Improve heap_lang notations a bit.

heap_lang/sugar.v

@@ -32,7 +32,7 @@ Module notations.
   (* What about Arguments for hoare triples?. *)
   (* The colons indicate binders. "let" is not consistent here though,
      thing are only bound in the "in". *)
-  Notation "# n" := (Var n) (at level 1, format "# n") : lang_scope.
+  Notation "# n" := (ids (term:=expr) n) (at level 1, format "# n") : lang_scope.
   Notation "! e" := (Load e%L) (at level 10, format "! e") : lang_scope.
   Notation "'ref' e" := (Alloc e%L) (at level 30) : lang_scope.
   Notation "e1 + e2" := (Plus e1%L e2%L)
@@ -41,12 +41,14 @@ Module notations.
   Notation "e1 < e2" := (Lt e1%L e2%L) (at level 70) : lang_scope.
   (* The unicode ← is already part of the notation "_ ← _; _" for bind. *)
   Notation "e1 <- e2" := (Store e1%L e2%L) (at level 80) : lang_scope.
-  Notation "e1 ; e2" := (Seq e1%L e2%L) (at level 100) : lang_scope.
-  Notation "'let:' e1 'in' e2" := (Let e1%L e2%L) (at level 102) : lang_scope.
-  Notation "'λ:' e" := (Lam e%L) (at level 102) : lang_scope.
-  Notation "'rec::' e" := (Rec e%L) (at level 102) : lang_scope.
+  Notation "e1 ; e2" := (Seq e1%L e2%L)
+    (at level 100, e2 at level 200) : lang_scope.
+  Notation "'let:' e1 'in' e2" := (Let e1%L e2%L)
+    (at level 102, e2 at level 200) : lang_scope.
+  Notation "'λ:' e" := (Lam e%L) (at level 102, e at level 200) : lang_scope.
+  Notation "'rec::' e" := (Rec e%L) (at level 102, e at level 200) : lang_scope.
   Notation "'if' e1 'then' e2 'else' e3" := (If e1%L e2%L e3%L)
-    (at level 200, e1, e2, e3 at level 200, only parsing) : lang_scope.
+    (at level 200, e1, e2, e3 at level 200) : lang_scope.
 End notations.
 
 Section suger.

heap_lang/tests.v

@@ -54,16 +54,15 @@ Module LiftingTests.
   (* TODO: once asimpl preserves notation, we don't need
      FindPred' anymore. *)
   (* FIXME: fix notation so that we do not need parenthesis or %L *)
-  Definition FindPred' n1 Sn1 n2 f : expr :=
+  Definition FindPred' (n1 Sn1 n2 f : expr) : expr :=
     if Sn1 < n2 then f Sn1 else n1.
-  Definition FindPred n2 : expr :=
-    rec:: (let: #1 + 1 in FindPred' #2 #0 n2.[ren(+3)] #1)%L.
+  Definition FindPred (n2 : expr) : expr :=
+    rec:: let: #1 + 1 in FindPred' #2 #0 n2.[ren(+3)] #1.
   Definition Pred : expr :=
-    λ: (if #0 ≤ 0 then 0 else FindPred #0 0)%L.
+    λ: if #0 ≤ 0 then 0 else FindPred #0 0.
 
   Lemma FindPred_spec n1 n2 E Q :
-    (■(n1 < n2) ∧ Q (pred n2)) ⊑
-       wp E (FindPred n2 n1) Q.
+    (■ (n1 < n2) ∧ Q (pred n2)) ⊑ wp E (FindPred n2 n1) Q.
   Proof.
     revert n1. apply löb_all_1=>n1.
     rewrite -wp_rec //. asimpl.
@@ -71,8 +70,10 @@ Module LiftingTests.
     etransitivity; first (etransitivity; last eapply equiv_spec, later_and).
     { apply and_mono; first done. by rewrite -later_intro. }
     apply later_mono.
-    (* Go on. *)
-    rewrite -(wp_let _ _ (FindPred' n1 #0 n2 (FindPred n2))).
+    (* "rewrite -(wp_let _ _ (FindPred' n1 #0 n2 (FindPred n2)))" started to
+    fail after we changed # n to use ids instead of Var *)
+    pose proof (wp_let (Σ:=Σ) E (n1 + 1)%L (FindPred' n1 #0 n2 (FindPred n2)) Q).
+    unfold Let, Lam in H; rewrite -H.
     rewrite -wp_plus. asimpl.
     rewrite -(wp_bindi (CaseCtx _ _)).
     rewrite -!later_intro /=.