Commit c2c84732 authored by Ralf Jung's avatar Ralf Jung

rename CMRA unit -> core

parent 361c9fbf
Pipeline #284 passed with stage
......@@ -60,7 +60,7 @@ 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. naive_solver eauto using agree_valid_S, dist_S. Qed.
Instance agree_unit : Unit (agree A) := id.
Instance agree_core : Core (agree A) := id.
Instance agree_div : Div (agree A) := λ x y, x.
Instance: Comm () (@op (agree A) _).
......
......@@ -85,8 +85,8 @@ Instance auth_validN : ValidN (auth A) := λ n x,
| ExclBot => False
end.
Global Arguments auth_validN _ !_ /.
Instance auth_unit : Unit (auth A) := λ x,
Auth (unit (authoritative x)) (unit (own x)).
Instance auth_core : Core (auth A) := λ x,
Auth (core (authoritative x)) (core (own x)).
Instance auth_op : Op (auth A) := λ x y,
Auth (authoritative x authoritative y) (own x own y).
Instance auth_div : Div (auth A) := λ x y,
......@@ -117,10 +117,10 @@ Proof.
- intros n [[] ?] ?; naive_solver eauto using cmra_includedN_S, cmra_validN_S.
- by split; simpl; rewrite assoc.
- by split; simpl; rewrite comm.
- by split; simpl; rewrite ?cmra_unit_l.
- by split; simpl; rewrite ?cmra_unit_idemp.
- by split; simpl; rewrite ?cmra_core_l.
- by split; simpl; rewrite ?cmra_core_idemp.
- intros ??; rewrite! auth_included; intros [??].
by split; simpl; apply cmra_unit_preserving.
by split; simpl; apply cmra_core_preserving.
- assert ( n (a b1 b2 : A), b1 b2 {n} a b1 {n} a).
{ intros n a b1 b2 <-; apply cmra_includedN_l. }
intros n [[a1| |] b1] [[a2| |] b2];
......
From algebra Require Export cofe.
Class Unit (A : Type) := unit : A A.
Instance: Params (@unit) 2.
Class Core (A : Type) := core : A A.
Instance: Params (@core) 2.
Class Op (A : Type) := op : A A A.
Instance: Params (@op) 2.
......@@ -34,10 +34,10 @@ Instance: Params (@includedN) 4.
Hint Extern 0 (_ {_} _) => reflexivity.
Record CMRAMixin A
`{Dist A, Equiv A, Unit A, Op A, Valid A, ValidN A, Div A} := {
`{Dist A, Equiv A, Core A, Op A, Valid A, ValidN A, Div A} := {
(* setoids *)
mixin_cmra_op_ne n (x : A) : Proper (dist n ==> dist n) (op x);
mixin_cmra_unit_ne n : Proper (dist n ==> dist n) unit;
mixin_cmra_core_ne n : Proper (dist n ==> dist n) core;
mixin_cmra_validN_ne n : Proper (dist n ==> impl) (validN n);
mixin_cmra_div_ne n : Proper (dist n ==> dist n ==> dist n) div;
(* valid *)
......@@ -46,9 +46,9 @@ Record CMRAMixin A
(* monoid *)
mixin_cmra_assoc : Assoc () ();
mixin_cmra_comm : Comm () ();
mixin_cmra_unit_l x : unit x x x;
mixin_cmra_unit_idemp x : unit (unit x) unit x;
mixin_cmra_unit_preserving x y : x y unit x unit y;
mixin_cmra_core_l x : core x x x;
mixin_cmra_core_idemp x : core (core x) core x;
mixin_cmra_core_preserving x y : x y core x core y;
mixin_cmra_validN_op_l n x y : {n} (x y) {n} x;
mixin_cmra_op_div x y : x y x y ÷ x y;
mixin_cmra_extend n x y1 y2 :
......@@ -62,7 +62,7 @@ Structure cmraT := CMRAT {
cmra_equiv : Equiv cmra_car;
cmra_dist : Dist cmra_car;
cmra_compl : Compl cmra_car;
cmra_unit : Unit cmra_car;
cmra_core : Core cmra_car;
cmra_op : Op cmra_car;
cmra_valid : Valid cmra_car;
cmra_validN : ValidN cmra_car;
......@@ -75,7 +75,7 @@ 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_core : simpl never.
