Notation for disjointness: replace ⊥ with ##, so that ⊥ can be used for bottom.

This is to be used on top of stdpp's 4b5d254e.

CHANGELOG.md

@@ -113,7 +113,7 @@ sed 's/\bPersistentP\b/Persistent/g; s/\bTimelessP\b/Timeless/g; s/\bCMRADiscret
 * Slightly weaker notion of atomicity: an expression is atomic if it reduces in
   one step to something that does not reduce further.
 * Changed notation for embedding Coq assertions into Iris.  The new notation is
-  ⌜φ⌝.  Also removed `=` and `⊥` from the Iris scope.  (The old notations are
+  ⌜φ⌝.  Also removed `=` and `##` from the Iris scope.  (The old notations are
   provided in `base_logic.deprecated`.)
 * Up-closure of namespaces is now a notation (↑) instead of a coercion.
 * With invariants and the physical state being handled in the logic, there is no

opam

@@ -12,5 +12,5 @@ remove: ["rm" "-rf" "%{lib}%/coq/user-contrib/iris"]
 depends: [
   "coq" { >= "8.6.1" & < "8.8~" }
   "coq-mathcomp-ssreflect" { (>= "1.6.1" & < "1.7~") | (= "dev") }
-  "coq-stdpp" { (= "dev.2017-10-28.3") | (= "dev") }
+  "coq-stdpp" { (= "dev.2017-10-28.7") | (= "dev") }
 ]

theories/algebra/big_op.v

@@ -299,7 +299,7 @@ Section gset.
   Proof. apply (big_opS_fn_insert (λ y, id)). Qed.
 
   Lemma big_opS_union f X Y :
-    X ⊥ Y →
+    X ## Y →
     ([^o set] y ∈ X ∪ Y, f y) ≡ ([^o set] y ∈ X, f y) `o` ([^o set] y ∈ Y, f y).
   Proof.
     intros. induction X as [|x X ? IH] using collection_ind_L.

theories/algebra/coPset.v

@@ -74,7 +74,7 @@ Section coPset_disj.
   Instance coPset_disj_unit : Unit coPset_disj := CoPset ∅.
   Instance coPset_disj_op : Op coPset_disj := λ X Y,
     match X, Y with
-    | CoPset X, CoPset Y => if decide (X ⊥ Y) then CoPset (X ∪ Y) else CoPsetBot
+    | CoPset X, CoPset Y => if decide (X ## Y) then CoPset (X ∪ Y) else CoPsetBot
     | _, _ => CoPsetBot
     end.
   Instance coPset_disj_pcore : PCore coPset_disj := λ _, Some ε.
@@ -91,11 +91,11 @@ Section coPset_disj.
       exists (CoPset Z). coPset_disj_solve.
   Qed.
   Lemma coPset_disj_valid_inv_l X Y :
-    ✓ (CoPset X ⋅ Y) → ∃ Y', Y = CoPset Y' ∧ X ⊥ Y'.
+    ✓ (CoPset X ⋅ Y) → ∃ Y', Y = CoPset Y' ∧ X ## Y'.
   Proof. destruct Y; repeat (simpl || case_decide); by eauto. Qed.
-  Lemma coPset_disj_union X Y : X ⊥ Y → CoPset X ⋅ CoPset Y = CoPset (X ∪ Y).
+  Lemma coPset_disj_union X Y : X ## Y → CoPset X ⋅ CoPset Y = CoPset (X ∪ Y).
   Proof. intros. by rewrite /= decide_True. Qed.
-  Lemma coPset_disj_valid_op X Y : ✓ (CoPset X ⋅ CoPset Y) ↔ X ⊥ Y.
+  Lemma coPset_disj_valid_op X Y : ✓ (CoPset X ⋅ CoPset Y) ↔ X ## Y.
   Proof. simpl. case_decide; by split. Qed.
 
