Simplify stack programs

theories/logrel/F_mu_ref_conc/examples/lock.v

@@ -7,20 +7,26 @@ Definition newlock : expr := Alloc (#♭ false).
 Definition acquire : expr :=
   Rec (If (CAS (Var 1) (#♭ false) (#♭ true)) (Unit) (App (Var 0) (Var 1))).
 (** [release = λ x. x <- false] *)
-Definition release : expr := Rec (Store (Var 1) (#♭ false)).
+Definition release : expr := Lam (Store (Var 0) (#♭ false)).
 (** [with_lock e l = λ x. (acquire l) ;; e x ;; (release l)] *)
 Definition with_lock (e : expr) (l : expr) : expr :=
-  Rec
-    (App (Rec (App (Rec (App (Rec (Var 3)) (App release l.[ren (+6)])))
-                   (App e.[ren (+4)] (Var 3))))
-         (App acquire l.[ren (+2)])
+  Lam
+    (Seq
+       (App acquire l.[ren (+1)])
+       (LetIn
+          (App e.[ren (+1)] (Var 0))
+          (Seq (App release l.[ren (+2)]) (Var 0))
+       )
     ).
 
 Definition with_lockV (e l : expr) : val :=
-  RecV
-    (App (Rec (App (Rec (App (Rec (Var 3)) (App release l.[ren (+6)])))
-                   (App e.[ren (+4)] (Var 3))))
-         (App acquire l.[ren (+2)])
+  LamV
+    (Seq
+       (App acquire l.[ren (+1)])
+       (LetIn
+          (App e.[ren (+1)] (Var 0))
+          (Seq (App release l.[ren (+2)]) (Var 0))
+       )
     ).
 
 Lemma with_lock_to_val e l : to_val (with_lock e l) = Some (with_lockV e l).
@@ -29,6 +35,7 @@ Proof. trivial. Qed.
 Lemma with_lock_of_val e l : of_val (with_lockV e l) = with_lock e l.
 Proof. trivial. Qed.
 
+Global Typeclasses Opaque with_lockV.
 Global Opaque with_lockV.
 
 Lemma newlock_closed f : newlock.[f] = newlock.
@@ -43,15 +50,10 @@ Lemma release_closed f : release.[f] = release.
 Proof. by asimpl. Qed.
 Hint Rewrite release_closed : autosubst.
 
-Lemma with_lock_subst (e l : expr) f :  (with_lock e l).[f] = with_lock e.[f] l.[f].
+Lemma with_lock_subst (e l : expr) f :
+  (with_lock e l).[f] = with_lock e.[f] l.[f].
 Proof. unfold with_lock; asimpl; trivial. Qed.
 
-Lemma with_lock_closed e l:
-  (∀ f : var → expr, e.[f] = e) →
-  (∀ f : var → expr, l.[f] = l) →
-  ∀ f, (with_lock e l).[f] = with_lock e l.
-Proof. asimpl => H1 H2 f. unfold with_lock. by rewrite ?H1 ?H2. Qed.
-
 Definition LockType := Tref TBool.
 
 Lemma newlock_type Γ : typed Γ newlock LockType.
@@ -68,18 +70,20 @@ Lemma with_lock_type e l Γ τ τ' :
   typed Γ l LockType →
   typed Γ (with_lock e l) (TArrow τ τ').
 Proof.
-  intros H1 H2. do 3 econstructor; eauto.
-  - repeat (econstructor; eauto using release_type).
-    + eapply (context_weakening [_; _; _; _; _; _]); eauto.
-    + eapply (context_weakening [_; _; _; _]); eauto.
-  - eapply acquire_type.
-  - eapply (context_weakening [_; _]); eauto.
+  intros ??.
+  do 3 econstructor; eauto using acquire_type.
+  - eapply (context_weakening [_]); eauto.
+  - econstructor; eauto using typed.
+    eapply (context_weakening [_]); eauto.
+  - econstructor; eauto using typed.
+    econstructor; eauto using release_type.
+    eapply (context_weakening [_;_]); eauto.
 Qed.
 