Arguments cmra_op : simpl never.
Arguments cmra_valid : simpl never.
Arguments cmra_validN : simpl never.
......@@ -83,7 +83,7 @@ Arguments cmra_div : simpl never.
Arguments cmra_cofe_mixin : simpl never.
Arguments cmra_mixin : simpl never.
Add Printing Constructor cmraT.
Existing Instances cmra_unit cmra_op cmra_valid cmra_validN cmra_div.
Existing Instances cmra_core cmra_op cmra_valid cmra_validN cmra_div.
Coercion cmra_cofeC (A : cmraT) : cofeT := CofeT (cmra_cofe_mixin A).
Canonical Structure cmra_cofeC.
......@@ -93,8 +93,8 @@ Section cmra_mixin.
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_core_ne n : Proper (dist n ==> dist n) (@core A _).
Proof. apply (mixin_cmra_core_ne _ (cmra_mixin A)). Qed.
Global Instance cmra_validN_ne n : Proper (dist n ==> impl) (@validN A _ n).
Proof. apply (mixin_cmra_validN_ne _ (cmra_mixin A)). Qed.
Global Instance cmra_div_ne n :
......@@ -108,12 +108,12 @@ Section cmra_mixin.
Proof. apply (mixin_cmra_assoc _ (cmra_mixin A)). Qed.
Global Instance cmra_comm : Comm () (@op A _).
Proof. apply (mixin_cmra_comm _ (cmra_mixin A)). Qed.
Lemma cmra_unit_l x : unit x x x.
Proof. apply (mixin_cmra_unit_l _ (cmra_mixin A)). Qed.
Lemma cmra_unit_idemp x : unit (unit x) unit x.
Proof. apply (mixin_cmra_unit_idemp _ (cmra_mixin A)). Qed.
Lemma cmra_unit_preserving x y : x y unit x unit y.
Proof. apply (mixin_cmra_unit_preserving _ (cmra_mixin A)). Qed.
Lemma cmra_core_l x : core x x x.
Proof. apply (mixin_cmra_core_l _ (cmra_mixin A)). Qed.
Lemma cmra_core_idemp x : core (core x) core x.
Proof. apply (mixin_cmra_core_idemp _ (cmra_mixin A)). Qed.
Lemma cmra_core_preserving x y : x y core x core y.
Proof. apply (mixin_cmra_core_preserving _ (cmra_mixin A)). Qed.
Lemma cmra_validN_op_l n x y : {n} (x y) {n} x.
Proof. apply (mixin_cmra_validN_op_l _ (cmra_mixin A)). Qed.
Lemma cmra_op_div x y : x y x y ÷ x y.
......@@ -175,7 +175,7 @@ Implicit Types x y z : A.
Implicit Types xs ys zs : list A.
(** ** Setoids *)
Global Instance cmra_unit_proper : Proper (() ==> ()) (@unit A _).
Global Instance cmra_core_proper : Proper (() ==> ()) (@core A _).
Proof. apply (ne_proper _). Qed.
Global Instance cmra_op_ne' n : Proper (dist n ==> dist n ==> dist n) (@op A _).
Proof.
......@@ -236,15 +236,15 @@ Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed.
Lemma cmra_valid_op_r x y : (x y) y.
Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_r. Qed.
(** ** Units *)
Lemma cmra_unit_r x : x unit x x.
Proof. by rewrite (comm _ x) cmra_unit_l. Qed.
Lemma cmra_unit_unit x : unit x unit x unit x.
Proof. by rewrite -{2}(cmra_unit_idemp x) cmra_unit_r. Qed.
Lemma cmra_unit_validN n x : {n} x {n} unit x.
Proof. rewrite -{1}(cmra_unit_l x); apply cmra_validN_op_l. Qed.
Lemma cmra_unit_valid x : x unit x.
Proof. rewrite -{1}(cmra_unit_l x); apply cmra_valid_op_l. Qed.
(** ** Core *)
Lemma cmra_core_r x : x core x x.
