := λ x, Auth (unit (authoritative x)) (unit (own x)). Instance auth_op `{Op A} : Op (auth A) := λ x y, ... ... @@ -25,32 +69,82 @@ 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. Instance auth_ra `{RA A} : RA (auth A). Lemma auth_includedN `{Dist A, Op A} 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) : validN n x → validN n (authoritative x). Proof. by destruct x as [[]]. Qed. Lemma own_validN `{CMRA A} n (x : auth A) : validN n x → validN n (own x). Proof. destruct x as [[]]; naive_solver eauto using cmra_valid_includedN. Qed. Instance auth_cmra `{CMRA A} : CMRA (auth A). Proof. split. * split. + by intros ?; split. + by intros ?? [??]; split. + intros ??? [??] [??]; split; etransitivity; eauto. * by intros x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy, ?Hy'. * by intros y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy, ?Hy'. * by intros y1 y2 [Hy Hy'] [??]; split; simpl; rewrite <-?Hy, <-?Hy'. * by intros x1 x2 [Hx Hx'] y1 y2 [Hy Hy']; * 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 *; destruct Hx as [[][]| | |]; intros ?; cofe_subst; auto. * by intros n x1 x2 [Hx Hx'] y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy, ?Hy', ?Hx, ?Hx'. * by intros [[] ?]; simpl. * intros n [[] ?] ?; naive_solver eauto using cmra_included_S, cmra_valid_S. * destruct x as [[a| |] b]; simpl; rewrite ?cmra_included_includedN, ?cmra_valid_validN; [naive_solver|naive_solver|]. split; [done|intros Hn; discriminate (Hn 1)]. * by split; simpl; rewrite (associative _). * by split; simpl; rewrite (commutative _). * by split; simpl; rewrite ?(ra_unit_l _). * by split; simpl; rewrite ?(ra_unit_idempotent _). * intros ??; rewrite! auth_included; intros [??]. by split; simpl; apply ra_unit_preserving. * intros ?? [??]; split; [by apply ra_valid_op_l with (authoritative y)|]. by apply excl_above_weaken with (own x ⋅ own y) (authoritative x ⋅ authoritative y); try apply ra_included_l. * by intros ??; rewrite auth_included; intros [??]; split; simpl; apply ra_op_minus. Qed. Instance auth_ra_empty `{RA A, Empty A, !RAEmpty A} : RAEmpty (auth A). Proof. split. done. by intros x; constructor; simpl; rewrite (left_id _ _). Qed. Lemma auth_frag_op `{RA A} a b : ∘(a ⋅ b) ≡ ∘a ⋅ ∘b. Proof. done. Qed. \ No newline at end of file * 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). { 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). Proof. intros n x y1 y2 ? [??]; simpl in *. destruct (cmra_extend_op n (authoritative x) (authoritative y1) (authoritative y2)) as (z1&?&?&?); auto using authoritative_validN. destruct (cmra_extend_op n (own x) (own y1) (own y2)) 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, !RAEmpty A} : RAEmpty (auth A). Proof. split; [apply (ra_empty_valid (A:=A))|]. by intros x; constructor; simpl; rewrite (left_id _ _). Qed. Lemma auth_frag_op `{CMRA A} a b : ∘(a ⋅ b) ≡ ∘a ⋅ ∘b. Proof. done. Qed. (* 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 : 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) : (∀ 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]; naive_solver eauto using @includedN_preserving, @validN_preserving. Qed. Definition authRA_map {A B : cmraT} (f : A -n> B) : authRA A -n> authRA B := CofeMor (fmap f : authRA A → authRA B). Lemma authRA_map_ne A B n : Proper (dist n ==> dist n) (@authRA_map A B). Proof. intros f f' Hf [[a| |] b]; repeat constructor; apply Hf. Qed.
