Commit 5cfe326f authored by Ralf Jung's avatar Ralf Jung
Browse files

fix indentation and various nits

parent b0da646d
...@@ -66,11 +66,14 @@ Section ofe. ...@@ -66,11 +66,14 @@ Section ofe.
dyn_reservation_map_data_proj x {n} dyn_reservation_map_data_proj y dyn_reservation_map_data_proj x {n} dyn_reservation_map_data_proj y
dyn_reservation_map_token_proj x = dyn_reservation_map_token_proj y. dyn_reservation_map_token_proj x = dyn_reservation_map_token_proj y.
Global Instance DynReservationMap_ne : NonExpansive2 (@DynReservationMap A). Global Instance DynReservationMap_ne :
NonExpansive2 (@DynReservationMap A).
Proof. by split. Qed. Proof. by split. Qed.
Global Instance DynReservationMap_proper : Proper (() ==> (=) ==> ()) (@DynReservationMap A). Global Instance DynReservationMap_proper :
Proper (() ==> (=) ==> ()) (@DynReservationMap A).
Proof. by split. Qed. Proof. by split. Qed.
Global Instance dyn_reservation_map_data_proj_ne: NonExpansive (@dyn_reservation_map_data_proj A). Global Instance dyn_reservation_map_data_proj_ne :
NonExpansive (@dyn_reservation_map_data_proj A).
Proof. by destruct 1. Qed. Proof. by destruct 1. Qed.
Global Instance dyn_reservation_map_data_proj_proper : Global Instance dyn_reservation_map_data_proj_proper :
Proper (() ==> ()) (@dyn_reservation_map_data_proj A). Proper (() ==> ()) (@dyn_reservation_map_data_proj A).
...@@ -182,14 +185,10 @@ Section cmra. ...@@ -182,14 +185,10 @@ Section cmra.
rewrite {1}/op /cmra_op /=. case_decide; last done. rewrite {1}/op /cmra_op /=. case_decide; last done.
intros [Hm [Hinf Hdisj]]; split; first by eauto using cmra_validN_op_l. intros [Hm [Hinf Hdisj]]; split; first by eauto using cmra_validN_op_l.
split. split.
+ rewrite ->difference_union_distr_r in Hinf. + rewrite ->difference_union_distr_r_L in Hinf.
eapply set_infinite_subseteq; last done. eapply set_infinite_subseteq, Hinf. set_solver.
set_solver.
+ intros i. move: (Hdisj i). rewrite lookup_op. + intros i. move: (Hdisj i). rewrite lookup_op.
case: (m1 !! i)=> [a|]; last auto. case: (m1 !! i); case: (m2 !! i); set_solver.
move=> [].
{ by case: (m2 !! i). }
set_solver.
Qed. Qed.
Canonical Structure dyn_reservation_mapR := Canonical Structure dyn_reservation_mapR :=
...@@ -209,7 +208,7 @@ Section cmra. ...@@ -209,7 +208,7 @@ Section cmra.
Proof. Proof.
split; simpl. split; simpl.
- rewrite dyn_reservation_map_valid_eq /=. split; [apply ucmra_unit_valid|]. split. - rewrite dyn_reservation_map_valid_eq /=. split; [apply ucmra_unit_valid|]. split.
+ rewrite difference_empty. apply top_infinite. + rewrite difference_empty_L. apply top_infinite.
+ set_solver. + set_solver.
- split; simpl; [by rewrite left_id|by rewrite left_id_L]. - split; simpl; [by rewrite left_id|by rewrite left_id_L].
- do 2 constructor; [apply (core_id_core _)|done]. - do 2 constructor; [apply (core_id_core _)|done].
...@@ -227,7 +226,7 @@ Section cmra. ...@@ -227,7 +226,7 @@ Section cmra.
rewrite dyn_reservation_map_valid_eq /= singleton_valid. rewrite dyn_reservation_map_valid_eq /= singleton_valid.
split; first naive_solver. intros Ha. split; first naive_solver. intros Ha.
split; first done. split; last set_solver. split; first done. split; last set_solver.
rewrite difference_empty. apply top_infinite. rewrite difference_empty_L. apply top_infinite.
Qed. Qed.
Lemma dyn_reservation_map_token_valid E : Lemma dyn_reservation_map_token_valid E :
(dyn_reservation_map_token E) set_infinite ( E). (dyn_reservation_map_token E) set_infinite ( E).
