Make nclose an explicit coercion.

We do this by introducing a type class UpClose with notation ↑.

The reason for this change is as follows: since `nclose : namespace
→ coPset` is declared as a coercion, the notation `nclose N ⊆ E` was
pretty printed as `N ⊆ E`. However, `N ⊆ E` could not be typechecked
because type checking goes from left to right, and as such would look
for an instance `SubsetEq namespace`, which causes the right hand side
to be ill-typed.

base_logic/lib/auth.v

@@ -128,10 +128,10 @@ Section auth.
   Qed.
 
   Lemma auth_open E N γ a :
-    nclose N ⊆ E →
-    auth_ctx γ N f φ ∗ auth_own γ a ={E,E∖N}=∗ ∃ t,
+    ↑N ⊆ E →
+    auth_ctx γ N f φ ∗ auth_own γ a ={E,E∖↑N}=∗ ∃ t,
       ⌜a ≼ f t⌝ ∗ ▷ φ t ∗ ∀ u b,
-      ⌜(f t, a) ~l~> (f u, b)⌝ ∗ ▷ φ u ={E∖N,E}=∗ auth_own γ b.
+      ⌜(f t, a) ~l~> (f u, b)⌝ ∗ ▷ φ u ={E∖↑N,E}=∗ auth_own γ b.
   Proof.
     iIntros (?) "[#? Hγf]". rewrite /auth_ctx. iInv N as "Hinv" "Hclose".
     (* The following is essentially a very trivial composition of the accessors

base_logic/lib/boxes.v

@@ -105,7 +105,7 @@ Proof.
 Qed.
 
 Lemma box_delete E f P Q γ :
-  nclose N ⊆ E →
+  ↑N ⊆ E →
   f !! γ = Some false →
   slice N γ Q ∗ ▷ box N f P ={E}=∗ ∃ P',
     ▷ ▷ (P ≡ (Q ∗ P')) ∗ ▷ box N (delete γ f) P'.
@@ -125,7 +125,7 @@ Proof.
 Qed.
 
 Lemma box_fill E f γ P Q :
-  nclose N ⊆ E →
+  ↑N ⊆ E →
   f !! γ = Some false →
   slice N γ Q ∗ ▷ Q ∗ ▷ box N f P ={E}=∗ ▷ box N (<[γ:=true]> f) P.
 Proof.
@@ -144,7 +144,7 @@ Proof.
 Qed.
 
 Lemma box_empty E f P Q γ :
-  nclose N ⊆ E →
+  ↑N ⊆ E →
   f !! γ = Some true →
   slice N γ Q ∗ ▷ box N f P ={E}=∗ ▷ Q ∗ ▷ box N (<[γ:=false]> f) P.
 Proof.
@@ -164,7 +164,7 @@ Proof.
 Qed.
 
 Lemma box_fill_all E f P :
-  nclose N ⊆ E →
+  ↑N ⊆ E →
   box N f P ∗ ▷ P ={E}=∗ box N (const true <$> f) P.
 Proof.
   iIntros (?) "[H HP]"; iDestruct "H" as (Φ) "[#HeqP Hf]".
@@ -181,7 +181,7 @@ Proof.
 Qed.
 
 Lemma box_empty_all E f P :
-  nclose N ⊆ E →
+  ↑N ⊆ E →
   map_Forall (λ _, (true =)) f →
   box N f P ={E}=∗ ▷ P ∗ box N (const false <$> f) P.
 Proof.

base_logic/lib/cancelable_invariants.v

@@ -51,8 +51,7 @@ Section proofs.
     iMod (inv_alloc N _ (P ∨ own γ 1%Qp)%I with "[HP]"); eauto.
   Qed.
 
-  Lemma cinv_cancel E N γ P :
-    nclose N ⊆ E → cinv N γ P ⊢ cinv_own γ 1 ={E}=∗ ▷ P.
+  Lemma cinv_cancel E N γ P : ↑N ⊆ E → cinv N γ P ⊢ cinv_own γ 1 ={E}=∗ ▷ P.
   Proof.
     rewrite /cinv. iIntros (?) "#Hinv Hγ".
     iInv N as "[$|>Hγ']" "Hclose"; first iApply "Hclose"; eauto.
@@ -60,8 +59,8 @@ Section proofs.
   Qed.
 