 Section proof.
   Context `{cfgSG Σ}.
   Context `{heapIG Σ}.
-  
+
   Lemma steps_newlock E ρ j K :
     nclose specN ⊆ E →
     spec_ctx ρ ∗ j ⤇ fill K newlock
@@ -89,6 +93,7 @@ Section proof.
     by iMod (step_alloc _ _ j K with "[Hj]") as "Hj"; eauto.
   Qed.
 
+  Global Typeclasses Opaque newlock.
   Global Opaque newlock.
 
   Lemma steps_acquire E ρ j K l :
@@ -107,6 +112,7 @@ Section proof.
     Unshelve. all:trivial.
   Qed.
 
+  Global Typeclasses Opaque acquire.
   Global Opaque acquire.
 
   Lemma steps_release E ρ j K l b:
@@ -115,48 +121,45 @@ Section proof.
       ⊢ |={E}=> j ⤇ fill K Unit ∗ l ↦ₛ (#♭v false).
   Proof.
     iIntros (HNE) "[#Hspec [Hl Hj]]". unfold release.
-    iMod (step_rec _ _ j K with "[Hj]") as "Hj"; eauto; try done.
-    iMod (step_store _ _ j K _ _ _ _ _ with "[Hj Hl]") as "[Hj Hl]"; eauto.
-    { by iFrame. }
+    iMod (do_step_pure with "[$Hj]") as "Hj"; eauto; try done.
+    iMod (step_store with "[$Hj $Hl]") as "[Hj Hl]"; eauto.
     by iIntros "!> {$Hj $Hl}".
-    Unshelve. all: trivial.
   Qed.
 
+  Global Typeclasses Opaque release.
   Global Opaque release.
 
   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 *) →
+    (* (∀ 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).
   Proof.
-    iIntros (HNE H1 H2) "[#Hspec [HP [Hl Hj]]]".
-    iMod (step_rec _ _ j K _ _ _ _ with "[Hj]") as "Hj"; eauto.
-    iAsimpl. rewrite H1.
-    iMod (steps_acquire _ _ j ((AppRCtx (RecV _)) :: K)
-                   _ _ with "[Hj Hl]") as "[Hj Hl]"; eauto.
-    { simpl. iFrame "Hspec Hj Hl"; eauto. }
-    simpl.
-    iMod (step_rec _ _ j K _ _ _ _ with "[Hj]") as "Hj"; eauto.
-    iAsimpl. rewrite H1.
-    iMod (H2 ((AppRCtx (RecV _)) :: K) with "[Hj HP]") as "[Hj HQ]"; eauto.
-    { simpl. iFrame "Hspec Hj HP"; eauto. }
+    iIntros (HNE He) "[#Hspec [HP [Hl Hj]]]".
+    iMod (do_step_pure with "[$Hj]") as "Hj"; eauto.
+    iAsimpl.
+    iMod (steps_acquire _ _ j (SeqCtx _ :: K) with "[$Hj Hl]") as "[Hj Hl]";
+      auto. simpl.
+    iMod (do_step_pure with "[$Hj]") as "Hj"; eauto.
+    iMod (He (LetInCtx _ :: K) with "[$Hj HP]") as "[Hj HQ]"; eauto.
     simpl.
-    iMod (step_rec _ _ j K _ _ _ _ with "[Hj]") as "Hj"; eauto.
+    iMod (do_step_pure with "[$Hj]") as "Hj"; eauto.
     iAsimpl.
-    iMod (steps_release _ _ j ((AppRCtx (RecV _)) :: K) _ _ with "[Hj Hl]")
+    iMod (steps_release _ _ j (SeqCtx _ :: K) _ _ with "[$Hj $Hl]")
       as "[Hj Hl]"; eauto.
-    { simpl. by iFrame. }
-    rewrite ?fill_app /=.
-    iMod (step_rec _ _ j K _ _ _ _ with "[Hj]") as "Hj"; eauto.
-    iAsimpl. iModIntro; by iFrame.
-    Unshelve.
-    all: try match goal with |- to_val _ = _ => auto using to_of_val end.
-    trivial.
+    simpl.
+    iMod (do_step_pure with "[$Hj]") as "Hj"; eauto.
+    iModIntro; by iFrame.
   Qed.
 