Proof. by rewrite (comm _ x) cmra_core_l. Qed.
Lemma cmra_core_core x : core x core x core x.
Proof. by rewrite -{2}(cmra_core_idemp x) cmra_core_r. Qed.
Lemma cmra_core_validN n x : {n} x {n} core x.
Proof. rewrite -{1}(cmra_core_l x); apply cmra_validN_op_l. Qed.
Lemma cmra_core_valid x : x core x.
Proof. rewrite -{1}(cmra_core_l x); apply cmra_valid_op_l. Qed.
(** ** Div *)
Lemma cmra_op_div' n x y : x {n} y x y ÷ x {n} y.
......@@ -260,7 +260,7 @@ Qed.
Global Instance cmra_includedN_preorder n : PreOrder (@includedN A _ _ n).
Proof.
split.
- by intros x; exists (unit x); rewrite cmra_unit_r.
- by intros x; exists (core x); rewrite cmra_core_r.
- intros x y z [z1 Hy] [z2 Hz]; exists (z1 z2).
by rewrite assoc -Hy -Hz.
Qed.
......@@ -288,13 +288,13 @@ Proof. rewrite (comm op); apply cmra_includedN_l. Qed.
Lemma cmra_included_r x y : y x y.
Proof. rewrite (comm op); apply cmra_included_l. Qed.
Lemma cmra_unit_preservingN n x y : x {n} y unit x {n} unit y.
Lemma cmra_core_preservingN n x y : x {n} y core x {n} core y.
Proof.
intros [z ->].
apply cmra_included_includedN, cmra_unit_preserving, cmra_included_l.
apply cmra_included_includedN, cmra_core_preserving, cmra_included_l.
Qed.
Lemma cmra_included_unit x : unit x x.
Proof. by exists x; rewrite cmra_unit_l. Qed.
Lemma cmra_included_core x : core x x.
Proof. by exists x; rewrite cmra_core_l. Qed.
Lemma cmra_preservingN_l n x y z : x {n} y z x {n} z y.
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed.
Lemma cmra_preserving_l x y z : x y z x z y.
......@@ -358,8 +358,8 @@ Section identity.
Proof. by exists x; rewrite left_id. Qed.
Global Instance cmra_empty_right_id : RightId () ().
Proof. by intros x; rewrite (comm op) left_id. Qed.
Lemma cmra_unit_empty : unit .
Proof. by rewrite -{2}(cmra_unit_l ) right_id. Qed.
Lemma cmra_core_empty : core .
Proof. by rewrite -{2}(cmra_core_l ) right_id. Qed.
End identity.
(** ** Local updates *)
......@@ -468,7 +468,7 @@ Section cmra_transport.
Proof. by intros ???; destruct H. Qed.
Lemma cmra_transport_op x y : T (x y) = T x T y.
Proof. by destruct H. Qed.
Lemma cmra_transport_unit x : T (unit x) = unit (T x).
Lemma cmra_transport_core x : T (core x) = core (T x).
Proof. by destruct H. Qed.
Lemma cmra_transport_validN n x : {n} T x {n} x.
Proof. by destruct H. Qed.
......@@ -486,25 +486,25 @@ End cmra_transport.
(** * Instances *)
(** ** Discrete CMRA *)
Class RA A `{Equiv A, Unit A, Op A, Valid A, Div A} := {
Class RA A `{Equiv A, Core A, Op A, Valid A, Div A} := {
(* setoids *)
ra_op_ne (x : A) : Proper (() ==> ()) (op x);
ra_unit_ne :> Proper (() ==> ()) unit;
ra_core_ne :> Proper (() ==> ()) core;
ra_validN_ne :> Proper (() ==> impl) valid;
ra_div_ne :> Proper (() ==> () ==> ()) div;
(* monoid *)
ra_assoc :> Assoc () ();
ra_comm :> Comm () ();
ra_unit_l x : unit x x x;
ra_unit_idemp x : unit (unit x) unit x;
ra_unit_preserving x y : x y unit x unit y;
ra_core_l x : core x x x;
ra_core_idemp x : core (core x) core x;
ra_core_preserving x y : x y core x core y;
ra_valid_op_l x y : (x y) x;
ra_op_div x y : x y x y ÷ x y
}.
