Use notation for `own`.

theories/base_logic/lib/own.v

@@ -58,6 +58,8 @@ Definition own {Σ A i} := own_aux.(unseal) Σ A i.
 Definition own_eq : @own = @own_def := own_aux.(seal_eq).
 Instance: Params (@own) 4 := {}.
 Typeclasses Opaque own.
+Notation "👻{ γ } x" := (own γ x)
+  (at level 20, x at next level, format "👻{ γ }  x").
 
 (** * Properties about ghost ownership *)
 Section global.
@@ -80,15 +82,15 @@ Proof. rewrite !own_eq. solve_proper. Qed.
 Global Instance own_proper γ :
   Proper ((≡) ==> (⊣⊢)) (@own Σ A _ γ) := ne_proper _.
 
-Lemma own_op γ a1 a2 : own γ (a1 ⋅ a2) ⊣⊢ own γ a1 ∗ own γ a2.
+Lemma own_op γ a1 a2 : 👻{γ} (a1 ⋅ a2) ⊣⊢ 👻{γ} a1 ∗ 👻{γ} a2.
 Proof. by rewrite !own_eq /own_def -ownM_op iRes_singleton_op. Qed.
-Lemma own_mono γ a1 a2 : a2 ≼ a1 → own γ a1 ⊢ own γ a2.
+Lemma own_mono γ a1 a2 : a2 ≼ a1 → 👻{γ} a1 ⊢ 👻{γ} a2.
 Proof. move=> [c ->]. by rewrite own_op sep_elim_l. Qed.
 
 Global Instance own_mono' γ : Proper (flip (≼) ==> (⊢)) (@own Σ A _ γ).
 Proof. intros a1 a2. apply own_mono. Qed.
 
-Lemma own_valid γ a : own γ a ⊢ ✓ a.
+Lemma own_valid γ a : 👻{γ} a ⊢ ✓ a.
 Proof.
   rewrite !own_eq /own_def ownM_valid /iRes_singleton.
   rewrite ofe_fun_validI (forall_elim (inG_id _)) ofe_fun_lookup_singleton.
@@ -96,21 +98,21 @@ Proof.
   (* implicit arguments differ a bit *)
   by trans (✓ cmra_transport inG_prf a : iProp Σ)%I; last destruct inG_prf.
 Qed.
-Lemma own_valid_2 γ a1 a2 : own γ a1 -∗ own γ a2 -∗ ✓ (a1 ⋅ a2).
+Lemma own_valid_2 γ a1 a2 : 👻{γ} a1 -∗ 👻{γ} a2 -∗ ✓ (a1 ⋅ a2).
 Proof. apply wand_intro_r. by rewrite -own_op own_valid. Qed.
-Lemma own_valid_3 γ a1 a2 a3 : own γ a1 -∗ own γ a2 -∗ own γ a3 -∗ ✓ (a1 ⋅ a2 ⋅ a3).
+Lemma own_valid_3 γ a1 a2 a3 : 👻{γ} a1 -∗ 👻{γ} a2 -∗ 👻{γ} a3 -∗ ✓ (a1 ⋅ a2 ⋅ a3).
 Proof. do 2 apply wand_intro_r. by rewrite -!own_op own_valid. Qed.
-Lemma own_valid_r γ a : own γ a ⊢ own γ a ∗ ✓ a.
+Lemma own_valid_r γ a : 👻{γ} a ⊢ 👻{γ} a ∗ ✓ a.
 Proof. apply: bi.persistent_entails_r. apply own_valid. Qed.
-Lemma own_valid_l γ a : own γ a ⊢ ✓ a ∗ own γ a.
+Lemma own_valid_l γ a : 👻{γ} a ⊢ ✓ a ∗ 👻{γ} a.
 Proof. by rewrite comm -own_valid_r. Qed.
 