 ... ... @@ -7,6 +7,7 @@ Definition includedN `{Dist A, Op A} (n : nat) (x y : A) := ∃ z, y ={n}= x ⋅ 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 *) ... ... @@ -38,6 +39,8 @@ Class CMRAMonotone validN_preserving n x : validN n x → validN n (f x) }. Hint Extern 0 (validN 0 _) => apply cmra_valid_0. (** Bundeled version *) Structure cmraT := CMRAT { cmra_car :> Type; ... ... @@ -69,6 +72,17 @@ Existing Instances cmra_equiv cmra_dist cmra_compl cmra_unit cmra_op 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, validN n (x ⋅ z) → ∃ y, P y ∧ validN 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, validN n (x ⋅ z) → validN n (y ⋅ z). Infix "⇝" := cmra_update (at level 70). Instance: Params (@cmra_update) 3. (** Properties **) Section cmra. Context `{cmra : CMRA A}. Implicit Types x y z : A. ... ... @@ -80,9 +94,9 @@ Proof. symmetry; apply cmra_op_minus, Hxy. Qed. Global Instance cmra_valid_ne' : Proper (dist n ==> iff) (validN n). Global Instance cmra_valid_ne' : Proper (dist n ==> iff) (validN n) | 1. Proof. by split; apply cmra_valid_ne. Qed. Global Instance cmra_valid_proper : Proper ((≡) ==> iff) (validN n). Global Instance cmra_valid_proper : Proper ((≡) ==> iff) (validN n) | 1. Proof. by intros n x1 x2 Hx; apply cmra_valid_ne', equiv_dist. Qed. Global Instance cmra_ra : RA A. Proof. ... ... @@ -118,12 +132,13 @@ Qed. (** * Included *) Global Instance cmra_included_ne n : Proper (dist n ==> dist n ==> iff) (includedN 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). 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. ... ... @@ -154,6 +169,16 @@ 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). ... ... @@ -168,6 +193,7 @@ 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. ... ... @@ -197,6 +223,15 @@ Section discrete. by exists (x,unit x); simpl; rewrite ra_unit_r. Qed. Definition discreteRA : cmraT := CMRAT A. Lemma discrete_updateP (x : A) (P : A → Prop) `{!Inhabited (sig P)} : (∀ z, valid (x ⋅ z) → ∃ y, P y ∧ valid (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, valid (x ⋅ z) → valid (y ⋅ z)) → x ⇝ y. Proof. intros Hvalid z [|n]; [done|apply Hvalid]. Qed. End discrete. Arguments discreteRA _ {_ _ _ _ _ _}. ... ...
 ... ... @@ -66,7 +66,7 @@ Proof. * by exists (None,Some x); inversion Hx'; repeat constructor. * exists (None,None); repeat constructor. Qed. Instance option_fmap_cmra_preserving `{CMRA A, CMRA B} (f : A → B) Instance option_fmap_cmra_monotone `{CMRA A, CMRA B} (f : A → B) `{!CMRAMonotone f} : CMRAMonotone (fmap f : option A → option B). Proof. split. ... ... @@ -169,7 +169,7 @@ Section map. 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_mapcmra_monotone {A B : cmraT} (f : A -n> B) Global Instance mapRA_map_monotone {A B : cmraT} (f : A -n> B) `{!CMRAMonotone f} : CMRAMonotone (mapRA_map f) := _. End map. ... ...
 ... ... @@ -97,6 +97,12 @@ Section cofe. Proper ((≡) ==> (≡)) f | 100 := _. End cofe. (** Mapping a chain *) 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. (** Timeless elements *) Class Timeless `{Dist A, Equiv A} (x : A) := timeless y : x ={1}= y → x ≡ y. Arguments timeless {_ _ _} _ {_} _ _. ... ... @@ -212,14 +218,12 @@ Proof. by repeat split; try exists 0. Qed. (** Product *) Instance prod_dist `{Dist A, Dist B} : Dist (A * B) := λ n, prod_relation (dist n) (dist n). Program Definition fst_chain `{Dist A, Dist B} (c : chain (A * B)) : chain A := {| chain_car n := fst (c n) |}. Next Obligation. by intros A ? B ? c n i ?; apply (chain_cauchy c n). Qed. Program Definition snd_chain `{Dist A, Dist B} (c : chain (A * B)) : chain B := {| chain_car n := snd (c n) |}. Next Obligation. by intros A ? B ? c n i ?; apply (chain_cauchy c n). Qed. 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 (fst_chain c), compl (snd_chain c)). (compl (chain_map fst c), compl (chain_map snd c)). Instance prod_cofe `{Cofe A, Cofe B} : Cofe (A * B). Proof. split. ... ... @@ -228,8 +232,8 @@ Proof. * apply _. * by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. * by split. * intros c n; split. apply (conv_compl (fst_chain c) n). apply (conv_compl (snd_chain c) n). * 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). ... ... @@ -245,10 +249,6 @@ 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. 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) := _. Typeclasses Opaque prod_dist. (** Discrete cofe *) ... ... @@ -268,11 +268,11 @@ Section discrete_cofe. Qed. Global Instance discrete_timeless (x : A) : Timeless x. Proof. by intros y. Qed. Definition discrete_cofeC : cofeT := CofeT A. Definition discreteC : cofeT := CofeT A. End discrete_cofe. Arguments discrete_cofeC _ {_ _}. Arguments discreteC _ {_ _}. Definition leibniz_cofeC (A : Type) : cofeT := @discrete_cofeC A equivL _. Definition leibnizC (A : Type) : cofeT := @discreteC A equivL _. (** Later *) Inductive later (A : Type) : Type := Later { later_car : A }. ... ...
