Disjointness of sets.

algebra/sts.v

@@ -26,17 +26,16 @@ Context {sts : stsT}.
 (** ** Step relations *)
 Inductive step : relation (state sts * tokens sts) :=
   | Step s1 s2 T1 T2 :
-     (* TODO: This asks for ⊥ on sets: T1 ⊥ T2 := T1 ∩ T2 ⊆ ∅. *)
-     prim_step s1 s2 → tok s1 ∩ T1 ≡ ∅ → tok s2 ∩ T2 ≡ ∅ →
+     prim_step s1 s2 → tok s1 ⊥ T1 → tok s2 ⊥ T2 →
      tok s1 ∪ T1 ≡ tok s2 ∪ T2 → step (s1,T1) (s2,T2).
 Notation steps := (rtc step).
 Inductive frame_step (T : tokens sts) (s1 s2 : state sts) : Prop :=
   | Frame_step T1 T2 :
-     T1 ∩ (tok s1 ∪ T) ≡ ∅ → step (s1,T1) (s2,T2) → frame_step T s1 s2.
+     T1 ⊥ tok s1 ∪ T → step (s1,T1) (s2,T2) → frame_step T s1 s2.
 
 (** ** Closure under frame steps *)
 Record closed (S : states sts) (T : tokens sts) : Prop := Closed {
-  closed_disjoint s : s ∈ S → tok s ∩ T ≡ ∅;
+  closed_disjoint s : s ∈ S → tok s ⊥ T;
   closed_step s1 s2 : s1 ∈ S → frame_step T s1 s2 → s2 ∈ S
 }.
 Definition up (s : state sts) (T : tokens sts) : states sts :=
@@ -50,6 +49,7 @@ Hint Extern 50 (equiv (A:=set _) _ _) => set_solver : sts.
 Hint Extern 50 (¬equiv (A:=set _) _ _) => set_solver : sts.
 Hint Extern 50 (_ ∈ _) => set_solver : sts.
 Hint Extern 50 (_ ⊆ _) => set_solver : sts.
+Hint Extern 50 (_ ⊥ _) => set_solver : sts.
 