...@@ -284,15 +283,14 @@ Section cmra. ...@@ -284,15 +283,14 @@ Section cmra.
such that both that set [E1] and the remainder [E2] are infinite. *) such that both that set [E1] and the remainder [E2] are infinite. *)
edestruct (coPset_split_infinite ( (Ef dom coPset mf))) as edestruct (coPset_split_infinite ( (Ef dom coPset mf))) as
(E1 & E2 & HEunion & HEdisj & HE1inf & HE2inf). (E1 & E2 & HEunion & HEdisj & HE1inf & HE2inf).
{ rewrite -difference_difference. { rewrite -difference_difference_L.
apply difference_infinite; first done. by apply difference_infinite, dom_finite. }
apply gset_to_coPset_finite. }
exists (dyn_reservation_map_token E1). exists (dyn_reservation_map_token E1).
split; first by apply HQ. clear HQ. split; first by apply HQ. clear HQ.
rewrite dyn_reservation_map_validN_eq /=. rewrite dyn_reservation_map_validN_eq /=.
rewrite coPset_disj_union; last set_solver. rewrite coPset_disj_union; last set_solver.
split; first by rewrite left_id. split. split; first by rewrite left_id_L. split.
- eapply set_infinite_subseteq; last by apply HE2inf. set_solver. - eapply set_infinite_subseteq, HE2inf. set_solver.
- intros i. rewrite left_id_L. destruct (Hdisj i) as [?|Hi]; first by left. - intros i. rewrite left_id_L. destruct (Hdisj i) as [?|Hi]; first by left.
destruct (mf !! i) as [p|] eqn:Hp; last by left. destruct (mf !! i) as [p|] eqn:Hp; last by left.
apply elem_of_dom_2 in Hp. right. set_solver. apply elem_of_dom_2 in Hp. right. set_solver.
...@@ -307,13 +305,13 @@ Section cmra. ...@@ -307,13 +305,13 @@ Section cmra.
intros ??. apply cmra_total_update=> n [mf [Ef|]] //. intros ??. apply cmra_total_update=> n [mf [Ef|]] //.
rewrite dyn_reservation_map_validN_eq /= {1}/op /cmra_op /=. case_decide; last done. rewrite dyn_reservation_map_validN_eq /= {1}/op /cmra_op /=. case_decide; last done.
rewrite left_id_L {1}left_id. intros [Hmf [Hinf Hdisj]]; split; last split. rewrite left_id_L {1}left_id. intros [Hmf [Hinf Hdisj]]; split; last split.
- destruct (Hdisj (k)) as [Hmfi|]; last set_solver. - destruct (Hdisj k) as [Hmfi|]; last set_solver.
move: Hmfi. rewrite lookup_op lookup_empty left_id_L=> Hmfi. move: Hmfi. rewrite lookup_op lookup_empty left_id_L=> Hmfi.
intros j. rewrite lookup_op. intros j. rewrite lookup_op.
destruct (decide (k = j)) as [<-|]. destruct (decide (k = j)) as [<-|].
+ rewrite Hmfi lookup_singleton right_id_L. by apply cmra_valid_validN. + rewrite Hmfi lookup_singleton right_id_L. by apply cmra_valid_validN.
+ by rewrite lookup_singleton_ne // left_id_L. + by rewrite lookup_singleton_ne // left_id_L.
- eapply set_infinite_subseteq; last done. set_solver. - eapply set_infinite_subseteq, Hinf. set_solver.
- intros j. destruct (decide (k = j)); first set_solver. - intros j. destruct (decide (k = j)); first set_solver.
rewrite lookup_op lookup_singleton_ne //. rewrite lookup_op lookup_singleton_ne //.
destruct (Hdisj j) as [Hmfi|?]; last set_solver. destruct (Hdisj j) as [Hmfi|?]; last set_solver.
...@@ -321,7 +319,8 @@ Section cmra. ...@@ -321,7 +319,8 @@ Section cmra.
Qed. Qed.
Lemma dyn_reservation_map_updateP P (Q : dyn_reservation_map A Prop) k a : Lemma dyn_reservation_map_updateP P (Q : dyn_reservation_map A Prop) k a :
a ~~>: P a ~~>: P
( a', P a' Q (dyn_reservation_map_data k a')) dyn_reservation_map_data k a ~~>: Q. ( a', P a' Q (dyn_reservation_map_data k a'))
dyn_reservation_map_data k a ~~>: Q.