   Lemma cinv_open E N γ p P :
-    nclose N ⊆ E →
-    cinv N γ P ⊢ cinv_own γ p ={E,E∖N}=∗ ▷ P ∗ cinv_own γ p ∗ (▷ P ={E∖N,E}=∗ True).
+    ↑N ⊆ E →
+    cinv N γ P ⊢ cinv_own γ p ={E,E∖↑N}=∗ ▷ P ∗ cinv_own γ p ∗ (▷ P ={E∖↑N,E}=∗ True).
   Proof.
     rewrite /cinv. iIntros (?) "#Hinv Hγ".
     iInv N as "[$|>Hγ']" "Hclose".

base_logic/lib/invariants.v

@@ -6,7 +6,7 @@ Import uPred.
 
 (** Derived forms and lemmas about them. *)
 Definition inv_def `{invG Σ} (N : namespace) (P : iProp Σ) : iProp Σ :=
-  (∃ i, ⌜i ∈ nclose N⌝ ∧ ownI i P)%I.
+  (∃ i, ⌜i ∈ ↑N⌝ ∧ ownI i P)%I.
 Definition inv_aux : { x | x = @inv_def }. by eexists. Qed.
 Definition inv {Σ i} := proj1_sig inv_aux Σ i.
 Definition inv_eq : @inv = @inv_def := proj2_sig inv_aux.
@@ -30,8 +30,8 @@ Proof. rewrite inv_eq /inv; apply _. Qed.
 Lemma inv_alloc N E P : ▷ P ={E}=∗ inv N P.
 Proof.
   rewrite inv_eq /inv_def fupd_eq /fupd_def. iIntros "HP [Hw $]".
-  iMod (ownI_alloc (∈ nclose N) P with "[HP Hw]") as (i) "(% & $ & ?)"; auto.
-  - intros Ef. exists (coPpick (nclose N ∖ coPset.of_gset Ef)).
+  iMod (ownI_alloc (∈ ↑ N) P with "[HP Hw]") as (i) "(% & $ & ?)"; auto.
+  - intros Ef. exists (coPpick (↑ N ∖ coPset.of_gset Ef)).
     rewrite -coPset.elem_of_of_gset comm -elem_of_difference.
     apply coPpick_elem_of=> Hfin.
     eapply nclose_infinite, (difference_finite_inv _ _), Hfin.
@@ -41,19 +41,19 @@ Proof.
 Qed.
 
 Lemma inv_open E N P :
-  nclose N ⊆ E → inv N P ={E,E∖N}=∗ ▷ P ∗ (▷ P ={E∖N,E}=∗ True).
+  ↑N ⊆ E → inv N P ={E,E∖↑N}=∗ ▷ P ∗ (▷ P ={E∖↑N,E}=∗ True).
 Proof.
   rewrite inv_eq /inv_def fupd_eq /fupd_def; iDestruct 1 as (i) "[Hi #HiP]".
   iDestruct "Hi" as % ?%elem_of_subseteq_singleton.
-  rewrite {1 4}(union_difference_L (nclose N) E) // ownE_op; last set_solver.
-  rewrite {1 5}(union_difference_L {[ i ]} (nclose N)) // ownE_op; last set_solver.
+  rewrite {1 4}(union_difference_L (↑ N) E) // ownE_op; last set_solver.
+  rewrite {1 5}(union_difference_L {[ i ]} (↑ N)) // ownE_op; last set_solver.
   iIntros "(Hw & [HE $] & $) !> !>".
   iDestruct (ownI_open i P with "[$Hw $HE $HiP]") as "($ & $ & HD)".
   iIntros "HP [Hw $] !> !>". iApply ownI_close; by iFrame.
 Qed.
 