 (** ** Setoids *)
 Instance framestep_mono : Proper (flip (⊆) ==> (=) ==> (=) ==> impl) frame_step.
@@ -60,10 +60,7 @@ Qed.
 Global Instance framestep_proper : Proper ((≡) ==> (=) ==> (=) ==> iff) frame_step.
 Proof. by intros ?? [??] ??????; split; apply framestep_mono. Qed.
 Instance closed_proper' : Proper ((≡) ==> (≡) ==> impl) closed.
-Proof.
-  intros ?? HT ?? HS; destruct 1;
-    constructor; intros until 0; rewrite -?HS -?HT; eauto.
-Qed.
+Proof. destruct 3; constructor; intros until 0; setoid_subst; eauto. Qed.
 Global Instance closed_proper : Proper ((≡) ==> (≡) ==> iff) closed.
 Proof. by split; apply closed_proper'. Qed.
 Global Instance up_preserving : Proper ((=) ==> flip (⊆) ==> (⊆)) up.
@@ -95,16 +92,16 @@ Proof.
   - apply Hstep2 with s3, Frame_step with T3 T4; auto with sts.
 Qed.
 Lemma step_closed s1 s2 T1 T2 S Tf :
-  step (s1,T1) (s2,T2) → closed S Tf → s1 ∈ S → T1 ∩ Tf ≡ ∅ →
-  s2 ∈ S ∧ T2 ∩ Tf ≡ ∅ ∧ tok s2 ∩ T2 ≡ ∅.
+  step (s1,T1) (s2,T2) → closed S Tf → s1 ∈ S → T1 ⊥ Tf →
+  s2 ∈ S ∧ T2 ⊥ Tf ∧ tok s2 ⊥ T2.
 Proof.
   inversion_clear 1 as [???? HR Hs1 Hs2]; intros [? Hstep]??; split_and?; auto.
   - eapply Hstep with s1, Frame_step with T1 T2; auto with sts.
   - set_solver -Hstep Hs1 Hs2.
 Qed.
 Lemma steps_closed s1 s2 T1 T2 S Tf :
-  steps (s1,T1) (s2,T2) → closed S Tf → s1 ∈ S → T1 ∩ Tf ≡ ∅ →
-  tok s1 ∩ T1 ≡ ∅ → s2 ∈ S ∧ T2 ∩ Tf ≡ ∅ ∧ tok s2 ∩ T2 ≡ ∅.
+  steps (s1,T1) (s2,T2) → closed S Tf → s1 ∈ S → T1 ⊥ Tf →
+  tok s1 ⊥ T1 → s2 ∈ S ∧ T2 ⊥ Tf ∧ tok s2 ⊥ T2.
 Proof.
   remember (s1,T1) as sT1 eqn:HsT1; remember (s2,T2) as sT2 eqn:HsT2.
   intros Hsteps; revert s1 T1 HsT1 s2 T2 HsT2.
@@ -120,8 +117,7 @@ Lemma subseteq_up_set S T : S ⊆ up_set S T.
 Proof. intros s ?; apply elem_of_bind; eauto using elem_of_up. Qed.
 Lemma up_up_set s T : up s T ≡ up_set {[ s ]} T.
 Proof. by rewrite /up_set collection_bind_singleton. Qed.
-Lemma closed_up_set S T :
-  (∀ s, s ∈ S → tok s ∩ T ≡ ∅) → closed (up_set S T) T.
+Lemma closed_up_set S T : (∀ s, s ∈ S → tok s ⊥ T) → closed (up_set S T) T.
 Proof.
   intros HS; unfold up_set; split.
   - intros s; rewrite !elem_of_bind; intros (s'&Hstep&Hs').
@@ -131,7 +127,7 @@ Proof.
   - intros s1 s2; rewrite /up; set_unfold; intros (s&?&?) ?; exists s.
     split; [eapply rtc_r|]; eauto.
 Qed.
-Lemma closed_up s T : tok s ∩ T ≡ ∅ → closed (up s T) T.
+Lemma closed_up s T : tok s ⊥ T → closed (up s T) T.
 Proof.
   intros; rewrite -(collection_bind_singleton (λ s, up s T) s).
   apply closed_up_set; set_solver.
@@ -188,7 +184,7 @@ Inductive sts_equiv : Equiv (car sts) :=
 Global Existing Instance sts_equiv.
 Global Instance sts_valid : Valid (car sts) := λ x,
   match x with
-  | auth s T => tok s ∩ T ≡ ∅
+  | auth s T => tok s ⊥ T
   | frag S' T => closed S' T ∧ S' ≢ ∅
   end.
 Global Instance sts_core : Core (car sts) := λ x,
@@ -198,11 +194,9 @@ Global Instance sts_core : Core (car sts) := λ x,
   end.
 Inductive sts_disjoint : Disjoint (car sts) :=
   | frag_frag_disjoint S1 S2 T1 T2 :
-     S1 ∩ S2 ≢ ∅ → T1 ∩ T2 ≡ ∅ → frag S1 T1 ⊥ frag S2 T2
-  | auth_frag_disjoint s S T1 T2 :
-     s ∈ S → T1 ∩ T2 ≡ ∅ → auth s T1 ⊥ frag S T2
-  | frag_auth_disjoint s S T1 T2 :
-     s ∈ S → T1 ∩ T2 ≡ ∅ → frag S T1 ⊥ auth s T2.
+     S1 ∩ S2 ≢ ∅ → T1 ⊥ T2 → frag S1 T1 ⊥ frag S2 T2
+  | auth_frag_disjoint s S T1 T2 : s ∈ S → T1 ⊥ T2 → auth s T1 ⊥ frag S T2
+  | frag_auth_disjoint s S T1 T2 : s ∈ S → T1 ⊥ T2 → frag S T1 ⊥ auth s T2.
 Global Existing Instance sts_disjoint.
 Global Instance sts_op : Op (car sts) := λ x1 x2,
   match x1, x2 with
@@ -216,6 +210,8 @@ Hint Extern 50 (equiv (A:=set _) _ _) => set_solver : sts.
 Hint Extern 50 (¬equiv (A:=set _) _ _) => set_solver : sts.
 Hint Extern 50 (_ ∈ _) => set_solver : sts.
 Hint Extern 50 (_ ⊆ _) => set_solver : sts.
+Hint Extern 50 (_ ⊥ _) => set_solver : sts.
+
 Global Instance sts_equivalence: Equivalence ((≡) : relation (car sts)).
 Proof.
   split.
@@ -303,11 +299,11 @@ Global Instance sts_frag_up_proper s : Proper ((≡) ==> (≡)) (sts_frag_up s).
 Proof. intros T1 T2 HT. by rewrite /sts_frag_up HT. Qed.
 