Proof. Proof.
intros Hup HP. apply cmra_total_updateP=> n [mf [Ef|]] //. intros Hup HP. apply cmra_total_updateP=> n [mf [Ef|]] //.
rewrite dyn_reservation_map_validN_eq /= left_id_L. intros [Hmf [Hinf Hdisj]]. rewrite dyn_reservation_map_validN_eq /= left_id_L. intros [Hmf [Hinf Hdisj]].
...@@ -339,7 +338,8 @@ Section cmra. ...@@ -339,7 +338,8 @@ Section cmra.
rewrite !lookup_op !op_None !lookup_singleton_None. naive_solver. rewrite !lookup_op !op_None !lookup_singleton_None. naive_solver.
Qed. Qed.
Lemma dyn_reservation_map_update k a b : Lemma dyn_reservation_map_update k a b :
a ~~> b dyn_reservation_map_data k a ~~> dyn_reservation_map_data k b. a ~~> b
dyn_reservation_map_data k a ~~> dyn_reservation_map_data k b.
Proof. Proof.
rewrite !cmra_update_updateP. eauto using dyn_reservation_map_updateP with subst. rewrite !cmra_update_updateP. eauto using dyn_reservation_map_updateP with subst.
Qed. Qed.
......
...@@ -41,63 +41,66 @@ Global Instance: Params (@reservation_map_data) 2 := {}. ...@@ -41,63 +41,66 @@ Global Instance: Params (@reservation_map_data) 2 := {}.
(* Ofe *) (* Ofe *)
Section ofe. Section ofe.
Context {A : ofe}. Context {A : ofe}.
Implicit Types x y : reservation_map A. Implicit Types x y : reservation_map A.
Local Instance reservation_map_equiv : Equiv (reservation_map A) := λ x y, Local Instance reservation_map_equiv : Equiv (reservation_map A) := λ x y,
reservation_map_data_proj x reservation_map_data_proj y reservation_map_data_proj x reservation_map_data_proj y
reservation_map_token_proj x = reservation_map_token_proj y. reservation_map_token_proj x = reservation_map_token_proj y.
Local Instance reservation_map_dist : Dist (reservation_map A) := λ n x y, Local Instance reservation_map_dist : Dist (reservation_map A) := λ n x y,
reservation_map_data_proj x {n} reservation_map_data_proj y reservation_map_data_proj x {n} reservation_map_data_proj y
reservation_map_token_proj x = reservation_map_token_proj y. reservation_map_token_proj x = reservation_map_token_proj y.
Global Instance ReservationMap_ne : NonExpansive2 (@ReservationMap A). Global Instance ReservationMap_ne :
Proof. by split. Qed. NonExpansive2 (@ReservationMap A).
Global Instance ReservationMap_proper : Proper (() ==> (=) ==> ()) (@ReservationMap A). Proof. by split. Qed.
Proof. by split. Qed. Global Instance ReservationMap_proper :
Global Instance reservation_map_data_proj_ne: NonExpansive (@reservation_map_data_proj A). Proper (() ==> (=) ==> ()) (@ReservationMap A).
Proof. by destruct 1. Qed. Proof. by split. Qed.
Global Instance reservation_map_data_proj_proper : Global Instance reservation_map_data_proj_ne :
NonExpansive (@reservation_map_data_proj A).
Proof. by destruct 1. Qed.
Global Instance reservation_map_data_proj_proper :
Proper (() ==> ()) (@reservation_map_data_proj A). Proper (() ==> ()) (@reservation_map_data_proj A).
Proof. by destruct 1. Qed. Proof. by destruct 1. Qed.
Definition reservation_map_ofe_mixin : OfeMixin (reservation_map A). Definition reservation_map_ofe_mixin : OfeMixin (reservation_map A).
Proof. Proof.
by apply (iso_ofe_mixin by apply (iso_ofe_mixin
(λ x, (reservation_map_data_proj x, reservation_map_token_proj x))). (λ x, (reservation_map_data_proj x, reservation_map_token_proj x))).
Qed. Qed.
Canonical Structure reservation_mapO := Canonical Structure reservation_mapO :=
Ofe (reservation_map A) reservation_map_ofe_mixin. Ofe (reservation_map A) reservation_map_ofe_mixin.