 Require Export iris.ra. Require Export iris.ra iris.cmra. (** From disjoint pcm *) Record validity {A} (P : A → Prop) : Type := Validity { ... ... @@ -88,17 +88,20 @@ Hint Immediate dra_disjoint_move_l dra_disjoint_move_r. Hint Unfold dra_included. Notation T := (validity (valid : A → Prop)). Lemma validity_valid_car_valid (z : T) : V z → V (validity_car z). Proof. apply validity_prf. Qed. Hint Resolve validity_valid_car_valid. Program Instance validity_unit : Unit T := λ x, Validity (unit (validity_car x)) (V x) _. Next Obligation. by apply dra_unit_valid, validity_prf. Qed. Solve Obligations with naive_solver auto using dra_unit_valid. Program Instance validity_op : Op T := λ x y, Validity (validity_car x ⋅ validity_car y) (V x ∧ V y ∧ validity_car x ⊥ validity_car y) _. Next Obligation. by apply dra_op_valid; try apply validity_prf. Qed. Solve Obligations with naive_solver auto using dra_op_valid. Program Instance validity_minus : Minus T := λ x y, Validity (validity_car x ⩪ validity_car y) (V x ∧ V y ∧ validity_car y ≼ validity_car x) _. Next Obligation. by apply dra_minus_valid; try apply validity_prf. Qed. Solve Obligations with naive_solver auto using dra_minus_valid. Instance validity_ra : RA T. Proof. split. ... ... @@ -130,10 +133,11 @@ Proof. * intros [x px ?] [y py ?] [[z pz ?] [??]]; split; simpl in *; intuition eauto 10 using dra_disjoint_minus, dra_op_minus. Qed. Definition dra_update (x y : T) : Definition validityRA : cmraT := discreteRA T. Definition validity_update (x y : validityRA) : (∀ z, V x → V z → validity_car x ⊥ z → V y ∧ validity_car y ⊥ z) → x ⇝ y. Proof. intros Hxy z (?&?&?); split_ands'; auto; eapply Hxy; eauto; by eapply validity_prf. intros Hxy; apply discrete_update. intros z (?&?&?); split_ands'; try eapply Hxy; eauto. Qed. End dra.
 Require Export iris.cmra. Local Arguments disjoint _ _ !_ !_ /. Local Arguments validN _ _ _ !_ /. Local Arguments valid _ _ !_ /. Inductive excl (A : Type) := | Excl : A → excl A | ExclUnit : Empty (excl A) | ExclUnit : excl A | ExclBot : excl A. Arguments Excl {_} _. Arguments ExclUnit {_}. Arguments ExclBot {_}. Existing Instance ExclUnit. Instance maybe_Excl {A} : Maybe (@Excl A) := λ x, match x with Excl a => Some a | _ => None end. (* Cofe *) Inductive excl_equiv `{Equiv A} : Equiv (excl A) := | Excl_equiv (x y : A) : x ≡ y → Excl x ≡ Excl y | ExclUnit_equiv : ∅ ≡ ∅ | ExclUnit_equiv : ExclUnit ≡ ExclUnit | ExclBot_equiv : ExclBot ≡ ExclBot. Existing Instance excl_equiv. Inductive excl_dist `{Dist A} : Dist (excl A) := | excl_dist_0 (x y : excl A) : x ={0}= y | Excl_dist (x y : A) n : x ={n}= y → Excl x ={n}= Excl y | ExclUnit_dist n : ExclUnit ={n}= ExclUnit | ExclBot_dist n : ExclBot ={n}= ExclBot. Existing Instance excl_dist. Program Definition excl_chain `{Cofe A} (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'. 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, 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). Proof. split. * intros mx my; split; [by destruct 1; constructor; apply equiv_dist|]. intros Hxy; feed inversion (Hxy 1); subst; constructor; apply equiv_dist. by intros n; feed inversion (Hxy n). * intros n; split. + by intros [x| |]; constructor. + by destruct 1; constructor. + destruct 1; inversion_clear 1; constructor; etransitivity; eauto. * by inversion_clear 1; constructor; apply dist_S. * constructor. * intros c n; unfold compl, excl_compl. destruct (decide (n = 0)) as [->|]; [constructor|]. destruct (Some_dec (maybe Excl (c 1))) as [[x Hx]|]. { assert (c 1 = Excl x) by (by destruct (c 