 (** Validity *)
-Lemma sts_auth_valid s T : ✓ sts_auth s T ↔ tok s ∩ T ≡ ∅.
+Lemma sts_auth_valid s T : ✓ sts_auth s T ↔ tok s ⊥ T.
 Proof. done. Qed.
 Lemma sts_frag_valid S T : ✓ sts_frag S T ↔ closed S T ∧ S ≢ ∅.
 Proof. done. Qed.
-Lemma sts_frag_up_valid s T : tok s ∩ T ≡ ∅ → ✓ sts_frag_up s T.
+Lemma sts_frag_up_valid s T : tok s ⊥ T → ✓ sts_frag_up s T.
 Proof. intros. by apply sts_frag_valid; auto using closed_up, up_non_empty. Qed.
 
 Lemma sts_auth_frag_valid_inv s S T1 T2 :
@@ -335,7 +331,7 @@ Proof.
 Qed.
 
 Lemma sts_op_frag S1 S2 T1 T2 :
-  T1 ∩ T2 ≡ ∅ → sts.closed S1 T1 → sts.closed S2 T2 →
+  T1 ⊥ T2 → sts.closed S1 T1 → sts.closed S2 T2 →
   sts_frag (S1 ∩ S2) (T1 ∪ T2) ≡ sts_frag S1 T1 ⋅ sts_frag S2 T2.
 Proof.
   intros HT HS1 HS2. rewrite /sts_frag.
@@ -390,7 +386,7 @@ Qed.
 Lemma sts_frag_included S1 S2 T1 T2 :
   closed S2 T2 → S2 ≢ ∅ →
   (sts_frag S1 T1 ≼ sts_frag S2 T2) ↔
-  (closed S1 T1 ∧ S1 ≢ ∅ ∧ ∃ Tf, T2 ≡ T1 ∪ Tf ∧ T1 ∩ Tf ≡ ∅ ∧
+  (closed S1 T1 ∧ S1 ≢ ∅ ∧ ∃ Tf, T2 ≡ T1 ∪ Tf ∧ T1 ⊥ Tf ∧
                                  S2 ≡ S1 ∩ up_set S2 Tf).
 Proof.
   intros ??; split.

prelude/collections.v

@@ -5,9 +5,11 @@ importantly, it implements some tactics to automatically solve goals involving
 collections. *)
 From iris.prelude Require Export base tactics orders.
 
+Instance collection_disjoint `{ElemOf A C} : Disjoint C := λ X Y,
+  ∀ x, x ∈ X → x ∈ Y → False.
 Instance collection_subseteq `{ElemOf A C} : SubsetEq C := λ X Y,
   ∀ x, x ∈ X → x ∈ Y.
-Typeclasses Opaque collection_subseteq.
+Typeclasses Opaque collection_disjoint collection_subseteq.
 