Global Instance ReservationMap_discrete a b : Global Instance ReservationMap_discrete a b :
Discrete a Discrete b Discrete (ReservationMap a b). Discrete a Discrete b Discrete (ReservationMap a b).
Proof. intros ?? [??] [??]; split; unfold_leibniz; by eapply discrete. Qed. Proof. intros ?? [??] [??]; split; unfold_leibniz; by eapply discrete. Qed.
Global Instance reservation_map_ofe_discrete : Global Instance reservation_map_ofe_discrete :
OfeDiscrete A OfeDiscrete reservation_mapO. OfeDiscrete A OfeDiscrete reservation_mapO.
Proof. intros ? [??]; apply _. Qed. Proof. intros ? [??]; apply _. Qed.
End ofe. End ofe.
Global Arguments reservation_mapO : clear implicits. Global Arguments reservation_mapO : clear implicits.
(* Camera *) (* Camera *)
Section cmra. Section cmra.
Context {A : cmra}. Context {A : cmra}.
Implicit Types a b : A. Implicit Types a b : A.
Implicit Types x y : reservation_map A. Implicit Types x y : reservation_map A.
Implicit Types k : positive. Implicit Types k : positive.
Global Instance reservation_map_data_ne i : NonExpansive (@reservation_map_data A i). Global Instance reservation_map_data_ne i : NonExpansive (@reservation_map_data A i).
Proof. solve_proper. Qed. Proof. solve_proper. Qed.
Global Instance reservation_map_data_proper N : Global Instance reservation_map_data_proper N :
Proper (() ==> ()) (@reservation_map_data A N). Proper (() ==> ()) (@reservation_map_data A N).
Proof. solve_proper. Qed. Proof. solve_proper. Qed.
Global Instance reservation_map_data_discrete N a : Global Instance reservation_map_data_discrete N a :
Discrete a Discrete (reservation_map_data N a). Discrete a Discrete (reservation_map_data N a).
Proof. intros. apply ReservationMap_discrete; apply _. Qed. Proof. intros. apply ReservationMap_discrete; apply _. Qed.
Global Instance reservation_map_token_discrete E : Discrete (@reservation_map_token A E). Global Instance reservation_map_token_discrete E : Discrete (@reservation_map_token A E).
Proof. intros. apply ReservationMap_discrete; apply _. Qed. Proof. intros. apply ReservationMap_discrete; apply _. Qed.
Local Instance reservation_map_valid_instance : Valid (reservation_map A) := λ x, Local Instance reservation_map_valid_instance : Valid (reservation_map A) := λ x,
match reservation_map_token_proj x with match reservation_map_token_proj x with
| CoPset E => | CoPset E =>
(reservation_map_data_proj x) (reservation_map_data_proj x)
...@@ -105,8 +108,8 @@ Local Instance reservation_map_valid_instance : Valid (reservation_map A) := λ ...@@ -105,8 +108,8 @@ Local Instance reservation_map_valid_instance : Valid (reservation_map A) := λ
i, reservation_map_data_proj x !! i = None i E i, reservation_map_data_proj x !! i = None i E
| CoPsetBot => False | CoPsetBot => False
end. end.
Global Arguments reservation_map_valid_instance !_ /. Global Arguments reservation_map_valid_instance !_ /.
Local Instance reservation_map_validN_instance : ValidN (reservation_map A) := λ n x, Local Instance reservation_map_validN_instance : ValidN (reservation_map A) := λ n x,
match reservation_map_token_proj x with match reservation_map_token_proj x with
| CoPset E => | CoPset E =>
{n} (reservation_map_data_proj x) {n} (reservation_map_data_proj x)
...@@ -114,14 +117,14 @@ Local Instance reservation_map_validN_instance : ValidN (reservation_map A) := ...@@ -114,14 +117,14 @@ Local Instance reservation_map_validN_instance : ValidN (reservation_map A) :=
i, reservation_map_data_proj x !! i = None i E i, reservation_map_data_proj x !! i = None i E
| CoPsetBot => False | CoPsetBot => False
end. end.
Global Arguments reservation_map_validN_instance !_ /. Global Arguments reservation_map_validN_instance !_ /.
Local Instance reservation_map_pcore_instance : PCore (reservation_map A) := λ x, Local Instance reservation_map_pcore_instance : PCore (reservation_map A) := λ x,
Some (ReservationMap (core (reservation_map_data_proj x)) ε). Some (ReservationMap (core (reservation_map_data_proj x)) ε).