 Lemma inv_open_timeless E N P `{!TimelessP P} :
-  nclose N ⊆ E → inv N P ={E,E∖N}=∗ P ∗ (P ={E∖N,E}=∗ True).
+  ↑N ⊆ E → inv N P ={E,E∖↑N}=∗ P ∗ (P ={E∖↑N,E}=∗ True).
 Proof.
   iIntros (?) "Hinv". iMod (inv_open with "Hinv") as "[>HP Hclose]"; auto.
   iIntros "!> {$HP} HP". iApply "Hclose"; auto.

base_logic/lib/namespaces.v

@@ -16,7 +16,7 @@ Definition ndot_eq : @ndot = @ndot_def := proj2_sig ndot_aux.
 
 Definition nclose_def (N : namespace) : coPset := coPset_suffixes (encode N).
 Definition nclose_aux : { x | x = @nclose_def }. by eexists. Qed.
-Coercion nclose := proj1_sig nclose_aux.
+Instance nclose : UpClose namespace coPset := proj1_sig nclose_aux.
 Definition nclose_eq : @nclose = @nclose_def := proj2_sig nclose_aux.
 
 Infix ".@" := ndot (at level 19, left associativity) : C_scope.
@@ -27,31 +27,34 @@ Instance ndisjoint : Disjoint namespace := λ N1 N2, nclose N1 ⊥ nclose N2.
 Section namespace.
   Context `{Countable A}.
   Implicit Types x y : A.
+  Implicit Types N : namespace.
+  Implicit Types E : coPset.
 
   Global Instance ndot_inj : Inj2 (=) (=) (=) (@ndot A _ _).
   Proof. intros N1 x1 N2 x2; rewrite !ndot_eq=> ?; by simplify_eq. Qed.
 
-  Lemma nclose_nroot : nclose nroot = ⊤.
+  Lemma nclose_nroot : ↑nroot = ⊤.
   Proof. rewrite nclose_eq. by apply (sig_eq_pi _). Qed.
-  Lemma encode_nclose N : encode N ∈ nclose N.
+  Lemma encode_nclose N : encode N ∈ ↑N.
   Proof.
     rewrite nclose_eq.
     by apply elem_coPset_suffixes; exists xH; rewrite (left_id_L _ _).
   Qed.
 
-  Lemma nclose_subseteq N x : nclose (N .@ x) ⊆ nclose N.
+  Lemma nclose_subseteq N x : ↑N .@ x ⊆ (↑N : coPset).
   Proof.
     intros p; rewrite nclose_eq /nclose !ndot_eq !elem_coPset_suffixes.
     intros [q ->]. destruct (list_encode_suffix N (ndot_def N x)) as [q' ?].
     { by exists [encode x]. }
     by exists (q ++ q')%positive; rewrite <-(assoc_L _); f_equal.
   Qed.
-  Lemma nclose_subseteq' E N x : nclose N ⊆ E → nclose (N .@ x) ⊆ E.
+
+  Lemma nclose_subseteq' E N x : ↑N ⊆ E → ↑N .@ x ⊆ E.
   Proof. intros. etrans; eauto using nclose_subseteq. Qed.
 
-  Lemma ndot_nclose N x : encode (N .@ x) ∈ nclose N.
+  Lemma ndot_nclose N x : encode (N .@ x) ∈ ↑ N.
   Proof. apply nclose_subseteq with x, encode_nclose. Qed.
-  Lemma nclose_infinite N : ¬set_finite (nclose N).
+  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.
@@ -61,19 +64,18 @@ Section namespace.
     revert Hqy. by intros [= ?%encode_inj]%list_encode_suffix_eq.
   Qed.
 
-  Lemma ndot_preserve_disjoint_l N E x : nclose N ⊥ E → nclose (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 ⊥ nclose N → E ⊥ nclose (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 ⊥ nclose N → E ⊆ F → E ⊆ F ∖ nclose N.
+  Lemma ndisj_subseteq_difference N E F : E ⊥ ↑N → E ⊆ F → E ⊆ F ∖ ↑N.
   Proof. set_solver. Qed.
 End namespace.
 