 (** * Basic theorems *)
 Section simple_collection.
@@ -36,6 +38,9 @@ Section simple_collection.
   Proof. firstorder. Qed.
   Lemma elem_of_equiv_empty X : X ≡ ∅ ↔ ∀ x, x ∉ X.
   Proof. firstorder. Qed.
+  Lemma elem_of_disjoint X Y : X ⊥ Y ↔ ∀ x, x ∈ X → x ∈ Y → False.
+  Proof. done. Qed.
+
   Lemma collection_positive_l X Y : X ∪ Y ≡ ∅ → X ≡ ∅.
   Proof.
     rewrite !elem_of_equiv_empty. setoid_rewrite elem_of_union. naive_solver.
@@ -52,11 +57,14 @@ Section simple_collection.
     - intros ??. rewrite elem_of_singleton. by intros ->.
     - intros Ex. by apply (Ex x), elem_of_singleton.
   Qed.
+
   Global Instance singleton_proper : Proper ((=) ==> (≡)) (singleton (B:=C)).
   Proof. by repeat intro; subst. Qed.
   Global Instance elem_of_proper :
-    Proper ((=) ==> (≡) ==> iff) ((∈) : A → C → Prop) | 5.
+    Proper ((=) ==> (≡) ==> iff) (@elem_of A C _) | 5.
   Proof. intros ???; subst. firstorder. Qed.
+  Global Instance disjoint_prope : Proper ((≡) ==> (≡) ==> iff) (@disjoint C _).
+  Proof. intros ??????. by rewrite !elem_of_disjoint; setoid_subst. Qed.
   Lemma elem_of_union_list Xs x : x ∈ ⋃ Xs ↔ ∃ X, X ∈ Xs ∧ x ∈ X.
   Proof.
     split.
@@ -196,6 +204,10 @@ Section set_unfold_simple.
     constructor. rewrite subset_spec, elem_of_subseteq, elem_of_equiv.
     repeat f_equiv; naive_solver.
   Qed.
+  Global Instance set_unfold_disjoint (P Q : A → Prop) :
+    (∀ x, SetUnfold (x ∈ X) (P x)) → (∀ x, SetUnfold (x ∈ Y) (Q x)) →
+    SetUnfold (X ⊥ Y) (∀ x, P x → Q x → False).
+  Proof. constructor. rewrite elem_of_disjoint. naive_solver. Qed.
 
   Context `{!LeibnizEquiv C}.
   Global Instance set_unfold_equiv_same_L X : SetUnfold (X = X) True | 1.
@@ -387,7 +399,7 @@ Section collection.
   Proof. set_solver. Qed.
   Lemma difference_intersection_distr_l X Y Z : (X ∩ Y) ∖ Z ≡ X ∖ Z ∩ Y ∖ Z.
   Proof. set_solver. Qed.
-  Lemma disjoint_union_difference X Y : X ∩ Y ≡ ∅ → (X ∪ Y) ∖ X ≡ Y.
+  Lemma disjoint_union_difference X Y : X ⊥ Y → (X ∪ Y) ∖ X ≡ Y.
   Proof. set_solver. Qed.
 
   Section leibniz.
@@ -407,7 +419,7 @@ Section collection.
     Lemma difference_intersection_distr_l_L X Y Z :
       (X ∩ Y) ∖ Z = X ∖ Z ∩ Y ∖ Z.
     Proof. unfold_leibniz. apply difference_intersection_distr_l. Qed.
-    Lemma disjoint_union_difference_L X Y : X ∩ Y = ∅ → (X ∪ Y) ∖ X = Y.
+    Lemma disjoint_union_difference_L X Y : X ⊥ Y → (X ∪ Y) ∖ X = Y.
     Proof. unfold_leibniz. apply disjoint_union_difference. Qed.
   End leibniz.
 

prelude/fin_collections.v

@@ -92,9 +92,9 @@ Proof.
   - rewrite elem_of_singleton. eauto using size_singleton_inv.
   - set_solver.
 Qed.
-Lemma size_union X Y : X ∩ Y ≡ ∅ → size (X ∪ Y) = size X + size Y.
+Lemma size_union X Y : X ⊥ Y → size (X ∪ Y) = size X + size Y.
 Proof.
-  intros [E _]. unfold size, collection_size. simpl. rewrite <-app_length.
+  intros. unfold size, collection_size. simpl. rewrite <-app_length.
   apply Permutation_length, NoDup_Permutation.
   - apply NoDup_elements.
   - apply NoDup_app; repeat split; try apply NoDup_elements.