Local Instance reservation_map_op_instance : Op (reservation_map A) := λ x y, Local Instance reservation_map_op_instance : Op (reservation_map A) := λ x y,
ReservationMap (reservation_map_data_proj x reservation_map_data_proj y) ReservationMap (reservation_map_data_proj x reservation_map_data_proj y)
(reservation_map_token_proj x reservation_map_token_proj y). (reservation_map_token_proj x reservation_map_token_proj y).
Definition reservation_map_valid_eq : Definition reservation_map_valid_eq :
valid = λ x, match reservation_map_token_proj x with valid = λ x, match reservation_map_token_proj x with
| CoPset E => | CoPset E =>
(reservation_map_data_proj x) (reservation_map_data_proj x)
...@@ -129,7 +132,7 @@ Definition reservation_map_valid_eq : ...@@ -129,7 +132,7 @@ Definition reservation_map_valid_eq :
i, reservation_map_data_proj x !! i = None i E i, reservation_map_data_proj x !! i = None i E
| CoPsetBot => False | CoPsetBot => False
end := eq_refl _. end := eq_refl _.
Definition reservation_map_validN_eq : Definition reservation_map_validN_eq :
validN = λ n x, match reservation_map_token_proj x with validN = λ n x, match reservation_map_token_proj x with
| CoPset E => | CoPset E =>
{n} (reservation_map_data_proj x) {n} (reservation_map_data_proj x)
...@@ -138,22 +141,22 @@ Definition reservation_map_validN_eq : ...@@ -138,22 +141,22 @@ Definition reservation_map_validN_eq :
| CoPsetBot => False | CoPsetBot => False
end := eq_refl _. end := eq_refl _.
Lemma reservation_map_included x y : Lemma reservation_map_included x y :
x y x y
reservation_map_data_proj x reservation_map_data_proj y reservation_map_data_proj x reservation_map_data_proj y
reservation_map_token_proj x reservation_map_token_proj y. reservation_map_token_proj x reservation_map_token_proj y.
Proof. Proof.
split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|]. split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|].
intros [[z1 Hz1] [z2 Hz2]]; exists (ReservationMap z1 z2); split; auto. intros [[z1 Hz1] [z2 Hz2]]; exists (ReservationMap z1 z2); split; auto.
Qed. Qed.
Lemma reservation_map_data_proj_validN n x : {n} x {n} reservation_map_data_proj x. Lemma reservation_map_data_proj_validN n x : {n} x {n} reservation_map_data_proj x.
Proof. by destruct x as [? [?|]]=> // -[??]. Qed. Proof. by destruct x as [? [?|]]=> // -[??]. Qed.
Lemma reservation_map_token_proj_validN n x : {n} x {n} reservation_map_token_proj x. Lemma reservation_map_token_proj_validN n x : {n} x {n} reservation_map_token_proj x.
Proof. by destruct x as [? [?|]]=> // -[??]. Qed. Proof. by destruct x as [? [?|]]=> // -[??]. Qed.
Lemma reservation_map_cmra_mixin : CmraMixin (reservation_map A). Lemma reservation_map_cmra_mixin : CmraMixin (reservation_map A).
Proof. Proof.
apply cmra_total_mixin. apply cmra_total_mixin.
- eauto. - eauto.
- by intros n x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'. - by intros n x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
...@@ -175,10 +178,7 @@ Proof. ...@@ -175,10 +178,7 @@ Proof.
rewrite {1}/op /cmra_op /=. case_decide; last done. rewrite {1}/op /cmra_op /=. case_decide; last done.
intros [Hm Hdisj]; split; first by eauto using cmra_validN_op_l. intros [Hm Hdisj]; split; first by eauto using cmra_validN_op_l.
intros i. move: (Hdisj i). rewrite lookup_op. intros i. move: (Hdisj i). rewrite lookup_op.
case: (m1 !! i)=> [a|]; last auto. case: (m1 !! i); case: (m2 !! i); set_solver.
move=> [].
{ by case: (m2 !! i). }
set_solver.
- intros n x y1 y2 ? [??]; simpl in *. - intros n x y1 y2 ? [??]; simpl in *.
destruct (cmra_extend n (reservation_map_data_proj x) destruct (cmra_extend n (reservation_map_data_proj x)
(reservation_map_data_proj y1) (reservation_map_data_proj y2)) (reservation_map_data_proj y1) (reservation_map_data_proj y2))
...@@ -187,80 +187,80 @@ Proof. ...@@ -187,80 +187,80 @@ Proof.