Section discrete.
Context {A : cofeT} `{Discrete A}.
Context `{Unit A, Op A, Valid A, Div A} (ra : RA A).
Context `{Core A, Op A, Valid A, Div A} (ra : RA A).
Instance discrete_validN : ValidN A := λ n x, x.
Definition discrete_cmra_mixin : CMRAMixin A.
......@@ -523,7 +523,7 @@ End discrete.
(** ** CMRA for the unit type *)
Section unit.
Instance unit_valid : Valid () := λ x, True.
Instance unit_unit : Unit () := λ x, x.
Instance unit_core : Core () := λ x, x.
Instance unit_op : Op () := λ x y, ().
Instance unit_div : Div () := λ x y, ().
Global Instance unit_empty : Empty () := ().
......@@ -541,7 +541,7 @@ 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_core : Core (A * B) := λ x, (core (x.1), core (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_div : Div (A * B) := λ x y, (x.1 ÷ y.1, x.2 ÷ y.2).
......@@ -569,10 +569,10 @@ Section prod.
- by intros n x [??]; split; apply cmra_validN_S.
- by split; rewrite /= assoc.
- by split; rewrite /= comm.
- by split; rewrite /= cmra_unit_l.
- by split; rewrite /= cmra_unit_idemp.
- by split; rewrite /= cmra_core_l.
- by split; rewrite /= cmra_core_idemp.
- intros x y; rewrite !prod_included.
by intros [??]; split; apply cmra_unit_preserving.
by intros [??]; split; apply cmra_core_preserving.
- intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l.
- intros x y; rewrite prod_included; intros [??].
by split; apply cmra_op_div.
......
......@@ -2,7 +2,7 @@ From algebra Require Export cmra.
Local Arguments validN _ _ _ !_ /.
Local Arguments valid _ _ !_ /.
Local Arguments op _ _ _ !_ /.
Local Arguments unit _ _ !_ /.
Local Arguments core _ _ !_ /.
(* This is isomorphic to option, but has a very different RA structure. *)
Inductive dec_agree (A : Type) : Type :=
......@@ -26,7 +26,7 @@ Instance dec_agree_op : Op (dec_agree A) := λ x y,
| DecAgree a, DecAgree b => if decide (a = b) then DecAgree a else DecAgreeBot
| _, _ => DecAgreeBot
end.
Instance dec_agree_unit : Unit (dec_agree A) := id.
Instance dec_agree_core : Core (dec_agree A) := id.
Instance dec_agree_div : Div (dec_agree A) := λ x y, x.
Definition dec_agree_ra : RA (dec_agree A).
......
......@@ -18,17 +18,17 @@ Definition dra_included `{Equiv A, Valid A, Disjoint A, Op A} := λ x y,
Instance: Params (@dra_included) 4.
Local Infix "≼" := dra_included.
Class DRA A `{Equiv A, Valid A, Unit A, Disjoint A, Op A, Div A} := {
Class DRA A `{Equiv A, Valid A, Core A, Disjoint A, Op A, Div A} := {
(* setoids *)
dra_equivalence :> Equivalence (() : relation A);
dra_op_proper :> Proper (() ==> () ==> ()) ();
dra_unit_proper :> Proper (() ==> ()) unit;
dra_core_proper :> Proper (() ==> ()) core;
dra_valid_proper :> Proper (() ==> impl) valid;
dra_disjoint_proper :> x, Proper (() ==> impl) (disjoint x);
dra_div_proper :> Proper (() ==> () ==> ()) div;
(* validity *)
dra_op_valid x y : x y x y (x y);
dra_unit_valid x : x unit x;
dra_core_valid x : x core x;
dra_div_valid x y : x y x y (y ÷ x);
(* monoid *)
dra_assoc :> Assoc () ();
......@@ -36,10 +36,10 @@ Class DRA A `{Equiv A, Valid A, Unit A, Disjoint A, Op A, Div A} := {
dra_disjoint_move_l x y z : x y z x y x y z x y z;
dra_symmetric :> Symmetric (@disjoint A _);
dra_comm x y : x y x y x y y x;
dra_unit_disjoint_l x : x unit x x;
dra_unit_l x : x unit x x x;
dra_unit_idemp x : x unit (unit x) unit x;
dra_unit_preserving x y : x y x y unit x unit y;
dra_core_disjoint_l x : x core x x;
dra_core_l x : x core x x x;
dra_core_idemp x : x core (core x) core x;
dra_core_preserving x y : x y x y core x core y;
dra_disjoint_div x y : x y x y x y ÷ x;
dra_op_div x y : x y x y x y ÷ x y
}.
