Make the steps_ lemma curried in the stack example

F_mu_ref_conc/examples/counter.v

@@ -62,10 +62,11 @@ Section CG_Counter.
 
   Lemma steps_CG_increment E ρ j K x n:
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ x ↦ₛ (#nv n) ∗ j ⤇ fill K (App (CG_increment (Loc x)) Unit)
-      ⊢ |={E}=> j ⤇ fill K (Unit) ∗ x ↦ₛ (#nv (S n)).
+    spec_ctx ρ -∗ x ↦ₛ (#nv n)
+    -∗ j ⤇ fill K (App (CG_increment (Loc x)) Unit)
+    ={E}=∗ j ⤇ fill K (Unit) ∗ x ↦ₛ (#nv (S n)).
   Proof.
-    iIntros (HNE) "[#Hspec [Hx Hj]]". unfold CG_increment.
+    iIntros (HNE) "#Hspec Hx Hj". unfold CG_increment.
     tp_rec j.
     tp_load j.
     tp_op j. tp_normalise j.
@@ -111,16 +112,15 @@ Section CG_Counter.
 
   Lemma steps_CG_locked_increment E ρ j K x n l :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ x ↦ₛ (#nv n) ∗ l ↦ₛ (#♭v false)
-      ∗ j ⤇ fill K (App (CG_locked_increment (Loc x) (Loc l)) Unit)
+    spec_ctx ρ -∗ x ↦ₛ (#nv n) -∗ l ↦ₛ (#♭v false)
+    -∗ j ⤇ fill K (App (CG_locked_increment (Loc x) (Loc l)) Unit)
     ={E}=∗ j ⤇ fill K Unit ∗ x ↦ₛ (#nv S n) ∗ l ↦ₛ (#♭v false).
   Proof.
-    iIntros (HNE) "[#Hspec [Hx [Hl Hj]]]".
-    iMod (steps_with_lock _ _ j K _ _ _ _ UnitV UnitV _ _ with "[Hj Hx Hl]") as "Hj"; last by iFrame.
-    - iIntros (K') "[#Hspec [Hx Hj]]".
-      iApply steps_CG_increment; first done. iFrame "Hspec Hj Hx"; trivial.
-    - by iFrame "Hspec Hj Hx".
-      Unshelve. all: trivial.
+    iIntros (HNE) "#Hspec Hx Hl Hj".
+    iMod (steps_with_lock _ _ _ K _ _ _ _ UnitV UnitV _ _ with "Hspec Hx Hl Hj") as "Hj"; last by iFrame.
+    { iIntros (K') "#Hspec Hx Hj /=".
+      iApply (steps_CG_increment E with "Hspec Hx"); auto. }
+    Unshelve. all: trivial.
   Qed.
 
   Global Opaque CG_locked_increment.
@@ -150,11 +150,11 @@ Section CG_Counter.
 