-Global Instance own_timeless γ a : Discrete a → Timeless (own γ a).
+Global Instance own_timeless γ a : Discrete a → Timeless (👻{γ} a).
 Proof. rewrite !own_eq /own_def; apply _. Qed.
-Global Instance own_core_persistent γ a : CoreId a → Persistent (own γ a).
+Global Instance own_core_persistent γ a : CoreId a → Persistent (👻{γ} a).
 Proof. rewrite !own_eq /own_def; apply _. Qed.
 
-Lemma later_own γ a : ▷ own γ a -∗ ◇ (∃ b, own γ b ∧ ▷ (a ≡ b)).
+Lemma later_own γ a : ▷ 👻{γ} a -∗ ◇ (∃ b, 👻{γ} b ∧ ▷ (a ≡ b)).
 Proof.
   rewrite own_eq /own_def later_ownM. apply exist_elim=> r.
   assert (NonExpansive (λ r : iResUR Σ, r (inG_id Hin) !! γ)).
@@ -145,7 +147,7 @@ Qed.
    assertion. However, the map_updateP_alloc does not suffice to show this. *)
 Lemma own_alloc_strong a (P : gname → Prop) :
   pred_infinite P →
-  ✓ a → (|==> ∃ γ, ⌜P γ⌝ ∧ own γ a)%I.
+  ✓ a → (|==> ∃ γ, ⌜P γ⌝ ∧ 👻{γ} a)%I.
 Proof.
   intros HP Ha.
   rewrite -(bupd_mono (∃ m, ⌜∃ γ, P γ ∧ m = iRes_singleton γ a⌝ ∧ uPred_ownM m)%I).
@@ -157,7 +159,7 @@ Proof.
     by rewrite !own_eq /own_def -(exist_intro γ) pure_True // left_id.
 Qed.
 Lemma own_alloc_cofinite a (G : gset gname) :
-  ✓ a → (|==> ∃ γ, ⌜γ ∉ G⌝ ∧ own γ a)%I.
+  ✓ a → (|==> ∃ γ, ⌜γ ∉ G⌝ ∧ 👻{γ} a)%I.
 Proof.
   intros Ha.
   apply (own_alloc_strong a (λ γ, γ ∉ G))=> //.
@@ -165,14 +167,14 @@ Proof.
   intros E. set (i := fresh (G ∪ E)).
   exists i. apply not_elem_of_union, is_fresh.
 Qed.
-Lemma own_alloc a : ✓ a → (|==> ∃ γ, own γ a)%I.
+Lemma own_alloc a : ✓ a → (|==> ∃ γ, 👻{γ} a)%I.
 Proof.
   intros Ha. rewrite /uPred_valid /bi_emp_valid (own_alloc_cofinite a ∅) //; [].
   apply bupd_mono, exist_mono=>?. eauto using and_elim_r.
 Qed.
 
 (** ** Frame preserving updates *)
-Lemma own_updateP P γ a : a ~~>: P → own γ a ==∗ ∃ a', ⌜P a'⌝ ∧ own γ a'.
+Lemma own_updateP P γ a : a ~~>: P → 👻{γ} a ==∗ ∃ a', ⌜P a'⌝ ∧ 👻{γ} a'.
 Proof.
   intros Ha. rewrite !own_eq.
   rewrite -(bupd_mono (∃ m, ⌜∃ a', m = iRes_singleton γ a' ∧ P a'⌝ ∧ uPred_ownM m)%I).
@@ -183,16 +185,16 @@ Proof.
     rewrite -(exist_intro a'). by apply and_intro; [apply pure_intro|].
 Qed.
 