......@@ -88,9 +88,9 @@ Hint Unfold dra_included.
Lemma validity_valid_car_valid (z : T) : z 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)) ( x) _.
Solve Obligations with naive_solver auto using dra_unit_valid.
Program Instance validity_core : Core T := λ x,
Validity (core (validity_car x)) ( x) _.
Solve Obligations with naive_solver auto using dra_core_valid.
Program Instance validity_op : Op T := λ x y,
Validity (validity_car x validity_car y)
( x y validity_car x validity_car y) _.
......@@ -118,14 +118,14 @@ Proof.
|by intros; rewrite assoc].
- intros [x px ?] [y py ?]; split; naive_solver eauto using dra_comm.
- intros [x px ?]; split;
naive_solver eauto using dra_unit_l, dra_unit_disjoint_l.
- intros [x px ?]; split; naive_solver eauto using dra_unit_idemp.
- intros x y Hxy; exists (unit y ÷ unit x).
naive_solver eauto using dra_core_l, dra_core_disjoint_l.
- intros [x px ?]; split; naive_solver eauto using dra_core_idemp.
- intros x y Hxy; exists (core y ÷ core x).
destruct x as [x px ?], y as [y py ?], Hxy as [[z pz ?] [??]]; simpl in *.
assert (py unit x unit y)
by intuition eauto 10 using dra_unit_preserving.
assert (py core x core y)
by intuition eauto 10 using dra_core_preserving.
constructor; [|symmetry]; simpl in *;
intuition eauto using dra_op_div, dra_disjoint_div, dra_unit_valid.
intuition eauto using dra_op_div, dra_disjoint_div, dra_core_valid.
- by intros [x px ?] [y py ?] (?&?&?).
- intros [x px ?] [y py ?] [[z pz ?] [??]]; split; simpl in *;
intuition eauto 10 using dra_disjoint_div, dra_op_div.
......
......@@ -91,7 +91,7 @@ Instance excl_valid : Valid (excl A) := λ x,
Instance excl_validN : ValidN (excl A) := λ n x,
match x with Excl _ | ExclUnit => True | ExclBot => False end.
Global Instance excl_empty : Empty (excl A) := ExclUnit.
Instance excl_unit : Unit (excl A) := λ _, .
Instance excl_core : Core (excl A) := λ _, .
Instance excl_op : Op (excl A) := λ x y,
match x, y with
| Excl a, ExclUnit | ExclUnit, Excl a => Excl a
......
......@@ -93,7 +93,7 @@ Context `{Countable K} {A : cmraT}.
Implicit Types m : gmap K A.
Instance map_op : Op (gmap K A) := merge op.
Instance map_unit : Unit (gmap K A) := fmap unit.
Instance map_core : Core (gmap K A) := fmap core.
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_div : Div (gmap K A) := merge div.
......@@ -102,7 +102,7 @@ Lemma lookup_op m1 m2 i : (m1 ⋅ m2) !! i = m1 !! i ⋅ m2 !! i.
Proof. by apply lookup_merge. Qed.
Lemma lookup_div m1 m2 i : (m1 ÷ m2) !! i = m1 !! i ÷ m2 !! i.
Proof. by apply lookup_merge. Qed.
Lemma lookup_unit m i : unit m !! i = unit (m !! i).
Lemma lookup_core m i : core m !! i = core (m !! i).
Proof. by apply lookup_fmap. Qed.
Lemma map_included_spec (m1 m2 : gmap K A) : m1 m2 i, m1 !! i m2 !! i.