 (* The hope is that registering these will suffice to solve most goals
-of the form [N1 ⊥ N2] and those of the form [N1 ⊆ E ∖ N2 ∖ .. ∖ Nn]. *)
+of the form [N1 ⊥ N2] and those of the form [↑N1 ⊆ E ∖ ↑N2 ∖ .. ∖ ↑Nn]. *)
 Hint Resolve ndisj_subseteq_difference : ndisj.
 Hint Extern 0 (_ ⊥ _) => apply ndot_ne_disjoint; congruence : ndisj.
 Hint Resolve ndot_preserve_disjoint_l : ndisj.

base_logic/lib/sts.v

@@ -117,10 +117,10 @@ Section sts.
   Proof. by apply sts_accS. Qed.
 
   Lemma sts_openS E N γ S T :
-    nclose N ⊆ E →
-    sts_ctx γ N φ ∗ sts_ownS γ S T ={E,E∖N}=∗ ∃ s,
+    ↑N ⊆ E →
+    sts_ctx γ N φ ∗ sts_ownS γ S T ={E,E∖↑N}=∗ ∃ s,
       ⌜s ∈ S⌝ ∗ ▷ φ s ∗ ∀ s' T',
-      ⌜sts.steps (s, T) (s', T')⌝ ∗ ▷ φ s' ={E∖N,E}=∗ sts_own γ s' T'.
+      ⌜sts.steps (s, T) (s', T')⌝ ∗ ▷ φ s' ={E∖↑N,E}=∗ sts_own γ s' T'.
   Proof.
     iIntros (?) "[#? Hγf]". rewrite /sts_ctx. iInv N as "Hinv" "Hclose".
     (* The following is essentially a very trivial composition of the accessors
@@ -135,9 +135,9 @@ Section sts.
   Qed.
 
   Lemma sts_open E N γ s0 T :
-    nclose N ⊆ E →
-    sts_ctx γ N φ ∗ sts_own γ s0 T ={E,E∖N}=∗ ∃ s,
+    ↑N ⊆ E →
+    sts_ctx γ N φ ∗ sts_own γ s0 T ={E,E∖↑N}=∗ ∃ s,
       ⌜sts.frame_steps T s0 s⌝ ∗ ▷ φ s ∗ ∀ s' T',
-      ⌜sts.steps (s, T) (s', T')⌝ ∗ ▷ φ s' ={E∖N,E}=∗ sts_own γ s' T'.
+      ⌜sts.steps (s, T) (s', T')⌝ ∗ ▷ φ s' ={E∖↑N,E}=∗ sts_own γ s' T'.
   Proof. by apply sts_openS. Qed.
 End sts.

base_logic/lib/thread_local.v

@@ -16,7 +16,7 @@ Section defs.
     own tid (CoPset E, ∅).
 
   Definition tl_inv (tid : thread_id) (N : namespace) (P : iProp Σ) : iProp Σ :=
-    (∃ i, ⌜i ∈ nclose N⌝ ∧
+    (∃ i, ⌜i ∈ ↑N⌝ ∧
           inv tlN (P ∗ own tid (∅, GSet {[i]}) ∨ tl_own tid {[i]}))%I.
 End defs.
 
@@ -57,8 +57,8 @@ Section proofs.
     iMod (own_empty (prodUR coPset_disjUR (gset_disjUR positive)) tid) as "Hempty".
     iMod (own_updateP with "Hempty") as ([m1 m2]) "[Hm Hown]".
     { apply prod_updateP'. apply cmra_updateP_id, (reflexivity (R:=eq)).
-      apply (gset_disj_alloc_empty_updateP_strong' (λ i, i ∈ nclose N)).
-      intros Ef. exists (coPpick (nclose N ∖ coPset.of_gset Ef)).
+      apply (gset_disj_alloc_empty_updateP_strong' (λ i, i ∈ ↑N)).
+      intros Ef. exists (coPpick (↑ N ∖ coPset.of_gset Ef)).
       rewrite -coPset.elem_of_of_gset comm -elem_of_difference.
       apply coPpick_elem_of=> Hfin.
       eapply nclose_infinite, (difference_finite_inv _ _), Hfin.
@@ -70,14 +70,14 @@ Section proofs.
   Qed.
 