-Lemma own_update γ a a' : a ~~> a' → own γ a ==∗ own γ a'.
+Lemma own_update γ a a' : a ~~> a' → 👻{γ} a ==∗ 👻{γ} a'.
 Proof.
   intros; rewrite (own_updateP (a' =)); last by apply cmra_update_updateP.
   by apply bupd_mono, exist_elim=> a''; apply pure_elim_l=> ->.
 Qed.
 Lemma own_update_2 γ a1 a2 a' :
-  a1 ⋅ a2 ~~> a' → own γ a1 -∗ own γ a2 ==∗ own γ a'.
+  a1 ⋅ a2 ~~> a' → 👻{γ} a1 -∗ 👻{γ} a2 ==∗ 👻{γ} a'.
 Proof. intros. apply wand_intro_r. rewrite -own_op. by apply own_update. Qed.
 Lemma own_update_3 γ a1 a2 a3 a' :
-  a1 ⋅ a2 ⋅ a3 ~~> a' → own γ a1 -∗ own γ a2 -∗ own γ a3 ==∗ own γ a'.
+  a1 ⋅ a2 ⋅ a3 ~~> a' → 👻{γ} a1 -∗ 👻{γ} a2 -∗ 👻{γ} a3 ==∗ 👻{γ} a'.
 Proof. intros. do 2 apply wand_intro_r. rewrite -!own_op. by apply own_update. Qed.
 End global.
 
@@ -206,7 +208,7 @@ Arguments own_update {_ _} [_] _ _ _ _.
 Arguments own_update_2 {_ _} [_] _ _ _ _ _.
 Arguments own_update_3 {_ _} [_] _ _ _ _ _ _.
 
-Lemma own_unit A `{!inG Σ (A:ucmraT)} γ : (|==> own γ ε)%I.
+Lemma own_unit A `{!inG Σ (A:ucmraT)} γ : (|==> 👻{γ} ε)%I.
 Proof.
   rewrite /uPred_valid /bi_emp_valid (ownM_unit emp) !own_eq /own_def.
   apply bupd_ownM_update, ofe_fun_singleton_update_empty.
@@ -217,7 +219,7 @@ Qed.
 
 (** Big op class instances *)
 Instance own_cmra_sep_homomorphism `{!inG Σ (A:ucmraT)} :
-  WeakMonoidHomomorphism op uPred_sep (≡) (own γ).
+  WeakMonoidHomomorphism op uPred_sep (≡) (👻{γ}).
 Proof. split; try apply _. apply own_op. Qed.
 
 (** Proofmode class instances *)
@@ -225,19 +227,19 @@ Section proofmode_classes.
   Context `{!inG Σ A}.
   Implicit Types a b : A.
 
-  Global Instance into_sep_own γ a b1 b2 :
-    IsOp a b1 b2 → IntoSep (own γ a) (own γ b1) (own γ b2).
+  Global Instance into_sep_👻{γ} a b1 b2 :
+    IsOp a b1 b2 → IntoSep (👻{γ} a) (👻{γ} b1) (👻{γ} b2).
   Proof. intros. by rewrite /IntoSep (is_op a) own_op. Qed.
   Global Instance into_and_own p γ a b1 b2 :
-    IsOp a b1 b2 → IntoAnd p (own γ a) (own γ b1) (own γ b2).
+    IsOp a b1 b2 → IntoAnd p (👻{γ} a) (👻{γ} b1) (👻{γ} b2).
   Proof. intros. by rewrite /IntoAnd (is_op a) own_op sep_and. Qed.
 
-  Global Instance from_sep_own γ a b1 b2 :
-    IsOp a b1 b2 → FromSep (own γ a) (own γ b1) (own γ b2).
+  Global Instance from_sep_👻{γ} a b1 b2 :
+    IsOp a b1 b2 → FromSep (👻{γ} a) (👻{γ} b1) (👻{γ} b2).
   Proof. intros. by rewrite /FromSep -own_op -is_op. Qed.
   Global Instance from_and_own_persistent γ a b1 b2 :
     IsOp a b1 b2 → TCOr (CoreId b1) (CoreId b2) →
-    FromAnd (own γ a) (own γ b1) (own γ b2).
+    FromAnd (👻{γ} a) (👻{γ} b1) (👻{γ} b2).
   Proof.
     intros ? Hb. rewrite /FromAnd (is_op a) own_op.
     destruct Hb; by rewrite persistent_and_sep.