Don't use subnamespaces in the ticket lock refinement

theories/examples/ticket_lock.v

@@ -117,21 +117,15 @@ Section refinement.
   Program Definition lockInt := λne vv,
     (∃ (lo ln : loc) (γ : gname) (l' : loc),
         ⌜vv.1 = (#lo, #ln)%V⌝ ∗ ⌜vv.2 = #l'⌝
-      ∗ inv (N.@(lo,ln,l')) (lockInv lo ln γ l'))%I.
+      ∗ inv N (lockInv lo ln γ l'))%I.
   Next Obligation. solve_proper. Qed.
   
-  (* Program Definition lockInt (γp : gname) := λne vv, *)
-  (*   (∃ (lo ln : loc) (γ : gname) (l' : loc), *)
-  (*       ⌜vv.1 = (#lo, #ln)%V⌝ ∗ ⌜vv.2 = #l'⌝ *)
-  (*     ∗ inPool γp lo ln γ l')%I. *)
-  (* Next Obligation. solve_proper. Qed. *)
-
   Instance lockInt_persistent ww : Persistent (lockInt ww).
   Proof. apply _. Qed.
 
   (** * Refinement proofs *)
 
-  Local Ltac openI N := 
+  Local Ltac openI :=
     iInv N as (o n b) ">(Hlo & Hln & Hissued & Hl' & Hbticket)" "Hcl".
   Local Ltac closeI := iMod ("Hcl" with "[-]") as "_";
     first by (iNext; iExists _,_,_; iFrame).
@@ -154,7 +148,7 @@ Section refinement.
     iIntros (l') "Hl'".
     (* Establishing the invariant *)
     iMod newIssuedTickets as (γ) "Hγ".
-    iMod (inv_alloc (N.@(lo,ln,l')) _ (lockInv lo ln γ l') with "[-]") as "#Hinv".
+    iMod (inv_alloc N _ (lockInv lo ln γ l') with "[-]") as "#Hinv".
     { iNext. iExists _,_,_. iFrame. }
     rel_vals. iModIntro. iAlways.
     iExists _,_,_,_. iFrame "Hinv". eauto.
@@ -163,7 +157,7 @@ Section refinement.
   (* Acquiring a lock *)
   (* helper lemma *)
   Lemma wait_loop_refinement Δ Γ (lo ln : loc) γ (l' : loc) (m : nat) :
-    inv (N.@(lo,ln,l')) (lockInv lo ln γ l') -∗
+    inv N (lockInv lo ln γ l') -∗
     ticket γ m -∗
     {(lockInt :: Δ); ⤉Γ} ⊨
       wait_loop #m (#lo, #ln) ≤log≤ lock.acquire #l' : TUnit.
@@ -174,7 +168,7 @@ Section refinement.
     unlock {2}wait_loop. simpl.
     rel_let_l. rel_proj_l.
     rel_load_l_atomic.
-    openI (N.@(lo, ln, l')).
+    openI.
     iModIntro. iExists _; iFrame; iNext.
     iIntros "Hlo".
     rel_op_l.
@@ -263,7 +257,7 @@ Section refinement.
                              if b then ticket γ o else True)%I
                    True%I γ); first done.
     iAlways.
-    openI (N.@(lo, ln, l')).
+    openI.
     iModIntro. iExists _,_; iFrame.
     iSplitL "Hbticket Hl'".
     { iExists _. iFrame. }
@@ -296,7 +290,7 @@ Section refinement.
     rel_let_l. repeat rel_proj_l.
     rel_apply_l (bin_log_FG_increment_logatomic _ (issuedTickets γ)%I True%I); first done.
     iAlways.
-    openI (N.@(lo, ln, l')).
+    openI.
     iModIntro. iExists _; iFrame.
     iSplit.
     - iDestruct 1 as (m) "[Hln ?]".
@@ -368,7 +362,7 @@ Section refinement.
                  ∗ if b then ticket γ o else True)%I).
     rel_apply_l (wkincr_atomic_l R True%I); first done.
     iAlways.
-    openI (N.@(lo, ln, l')).
+    openI.
     iModIntro. iExists _; iFrame.
     rewrite {1}/R. iSplitR "Hcl".
     { iExists _,_; by iFrame. } clear o n b.