(reservation_map_token_proj y1) (reservation_map_token_proj y2)) (reservation_map_token_proj y1) (reservation_map_token_proj y2))
as (E1&E2&?&?&?); auto using reservation_map_token_proj_validN. as (E1&E2&?&?&?); auto using reservation_map_token_proj_validN.
by exists (ReservationMap m1 E1), (ReservationMap m2 E2). by exists (ReservationMap m1 E1), (ReservationMap m2 E2).
Qed. Qed.
Canonical Structure reservation_mapR := Canonical Structure reservation_mapR :=
Cmra (reservation_map A) reservation_map_cmra_mixin. Cmra (reservation_map A) reservation_map_cmra_mixin.
Global Instance reservation_map_cmra_discrete : Global Instance reservation_map_cmra_discrete :
CmraDiscrete A CmraDiscrete reservation_mapR. CmraDiscrete A CmraDiscrete reservation_mapR.
Proof. Proof.
split; first apply _. split; first apply _.
intros [m [E|]]; rewrite reservation_map_validN_eq reservation_map_valid_eq //=. intros [m [E|]]; rewrite reservation_map_validN_eq reservation_map_valid_eq //=.
by intros [?%cmra_discrete_valid ?]. by intros [?%cmra_discrete_valid ?].
Qed. Qed.
Local Instance reservation_map_empty_instance : Unit (reservation_map A) := ReservationMap ε ε. Local Instance reservation_map_empty_instance : Unit (reservation_map A) := ReservationMap ε ε.
Lemma reservation_map_ucmra_mixin : UcmraMixin (reservation_map A). Lemma reservation_map_ucmra_mixin : UcmraMixin (reservation_map A).
Proof. Proof.
split; simpl. split; simpl.
- rewrite reservation_map_valid_eq /=. split; [apply ucmra_unit_valid|]. set_solver. - rewrite reservation_map_valid_eq /=. split; [apply ucmra_unit_valid|]. set_solver.
- split; simpl; [by rewrite left_id|by rewrite left_id_L]. - split; simpl; [by rewrite left_id|by rewrite left_id_L].
- do 2 constructor; [apply (core_id_core _)|done]. - do 2 constructor; [apply (core_id_core _)|done].
Qed. Qed.
Canonical Structure reservation_mapUR := Canonical Structure reservation_mapUR :=
Ucmra (reservation_map A) reservation_map_ucmra_mixin. Ucmra (reservation_map A) reservation_map_ucmra_mixin.
Global Instance reservation_map_data_core_id N a : Global Instance reservation_map_data_core_id N a :
CoreId a CoreId (reservation_map_data N a). CoreId a CoreId (reservation_map_data N a).
Proof. do 2 constructor; simpl; auto. apply core_id_core, _. Qed. Proof. do 2 constructor; simpl; auto. apply core_id_core, _. Qed.
Lemma reservation_map_data_valid N a : (reservation_map_data N a) a. Lemma reservation_map_data_valid N a : (reservation_map_data N a) a.
Proof. rewrite reservation_map_valid_eq /= singleton_valid. set_solver. Qed. Proof. rewrite reservation_map_valid_eq /= singleton_valid. set_solver. Qed.
Lemma reservation_map_token_valid E : (reservation_map_token E). Lemma reservation_map_token_valid E : (reservation_map_token E).
Proof. rewrite reservation_map_valid_eq /=. split; first done. by left. Qed. Proof. rewrite reservation_map_valid_eq /=. split; first done. by left. Qed.
Lemma reservation_map_data_op N a b : Lemma reservation_map_data_op N a b :
reservation_map_data N (a b) = reservation_map_data N a reservation_map_data N b. reservation_map_data N (a b) = reservation_map_data N a reservation_map_data N b.
Proof. Proof.
by rewrite {2}/op /reservation_map_op_instance /reservation_map_data /= singleton_op left_id_L. by rewrite {2}/op /reservation_map_op_instance /reservation_map_data /= singleton_op left_id_L.
Qed. Qed.
Lemma reservation_map_data_mono N a b : Lemma reservation_map_data_mono N a b :
a b reservation_map_data N a reservation_map_data N b. a b reservation_map_data N a reservation_map_data N b.