......@@ -125,7 +125,7 @@ Definition map_cmra_mixin : CMRAMixin (gmap K A).
Proof.
split.
- 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 !lookup_core (Hm i).
- by intros n m1 m2 Hm ? i; rewrite -(Hm i).
- by intros n m1 m1' Hm1 m2 m2' Hm2 i; rewrite !lookup_div (Hm1 i) (Hm2 i).
- intros m; split.
......@@ -134,10 +134,10 @@ Proof.
- intros n m Hm i; apply cmra_validN_S, Hm.
- by intros m1 m2 m3 i; rewrite !lookup_op assoc.
- by intros m1 m2 i; rewrite !lookup_op comm.
- by intros m i; rewrite lookup_op !lookup_unit cmra_unit_l.
- by intros m i; rewrite !lookup_unit cmra_unit_idemp.
- by intros m i; rewrite lookup_op !lookup_core cmra_core_l.
- by intros m i; rewrite !lookup_core cmra_core_idemp.
- intros x y; rewrite !map_included_spec; intros Hm i.
by rewrite !lookup_unit; apply cmra_unit_preserving.
by rewrite !lookup_core; apply cmra_core_preserving.
- intros n m1 m2 Hm i; apply cmra_validN_op_l with (m2 !! i).
by rewrite -lookup_op.
- intros x y; rewrite map_included_spec=> ? i.
......@@ -201,21 +201,21 @@ Lemma map_singleton_valid i x : ✓ ({[ i := x ]} : gmap K A) ↔ ✓ x.
Proof. rewrite !cmra_valid_validN. by setoid_rewrite map_singleton_validN. Qed.
Lemma map_insert_singleton_opN n m i x :
m !! i = None m !! i {n} Some (unit x) <[i:=x]> m {n} {[ i := x ]} m.
m !! i = None m !! i {n} Some (core x) <[i:=x]> m {n} {[ i := x ]} m.
Proof.
intros Hi j; destruct (decide (i = j)) as [->|];
[|by rewrite lookup_op lookup_insert_ne // lookup_singleton_ne // left_id].
rewrite lookup_op lookup_insert lookup_singleton.
by destruct Hi as [->| ->]; constructor; rewrite ?cmra_unit_r.
by destruct Hi as [->| ->]; constructor; rewrite ?cmra_core_r.
Qed.
Lemma map_insert_singleton_op m i x :
m !! i = None m !! i Some (unit x) <[i:=x]> m {[ i := x ]} m.
m !! i = None m !! i Some (core x) <[i:=x]> m {[ i := x ]} m.
Proof.
rewrite !equiv_dist; naive_solver eauto using map_insert_singleton_opN.
Qed.
Lemma map_unit_singleton (i : K) (x : A) :
unit ({[ i := x ]} : gmap K A) = {[ i := unit x ]}.
Lemma map_core_singleton (i : K) (x : A) :
core ({[ i := x ]} : gmap K A) = {[ i := core 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).
......@@ -315,7 +315,7 @@ End freshness.
(* Allocation is a local update: Just use composition with a singleton map. *)
(* Deallocation is *not* a local update. The trouble is that if we
own {[ i ↦ x ]}, then the frame could always own "unit x", and prevent
own {[ i ↦ x ]}, then the frame could always own "core x", and prevent
deallocation. *)
(* Applying a local update at a position we own is a local update. *)
......
......@@ -122,7 +122,7 @@ Instance frac_valid : Valid (frac A) := λ x,
Instance frac_validN : ValidN (frac A) := λ n x,
match x with Frac q a => (q 1)%Qc {n} a | FracUnit => True end.
Global Instance frac_empty : Empty (frac A) := FracUnit.
Instance frac_unit : Unit (frac A) := λ _, .
Instance frac_core : Core (frac A) := λ _, .
Instance frac_op : Op (frac A) := λ x y,
match x, y with
| Frac q1 a, Frac q2 b => Frac (q1 + q2) (a b)
......@@ -225,7 +225,7 @@ Qed.
Lemma frac_update (a1 a2 : A) p : a1 ~~> a2 Frac p a1 ~~> Frac p a2.