   Lemma coPset_disj_ra_mixin : RAMixin coPset_disj.

theories/algebra/dra.v

@@ -9,21 +9,21 @@ Record DraMixin A `{Equiv A, Core A, Disjoint A, Op A, Valid A} := {
   mixin_dra_valid_proper : Proper ((≡) ==> impl) valid;
   mixin_dra_disjoint_proper x : Proper ((≡) ==> impl) (disjoint x);
   (* validity *)
-  mixin_dra_op_valid x y : ✓ x → ✓ y → x ⊥ y → ✓ (x ⋅ y);
+  mixin_dra_op_valid x y : ✓ x → ✓ y → x ## y → ✓ (x ⋅ y);
   mixin_dra_core_valid x : ✓ x → ✓ core x;
   (* monoid *)
   mixin_dra_assoc x y z :
-    ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⋅ (y ⋅ z) ≡ (x ⋅ y) ⋅ z;
-  mixin_dra_disjoint_ll x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ z;
+    ✓ x → ✓ y → ✓ z → x ## y → x ⋅ y ## z → x ⋅ (y ⋅ z) ≡ (x ⋅ y) ⋅ z;
+  mixin_dra_disjoint_ll x y z : ✓ x → ✓ y → ✓ z → x ## y → x ⋅ y ## z → x ## z;
   mixin_dra_disjoint_move_l x y z :
-    ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ y ⋅ z;
+    ✓ x → ✓ y → ✓ z → x ## y → x ⋅ y ## z → x ## y ⋅ z;
   mixin_dra_symmetric : Symmetric (@disjoint A _);
-  mixin_dra_comm x y : ✓ x → ✓ y → x ⊥ y → x ⋅ y ≡ y ⋅ x;
-  mixin_dra_core_disjoint_l x : ✓ x → core x ⊥ x;
+  mixin_dra_comm x y : ✓ x → ✓ y → x ## y → x ⋅ y ≡ y ⋅ x;
+  mixin_dra_core_disjoint_l x : ✓ x → core x ## x;
   mixin_dra_core_l x : ✓ x → core x ⋅ x ≡ x;
   mixin_dra_core_idemp x : ✓ x → core (core x) ≡ core x;
   mixin_dra_core_mono x y : 
-    ∃ z, ✓ x → ✓ y → x ⊥ y → core (x ⋅ y) ≡ core x ⋅ z ∧ ✓ z ∧ core x ⊥ z
+    ∃ z, ✓ x → ✓ y → x ## y → core (x ⋅ y) ≡ core x ⋅ z ∧ ✓ z ∧ core x ## z
 }.
 Structure draT := DraT {
   dra_car :> Type;
@@ -59,30 +59,30 @@ Section dra_mixin.
   Proof. apply (mixin_dra_valid_proper _ (dra_mixin A)). Qed.
   Global Instance dra_disjoint_proper x : Proper ((≡) ==> impl) (disjoint x).
   Proof. apply (mixin_dra_disjoint_proper _ (dra_mixin A)). Qed.
-  Lemma dra_op_valid x y : ✓ x → ✓ y → x ⊥ y → ✓ (x ⋅ y).
+  Lemma dra_op_valid x y : ✓ x → ✓ y → x ## y → ✓ (x ⋅ y).
   Proof. apply (mixin_dra_op_valid _ (dra_mixin A)). Qed.
   Lemma dra_core_valid x : ✓ x → ✓ core x.
   Proof. apply (mixin_dra_core_valid _ (dra_mixin A)). Qed.
   Lemma dra_assoc x y z :
-    ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⋅ (y ⋅ z) ≡ (x ⋅ y) ⋅ z.
+    ✓ x → ✓ y → ✓ z → x ## y → x ⋅ y ## z → x ⋅ (y ⋅ z) ≡ (x ⋅ y) ⋅ z.
   Proof. apply (mixin_dra_assoc _ (dra_mixin A)). Qed.
-  Lemma dra_disjoint_ll x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ z.
+  Lemma dra_disjoint_ll x y z : ✓ x → ✓ y → ✓ z → x ## y → x ⋅ y ## z → x ## z.
   Proof. apply (mixin_dra_disjoint_ll _ (dra_mixin A)). Qed.
   Lemma dra_disjoint_move_l x y z :
-    ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ y ⋅ z.
+    ✓ x → ✓ y → ✓ z → x ## y → x ⋅ y ## z → x ## y ⋅ z.
   Proof. apply (mixin_dra_disjoint_move_l _ (dra_mixin A)). Qed.
   Global Instance  dra_symmetric : Symmetric (@disjoint A _).
   Proof. apply (mixin_dra_symmetric _ (dra_mixin A)). Qed.
-  Lemma dra_comm x y : ✓ x → ✓ y → x ⊥ y → x ⋅ y ≡ y ⋅ x.
+  Lemma dra_comm x y : ✓ x → ✓ y → x ## y → x ⋅ y ≡ y ⋅ x.
   Proof. apply (mixin_dra_comm _ (dra_mixin A)). Qed.
-  Lemma dra_core_disjoint_l x : ✓ x → core x ⊥ x.
+  Lemma dra_core_disjoint_l x : ✓ x → core x ## x.
   Proof. apply (mixin_dra_core_disjoint_l _ (dra_mixin A)). Qed.
   Lemma dra_core_l x : ✓ x → core x ⋅ x ≡ x.
   Proof. apply (mixin_dra_core_l _ (dra_mixin A)). Qed.
   Lemma dra_core_idemp x : ✓ x → core (core x) ≡ core x.
   Proof. apply (mixin_dra_core_idemp _ (dra_mixin A)). Qed.
   Lemma dra_core_mono x y : 
-    ∃ z, ✓ x → ✓ y → x ⊥ y → core (x ⋅ y) ≡ core x ⋅ z ∧ ✓ z ∧ core x ⊥ z.
+    ∃ z, ✓ x → ✓ y → x ## y → core (x ⋅ y) ≡ core x ⋅ z ∧ ✓ z ∧ core x ## z.
   Proof. apply (mixin_dra_core_mono _ (dra_mixin A)). Qed.
 End dra_mixin.
 