   Lemma tl_inv_open tid tlE E N P :
-    nclose tlN ⊆ tlE → nclose N ⊆ E →
-    tl_inv tid N P ⊢ tl_own tid E ={tlE}=∗ ▷ P ∗ tl_own tid (E ∖ N) ∗
-                       (▷ P ∗ tl_own tid (E ∖ N) ={tlE}=∗ tl_own tid E).
+    ↑tlN ⊆ tlE → ↑N ⊆ E →
+    tl_inv tid N P ⊢ tl_own tid E ={tlE}=∗ ▷ P ∗ tl_own tid (E∖↑N) ∗
+                       (▷ P ∗ tl_own tid (E∖↑N) ={tlE}=∗ tl_own tid E).
   Proof.
     rewrite /tl_inv. iIntros (??) "#Htlinv Htoks".
     iDestruct "Htlinv" as (i) "[% Hinv]".
-    rewrite {1 4}(union_difference_L (nclose N) E) //.
-    rewrite {1 5}(union_difference_L {[i]} (nclose N)) ?tl_own_union; [|set_solver..].
+    rewrite {1 4}(union_difference_L (↑N) E) //.
+    rewrite {1 5}(union_difference_L {[i]} (↑N)) ?tl_own_union; [|set_solver..].
     iDestruct "Htoks" as "[[Htoki $] $]".
     iInv tlN as "[[$ >Hdis]|>Htoki2]" "Hclose".
     - iMod ("Hclose" with "[Htoki]") as "_"; first auto.

base_logic/lib/viewshifts.v

@@ -69,13 +69,13 @@ Proof.
 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.
+  ↑N ⊆ E → inv N R ∗ (▷ R ∗ P ={E∖↑N}=> ▷ R ∗ Q) ⊢ P ={E}=> Q.
 Proof.
   iIntros (?) "#[? Hvs] !# HP". iInv N as "HR" "Hclose".
   iMod ("Hvs" with "[HR HP]") as "[? $]"; first by iFrame.
   by iApply "Hclose".
 Qed.
 
-Lemma vs_alloc N P : ▷ P ={N}=> inv N P.
+Lemma vs_alloc N P : ▷ P ={↑N}=> inv N P.
 Proof. iIntros "!# HP". by iApply inv_alloc. Qed.
 End vs.

heap_lang/heap.v

@@ -130,7 +130,7 @@ Section heap.
 
   (** Weakest precondition *)
   Lemma wp_alloc E e v :
-    to_val e = Some v → nclose heapN ⊆ E →
+    to_val e = Some v → ↑heapN ⊆ E →
     {{{ heap_ctx }}} Alloc e @ E {{{ l, RET LitV (LitLoc l); l ↦ v }}}.
   Proof.
     iIntros (<-%of_to_val ? Φ) "#Hinv HΦ". rewrite /heap_ctx.
@@ -144,7 +144,7 @@ Section heap.
   Qed.
 
   Lemma wp_load E l q v :
-    nclose heapN ⊆ E →
+    ↑heapN ⊆ E →
     {{{ heap_ctx ∗ ▷ l ↦{q} v }}} Load (Lit (LitLoc l)) @ E
     {{{ RET v; l ↦{q} v }}}.
   Proof.
@@ -157,7 +157,7 @@ Section heap.
   Qed.
 
   Lemma wp_store E l v' e v :
-    to_val e = Some v → nclose heapN ⊆ E →
+    to_val e = Some v → ↑heapN ⊆ E →
     {{{ heap_ctx ∗ ▷ l ↦ v' }}} Store (Lit (LitLoc l)) e @ E
     {{{ RET LitV LitUnit; l ↦ v }}}.
   Proof.
@@ -173,7 +173,7 @@ Section heap.
   Qed.
 