   Lemma steps_counter_read E ρ j K x n :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ x ↦ₛ (#nv n)
-               ∗ j ⤇ fill K (App (counter_read (Loc x)) Unit)
+    spec_ctx ρ -∗ x ↦ₛ (#nv n)
+    -∗ j ⤇ fill K (App (counter_read (Loc x)) Unit)
     ={E}=∗ j ⤇ fill K (#n n) ∗ x ↦ₛ (#nv n).
   Proof.
-    intros HNE. iIntros "[#Hspec [Hx Hj]]". unfold counter_read.
+    intros HNE. iIntros "#Hspec Hx Hj". unfold counter_read.
     tp_rec j.
     tp_load j. tp_normalise j.
     by iFrame.
@@ -250,10 +250,8 @@ Section CG_Counter.
     iClear "HΓ". cbn -[FG_counter CG_counter].
     rewrite ?empty_env_subst /CG_counter /FG_counter.
     iApply fupd_wp.
-    iMod (steps_newlock _ _ j (K ++ [AppRCtx (RecV _)]) _ with "[Hj]")
-      as (l) "[Hj Hl]"; eauto.
-    { rewrite fill_app /=. by iFrame. }
-    rewrite fill_app /=.
+    tp_bind j newlock.
+    iMod (steps_newlock with "Hspec Hj") as (l) "[Hj Hl]"; eauto.
     tp_rec j.
     asimpl. rewrite CG_locked_increment_subst /=.
     rewrite counter_read_subst /=.
@@ -306,9 +304,7 @@ Section CG_Counter.
       destruct (decide (n = n')) as [|Hneq]; subst.
       + (* CAS succeeds *)
         (* In this case, we perform increment in the coarse-grained one *)
-        iMod (steps_CG_locked_increment
-                _ _ _ _ _ _ _ _ with "[Hj Hl Hcnt']") as "[Hj [Hcnt' Hl]]".
-        { iFrame "Hspec Hcnt' Hl Hj"; trivial. }
+        iMod (steps_CG_locked_increment with "Hspec Hcnt' Hl Hj") as "[Hj [Hcnt' Hl]]"; first by solve_ndisj.
         iApply (wp_cas_suc with "[Hcnt]"); auto.
         iNext. iIntros "Hcnt".
         iMod ("Hclose" with "[Hl Hcnt Hcnt']").
@@ -334,13 +330,12 @@ Section CG_Counter.
       iNext. 
       iApply wp_atomic; eauto.
       iInv counterN as (n) ">[Hl [Hcnt Hcnt']]" "Hclose".
-      iMod (steps_counter_read with "[$Hspec $Hj $Hcnt']") as "[Hj Hcnt']"; first by solve_ndisj.
+      iMod (steps_counter_read with "Hspec Hcnt' Hj") as "[Hj Hcnt']"; first by solve_ndisj.
       iApply (wp_load with "[Hcnt]"); eauto. 
       iNext. iIntros "Hcnt".
       iMod ("Hclose" with "[Hl Hcnt Hcnt']").
       { iNext. iExists _; iFrame "Hl Hcnt Hcnt'". }
       iExists (#nv _); eauto.
-      Unshelve. all: solve_ndisj.
   Qed.
 End CG_Counter.
 

F_mu_ref_conc/examples/lock.v

@@ -84,10 +84,10 @@ Section proof.
   
   Lemma steps_newlock E ρ j K :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ j ⤇ fill K newlock
-      ⊢ |={E}=> ∃ l, j ⤇ fill K (Loc l) ∗ l ↦ₛ (#♭v false).
+    spec_ctx ρ -∗ j ⤇ fill K newlock
+    ={E}=∗ ∃ l, j ⤇ fill K (Loc l) ∗ l ↦ₛ (#♭v false).
   Proof.
-    iIntros (HNE) "[#Hspec Hj]".
+    iIntros (HNE) "#Hspec Hj".
     tp_alloc j as l "Hl". tp_normalise j.
     by iExists _; iFrame.
   Qed.
@@ -96,10 +96,10 @@ Section proof.
 
   Lemma steps_acquire E ρ j K l :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ l ↦ₛ (#♭v false) ∗ j ⤇ fill K (App acquire (Loc l))
-      ⊢ |={E}=> j ⤇ fill K Unit ∗ l ↦ₛ (#♭v true).
+    spec_ctx ρ -∗ l ↦ₛ (#♭v false) -∗ j ⤇ fill K (App acquire (Loc l))
+    ={E}=∗ j ⤇ fill K Unit ∗ l ↦ₛ (#♭v true).
   Proof.
-    iIntros (HNE) "[#Hspec [Hl Hj]]". unfold acquire.
+    iIntros (HNE) "#Hspec Hl Hj". unfold acquire.
     tp_rec j. 
     tp_cas_suc j. 
     tp_if_true j. tp_normalise j.    
@@ -110,10 +110,10 @@ Section proof.
 