@@ -126,16 +126,16 @@ Proof. by intros x1 x2 Hx; split; rewrite /= Hx. Qed.
 Instance: Proper ((≡) ==> (≡) ==> iff) (disjoint : relation A).
 Proof.
   intros x1 x2 Hx y1 y2 Hy; split.
-  - by rewrite Hy (symmetry_iff (⊥) x1) (symmetry_iff (⊥) x2) Hx.
-  - by rewrite -Hy (symmetry_iff (⊥) x2) (symmetry_iff (⊥) x1) -Hx.
+  - by rewrite Hy (symmetry_iff (##) x1) (symmetry_iff (##) x2) Hx.
+  - by rewrite -Hy (symmetry_iff (##) x2) (symmetry_iff (##) x1) -Hx.
 Qed.
 
-Lemma dra_disjoint_rl a b c : ✓ a → ✓ b → ✓ c → b ⊥ c → a ⊥ b ⋅ c → a ⊥ b.
+Lemma dra_disjoint_rl a b c : ✓ a → ✓ b → ✓ c → b ## c → a ## b ⋅ c → a ## b.
 Proof. intros ???. rewrite !(symmetry_iff _ a). by apply dra_disjoint_ll. Qed.
-Lemma dra_disjoint_lr a b c : ✓ a → ✓ b → ✓ c → a ⊥ b → a ⋅ b ⊥ c → b ⊥ c.
+Lemma dra_disjoint_lr a b c : ✓ a → ✓ b → ✓ c → a ## b → a ⋅ b ## c → b ## c.
 Proof. intros ????. rewrite dra_comm //. by apply dra_disjoint_ll. Qed.
 Lemma dra_disjoint_move_r a b c :
-  ✓ a → ✓ b → ✓ c → b ⊥ c → a ⊥ b ⋅ c → a ⋅ b ⊥ c.
+  ✓ a → ✓ b → ✓ c → b ## c → a ## b ⋅ c → a ⋅ b ## c.
 Proof.
   intros; symmetry; rewrite dra_comm; eauto using dra_disjoint_rl.
   apply dra_disjoint_move_l; auto; by rewrite dra_comm.
@@ -150,7 +150,7 @@ Program Instance validity_pcore : PCore (validity A) := λ x,
 Solve Obligations with naive_solver eauto using dra_core_valid.
 Program Instance validity_op : Op (validity A) := λ x y,
   Validity (validity_car x ⋅ validity_car y)
-           (✓ x ∧ ✓ y ∧ validity_car x ⊥ validity_car y) _.
+           (✓ x ∧ ✓ y ∧ validity_car x ## validity_car y) _.
 Solve Obligations with naive_solver eauto using dra_op_valid.
 