prelude/fin_map_dom.v

@@ -74,15 +74,14 @@ Proof. rewrite not_elem_of_dom. apply delete_partial_alter. Qed.
 Lemma delete_insert_dom {A} (m : M A) i x :
   i ∉ dom D m → delete i (<[i:=x]>m) = m.
 Proof. rewrite not_elem_of_dom. apply delete_insert. Qed.
-Lemma map_disjoint_dom {A} (m1 m2 : M A) : m1 ⊥ₘ m2 ↔ dom D m1 ∩ dom D m2 ≡ ∅.
+Lemma map_disjoint_dom {A} (m1 m2 : M A) : m1 ⊥ₘ m2 ↔ dom D m1 ⊥ dom D m2.
 Proof.
-  rewrite map_disjoint_spec, elem_of_equiv_empty.
-  setoid_rewrite elem_of_intersection.
+  rewrite map_disjoint_spec, elem_of_disjoint.
   setoid_rewrite elem_of_dom. unfold is_Some. naive_solver.
 Qed.
-Lemma map_disjoint_dom_1 {A} (m1 m2 : M A) : m1 ⊥ₘ m2 → dom D m1 ∩ dom D m2 ≡ ∅.
+Lemma map_disjoint_dom_1 {A} (m1 m2 : M A) : m1 ⊥ₘ m2 → dom D m1 ⊥ dom D m2.
 Proof. apply map_disjoint_dom. Qed.
-Lemma map_disjoint_dom_2 {A} (m1 m2 : M A) : dom D m1 ∩ dom D m2 ≡ ∅ → m1 ⊥ₘ m2.
+Lemma map_disjoint_dom_2 {A} (m1 m2 : M A) : dom D m1 ⊥ dom D m2 → m1 ⊥ₘ m2.
 Proof. apply map_disjoint_dom. Qed.
 Lemma dom_union {A} (m1 m2 : M A) : dom D (m1 ∪ m2) ≡ dom D m1 ∪ dom D m2.
 Proof.
@@ -90,8 +89,7 @@ Proof.
   unfold is_Some. setoid_rewrite lookup_union_Some_raw.
   destruct (m1 !! i); naive_solver.
 Qed.
-Lemma dom_intersection {A} (m1 m2 : M A) :
-  dom D (m1 ∩ m2) ≡ dom D m1 ∩ dom D m2.
+Lemma dom_intersection {A} (m1 m2: M A) : dom D (m1 ∩ m2) ≡ dom D m1 ∩ dom D m2.
 Proof.
   apply elem_of_equiv. intros i. rewrite elem_of_intersection, !elem_of_dom.
   unfold is_Some. setoid_rewrite lookup_intersection_Some. naive_solver.

program_logic/adequacy.v

 From iris.program_logic Require Export hoare.
 From iris.program_logic Require Import wsat ownership.
 Local Hint Extern 10 (_ ≤ _) => omega.
-Local Hint Extern 100 (@eq coPset _ _) => eassumption || set_solver.
+Local Hint Extern 100 (_ ⊥ _) => set_solver.
 Local Hint Extern 10 (✓{_} _) =>
   repeat match goal with
   | H : wsat _ _ _ _ |- _ => apply wsat_valid in H; last omega
@@ -151,5 +151,4 @@ Proof.
   intros ? Hht. eapply wp_adequacy_safe with (E:=E) (Φ:=Φ); first done.
   move:Hht. by rewrite /ht uPred.always_elim=>/uPred.impl_entails.
 Qed.
-
 End adequacy.