Proof.
intros Ha n [q b|] [??]; split; auto.
apply cmra_validN_op_l with (unit a1), Ha. by rewrite cmra_unit_r.
apply cmra_validN_op_l with (core a1), Ha. by rewrite cmra_core_r.
Qed.
End cmra.
......
......@@ -117,13 +117,13 @@ Section iprod_cmra.
Implicit Types f g : iprod B.
Instance iprod_op : Op (iprod B) := λ f g x, f x g x.
Instance iprod_unit : Unit (iprod B) := λ f x, unit (f x).
Instance iprod_core : Core (iprod B) := λ f x, core (f x).
Instance iprod_valid : Valid (iprod B) := λ f, x, f x.
Instance iprod_validN : ValidN (iprod B) := λ n f, x, {n} f x.
Instance iprod_div : Div (iprod B) := λ f g x, f x ÷ g x.
Definition iprod_lookup_op f g x : (f g) x = f x g x := eq_refl.
Definition iprod_lookup_unit f x : (unit f) x = unit (f x) := eq_refl.
Definition iprod_lookup_core f x : (core f) x = core (f x) := eq_refl.
Definition iprod_lookup_div f g x : (f ÷ g) x = f x ÷ g x := eq_refl.
Lemma iprod_included_spec (f g : iprod B) : f g x, f x g x.
......@@ -138,7 +138,7 @@ Section iprod_cmra.
Proof.
split.
- by intros n f1 f2 f3 Hf x; rewrite iprod_lookup_op (Hf x).
- by intros n f1 f2 Hf x; rewrite iprod_lookup_unit (Hf x).
- by intros n f1 f2 Hf x; rewrite iprod_lookup_core (Hf x).
- by intros n f1 f2 Hf ? x; rewrite -(Hf x).
- by intros n f f' Hf g g' Hg i; rewrite iprod_lookup_div (Hf i) (Hg i).
- intros g; split.
......@@ -147,10 +147,10 @@ Section iprod_cmra.
- intros n f Hf x; apply cmra_validN_S, Hf.
- by intros f1 f2 f3 x; rewrite iprod_lookup_op assoc.
- by intros f1 f2 x; rewrite iprod_lookup_op comm.
- by intros f x; rewrite iprod_lookup_op iprod_lookup_unit cmra_unit_l.
- by intros f x; rewrite iprod_lookup_unit cmra_unit_idemp.
- by intros f x; rewrite iprod_lookup_op iprod_lookup_core cmra_core_l.
- by intros f x; rewrite iprod_lookup_core cmra_core_idemp.
- intros f1 f2; rewrite !iprod_included_spec=> Hf x.
by rewrite iprod_lookup_unit; apply cmra_unit_preserving, Hf.
by rewrite iprod_lookup_core; apply cmra_core_preserving, Hf.
- intros n f1 f2 Hf x; apply cmra_validN_op_l with (f2 x), Hf.
- intros f1 f2; rewrite iprod_included_spec=> Hf x.
by rewrite iprod_lookup_op iprod_lookup_div cmra_op_div; try apply Hf.
......@@ -211,12 +211,12 @@ Section iprod_cmra.
by apply cmra_empty_validN.
Qed.
Lemma iprod_unit_singleton x (y : B x) :
unit (iprod_singleton x y) iprod_singleton x (unit y).
Lemma iprod_core_singleton x (y : B x) :
core (iprod_singleton x y) iprod_singleton x (core y).
Proof.
by move=>x'; destruct (decide (x = x')) as [->|];
rewrite iprod_lookup_unit ?iprod_lookup_singleton
?iprod_lookup_singleton_ne // cmra_unit_empty.
rewrite iprod_lookup_core ?iprod_lookup_singleton
?iprod_lookup_singleton_ne // cmra_core_empty.
Qed.
Lemma iprod_op_singleton (x : A) (y1 y2 : B x) :
......
......@@ -67,7 +67,7 @@ Instance option_valid : Valid (option A) := λ mx,
Instance option_validN : ValidN (option A) := λ n mx,
match mx with Some x => {n} x | None => True end.
Global Instance option_empty : Empty (option A) := None.