 Definition validity_ra_mixin : RAMixin (validity A).
@@ -185,14 +185,14 @@ Global Instance validity_cmra_total : CmraTotal validityR.
 Proof. rewrite /CmraTotal; eauto. Qed.
 
 Lemma validity_update x y :
-  (∀ c, ✓ x → ✓ c → validity_car x ⊥ c → ✓ y ∧ validity_car y ⊥ c) → x ~~> y.
+  (∀ c, ✓ x → ✓ c → validity_car x ## c → ✓ y ∧ validity_car y ## c) → x ~~> y.
 Proof.
   intros Hxy; apply cmra_discrete_update=> z [?[??]].
   split_and!; try eapply Hxy; eauto.
 Qed.
 
 Lemma to_validity_op a b :
-  (✓ (a ⋅ b) → ✓ a ∧ ✓ b ∧ a ⊥ b) →
+  (✓ (a ⋅ b) → ✓ a ∧ ✓ b ∧ a ## b) →
   to_validity (a ⋅ b) ≡ to_validity a ⋅ to_validity b.
 Proof. split; naive_solver eauto using dra_op_valid. Qed.
 

theories/algebra/gset.v

@@ -86,7 +86,7 @@ Section gset_disj.
   Instance gset_disj_unit : Unit (gset_disj K) := GSet ∅.
   Instance gset_disj_op : Op (gset_disj K) := λ X Y,
     match X, Y with
-    | GSet X, GSet Y => if decide (X ⊥ Y) then GSet (X ∪ Y) else GSetBot
+    | GSet X, GSet Y => if decide (X ## Y) then GSet (X ∪ Y) else GSetBot
     | _, _ => GSetBot
     end.
   Instance gset_disj_pcore : PCore (gset_disj K) := λ _, Some ε.
@@ -102,11 +102,11 @@ Section gset_disj.
     - intros (Z&->&?)%subseteq_disjoint_union_L.
       exists (GSet Z). gset_disj_solve.
   Qed.
-  Lemma gset_disj_valid_inv_l X Y : ✓ (GSet X ⋅ Y) → ∃ Y', Y = GSet Y' ∧ X ⊥ Y'.
+  Lemma gset_disj_valid_inv_l X Y : ✓ (GSet X ⋅ Y) → ∃ Y', Y = GSet Y' ∧ X ## Y'.
   Proof. destruct Y; repeat (simpl || case_decide); by eauto. Qed.
-  Lemma gset_disj_union X Y : X ⊥ Y → GSet X ⋅ GSet Y = GSet (X ∪ Y).
+  Lemma gset_disj_union X Y : X ## Y → GSet X ⋅ GSet Y = GSet (X ∪ Y).
   Proof. intros. by rewrite /= decide_True. Qed.
-  Lemma gset_disj_valid_op X Y : ✓ (GSet X ⋅ GSet Y) ↔ X ⊥ Y.
+  Lemma gset_disj_valid_op X Y : ✓ (GSet X ⋅ GSet Y) ↔ X ## Y.
   Proof. simpl. case_decide; by split. Qed.
 
   Lemma gset_disj_ra_mixin : RAMixin (gset_disj K).
@@ -207,19 +207,19 @@ Section gset_disj.
   Qed.
 
