Tweak some notations.

theories/base.v

@@ -3,9 +3,9 @@ From iris.heap_lang Require Export lang notation metatheory.
 
 (** We model type-level lambdas and applications as thunks since heap_lang does
 not have them. *)
-Notation "Λ: e" := (λ: <>, e)%E (at level 200, only parsing).
+Notation "Λ: e" := (λ: <>, e)%E (at level 200, only parsing) : expr_scope.
 Notation "Λ: e" := (λ: <>, e)%V (at level 200, only parsing) : val_scope.
-Notation "'TApp' e" := (App e%E #()) (at level 200, only parsing).
+Notation "e !" := (App e%E #()) (at level 10, only parsing) : expr_scope.
 
 (** We wrap unpack into an explicitly function so that we can have a nice
 notation for it. *)

theories/sem_typing.v

@@ -191,12 +191,12 @@ Section typed_properties.
       wp_apply (wp_wand with "(H2 [//])"); iIntros (v) "#HAB". by iApply "HAB".
   Qed.
 
-  Lemma sem_typed_tlam Γ e C : (∀ A, Γ ⊨ e : C A) -∗ Γ ⊨ (λ: <>, e) : ∀ A, C A.
+  Lemma sem_typed_tlam Γ e C : (∀ A, Γ ⊨ e : C A) -∗ Γ ⊨ (Λ: e) : ∀ A, C A.
   Proof.
     iIntros "#H" (vs) "!# #HΓ /=". wp_pures.
     iIntros "!#" (A) "/=". wp_pures. by iApply ("H" $! A).
   Qed.
-  Lemma sem_typed_tapp Γ e C A : (Γ ⊨ e : ∀ A, C A) -∗ Γ ⊨ e #() : C A.
+  Lemma sem_typed_tapp Γ e C A : (Γ ⊨ e : ∀ A, C A) -∗ Γ ⊨ e ! : C A.
   Proof.
     iIntros "#H" (vs) "!# #HΓ /=".
     wp_apply (wp_wand with "(H [//])"); iIntros (w) "#HB". by iApply ("HB" $! A).

theories/typing.v

@@ -73,7 +73,7 @@ Inductive typed (Γ : gmap string ty) : expr → ty → Prop :=
      Γ ⊢ₜ (Λ: e) : TForall τ
   | TApp_typed e τ τ' :
      Γ ⊢ₜ e : TForall τ →
-     Γ ⊢ₜ e #() : ty_subst 0 τ' τ
+     Γ ⊢ₜ e ! : ty_subst 0 τ' τ
   | TPack e τ τ' :
      Γ ⊢ₜ e : ty_subst 0 τ' τ →
      Γ ⊢ₜ e : TExist τ