program_logic/lifting.v

 From iris.program_logic Require Export weakestpre.
 From iris.program_logic Require Import wsat ownership.
 Local Hint Extern 10 (_ ≤ _) => omega.
-Local Hint Extern 100 (@eq coPset _ _) => set_solver.
+Local Hint Extern 100 (_ ⊥ _) => set_solver.
 Local Hint Extern 10 (✓{_} _) =>
   repeat match goal with
   | H : wsat _ _ _ _ |- _ => apply wsat_valid in H; last omega

program_logic/namespaces.v

@@ -49,13 +49,11 @@ Section ndisjoint.
   Lemma ndot_preserve_disjoint_r N1 N2 x : N1 ⊥ N2 → N1 ⊥ N2 .@ x .
   Proof. rewrite ![N1 ⊥ _]comm. apply ndot_preserve_disjoint_l. Qed.
 
-  Lemma ndisj_disjoint N1 N2 : N1 ⊥ N2 → nclose N1 ∩ nclose N2 = ∅.
+  Lemma ndisj_disjoint N1 N2 : N1 ⊥ N2 → nclose N1 ⊥ nclose N2.
   Proof.
-    intros (N1' & N2' & [N1'' ->] & [N2'' ->] & Hl & Hne).
-    apply elem_of_equiv_empty_L=> p; unfold nclose.
-    rewrite elem_of_intersection !elem_coPset_suffixes; intros [[q ->] [q' Hq]].
-    rewrite !list_encode_app !assoc in Hq.
-    by eapply Hne, list_encode_suffix_eq.
+    intros (N1' & N2' & [N1'' ->] & [N2'' ->] & Hl & Hne) p; unfold nclose.
+    rewrite !elem_coPset_suffixes; intros [q ->] [q' Hq]; destruct Hne.
+    by rewrite !list_encode_app !assoc in Hq; apply list_encode_suffix_eq in Hq.
   Qed.
 End ndisjoint.
 
@@ -63,20 +61,19 @@ End ndisjoint.
    of masks (i.e., coPsets) with set_solver, taking
    disjointness of namespaces into account. *)
 (* TODO: This tactic is by far now yet as powerful as it should be.
-   For example, given N1 ⊥ N2, it should be able to solve
-   nclose (ndot N1 x) ∩ N2 ≡ ∅. It should also solve
-   (ndot N x) ∩ (ndot N y) ≡ ∅ if x ≠ y is in the context or
+   For example, given [N1 ⊥ N2], it should be able to solve
+   [nclose (ndot N1 x) ⊥ N2]. It should also solve
+   [ndot N x ⊥ ndot N y] if x ≠ y is in the context or
    follows from [discriminate]. *)
 Ltac set_solver_ndisj :=
   repeat match goal with
-         (* TODO: Restrict these to have type namespace *)
-         | [ H : (?N1 ⊥ ?N2) |-_ ] => apply ndisj_disjoint in H
-         end;
-  set_solver.
+  (* TODO: Restrict these to have type namespace *)
+  | [ H : ?N1 ⊥ ?N2 |-_ ] => apply ndisj_disjoint in H
+  end; set_solver.
 (* TODO: restrict this to match only if this is ⊆ of coPset *)
 Hint Extern 500 (_ ⊆ _) => set_solver_ndisj : ndisj.
 (* The hope is that registering these will suffice to solve most goals
-   of the form N1 ⊥ N2.
+   of the form [N1 ⊥ N2].
    TODO: Can this prove x ≠ y if discriminate can? *)
 Hint Resolve ndot_ne_disjoint : ndisj.
 Hint Resolve ndot_preserve_disjoint_l : ndisj.

program_logic/pviewshifts.v

@@ -2,7 +2,7 @@ From iris.prelude Require Export co_pset.
 From iris.program_logic Require Export model.
 From iris.program_logic Require Import ownership wsat.
 Local Hint Extern 10 (_ ≤ _) => omega.
-Local Hint Extern 100 (@eq coPset _ _) => set_solver.
+Local Hint Extern 100 (_ ⊥ _) => set_solver.
 Local Hint Extern 100 (_ ∉ _) => set_solver.
 Local Hint Extern 10 (✓{_} _) =>
   repeat match goal with
@@ -11,7 +11,7 @@ Local Hint Extern 10 (✓{_} _) =>
 