   Lemma gset_disj_alloc_op_local_update X Y Z :
-    Z ⊥ X → (GSet X,GSet Y) ~l~> (GSet Z ⋅ GSet X, GSet Z ⋅ GSet Y).
+    Z ## X → (GSet X,GSet Y) ~l~> (GSet Z ⋅ GSet X, GSet Z ⋅ GSet Y).
   Proof.
     intros. apply op_local_update_discrete. by rewrite gset_disj_valid_op.
   Qed.
   Lemma gset_disj_alloc_local_update X Y Z :
-    Z ⊥ X → (GSet X,GSet Y) ~l~> (GSet (Z ∪ X), GSet (Z ∪ Y)).
+    Z ## X → (GSet X,GSet Y) ~l~> (GSet (Z ∪ X), GSet (Z ∪ Y)).
   Proof.
     intros. apply local_update_total_valid=> _ _ /gset_disj_included ?.
     rewrite -!gset_disj_union //; last set_solver.
     auto using gset_disj_alloc_op_local_update.
   Qed.
   Lemma gset_disj_alloc_empty_local_update X Z :
-    Z ⊥ X → (GSet X, GSet ∅) ~l~> (GSet (Z ∪ X), GSet Z).
+    Z ## X → (GSet X, GSet ∅) ~l~> (GSet (Z ∪ X), GSet Z).
   Proof.
     intros. rewrite -{2}(right_id_L _ union Z).
     apply gset_disj_alloc_local_update; set_solver.

theories/algebra/sts.v

@@ -27,18 +27,18 @@ Context {sts : stsT}.
 (** ** Step relations *)
 Inductive step : relation (state sts * tokens sts) :=
   | Step s1 s2 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 :=
-  (* Probably equivalent definition: (\mathcal{L}(s') ⊥ T) ∧ s \rightarrow s' *)
+  (* Probably equivalent definition: (\mathcal{L}(s') ## T) ∧ s \rightarrow s' *)
   | 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.
 Notation frame_steps T := (rtc (frame_step T)).
 
 (** ** 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 :=
@@ -52,7 +52,7 @@ 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.
+Hint Extern 50 (_ ## _) => set_solver : sts.
 
 (** ** Setoids *)
 Instance frame_step_mono : Proper (flip (⊆) ==> (=) ==> (=) ==> impl) frame_step.
@@ -100,16 +100,16 @@ 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.
@@ -127,7 +127,7 @@ Lemma elem_of_up_set S T s : s ∈ S → s ∈ up_set S T.
 Proof. apply subseteq_up_set. 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').
@@ -137,7 +137,7 @@ 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.
@@ -192,7 +192,7 @@ Inductive sts_equiv : Equiv (car sts) :=
 Existing Instance sts_equiv.
 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, s ∈ S'
   end.
 Instance sts_core : Core (car sts) := λ x,
@@ -202,9 +202,9 @@ Instance sts_core : Core (car sts) := λ x,
   end.
 Inductive sts_disjoint : Disjoint (car sts) :=
   | frag_frag_disjoint S1 S2 T1 T2 :
-     (∃ s, s ∈ 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.
+     (∃ s, s ∈ 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.
 Existing Instance sts_disjoint.
 Instance sts_op : Op (car sts) := λ x1 x2,
   match x1, x2 with
@@ -218,7 +218,7 @@ Hint Extern 50 (equiv (A:=set _) _ _) => set_solver : sts.
 Hint Extern 50 (∃ s : state sts, _) => set_solver : sts.
 Hint Extern 50 (_ ∈ _) => set_solver : sts.
 Hint Extern 50 (_ ⊆ _) => set_solver : sts.
-Hint Extern 50 (_ ⊥ _) => set_solver : sts.
+Hint Extern 50 (_ ## _) => set_solver : sts.
 