   Lemma steps_release E ρ j K l b:
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ l ↦ₛ (#♭v b) ∗ j ⤇ fill K (App release (Loc l))
-      ⊢ |={E}=> j ⤇ fill K Unit ∗ l ↦ₛ (#♭v false).
+    spec_ctx ρ -∗ l ↦ₛ (#♭v b) -∗ j ⤇ fill K (App release (Loc l))
+    ={E}=∗ j ⤇ fill K Unit ∗ l ↦ₛ (#♭v false).
   Proof.
-    iIntros (HNE) "[#Hspec [Hl Hj]]". unfold release.
+    iIntros (HNE) "#Hspec Hl Hj". unfold release.
     tp_rec j.
     tp_store j. tp_normalise j.
     by iFrame.
@@ -124,27 +124,27 @@ Section proof.
   Lemma steps_with_lock E ρ j K e l P Q v w:
     nclose specN ⊆ E →
     (∀ f, e.[f] = e) (* e is a closed term *) →
-    (∀ K', spec_ctx ρ ∗ P ∗ j ⤇ fill K' (App e (of_val w))
-            ⊢ |={E}=> j ⤇ fill K' (of_val v) ∗ Q) →
-    spec_ctx ρ ∗ P ∗ l ↦ₛ (#♭v false)
-                ∗ j ⤇ fill K (App (with_lock e (Loc l)) (of_val w))
-      ⊢ |={E}=> j ⤇ fill K (of_val v) ∗ Q ∗ l ↦ₛ (#♭v false).
+    (∀ K', spec_ctx ρ -∗ P -∗ j ⤇ fill K' (App e (of_val w))
+            ={E}=∗ j ⤇ fill K' (of_val v) ∗ Q) →
+    spec_ctx ρ -∗ P -∗ l ↦ₛ (#♭v false)
+    -∗ j ⤇ fill K (App (with_lock e (Loc l)) (of_val w))
+    ={E}=∗ j ⤇ fill K (of_val v) ∗ Q ∗ l ↦ₛ (#♭v false).
   Proof.
-    iIntros (HNE H1 H2) "[#Hspec [HP [Hl Hj]]]".
+    iIntros (HNE H1 H2) "#Hspec HP Hl Hj".
     tp_rec j; eauto using to_of_val.
     asimpl. rewrite H1.
     (* TODO: a tp_apply tactic similar to that of iApply *)
     tp_bind j (App acquire (Loc l)).
-    iMod (steps_acquire _ _ j with "[$Hspec $Hj $Hl]") as "[Hj Hl]"; eauto.
+    iMod (steps_acquire _ _ j with "Hspec Hl Hj") as "[Hj Hl]"; eauto.
     tp_rec j.
     asimpl. rewrite H1.
     tp_bind j (App e _).
-    iMod (H2 with "[$Hspec $Hj $HP]") as "[Hj HQ]". 
+    iMod (H2 with "Hspec HP Hj") as "[Hj HQ]". 
     tp_normalise j.
     tp_rec j; eauto using to_of_val.
     asimpl.
     tp_bind j (App release _).
-    iMod (steps_release with "[$Hspec $Hj $Hl]") as "[Hj Hl]"; auto.
+    iMod (steps_release with "Hspec Hl Hj") as "[Hj Hl]"; auto.
     tp_rec j.
     tp_normalise j. asimpl.
     by iFrame.

F_mu_ref_conc/examples/stack/CG_stack.v

@@ -98,10 +98,10 @@ Section CG_Stack.
 
   Lemma steps_CG_push E ρ j K st v w :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ st ↦ₛ v ∗ j ⤇ fill K (App (CG_push (Loc st)) (of_val w))
-    ⊢ |={E}=> j ⤇ fill K Unit ∗ st ↦ₛ FoldV (InjRV (PairV w v)).
+    spec_ctx ρ -∗ st ↦ₛ v -∗ j ⤇ fill K (App (CG_push (Loc st)) (of_val w))
+    ={E}=∗ j ⤇ fill K Unit ∗ st ↦ₛ FoldV (InjRV (PairV w v)).
   Proof.
-    intros HNE. iIntros "[#Hspec [Hx Hj]]". unfold CG_push.
+    intros HNE. iIntros "#Hspec Hx Hj". unfold CG_push.
     tp_rec j; eauto using to_of_val.
     tp_normalise j.
     tp_load j. tp_normalise j.
@@ -144,19 +144,16 @@ Section CG_Stack.
 