   Lemma wp_cas_fail E l q v' e1 v1 e2 v2 :
-    to_val e1 = Some v1 → to_val e2 = Some v2 → v' ≠ v1 → nclose heapN ⊆ E →
+    to_val e1 = Some v1 → to_val e2 = Some v2 → v' ≠ v1 → ↑heapN ⊆ E →
     {{{ heap_ctx ∗ ▷ l ↦{q} v' }}} CAS (Lit (LitLoc l)) e1 e2 @ E
     {{{ RET LitV (LitBool false); l ↦{q} v' }}}.
   Proof.
@@ -186,7 +186,7 @@ Section heap.
   Qed.
 
   Lemma wp_cas_suc E l e1 v1 e2 v2 :
-    to_val e1 = Some v1 → to_val e2 = Some v2 → nclose heapN ⊆ E →
+    to_val e1 = Some v1 → to_val e2 = Some v2 → ↑heapN ⊆ E →
     {{{ heap_ctx ∗ ▷ l ↦ v1 }}} CAS (Lit (LitLoc l)) e1 e2 @ E
     {{{ RET LitV (LitBool true); l ↦ v2 }}}.
   Proof.

heap_lang/lib/barrier/proof.v

@@ -162,7 +162,7 @@ Proof.
 Qed.
 
 Lemma recv_split E l P1 P2 :
-  nclose N ⊆ E → recv l (P1 ∗ P2) ={E}=∗ recv l P1 ∗ recv l P2.
+  ↑N ⊆ E → recv l (P1 ∗ P2) ={E}=∗ recv l P1 ∗ recv l P2.
 Proof.
   rename P1 into R1; rename P2 into R2. rewrite {1}/recv /barrier_ctx.
   iIntros (?). iDestruct 1 as (γ P Q i) "(#(%&Hh&Hsts)&Hγ&#HQ&HQR)".

heap_lang/lib/barrier/specification.v

@@ -14,7 +14,7 @@ Lemma barrier_spec (N : namespace) :
                      {{ v, ∃ l : loc, ⌜v = #l⌝ ∗ recv l P ∗ send l P }}) ∧
     (∀ l P, {{ send l P ∗ P }} signal #l {{ _, True }}) ∧
     (∀ l P, {{ recv l P }} wait #l {{ _, P }}) ∧
-    (∀ l P Q, recv l (P ∗ Q) ={N}=> recv l P ∗ recv l Q) ∧
+    (∀ l P Q, recv l (P ∗ Q) ={↑N}=> recv l P ∗ recv l Q) ∧
     (∀ l P Q, (P -∗ Q) ⊢ recv l P -∗ recv l Q).
 Proof.
   intros HN.

heap_lang/proofmode.v

@@ -18,7 +18,7 @@ Proof. by rewrite /IntoAnd heap_mapsto_op_eq Qp_div_2. Qed.
 
 Lemma tac_wp_alloc Δ Δ' E j e v Φ :
   to_val e = Some v →
-  (Δ ⊢ heap_ctx) → nclose heapN ⊆ E →
+  (Δ ⊢ heap_ctx) → ↑heapN ⊆ E →
   IntoLaterEnvs Δ Δ' →
   (∀ l, ∃ Δ'',
     envs_app false (Esnoc Enil j (l ↦ v)) Δ' = Some Δ'' ∧
@@ -33,7 +33,7 @@ Proof.
 Qed.
 
 Lemma tac_wp_load Δ Δ' E i l q v Φ :
-  (Δ ⊢ heap_ctx) → nclose heapN ⊆ E →
+  (Δ ⊢ heap_ctx) → ↑heapN ⊆ E →
   IntoLaterEnvs Δ Δ' →
   envs_lookup i Δ' = Some (false, l ↦{q} v)%I →
   (Δ' ⊢ Φ v) →
@@ -47,7 +47,7 @@ Qed.
 