 Global Instance auth_proper s : Proper ((≡) ==> (≡)) (@auth sts s).
 Proof. by constructor. Qed.
@@ -304,11 +304,11 @@ Global Instance sts_frag_up_proper s : Proper ((≡) ==> (≡)) (sts_frag_up s).
 Proof. solve_proper. 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, s ∈ S.
 Proof. done. Qed.
-Lemma sts_frag_up_valid s T : ✓ sts_frag_up s T ↔ tok s ⊥ T.
+Lemma sts_frag_up_valid s T : ✓ sts_frag_up s T ↔ tok s ## T.
 Proof.
   split.
   - move=>/sts_frag_valid [H _]. apply H, elem_of_up. 
@@ -340,7 +340,7 @@ 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 -to_validity_op //.
@@ -351,7 +351,7 @@ Qed.
    two closures is weaker than the closure with the itnersected token
    set.  Also see up_op.
 Lemma sts_op_frag_up s T1 T2 :
-  T1 ⊥ T2 → sts_frag_up s (T1 ∪ T2) ≡ sts_frag_up s T1 ⋅ sts_frag_up s T2.
+  T1 ## T2 → sts_frag_up s (T1 ∪ T2) ≡ sts_frag_up s T1 ⋅ sts_frag_up s T2.
 *)
 
 (** Frame preserving updates *)
@@ -375,7 +375,7 @@ Proof.
 Qed.
 
 Lemma sts_update_frag_up s1 S2 T1 T2 :
-  (tok s1 ⊥ T1 → closed S2 T2) → s1 ∈ S2 → T2 ⊆ T1 → sts_frag_up s1 T1 ~~> sts_frag S2 T2.
+  (tok s1 ## T1 → closed S2 T2) → s1 ∈ S2 → T2 ⊆ T1 → sts_frag_up s1 T1 ~~> sts_frag S2 T2.
 Proof.
   intros HC ? HT; apply sts_update_frag. intros HC1; split; last split; eauto using closed_steps.
   - eapply HC, HC1, elem_of_up.
@@ -405,7 +405,7 @@ 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.

theories/base_logic/big_op.v

@@ -415,7 +415,7 @@ Section gset.
   Proof. apply big_opS_fn_insert'. Qed.
 
   Lemma big_sepS_union Φ X Y :
-    X ⊥ Y →
+    X ## Y →
     ([∗ set] y ∈ X ∪ Y, Φ y) ⊣⊢ ([∗ set] y ∈ X, Φ y) ∗ ([∗ set] y ∈ Y, Φ y).
   Proof. apply big_opS_union. Qed.
 

theories/base_logic/deprecated.v

@@ -8,5 +8,5 @@ Notation "■ φ" := (uPred_pure φ%C%type)
 (* Deprecated 2016-11-22. Use ⌜x = y⌝ instead. *)
 Notation "x = y" := (uPred_pure (x%C%type = y%C%type)) (only parsing) : uPred_scope.
 
-(* Deprecated 2016-11-22. Use ⌜x ⊥ y ⌝ instead. *)
-Notation "x ⊥ y" := (uPred_pure (x%C%type ⊥ y%C%type)) (only parsing) : uPred_scope.
+(* Deprecated 2016-11-22. Use ⌜x ## y ⌝ instead. *)
+Notation "x ## y" := (uPred_pure (x%C%type ## y%C%type)) (only parsing) : uPred_scope.

theories/base_logic/lib/counter_examples.v

@@ -79,7 +79,7 @@ Module inv. Section inv.
      * Using the STS monoid of a two-state STS, where [start] is the
        authoritative saying the state is exactly [start], and [finish]
        is the "we are at least in state [finish]" typically owned by threads.
-     * Ex () +_⊥ ()
+     * Ex () +_## ()
   *)
   Context (gname : Type).
   Context (start finished : gname → iProp).

theories/base_logic/lib/fancy_updates.v

@@ -70,7 +70,7 @@ Proof.
 Qed.
 
 Lemma fupd_mask_frame_r' E1 E2 Ef P :
-  E1 ⊥ Ef → (|={E1,E2}=> ⌜E2 ⊥ Ef⌝ → P) ={E1 ∪ Ef,E2 ∪ Ef}=∗ P.
+  E1 ## Ef → (|={E1,E2}=> ⌜E2 ## Ef⌝ → P) ={E1 ∪ Ef,E2 ∪ Ef}=∗ P.
 Proof.
   intros. rewrite fupd_eq /fupd_def ownE_op //. iIntros "Hvs (Hw & HE1 &HEf)".
   iMod ("Hvs" with "[Hw HE1]") as ">($ & HE2 & HP)"; first by iFrame.
@@ -110,7 +110,7 @@ Proof.
 Qed.
 