1); simplify_equality'). assert (∃ y, c n = Excl y) as [y Hy]. { feed inversion (chain_cauchy c 1 n); try congruence; eauto with lia. } rewrite Hy; constructor. by rewrite (conv_compl (excl_chain c x Hx) n); simpl; rewrite Hy. } feed inversion (chain_cauchy c 1 n); auto with lia; constructor. destruct (c 1); simplify_equality'. Qed. (* CMRA *) Instance excl_valid {A} : Valid (excl A) := λ x, match x with Excl _ | ExclUnit => True | ExclBot => False end. Instance excl_validN {A} : 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, ... ... @@ -31,43 +81,60 @@ Instance excl_minus {A} : Minus (excl A) := λ x y, | Excl _, Excl _ => ExclUnit | _, _ => ExclBot end. Instance excl_ra `{Equiv A, !Equivalence (@equiv A _)} : RA (excl A). Instance excl_cmra `{Cofe A} : CMRA (excl A). Proof. split. * split. + by intros []; constructor. + by destruct 1; constructor. + destruct 1; inversion 1; constructor; etransitivity; eauto. * by intros []; destruct 1; constructor. * apply _. * by intros n []; destruct 1; constructor. * constructor. * by destruct 1. * by do 2 destruct 1; constructor. * by destruct 1 as [? []| | |]; intros ?. * by destruct 1; inversion_clear 1; constructor. * by intros []. * intros n [?| |]; simpl; auto with lia. * intros x; split; [by intros ? [|n]; destruct x|]. by intros Hx; specialize (Hx 1); destruct x. * by intros [?| |] [?| |] [?| |]; constructor. * by intros [?| |] [?| |]; constructor. * by intros [?| |]; constructor. * constructor. * intros [?| |] [?| |]; exists ∅; constructor. * by intros [?| |] [?| |]. * by intros [?| |] [?| |] [[?| |] Hz]; inversion_clear Hz; constructor. * by intros n [?| |] [?| |]; exists ∅. * by intros n [?| |] [?| |]. * by intros n [?| |] [?| |] [[?| |] Hz]; inversion_clear Hz; constructor. Qed. Instance excl_empty_ra `{Equiv A, !Equivalence (@equiv A _)} : RAEmpty (excl A). Instance excl_empty_ra `{Cofe A} : RAEmpty (excl A). Proof. split. done. by intros []. Qed. Instance excl_extend `{Cofe A} : CMRAExtend (excl A). Proof. intros [|n] x y1 y2 ? Hx; [by exists (x,∅); destruct x|]. by exists match y1, y2 with | Excl a1, Excl a2 => (Excl a1, Excl a2) | ExclBot, _ => (ExclBot, y2) | _, ExclBot => (y1, ExclBot) | ExclUnit, _ => (ExclUnit, x) | _, ExclUnit => (x, ExclUnit) end; destruct y1, y2; inversion_clear Hx; repeat constructor. Qed. (* Updates *) Lemma excl_update {A} (x : A) y : valid y → Excl x ⇝ y. Proof. by destruct y; intros ? [?| |]. Qed. Definition excl_above `{Equiv A, Op A} (x : A) (y : excl A) : Prop := match y with Excl y' => x ≼ y' | ExclUnit => True | ExclBot => False end. Instance: Params (@excl_above) 3. Section excl_above. Context `{RA A}. Global Instance excl_above_proper : Proper ((≡) ==> (≡) ==> iff) excl_above. Proof. by intros ?? Hx; destruct 1 as [?? Hy| |]; simpl; rewrite ?Hx,?Hy. Qed. Lemma excl_above_weaken (a b : A) x y : excl_above b y → a ≼ b → x ≼ y → excl_above a x. Proof. destruct x as [a'| |], y as [b'| |]; intros ?? [[] Hz]; inversion_clear Hz; simpl in *; try done. by setoid_subst; transitivity b. Qed. End excl_above. (* 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 : 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} : (∀ n, Proper (dist n ==> dist n) f) → CMRAMonotone (fmap f : excl A → excl B). Proof. split. * intros n x y [z Hy]; exists (f <\$> z); rewrite Hy. 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 := 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.
 ... ... @@ -113,10 +113,6 @@ Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1, (associative (⋅)). Qed. Lemma ra_preserving_r x y z : x ≼ y → x ⋅ z ≼ y ⋅ z.