More robust solve_closed tactic

Add a pre_solve_closed ltac that basically changes the goal from
`Closed X e` to `Closed ∅ e`. It is actually OK in practice.

theories/F_mu_ref_conc/reflection.v

@@ -238,7 +238,18 @@ Ltac solve_to_val :=
   end.
 Hint Extern 0 (IntoVal _ _) => solve_to_val : typeclass_instances.
 
+(* TODO:
+DF: this is also kind of a hack.
+Basically, we canno't really solve goals of the form `Closed X e` for an abstract `X`.
+`vm_compute` loops on goals like `bool_decide (x ∈ {[x]} ∪ X)`. *)
+Ltac pre_solve_closed :=
+  lazymatch goal with
+  | [ |- Closed ?X ?e ] =>
+    apply (Closed_mono e ∅ X); [| apply empty_subseteq]
+  end.
+
 Ltac solve_closed :=
+  pre_solve_closed;
   match goal with
   | [ |- Closed ?X ?e] =>
      let e' := R.of_expr e in change (is_closed X (R.to_expr e'));

theories/examples/generative.v

@@ -222,27 +222,6 @@ Section cell_refinement.
   Proof.
     iIntros (Δ).
     unlock cell2 cell1 cellτ.
-    assert (Closed (dom _ Γ) (Pack
-          (λ: "x", (ref #false, ref "x", ref "x", newlock #()),
-          λ: "r",
-            let: "l" := Snd "r" in
-            acquire "l" ;;
-            let: "v" := if: ! (Fst (Fst (Fst "r"))) then
-                        ! (Snd (Fst "r")) else ! (Snd (Fst (Fst "r"))) in
-            release "l" ;; "v",
-          λ: "r" "x",
-            let: "l" := Snd "r" in
-            acquire "l" ;;
-            (if: ! (Fst (Fst (Fst "r")))
-             then Snd (Fst (Fst "r")) <- "x" ;;
-                  Fst (Fst (Fst "r")) <- #false
-             else Snd (Fst "r") <- "x" ;; Fst (Fst (Fst "r")) <- #true) ;;
-            release "l"))%E).
-    { apply Closed_mono with ∅; eauto.
-      set_solver. }
-    assert (Closed (dom _ Γ) (Pack (λ: "x", ref "x", λ: "r", ! "r", λ: "r" "x", "r" <- "x"))).
-    { apply Closed_mono with ∅; eauto.
-      set_solver. }
     iApply bin_log_related_tlam; auto.
     iIntros (R HR) "!#".
     iApply (bin_log_related_pack _ (cellR R)).

theories/examples/stack/helping.v

@@ -368,12 +368,6 @@ Section refinement.
   Proof.
     iIntros (Δ).
     unlock CG_mkstack.
-    assert (Closed (dom (gset string) Γ) (mk_stack #())).
-    { eapply Closed_mono with ∅; eauto. set_solver. }
-    assert (Closed (dom (gset string) Γ) (let: "l" := ref #false in
-      let: "st" := ref Nile in
-      ((CG_locked_pop "st") "l", (CG_locked_push "st") "l"))).
-    { eapply Closed_mono with ∅; eauto. set_solver. }
     rel_tlam_r.
     rel_alloc_r as l' "Hl'". rel_let_r.
     rel_alloc_r as st' "Hst'". rel_let_r.

theories/examples/stack/refinement.v

@@ -19,7 +19,7 @@ Section Stack_refinement.
         ∗ l' ↦ₛ #false)%I.
 
   Context `{stackG Σ}.
-  Definition sinv τi stk stk' l : iProp Σ := 
+  Definition sinv τi stk stk' l : iProp Σ :=
     (∃ (istk : loc) v h, (stack_owns h)
         ∗ stk' ↦ₛ v
         ∗ stk ↦ᵢ (FoldV #istk)
@@ -68,7 +68,7 @@ Section Stack_refinement.
       { iNext. rewrite {2}/sinv. iExists _,_,_.
         iFrame "Hoe Hst' Hst Hl".
         rewrite (StackLink_unfold _ (# nstk, _)).
-        iExists _, _. iSplitR; auto. 
+        iExists _, _. iSplitR; auto.
         iFrame "Hnstk".
         iRight. iExists _, _, _, _. auto. }
       rel_if_true_l.
@@ -248,7 +248,7 @@ replace ((rec: "pop" "st" <> :=
       rel_rec_r.
 
       rel_load_l_atomic.
-      iInv stackN as (istk v h) "[Hoe [Hst' [Hst [#HLK Hl]]]]" "Hclose".      
+      iInv stackN as (istk v h) "[Hoe [Hst' [Hst [#HLK Hl]]]]" "Hclose".
       iDestruct (stack_owns_later_open_close with "Hoe Histk") as "[Histk_i Hoe]".
       iExists _. iFrame "Histk_i".
       iModIntro. iNext. iIntros "Histk_i /=".
@@ -277,7 +277,7 @@ replace ((rec: "pop" "st" <> :=
       rel_rec_l.
       rel_snd_l.
       rel_rec_l.
-      close_sinv "Hclose" "[Hoe Hst' Hst Hl HLK]".      
+      close_sinv "Hclose" "[Hoe Hst' Hst Hl HLK]".
       simpl.
       iApply (bin_log_related_app _ _ _ _ _ _ _ TUnit with "[] [Hτ]"). (* TODO: abbreivate this as related_let *)
       + iApply bin_log_related_arrow; eauto.
@@ -294,7 +294,7 @@ replace ((rec: "pop" "st" <> :=
       + clear.
         iClear "IH Histk HLK_tail HLK HLK'".
         iSpecialize ("Hff" $! (y1,y2) with "Hτ").
-        iApply (related_ret with "[Hff]"). 
+        iApply (related_ret with "[Hff]").
         done.
   Qed.
 
@@ -320,11 +320,6 @@ Section Full_refinement.
            (TArrow (TArrow (TVar 0) TUnit) TUnit)).
   Proof.
     unfold FG_stack. unfold CG_stack.
-    assert (Closed (dom _ Γ) ((λ: "st", FG_stack_body "st") (ref Fold (ref InjL #())))%E).
-    { apply Closed_mono with ∅; last done. solve_closed. }
-    assert (Closed (dom _ Γ) ((λ: "l", (λ: "st", (CG_stack_body "st") "l") (ref Fold (InjL #()))) (ref #false))%E).
-    { apply Closed_mono with ∅; last done. solve_closed. }
-
     iApply bin_log_related_tlam; auto.
     iIntros (τi) "% !#".
     rel_alloc_r as l "Hl".

theories/examples/ticket_lock.v

@@ -283,4 +283,5 @@ Section refinement.
         iApply "Hpool". iExists _,_,_; iFrame. }
       iApply bin_log_related_unit.
   Qed.
+
 End refinement.