   Lemma steps_CG_locked_push E ρ j K st w v l :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ st ↦ₛ v ∗ l ↦ₛ (#♭v false)
-      ∗ j ⤇ fill K (App (CG_locked_push (Loc st) (Loc l)) (of_val w))
-    ⊢ |={E}=> j ⤇ fill K Unit ∗ st ↦ₛ FoldV (InjRV (PairV w v)) ∗ l ↦ₛ (#♭v false).
+    spec_ctx ρ -∗ st ↦ₛ v -∗ l ↦ₛ (#♭v false)
+    -∗ j ⤇ fill K (App (CG_locked_push (Loc st) (Loc l)) (of_val w))
+    ={E}=∗ j ⤇ fill K Unit ∗ st ↦ₛ FoldV (InjRV (PairV w v)) ∗ l ↦ₛ (#♭v false).
   Proof.
-    iIntros (?) "(#Hspec & Hst & Hl & Hj)".
+    iIntros (?) "#Hspec Hst Hl Hj".
     unfold CG_locked_push.
     (* TODO would be nice to be able to determine that e := Loc l for instance *)
-    iMod (steps_with_lock _ _ j K (CG_push (Loc st)) l _ _ UnitV _ _ _ with "[Hspec Hst Hj Hl]") as "Hj"; last done. 
-    - iIntros (K') "(#Hspec & HQ & Hj)".
-      iApply steps_CG_push; first eauto.
-      iFrame "Hspec Hj". iFrame "HQ".
-    - by iFrame. 
-      Unshelve. all: trivial.
+    iMod (steps_with_lock _ _ _ _ _ _ _ _ UnitV with "Hspec Hst Hl Hj") as "Hj"; auto. 
+    iIntros (K') "#Hspec HQ Hj".
+    iApply (steps_CG_push with "Hspec HQ"); auto. 
   Qed.
 
   Global Opaque CG_locked_push.
@@ -187,11 +184,11 @@ Section CG_Stack.
 
   Lemma steps_CG_pop_suc E ρ j K st v w :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ st ↦ₛ FoldV (InjRV (PairV w v)) ∗
-               j ⤇ fill K (App (CG_pop (Loc st)) Unit)
-      ⊢ |={E}=> j ⤇ fill K (InjR (of_val w)) ∗ st ↦ₛ v.
+    spec_ctx ρ -∗ st ↦ₛ FoldV (InjRV (PairV w v))
+    -∗ j ⤇ fill K (App (CG_pop (Loc st)) Unit)
+    ={E}=∗ j ⤇ fill K (InjR (of_val w)) ∗ st ↦ₛ v.
   Proof.
-    intros HNE. iIntros "[#Hspec [Hx Hj]]". unfold CG_pop.
+    intros HNE. iIntros "#Hspec Hx Hj". unfold CG_pop.
     tp_rec j. asimpl.
     tp_load j. tp_normalise j.
     tp_fold j; simpl; first by rewrite ?to_of_val /=.
@@ -206,11 +203,11 @@ Section CG_Stack.
 
   Lemma steps_CG_pop_fail E ρ j K st :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ st ↦ₛ FoldV (InjLV UnitV) ∗
-               j ⤇ fill K (App (CG_pop (Loc st)) Unit)
-      ⊢ |={E}=> j ⤇ fill K (InjL Unit) ∗ st ↦ₛ FoldV (InjLV UnitV).
+    spec_ctx ρ -∗ st ↦ₛ FoldV (InjLV UnitV)
+    -∗ j ⤇ fill K (App (CG_pop (Loc st)) Unit)
+    ={E}=∗ j ⤇ fill K (InjL Unit) ∗ st ↦ₛ FoldV (InjLV UnitV).
   Proof.
-    iIntros (HNE) "[#Hspec [Hx Hj]]". unfold CG_pop.
+    iIntros (HNE) "#Hspec Hx Hj". unfold CG_pop.
     tp_rec j.
     tp_load j. asimpl. tp_normalise j.
     tp_fold j.
@@ -253,32 +250,30 @@ Section CG_Stack.
 