 Lemma fupd_mask_frame_r E1 E2 Ef P :
-  E1 ⊥ Ef → (|={E1,E2}=> P) ={E1 ∪ Ef,E2 ∪ Ef}=∗ P.
+  E1 ## Ef → (|={E1,E2}=> P) ={E1 ∪ Ef,E2 ∪ Ef}=∗ P.
 Proof.
   iIntros (?) "H". iApply fupd_mask_frame_r'; auto.
   iApply fupd_wand_r; iFrame "H"; eauto.
@@ -222,7 +222,7 @@ Lemma step_fupd_wand E1 E2 P Q : (|={E1,E2}▷=> P) -∗ (P -∗ Q) -∗ |={E1,E
 Proof. iIntros "HP HPQ". by iApply "HPQ". Qed.
 
 Lemma step_fupd_mask_frame_r E1 E2 Ef P :
-  E1 ⊥ Ef → E2 ⊥ Ef → (|={E1,E2}▷=> P) ⊢ |={E1 ∪ Ef,E2 ∪ Ef}▷=> P.
+  E1 ## Ef → E2 ## Ef → (|={E1,E2}▷=> P) ⊢ |={E1 ∪ Ef,E2 ∪ Ef}▷=> P.
 Proof.
   iIntros (??) "HP". iApply fupd_mask_frame_r. done. iMod "HP". iModIntro.
   iNext. by iApply fupd_mask_frame_r.

theories/base_logic/lib/fancy_updates_from_vs.v

@@ -19,7 +19,7 @@ Context (vs_impl : ∀ E P Q, □ (P → Q) ⊢ P ={E,E}=> Q).
 Context (vs_transitive : ∀ E1 E2 E3 P Q R,
   (P ={E1,E2}=> Q) ∧ (Q ={E2,E3}=> R) ⊢ P ={E1,E3}=> R).
 Context (vs_mask_frame_r : ∀ E1 E2 Ef P Q,
-  E1 ⊥ Ef → (P ={E1,E2}=> Q) ⊢ P ={E1 ∪ Ef,E2 ∪ Ef}=> Q).
+  E1 ## Ef → (P ={E1,E2}=> Q) ⊢ P ={E1 ∪ Ef,E2 ∪ Ef}=> Q).
 Context (vs_frame_r : ∀ E1 E2 P Q R, (P ={E1,E2}=> Q) ⊢ P ∗ R ={E1,E2}=> Q ∗ R).
 Context (vs_exists : ∀ {A} E1 E2 (Φ : A → uPred M) Q,
   (∀ x, Φ x ={E1,E2}=> Q) ⊢ (∃ x, Φ x) ={E1,E2}=> Q).
@@ -56,7 +56,7 @@ Proof.
 Qed.
 
 Lemma fupd_mask_frame_r E1 E2 Ef P :
-  E1 ⊥ Ef → (|={E1,E2}=> P) ⊢ |={E1 ∪ Ef,E2 ∪ Ef}=> P.
+  E1 ## Ef → (|={E1,E2}=> P) ⊢ |={E1 ∪ Ef,E2 ∪ Ef}=> P.
 Proof.
   iIntros (HE); iDestruct 1 as (R) "[HR Hvs]". iExists R; iFrame "HR".
   by iApply vs_mask_frame_r.

theories/base_logic/lib/na_invariants.v

@@ -46,14 +46,14 @@ Section proofs.
   Lemma na_alloc : (|==> ∃ p, na_own p ⊤)%I.
   Proof. by apply own_alloc. Qed.
 