+  Global Typeclasses Opaque with_lock.
   Global Opaque with_lock.
 End proof.
+
+Global Hint Rewrite newlock_closed : autosubst.
+Global Hint Rewrite acquire_closed : autosubst.
+Global Hint Rewrite release_closed : autosubst.
+Global Hint Rewrite with_lock_subst : autosubst.

theories/logrel/F_mu_ref_conc/examples/stack/CG_stack.v

@@ -7,33 +7,34 @@ Definition CG_StackType τ :=
 
 (* Coarse-grained push *)
 Definition CG_push (st : expr) : expr :=
-  Rec (Store
-         (st.[ren (+2)]) (Fold (InjR (Pair (Var 1) (Load st.[ren (+ 2)]))))).
+  Lam (Store (st.[ren (+1)]) (Fold (InjR (Pair (Var 0) (Load st.[ren (+ 1)]))))).
 
 Definition CG_locked_push (st l : expr) := with_lock (CG_push st) l.
 Definition CG_locked_pushV (st l : expr) : val := with_lockV (CG_push st) l.
 
 Definition CG_pop (st : expr) : expr :=
-  Rec (Case (Unfold (Load st.[ren (+ 2)]))
+  Lam (Case (Unfold (Load st.[ren (+ 1)]))
             (InjL Unit)
             (
-              App (Rec (InjR (Fst (Var 2))))
-                  (Store st.[ren (+ 3)] (Snd (Var 0)))
+              Seq
+                (Store st.[ren (+ 2)] (Snd (Var 0)))
+                (InjR (Fst (Var 0)))
             )
       ).
 
 Definition CG_locked_pop (st l : expr) := with_lock (CG_pop st) l.
 Definition CG_locked_popV (st l : expr) : val := with_lockV (CG_pop st) l.
 
-Definition CG_snap (st l : expr) :=  with_lock (Rec (Load st.[ren (+2)])) l.
-Definition CG_snapV (st l : expr) : val := with_lockV (Rec (Load st.[ren (+2)])) l.
+Definition CG_snap (st l : expr) :=  with_lock (Lam (Load st.[ren (+1)])) l.
+Definition CG_snapV (st l : expr) : val := with_lockV (Lam (Load st.[ren (+1)])) l.
 
 Definition CG_iter (f : expr) : expr :=
   Rec (Case (Unfold (Var 1))
             Unit
             (
-              App (Rec (App (Var 3) (Snd (Var 2))))
-                  (App f.[ren (+3)] (Fst (Var 0)))
+              Seq
+                (App f.[ren (+3)] (Fst (Var 0)))
+                (App (Var 1) (Snd (Var 0)))
             )
       ).
 
@@ -41,20 +42,26 @@ Definition CG_iterV (f : expr) : val :=
   RecV (Case (Unfold (Var 1))
             Unit
             (
-              App (Rec (App (Var 3) (Snd (Var 2))))
-                  (App f.[ren (+3)] (Fst (Var 0)))
+              Seq
+                (App f.[ren (+3)] (Fst (Var 0)))
+                (App (Var 1) (Snd (Var 0)))
             )
       ).
 
 Definition CG_snap_iter (st l : expr) : expr :=
-  Rec (App (CG_iter (Var 1)) (App (CG_snap st.[ren (+2)] l.[ren (+2)]) Unit)).
+  Lam (App (CG_iter (Var 0)) (App (CG_snap st.[ren (+1)] l.[ren (+1)]) Unit)).
+
 Definition CG_stack_body (st l : expr) : expr :=
   Pair (Pair (CG_locked_push st l) (CG_locked_pop st l))
        (CG_snap_iter st l).
 