   Lemma steps_CG_locked_pop_suc E ρ j K st v w l :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ st ↦ₛ FoldV (InjRV (PairV w v)) ∗ l ↦ₛ (#♭v false)
-               ∗ j ⤇ fill K (App (CG_locked_pop (Loc st) (Loc l)) Unit)
-      ⊢ |={E}=> j ⤇ fill K (InjR (of_val w)) ∗ st ↦ₛ v ∗ l ↦ₛ (#♭v false).
+    spec_ctx ρ -∗ st ↦ₛ FoldV (InjRV (PairV w v)) -∗ l ↦ₛ (#♭v false)
+    -∗ j ⤇ fill K (App (CG_locked_pop (Loc st) (Loc l)) Unit)
+    ={E}=∗ j ⤇ fill K (InjR (of_val w)) ∗ st ↦ₛ v ∗ l ↦ₛ (#♭v false).
   Proof.
-    iIntros (HNE) "[#Hspec [Hx [Hl Hj]]]". unfold CG_locked_pop.
-    iMod (steps_with_lock _ _ j K _ _ _ _ (InjRV w) UnitV _ _
-          with "[Hj Hx Hl]") as "Hj"; last done.
-    - iIntros (K') "[#Hspec [Hx Hj]]".
-      iApply steps_CG_pop_suc; first done. iFrame "Hspec Hj Hx"; trivial.
-    - iFrame "Hspec Hj Hx"; trivial.
-      Unshelve. all: trivial.
+    iIntros (HNE) "#Hspec Hx Hl Hj". unfold CG_locked_pop.
+    iMod (steps_with_lock _ _ _ _ _ _ _ _ (InjRV w) UnitV _ _
+          with "Hspec Hx Hl Hj") as "Hj"; auto.
+    iIntros (K') "#Hspec Hx Hj".
+    iApply (steps_CG_pop_suc with "Hspec Hx Hj"). trivial. 
+    Unshelve. all: trivial.
   Qed.
 
   Lemma steps_CG_locked_pop_fail E ρ j K st l :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ st ↦ₛ FoldV (InjLV UnitV) ∗ l ↦ₛ (#♭v false)
-               ∗ j ⤇ fill K (App (CG_locked_pop (Loc st) (Loc l)) Unit)
-      ⊢ |={E}=> j ⤇ fill K (InjL Unit) ∗ st ↦ₛ FoldV (InjLV UnitV) ∗ l ↦ₛ (#♭v false).
+    spec_ctx ρ -∗ st ↦ₛ FoldV (InjLV UnitV) -∗ l ↦ₛ (#♭v false)
+    -∗ j ⤇ fill K (App (CG_locked_pop (Loc st) (Loc l)) Unit)
+    ={E}=∗ j ⤇ fill K (InjL Unit) ∗ st ↦ₛ FoldV (InjLV UnitV) ∗ l ↦ₛ (#♭v false).
   Proof.
-    iIntros (HNE) "[#Hspec [Hx [Hl Hj]]]". unfold CG_locked_pop.
-    iMod (steps_with_lock _ _ j K _ _ _ _ (InjLV UnitV) UnitV _ _
-          with "[Hj Hx Hl]") as "Hj"; last done.
-    - iIntros (K') "[#Hspec [Hx Hj]] /=".
-      iApply steps_CG_pop_fail; first done. iFrame "Hspec Hj Hx"; trivial.
-    - iFrame "Hspec Hj Hx"; trivial.
-      Unshelve. all: trivial.
+    iIntros (HNE) "#Hspec Hx Hl Hj". unfold CG_locked_pop.
+    iMod (steps_with_lock _ _ _ _ _ _ _ _ (InjLV UnitV) UnitV _ _
+          with "Hspec Hx Hl Hj") as "Hj"; auto.
+    iIntros (K') "#Hspec Hx Hj /=".
+    iApply (steps_CG_pop_fail with "Hspec Hx Hj"). trivial. 
+    Unshelve. all: trivial.
   Qed.
 
   Global Opaque CG_locked_pop.
@@ -316,14 +311,14 @@ Section CG_Stack.
 