-  Lemma na_own_disjoint p E1 E2 : na_own p E1 -∗ na_own p E2 -∗ ⌜E1 ⊥ E2⌝.
+  Lemma na_own_disjoint p E1 E2 : na_own p E1 -∗ na_own p E2 -∗ ⌜E1 ## E2⌝.
   Proof.
     apply wand_intro_r.
     rewrite /na_own -own_op own_valid -coPset_disj_valid_op. by iIntros ([? _]).
   Qed.
 
   Lemma na_own_union p E1 E2 :
-    E1 ⊥ E2 → na_own p (E1 ∪ E2) ⊣⊢ na_own p E1 ∗ na_own p E2.
+    E1 ## E2 → na_own p (E1 ∪ E2) ⊣⊢ na_own p E1 ∗ na_own p E2.
   Proof.
     intros ?. by rewrite /na_own -own_op pair_op left_id coPset_disj_union.
   Qed.

theories/base_logic/lib/namespaces.v

@@ -24,7 +24,7 @@ Notation "N .@ x" := (ndot N x)
   (at level 19, left associativity, format "N .@ x") : C_scope.
 Notation "(.@)" := ndot (only parsing) : C_scope.
 
-Instance ndisjoint : Disjoint namespace := λ N1 N2, nclose N1 ⊥ nclose N2.
+Instance ndisjoint : Disjoint namespace := λ N1 N2, nclose N1 ## nclose N2.
 
 Section namespace.
   Context `{Countable A}.
@@ -59,37 +59,37 @@ Section namespace.
   Lemma nclose_infinite N : ¬set_finite (↑ N : coPset).
   Proof. rewrite nclose_eq. apply coPset_suffixes_infinite. Qed.
 
-  Lemma ndot_ne_disjoint N x y : x ≠ y → N.@x ⊥ N.@y.
+  Lemma ndot_ne_disjoint N x y : x ≠ y → N.@x ## N.@y.
   Proof.
     intros Hxy a. rewrite !nclose_eq !elem_coPset_suffixes !ndot_eq.
     intros [qx ->] [qy Hqy].
     revert Hqy. by intros [= ?%encode_inj]%list_encode_suffix_eq.
   Qed.
 
-  Lemma ndot_preserve_disjoint_l N E x : ↑N ⊥ E → ↑N.@x ⊥ E.
+  Lemma ndot_preserve_disjoint_l N E x : ↑N ## E → ↑N.@x ## E.
   Proof. intros. pose proof (nclose_subseteq N x). set_solver. Qed.
 
-  Lemma ndot_preserve_disjoint_r N E x : E ⊥ ↑N → E ⊥ ↑N.@x.
+  Lemma ndot_preserve_disjoint_r N E x : E ## ↑N → E ## ↑N.@x.
   Proof. intros. by apply symmetry, ndot_preserve_disjoint_l. Qed.
 
-  Lemma ndisj_subseteq_difference N E F : E ⊥ ↑N → E ⊆ F → E ⊆ F ∖ ↑N.
+  Lemma ndisj_subseteq_difference N E F : E ## ↑N → E ⊆ F → E ⊆ F ∖ ↑N.
   Proof. set_solver. Qed.
 
   Lemma namespace_subseteq_difference_l E1 E2 E3 : E1 ⊆ E3 → E1 ∖ E2 ⊆ E3.
   Proof. set_solver. Qed.
 
-  Lemma ndisj_difference_l E N1 N2 : ↑N2 ⊆ (↑N1 : coPset) → E ∖ ↑N1 ⊥ ↑N2.
+  Lemma ndisj_difference_l E N1 N2 : ↑N2 ⊆ (↑N1 : coPset) → E ∖ ↑N1 ## ↑N2.
   Proof. set_solver. Qed.
 End namespace.
 
 (* The hope is that registering these will suffice to solve most goals
 of the forms:
-- [N1 ⊥ N2] 
+- [N1 ## N2] 
 - [↑N1 ⊆ E ∖ ↑N2 ∖ .. ∖ ↑Nn]
 - [E1 ∖ ↑N1 ⊆ E2 ∖ ↑N2 ∖ .. ∖ ↑Nn] *)