 Definition CG_stack : expr :=
-  TLam (App (Rec (App (Rec (CG_stack_body (Var 1) (Var 3)))
-                (Alloc (Fold (InjL Unit))))) newlock).
+  TLam (
+      LetIn
+        newlock
+        (LetIn
+           (Alloc (Fold (InjL Unit)))
+           (CG_stack_body (Var 0) (Var 1)))).
 
 Section CG_Stack.
   Context `{heapIG Σ, cfgSG Σ}.
@@ -64,37 +71,33 @@ Section CG_Stack.
     typed Γ (CG_push st) (TArrow τ TUnit).
   Proof.
     intros H1. repeat econstructor.
-    eapply (context_weakening [_; _]); eauto.
+    eapply (context_weakening [_]); eauto.
     repeat constructor; asimpl; trivial.
-    eapply (context_weakening [_; _]); eauto.
+    eapply (context_weakening [_]); eauto.
   Qed.
 
-  Lemma CG_push_closed (st : expr) :
-    (∀ f, st.[f] = st) → ∀ f, (CG_push st).[f] = CG_push st.
-  Proof. intros Hst f. unfold CG_push. asimpl. rewrite ?Hst; trivial. Qed.
-
   Lemma CG_push_subst (st : expr) f : (CG_push st).[f] = CG_push st.[f].
   Proof. unfold CG_push; asimpl; trivial. Qed.
 
+  Global Hint Rewrite CG_push_subst : autosubst.
+
   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)).
   Proof.
     intros HNE. iIntros "[#Hspec [Hx Hj]]". unfold CG_push.
-    iMod (step_rec _ _ j K _ _ _ _ with "[Hj]") as "Hj"; eauto.
+    iMod (do_step_pure with "[$Hj]") as "Hj"; eauto.
     iAsimpl.
     iMod (step_load _ _ j (PairRCtx _ :: InjRCtx :: FoldCtx :: StoreRCtx (LocV _) :: K)
-                    _ _ _ with "[Hj Hx]") as "[Hj Hx]"; eauto.
-    simpl. iFrame "Hspec Hj"; trivial. simpl.
-    iMod (step_store _ _ j K _ _ _ _ _ with "[Hj Hx]") as "[Hj Hx]"; eauto.
-    { by iFrame. }
+            with "[$Hj $Hx]") as "[Hj Hx]"; eauto.
+    simpl.
+    iMod (step_store _ _ j K with "[$Hj $Hx]") as "[Hj Hx]"; eauto.
+    { rewrite /= !to_of_val //. }
     iModIntro. by iFrame.
-    Unshelve.
-    all: try match goal with |- to_val _ = _ => auto using to_of_val end.
-    simpl; by rewrite ?to_of_val.
   Qed.
 
+  Global Typeclasses Opaque CG_push.
   Global Opaque CG_push.
 
   Lemma CG_locked_push_to_val st l :
@@ -106,7 +109,6 @@ Section CG_Stack.
   Proof. trivial. Qed.
 
   Global Opaque CG_locked_pushV.
-
   Lemma CG_locked_push_type st l Γ τ :
     typed Γ st (Tref (CG_StackType τ)) →
     typed Γ l LockType →
@@ -116,17 +118,11 @@ Section CG_Stack.
     eapply with_lock_type; auto using CG_push_type.
   Qed.
 
-  Lemma CG_locked_push_closed (st l : expr) :
-    (∀ f, st.[f] = st) → (∀ f, l.[f] = l) →
-    ∀ f, (CG_locked_push st l).[f] = CG_locked_push st l.
-  Proof.
-    intros H1 H2 f. asimpl. unfold CG_locked_push.
-    rewrite with_lock_closed; trivial. apply CG_push_closed; trivial.
-  Qed.
-
   Lemma CG_locked_push_subst (st l : expr) f :
     (CG_locked_push st l).[f] = CG_locked_push st.[f] l.[f].
-  Proof. by rewrite with_lock_subst CG_push_subst. Qed.
+  Proof. by rewrite /CG_locked_push; asimpl. Qed.
+
+  Hint Rewrite CG_locked_push_subst : autosubst.
 