   Lemma steps_CG_snap E ρ j K st v l :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ st ↦ₛ v ∗ l ↦ₛ (#♭v false)
-               ∗ j ⤇ fill K (App (CG_snap (Loc st) (Loc l)) Unit)
-      ⊢ |={E}=> j ⤇ (fill K (of_val v)) ∗ st ↦ₛ v ∗ l ↦ₛ (#♭v false).
+    spec_ctx ρ -∗ st ↦ₛ v -∗ l ↦ₛ (#♭v false)
+    -∗ j ⤇ fill K (App (CG_snap (Loc st) (Loc l)) Unit)
+    ={E}=∗ j ⤇ (fill K (of_val v)) ∗ st ↦ₛ v ∗ l ↦ₛ (#♭v false).
   Proof.
-    iIntros (HNE) "[#Hspec [Hx [Hl Hj]]]". unfold CG_snap.
+    iIntros (HNE) "#Hspec Hx Hl Hj". unfold CG_snap.
     iMod (steps_with_lock _ _ j K _ _ _ _ v UnitV _ _
-          with "[Hj Hx Hl]") as "Hj"; last done; [|by iFrame "Hspec Hx Hl Hj"].
-    iIntros (K') "[#Hspec [Hx Hj]]".
+          with "Hspec Hx Hl Hj") as "Hj"; auto. 
+    iIntros (K') "#Hspec Hx Hj".
     tp_rec j.
     tp_load j. tp_normalise j.
     by iFrame.
@@ -377,17 +372,16 @@ Section CG_Stack.
   Lemma steps_CG_iter E ρ j K f v w :
     nclose specN ⊆ E →
     spec_ctx ρ
-             ∗ j ⤇ fill K (App (CG_iter (of_val f))
-                               (Fold (InjR (Pair (of_val w) (of_val v)))))
-      ⊢ |={E}=>
-    j ⤇ fill K
-          (App
-             (Rec
-                (App ((CG_iter (of_val f)).[ren (+2)])
-                     (Snd (Pair ((of_val w).[ren (+2)]) (of_val v).[ren (+2)]))))
-             (App (of_val f) (of_val w))).
+    -∗ j ⤇ fill K (App (CG_iter (of_val f))
+                       (Fold (InjR (Pair (of_val w) (of_val v)))))
+    ={E}=∗ j ⤇ fill K
+        (App
+           (Rec
+              (App ((CG_iter (of_val f)).[ren (+2)])
+                   (Snd (Pair ((of_val w).[ren (+2)]) (of_val v).[ren (+2)]))))
+           (App (of_val f) (of_val w))).
   Proof.
-    iIntros (HNE) "[#Hspec Hj]". unfold CG_iter.
+    iIntros (HNE) "#Hspec Hj". unfold CG_iter.
     tp_rec j; first by (rewrite /= ?to_of_val /=).
     rewrite -CG_iter_folding. Opaque CG_iter.
     tp_fold j; first by (rewrite /= ?to_of_val /=). 
@@ -402,10 +396,10 @@ Section CG_Stack.
 
   Lemma steps_CG_iter_end E ρ j K f :
     nclose specN ⊆ E →
-    spec_ctx ρ ∗ j ⤇ fill K (App (CG_iter (of_val f)) (Fold (InjL Unit)))
-      ⊢ |={E}=> j ⤇ fill K Unit.
+    spec_ctx ρ -∗ j ⤇ fill K (App (CG_iter (of_val f)) (Fold (InjL Unit)))
+    ={E}=∗ j ⤇ fill K Unit.
   Proof.
-    iIntros (HNE) "[#Hspec Hj]". unfold CG_iter.
+    iIntros (HNE) "#Hspec Hj". unfold CG_iter.
     tp_rec j. 
     tp_fold j.
     tp_case_inl j. tp_normalise j. 

F_mu_ref_conc/examples/stack/refinement.v

@@ -63,9 +63,8 @@ Section Stack_refinement.
        end) with (⊤ ∖ ↑stackN) by reflexivity.
     replace (CG_iter (of_val f2)) with (of_val (CG_iterV (of_val f2))) by (rewrite CG_iter_of_val; done).        
     tp_bind j (App (CG_snap _ _) ())%E.
-    iMod (steps_CG_snap with "[$Hs Hst' Hl Hj]") as "(Hj & Hst' & Hl)";  
+    iMod (steps_CG_snap with "Hs Hst' Hl Hj") as "(Hj & Hst' & Hl)";
       first solve_ndisj.
-    { iFrame "Hst'". iFrame. }
     tp_normalise j.
     close_sinv "Hclose" "[Hoe Hst' Hst HLK Hl]".
 