 Program Definition pvs_def {Λ Σ} (E1 E2 : coPset) (P : iProp Λ Σ) : iProp Λ Σ :=
   {| uPred_holds n r1 := ∀ rf k Ef σ,
-       0 < k ≤ n → (E1 ∪ E2) ∩ Ef = ∅ →
+       0 < k ≤ n → E1 ∪ E2 ⊥ Ef →
        wsat k (E1 ∪ Ef) σ (r1 ⋅ rf) →
        ∃ r2, P k r2 ∧ wsat k (E2 ∪ Ef) σ (r2 ⋅ rf) |}.
 Next Obligation.
@@ -84,7 +84,7 @@ Proof.
   destruct (HP1 rf k Ef σ) as (r2&HP2&?); auto.
 Qed.
 Lemma pvs_mask_frame E1 E2 Ef P :
-  Ef ∩ (E1 ∪ E2) = ∅ → (|={E1,E2}=> P) ⊢ (|={E1 ∪ Ef,E2 ∪ Ef}=> P).
+  Ef ⊥ E1 ∪ E2 → (|={E1,E2}=> P) ⊢ (|={E1 ∪ Ef,E2 ∪ Ef}=> P).
 Proof.
   rewrite pvs_eq. intros ?; split=> n r ? HP rf k Ef' σ ???.
   destruct (HP rf k (Ef∪Ef') σ) as (r'&?&?); rewrite ?(assoc_L _); eauto.
@@ -244,6 +244,5 @@ Proof.
 Qed.
 
 Lemma pvs_mk_fsa {Λ Σ} E (P Q : iProp Λ Σ) :
-  P ⊢ pvs_fsa E (λ _, Q) →
-  P ⊢ |={E}=> Q.
+  P ⊢ pvs_fsa E (λ _, Q) → P ⊢ |={E}=> Q.
 Proof. by intros ?. Qed.

program_logic/sts.v

@@ -72,7 +72,7 @@ Section sts.
   Proof. intros ???. by apply own_update, sts_update_frag_up. Qed.
 
   Lemma sts_ownS_op γ S1 S2 T1 T2 :
-    T1 ∩ T2 ⊆ ∅ → sts.closed S1 T1 → sts.closed S2 T2 →
+    T1 ⊥ T2 → sts.closed S1 T1 → sts.closed S2 T2 →
     sts_ownS γ (S1 ∩ S2) (T1 ∪ T2) ⊣⊢ (sts_ownS γ S1 T1 ★ sts_ownS γ S2 T2).
   Proof. intros. by rewrite /sts_ownS -own_op sts_op_frag. Qed.
 

program_logic/viewshifts.v

@@ -78,18 +78,17 @@ Lemma vs_frame_r E1 E2 P Q R : (P ={E1,E2}=> Q) ⊢ (P ★ R ={E1,E2}=> Q ★ R)
 Proof. rewrite !(comm _ _ R); apply vs_frame_l. Qed.
 
 Lemma vs_mask_frame E1 E2 Ef P Q :
-  Ef ∩ (E1 ∪ E2) = ∅ → (P ={E1,E2}=> Q) ⊢ (P ={E1 ∪ Ef,E2 ∪ Ef}=> Q).
+  Ef ⊥ E1 ∪ E2 → (P ={E1,E2}=> Q) ⊢ (P ={E1 ∪ Ef,E2 ∪ Ef}=> Q).
 Proof.
   intros ?; apply always_intro', impl_intro_l; rewrite (pvs_mask_frame _ _ Ef)//.
   by rewrite always_elim impl_elim_r.
 Qed.
 
-Lemma vs_mask_frame' E Ef P Q : Ef ∩ E = ∅ → (P ={E}=> Q) ⊢ (P ={E ∪ Ef}=> Q).
+Lemma vs_mask_frame' E Ef P Q : Ef ⊥ E → (P ={E}=> Q) ⊢ (P ={E ∪ Ef}=> Q).
 Proof. intros; apply vs_mask_frame; set_solver. Qed.
 