   Lemma steps_CG_locked_push E ρ j K st w v l :
     nclose specN ⊆ E →
@@ -135,14 +131,13 @@ Section CG_Stack.
     ⊢ |={E}=> j ⤇ fill K Unit ∗ st ↦ₛ FoldV (InjRV (PairV w v)) ∗ l ↦ₛ (#♭v false).
   Proof.
     intros HNE. iIntros "[#Hspec [Hx [Hl Hj]]]". unfold CG_locked_push.
-    iMod (steps_with_lock
-            _ _ j K _ _ _ _ UnitV _ _ _ with "[Hj Hx Hl]") as "Hj"; last done.
-    - iIntros (K') "[#Hspec Hxj]".
-      iApply steps_CG_push; first done. iFrame. trivial.
-    - iFrame "Hspec Hj Hx"; trivial.
-      Unshelve. all: trivial.
+    iMod (steps_with_lock _ _ _ _ _ _ (st ↦ₛ v)%I _ UnitV with "[$Hj $Hx $Hl]")
+      as "Hj"; auto.
+    iIntros (K') "[#Hspec Hxj]".
+    iApply steps_CG_push; first done. by iFrame.
   Qed.
 
+  Global Typeclasses Opaque CG_locked_push.
   Global Opaque CG_locked_push.
 
   (* Coarse-grained pop *)
@@ -152,22 +147,22 @@ Section CG_Stack.
   Proof.
     intros H1.
     econstructor.
-    eapply (Case_typed _ _ _ _ TUnit);
-      [| repeat constructor
-       | repeat econstructor; eapply (context_weakening [_; _; _]); eauto].
-    replace (TSum TUnit (TProd τ (CG_StackType τ))) with
-    ((TSum TUnit (TProd τ.[ren (+1)] (TVar 0))).[(CG_StackType τ)/])
-      by (by asimpl).
-    repeat econstructor.
-    eapply (context_weakening [_; _]); eauto.
+    eapply (Case_typed _ _ _ _ TUnit); eauto using typed.
+    - replace (TSum TUnit (TProd τ (CG_StackType τ))) with
+          ((TSum TUnit (TProd τ.[ren (+1)] (TVar 0))).[(CG_StackType τ)/])
+        by (by asimpl).
+      repeat econstructor.
+      eapply (context_weakening [_]); eauto.
+    - econstructor; eauto using typed.
+      econstructor; eauto using typed.
+      eapply (context_weakening [_; _]); eauto.
   Qed.
 
-  Lemma CG_pop_closed (st : expr) :
-    (∀ f, st.[f] = st) → ∀ f, (CG_pop st).[f] = CG_pop st.
-  Proof. intros Hst f. unfold CG_pop. asimpl. rewrite ?Hst; trivial. Qed.
+  Lemma CG_pop_subst (st : expr) f :
+    (CG_pop st).[f] = CG_pop st.[f].
+  Proof. by rewrite /CG_pop; asimpl. Qed.
 
-  Lemma CG_pop_subst (st : expr) f : (CG_pop st).[f] = CG_pop st.[f].
-  Proof. unfold CG_pop; asimpl; trivial. Qed.
+  Hint Rewrite CG_pop_subst : autosubst.
 