@@ -89,7 +88,7 @@ Section Stack_refinement.
       iNext.
       iApply fupd_wp.
       rewrite CG_iter_of_val /=.
-      iMod (steps_CG_iter_end with "[$Hs $Hj]") as "Hj"; first solve_ndisj.
+      iMod (steps_CG_iter_end with "Hs Hj") as "Hj"; first solve_ndisj.
       iModIntro.
       iApply wp_value; auto.
       iExists UnitV. iFrame. eauto.
@@ -101,7 +100,7 @@ Section Stack_refinement.
       iNext. iModIntro.
       wp_bind (App (of_val f1) _).
       rewrite CG_iter_of_val.
-      iMod (steps_CG_iter with "[$Hs $Hj]") as "Hj"; first solve_ndisj.
+      iMod (steps_CG_iter with "Hs Hj") as "Hj"; first solve_ndisj.
       rewrite CG_iter_subst.
       tp_bind j (App (of_val f2) _).
       iSpecialize ("Hff" $! (y1, y2) with "Hy"). 
@@ -153,7 +152,7 @@ Section Stack_refinement.
     iInv stackN as (istk2 v h) "[Hoe [>Hst' [Hst [HLK >Hl]]]]" "Hclose".    (* TODO : why can we remove the later here?*) 
     destruct (decide (istk = istk2)) as [Heq|Hneq]; first subst istk.
     * (* Case CAS succeeds *)
-      iMod (steps_CG_locked_push _ _ j K st' v2 with "[$Hj $Hs $Hst' $Hl]") as "[Hj [Hst' Hl]]"; first solve_ndisj.
+      iMod (steps_CG_locked_push _ _ j K st' v2 with "Hs Hst' Hl Hj") as "[Hj [Hst' Hl]]"; first solve_ndisj.
       iApply (wp_cas_suc with "Hst"); auto.
       iNext. iIntros "Hst".
 
@@ -196,7 +195,7 @@ Section Stack_refinement.
     (* Checking whether the stack is empty *)
     rewrite {2}StackLink_unfold.
     iDestruct "HLK'" as (istk2 w) "[% [Hmpt [[% %]|HLK']]]"; simplify_eq/=.
-    + iMod (steps_CG_locked_pop_fail with "[$Hs $Hst' $Hl $Hj]")
+    + iMod (steps_CG_locked_pop_fail with "Hs Hst' Hl Hj")
         as "(Hj & Hstk' & Hl)"; first solve_ndisj.
       close_sinv "Hclose" "[Hoe Hstk' Hst Hl]".
       wp_bind (Unfold _). iApply wp_fold; first by auto using to_of_val.    iNext.
@@ -249,7 +248,7 @@ Section Stack_refinement.
         iDestruct "HLK'" as (yn1 yn2 zn1 zn2)
                                    "[% [% [#Hrel HLK'']]]"; simplify_eq/=.
         (* Now we have proven that specification can also pop. *)
-        iMod (steps_CG_locked_pop_suc with "[$Hs $Hst' $Hl $Hj]")
+        iMod (steps_CG_locked_pop_suc with "Hs Hst' Hl Hj")
                 as "[Hj [Hst' Hl]]"; first solve_ndisj.
         iMod ("Hclose" with "[-Hj]") as "_".
         { iNext. 
@@ -297,7 +296,7 @@ Section Stack_refinement.
     iApply fupd_wp.
     tp_tlam j.
     tp_bind j newlock.
-    iMod (steps_newlock with "[$Hj]") as (l') "[Hj Hl']"; eauto.
+    iMod (steps_newlock with "Hspec Hj") as (l') "[Hj Hl']"; eauto.
     tp_normalise j.
     tp_rec j.
     tp_alloc j as stk' "Hstk'".