 Lemma tac_wp_store Δ Δ' Δ'' E i l v e v' Φ :
   to_val e = Some v' →
-  (Δ ⊢ heap_ctx) → nclose heapN ⊆ E →
+  (Δ ⊢ heap_ctx) → ↑heapN ⊆ E →
   IntoLaterEnvs Δ Δ' →
   envs_lookup i Δ' = Some (false, l ↦ v)%I →
   envs_simple_replace i false (Esnoc Enil i (l ↦ v')) Δ' = Some Δ'' →
@@ -62,7 +62,7 @@ Qed.
 
 Lemma tac_wp_cas_fail Δ Δ' E i l q v e1 v1 e2 v2 Φ :
   to_val e1 = Some v1 → to_val e2 = Some v2 →
-  (Δ ⊢ heap_ctx) → nclose heapN ⊆ E →
+  (Δ ⊢ heap_ctx) → ↑heapN ⊆ E →
   IntoLaterEnvs Δ Δ' →
   envs_lookup i Δ' = Some (false, l ↦{q} v)%I → v ≠ v1 →
   (Δ' ⊢ Φ (LitV (LitBool false))) →
@@ -76,7 +76,7 @@ Qed.
 
 Lemma tac_wp_cas_suc Δ Δ' Δ'' E i l v e1 v1 e2 v2 Φ :
   to_val e1 = Some v1 → to_val e2 = Some v2 →
-  (Δ ⊢ heap_ctx) → nclose heapN ⊆ E →
+  (Δ ⊢ heap_ctx) → ↑heapN ⊆ E →
   IntoLaterEnvs Δ Δ' →
   envs_lookup i Δ' = Some (false, l ↦ v)%I → v = v1 →
   envs_simple_replace i false (Esnoc Enil i (l ↦ v2)) Δ' = Some Δ'' →

prelude/base.v

@@ -726,6 +726,8 @@ End disjoint_list.
 
 Class Filter A B := filter: ∀ (P : A → Prop) `{∀ x, Decision (P x)}, B → B.
 
+Class UpClose A B := up_close : A → B.
+Notation "↑ x" := (up_close x) (at level 20, format "↑ x").
 
 (** * Monadic operations *)
 (** We define operational type classes for the monadic operations bind, join 

tests/heap_lang.v

@@ -12,7 +12,7 @@ Section LiftingTests.
   Definition heap_e  : expr :=
     let: "x" := ref #1 in "x" <- !"x" + #1 ;; !"x".
   Lemma heap_e_spec E :
-     nclose heapN ⊆ E → heap_ctx ⊢ WP heap_e @ E {{ v, ⌜v = #2⌝ }}.
+    ↑heapN ⊆ E → heap_ctx ⊢ WP heap_e @ E {{ v, ⌜v = #2⌝ }}.
   Proof.
     iIntros (HN) "#?". rewrite /heap_e.
     wp_alloc l. wp_let. wp_load. wp_op. wp_store. by wp_load.
@@ -23,7 +23,7 @@ Section LiftingTests.
     let: "y" := ref #1 in
     "x" <- !"x" + #1 ;; !"x".
   Lemma heap_e2_spec E :
-     nclose heapN ⊆ E → heap_ctx ⊢ WP heap_e2 @ E {{ v, ⌜v = #2⌝ }}.
+    ↑heapN ⊆ E → heap_ctx ⊢ WP heap_e2 @ E {{ v, ⌜v = #2⌝ }}.
   Proof.
     iIntros (HN) "#?". rewrite /heap_e2.
     wp_alloc l. wp_let. wp_alloc l'. wp_let.

tests/proofmode.v

@@ -90,7 +90,7 @@ Section iris.
   Implicit Types P Q : iProp Σ.
 
   Lemma demo_8 N E P Q R :
-    nclose N ⊆ E →
+    ↑N ⊆ E →
     (True -∗ P -∗ inv N Q -∗ True -∗ R) ⊢ P -∗ ▷ Q ={E}=∗ R.
   Proof.
     iIntros (?) "H HP HQ".