   Lemma steps_CG_pop_suc E ρ j K st v w :
     nclose specN ⊆ E →
@@ -176,35 +171,25 @@ Section CG_Stack.
       ⊢ |={E}=> j ⤇ fill K (InjR (of_val w)) ∗ st ↦ₛ v.
   Proof.
     intros HNE. iIntros "[#Hspec [Hx Hj]]". unfold CG_pop.
-    iMod (step_rec _ _ j K _ _ _ _ with "[Hj]") as "Hj"; eauto.
+    iMod (do_step_pure with "[$Hj]") as "Hj"; eauto.
     iAsimpl.
-    iMod (step_load _ _ j (UnfoldCtx :: CaseCtx _ _ :: K)
-                    _ _ _ with "[Hj Hx]") as "[Hj Hx]"; eauto.
-    rewrite ?fill_app. simpl.
-    iFrame "Hspec Hj"; trivial.
-    rewrite ?fill_app. simpl.
-    iMod (step_Fold  _ _ j (CaseCtx _ _ :: K)
-                    _ _ _ _ with "[Hj]") as "Hj"; eauto.
+    iMod (step_load _ _ _ (UnfoldCtx :: CaseCtx _ _ :: K)  with "[$Hj $Hx]")
+      as "[Hj Hx]"; eauto.
     simpl.
-    iMod (step_case_inr _ _ j K _ _ _ _ _ with "[Hj]") as "Hj"; eauto.
-    iAsimpl.
-    iMod (step_snd _ _ j (StoreRCtx (LocV _) :: AppRCtx (RecV _) :: K) _ _ _ _
-                   _ _ with "[Hj]") as "Hj"; eauto.
+    iMod (do_step_pure  _ _ _ (CaseCtx _ _ :: K) with "[$Hj]") as "Hj"; eauto.
     simpl.
-    iMod (step_store _ _ j (AppRCtx (RecV _) :: K) _ _ _ _ _ _
-          with "[Hj Hx]") as "[Hj Hx]"; eauto.
-    rewrite ?fill_app. simpl.
-    iFrame "Hspec Hj"; trivial.
-    rewrite ?fill_app. simpl.
-    iMod (step_rec _ _ j K _ _ _ _ with "[Hj]") as "Hj"; eauto.
+    iMod (do_step_pure with "[$Hj]") as "Hj"; eauto.
     iAsimpl.
-    iMod (step_fst _ _ j (InjRCtx :: K) _ _ _ _ _ _
-          with "[Hj]") as "Hj"; eauto.
+    iMod (do_step_pure _ _ _ (StoreRCtx (LocV _) :: SeqCtx _ :: K)
+            with "[$Hj]") as "Hj"; eauto.
     simpl.
-    iModIntro. iFrame "Hj Hx"; trivial.
-    Unshelve.
-    all: try match goal with |- to_val _ = _ => simpl; by rewrite ?to_of_val end.
-    all: trivial.
+    iMod (step_store _ _ j (SeqCtx _ :: K) with "[$Hj $Hx]") as "[Hj Hx]";
+      eauto using to_of_val.
+    simpl.
+    iMod (do_step_pure with "[$Hj]") as "Hj"; eauto.
+    iMod (do_step_pure _ _ _ (InjRCtx :: K) with "[$Hj]") as "Hj"; eauto.
+    simpl.
+    by iFrame "Hj Hx".
   Qed.
 
   Lemma steps_CG_pop_fail E ρ j K st :
@@ -214,21 +199,17 @@ Section CG_Stack.
       ⊢ |={E}=> j ⤇ fill K (InjL Unit) ∗ st ↦ₛ FoldV (InjLV UnitV).
   Proof.
     iIntros (HNE) "[#Hspec [Hx Hj]]". unfold CG_pop.
-    iMod (step_rec _ _ j K _ _ _ _ with "[Hj]") as "Hj"; eauto.
+    iMod (do_step_pure with "[$Hj]") as "Hj"; eauto.
     iAsimpl.
     iMod (step_load _ _ j (UnfoldCtx :: CaseCtx _ _ :: K)
-                    _ _ _ with "[Hj Hx]") as "[Hj Hx]"; eauto.
-    simpl. iFrame "Hspec Hj"; trivial. simpl.
-    iMod (step_Fold _ _ j (CaseCtx _ _ :: K)
-                    _ _ _ _ with "[Hj]") as "Hj"; eauto.
-    iMod (step_case_inl _ _ j K _ _ _ _ _ with "[Hj]") as "Hj"; eauto.
-    iAsimpl.
-    iModIntro. iFrame "Hj Hx"; trivial.
-    Unshelve.
-    all: try match goal with |- to_val _ = _ => simpl; by rewrite ?to_of_val end.
-    all: trivial.
+                    _ _ _ with "[$Hj $Hx]") as "[Hj Hx]"; eauto.
+    iMod (do_step_pure _ _ _ (CaseCtx _ _ :: K) with "[$Hj]") as "Hj"; eauto.
+    simpl.
+    iMod (do_step_pure with "[$Hj]") as "Hj"; eauto.
+    by iFrame "Hj Hx"; trivial.
   Qed.
 