Proof. intros [c ->]. rewrite reservation_map_data_op. apply cmra_included_l. Qed. Proof. intros [c ->]. rewrite reservation_map_data_op. apply cmra_included_l. Qed.
Global Instance reservation_map_data_is_op N a b1 b2 : Global Instance reservation_map_data_is_op N a b1 b2 :
IsOp a b1 b2 IsOp a b1 b2
IsOp' (reservation_map_data N a) (reservation_map_data N b1) (reservation_map_data N b2). IsOp' (reservation_map_data N a) (reservation_map_data N b1) (reservation_map_data N b2).
Proof. rewrite /IsOp' /IsOp=> ->. by rewrite reservation_map_data_op. Qed. Proof. rewrite /IsOp' /IsOp=> ->. by rewrite reservation_map_data_op. Qed.
Lemma reservation_map_token_union E1 E2 : Lemma reservation_map_token_union E1 E2 :
E1 ## E2 E1 ## E2
reservation_map_token (E1 E2) = reservation_map_token E1 reservation_map_token E2. reservation_map_token (E1 E2) = reservation_map_token E1 reservation_map_token E2.
Proof. Proof.
intros. by rewrite /op /reservation_map_op_instance intros. by rewrite /op /reservation_map_op_instance
/reservation_map_token /= coPset_disj_union // left_id_L. /reservation_map_token /= coPset_disj_union // left_id_L.
Qed. Qed.
Lemma reservation_map_token_difference E1 E2 : Lemma reservation_map_token_difference E1 E2 :
E1 E2 E1 E2
reservation_map_token E2 = reservation_map_token E1 reservation_map_token (E2 E1). reservation_map_token E2 = reservation_map_token E1 reservation_map_token (E2 E1).
Proof. Proof.
intros. rewrite -reservation_map_token_union; last set_solver. intros. rewrite -reservation_map_token_union; last set_solver.
by rewrite -union_difference_L. by rewrite -union_difference_L.
Qed. Qed.
Lemma reservation_map_token_valid_op E1 E2 : Lemma reservation_map_token_valid_op E1 E2 :
(reservation_map_token E1 reservation_map_token E2) E1 ## E2. (reservation_map_token E1 reservation_map_token E2) E1 ## E2.
Proof. Proof.
rewrite reservation_map_valid_eq /= {1}/op /cmra_op /=. case_decide; last done. rewrite reservation_map_valid_eq /= {1}/op /cmra_op /=. case_decide; last done.
split; [done|]; intros _. split. split; [done|]; intros _. split.
- by rewrite left_id. - by rewrite left_id.
- intros i. rewrite lookup_op lookup_empty. auto. - intros i. rewrite lookup_op lookup_empty. auto.
Qed. Qed.
Lemma reservation_map_alloc E k a : Lemma reservation_map_alloc E k a :
k E a reservation_map_token E ~~> reservation_map_data k a. k E a reservation_map_token E ~~> reservation_map_data k a.
Proof. Proof.
intros ??. apply cmra_total_update=> n [mf [Ef|]] //. intros ??. apply cmra_total_update=> n [mf [Ef|]] //.
rewrite reservation_map_validN_eq /= {1}/op /cmra_op /=. case_decide; last done. rewrite reservation_map_validN_eq /= {1}/op /cmra_op /=. case_decide; last done.
rewrite left_id_L {1}left_id. intros [Hmf Hdisj]; split. rewrite left_id_L {1}left_id. intros [Hmf Hdisj]; split.
- destruct (Hdisj (k)) as [Hmfi|]; last set_solver. - destruct (Hdisj k) as [Hmfi|]; last set_solver.
move: Hmfi. rewrite lookup_op lookup_empty left_id_L=> Hmfi. move: Hmfi. rewrite lookup_op lookup_empty left_id_L=> Hmfi.
intros j. rewrite lookup_op. intros j. rewrite lookup_op.
destruct (decide (k = j)) as [<-|]. destruct (decide (k = j)) as [<-|].
...@@ -270,11 +270,12 @@ Proof. ...@@ -270,11 +270,12 @@ Proof.
rewrite lookup_op lookup_singleton_ne //. rewrite lookup_op lookup_singleton_ne //.
destruct (Hdisj j) as [Hmfi|?]; last set_solver. destruct (Hdisj j) as [Hmfi|?]; last set_solver.
move: Hmfi. rewrite lookup_op lookup_empty; auto. move: Hmfi. rewrite lookup_op lookup_empty; auto.
Qed.