 Lemma vs_inv N E P Q R :
-  nclose N ⊆ E →
-  (inv N R ★ (▷ R ★ P ={E ∖ nclose N}=> ▷ R ★ Q)) ⊢ (P ={E}=> Q).
+  nclose N ⊆ E → (inv N R ★ (▷ R ★ P ={E ∖ nclose N}=> ▷ R ★ Q)) ⊢ (P ={E}=> Q).
 Proof.
   intros; apply: always_intro. apply impl_intro_l.
   rewrite always_and_sep_r assoc [(P ★ _)%I]comm -assoc.

program_logic/weakestpre.v

 From iris.program_logic Require Export pviewshifts.
 From iris.program_logic Require Import wsat.
 Local Hint Extern 10 (_ ≤ _) => omega.
-Local Hint Extern 100 (@eq coPset _ _) => eassumption || set_solver.
-Local Hint Extern 100 (_ ∉ _) => set_solver.
+Local Hint Extern 100 (_ ⊥ _) => set_solver.
+Local Hint Extern 100 (_ ∉_) => set_solver.
 Local Hint Extern 100 (@subseteq coPset _ _ _) => set_solver.
 Local Hint Extern 10 (✓{_} _) =>
   repeat match goal with
@@ -25,7 +25,7 @@ CoInductive wp_pre {Λ Σ} (E : coPset)
   | wp_pre_step n r1 e1 :
      to_val e1 = None →
      (∀ rf k Ef σ1,
-       0 < k < n → E ∩ Ef = ∅ →
+       0 < k < n → E ⊥ Ef →
        wsat (S k) (E ∪ Ef) σ1 (r1 ⋅ rf) →
        wp_go (E ∪ Ef) (wp_pre E Φ)
                       (wp_pre ⊤ (λ _, True%I)) k rf e1 σ1) →
@@ -122,7 +122,7 @@ Proof.
   by inversion 1 as [|??? He]; [|rewrite ?to_of_val in He]; simplify_eq.
 Qed.
 Lemma wp_step_inv E Ef Φ e k n σ r rf :
-  to_val e = None → 0 < k < n → E ∩ Ef = ∅ →
+  to_val e = None → 0 < k < n → E ⊥ Ef →
   wp_def E e Φ n r → wsat (S k) (E ∪ Ef) σ (r ⋅ rf) →
   wp_go (E ∪ Ef) (λ e, wp_def E e Φ) (λ e, wp_def ⊤ e (λ _, True%I)) k rf e σ.
 Proof.

program_logic/weakestpre_fix.v

 From Coq Require Import Wf_nat.
 From iris.program_logic Require Import weakestpre wsat.
 Local Hint Extern 10 (_ ≤ _) => omega.
-Local Hint Extern 10 (@eq coPset _ _) => set_solver.
+Local Hint Extern 10 (_ ⊥ _) => set_solver.
 Local Hint Extern 10 (✓{_} _) =>
   repeat match goal with
   | H : wsat _ _ _ _ |- _ => apply wsat_valid in H; last omega
@@ -19,7 +19,7 @@ Program Definition wp_pre
     (wp : coPsetC -n> exprC Λ -n> (valC Λ -n> iProp) -n> iProp)
     (E : coPset) (e1 : expr Λ) (Φ : valC Λ -n> iProp) :  iProp :=
   {| uPred_holds n r1 := ∀ rf k Ef σ1,
-       0 ≤ k < n → E ∩ Ef = ∅ →
+       0 ≤ k < n → E ⊥ Ef →
        wsat (S k) (E ∪ Ef) σ1 (r1 ⋅ rf) →
        (∀ v, to_val e1 = Some v →
          ∃ r2, Φ v (S k) r2 ∧ wsat (S k) (E ∪ Ef) σ1 (r2 ⋅ rf)) ∧