+  Global Typeclasses Opaque CG_pop.
   Global Opaque CG_pop.
 
   Lemma CG_locked_pop_to_val st l :
@@ -250,17 +231,11 @@ Section CG_Stack.
     eapply with_lock_type; auto using CG_pop_type.
   Qed.
 
-  Lemma CG_locked_pop_closed (st l : expr) :
-    (∀ f, st.[f] = st) → (∀ f, l.[f] = l) →
-    ∀ f, (CG_locked_pop st l).[f] = CG_locked_pop st l.
-  Proof.
-    intros H1 H2 f. asimpl. unfold CG_locked_pop.
-    rewrite with_lock_closed; trivial. apply CG_pop_closed; trivial.
-  Qed.
-
   Lemma CG_locked_pop_subst (st l : expr) f :
-  (CG_locked_pop st l).[f] = CG_locked_pop st.[f] l.[f].
-  Proof. by rewrite with_lock_subst CG_pop_subst. Qed.
+    (CG_locked_pop st l).[f] = CG_locked_pop st.[f] l.[f].
+  Proof. by rewrite /CG_locked_pop; asimpl. Qed.
+
+  Hint Rewrite CG_locked_pop_subst : autosubst.
 
   Lemma steps_CG_locked_pop_suc E ρ j K st v w l :
     nclose specN ⊆ E →
@@ -269,12 +244,11 @@ Section CG_Stack.
       ⊢ |={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 Hxj]".
-      iApply steps_CG_pop_suc; first done. by iFrame.
-    - iFrame "Hspec Hj Hx"; trivial.
-      Unshelve. all: trivial.
+    iMod (steps_with_lock _ _ _ _ _ _ (st ↦ₛ FoldV (InjRV (PairV w v)))%I
+                          _ (InjRV w) UnitV
+            with "[$Hj $Hx $Hl]") as "Hj"; eauto.
+    iIntros (K') "[#Hspec Hxj]".
+    iApply steps_CG_pop_suc; eauto.
   Qed.
 
   Lemma steps_CG_locked_pop_fail E ρ j K st l :
@@ -284,14 +258,14 @@ Section CG_Stack.
       ⊢ |={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 Hxj] /=".
-      iApply steps_CG_pop_fail; first done. by iFrame.
-    - iFrame "Hspec Hj Hx"; trivial.
-      Unshelve. all: trivial.
+    iMod (steps_with_lock _ _ _ _ _ _ (st ↦ₛ FoldV (InjLV UnitV))%I _
+                          (InjLV UnitV) UnitV
+          with "[$Hj $Hx $Hl]") as "Hj"; eauto.
+    iIntros (K') "[#Hspec Hxj] /=".
+      iApply steps_CG_pop_fail; eauto.
   Qed.
 
+  Global Typeclasses Opaque CG_locked_pop.
   Global Opaque CG_locked_pop.
 
   Lemma CG_snap_to_val st l : to_val (CG_snap st l) = Some (CG_snapV st l).
@@ -300,6 +274,7 @@ Section CG_Stack.
   Lemma CG_snap_of_val st l : of_val (CG_snapV st l) = CG_snap st l.
   Proof. trivial. Qed.
 
+  Global Typeclasses Opaque CG_snapV.
   Global Opaque CG_snapV.
 
   Lemma CG_snap_type st l Γ τ :
@@ -309,21 +284,14 @@ Section CG_Stack.
   Proof.
     intros H1 H2. repeat econstructor.
     eapply with_lock_type; trivial. do 2 constructor.
-    eapply (context_weakening [_; _]); eauto.