Better notation for type application.

theories/base.v

@@ -6,7 +6,7 @@ From iris.heap_lang.lib Require Export assert.
 not have them. *)
 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 "e !" := (App e%E #()) (at level 10, only parsing) : expr_scope.
+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/compatibility.v

@@ -92,7 +92,7 @@ Section compatibility.
     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/parametricity.v

@@ -8,7 +8,7 @@ Section parametricity.
 
   Lemma exercise5 `{!heapPreG Σ} e (v : val) σ w es σ' :
     (∀ `{heapG Σ}, ∅ ⊨ e : ∀ A, A → A) →
-    rtc erased_step ([e ! v]%E, σ) (of_val w :: es, σ') → w = v.
+    rtc erased_step ([e <_> v]%E, σ) (of_val w :: es, σ') → w = v.
   (* REMOVE *) Proof.
     intros He.
     apply sem_gen_type_safety with (φ := λ u, u = v)=> ?.
@@ -27,10 +27,10 @@ Section parametricity.
 
   Lemma exercise6 `{!heapPreG Σ} e (v : val) σ w es σ' :
     (∀ `{heapG Σ}, ∅ ⊨ e : ∀ A, A) →
-    ¬ rtc erased_step ([e !]%E, σ) (of_val w :: es, σ').
+    rtc erased_step ([e <_>]%E, σ) (of_val w :: es, σ') →
+    False.
   (* REMOVE *) Proof.
     intros He.
-    unfold not.
     change False with ((λ _, False) w).
     apply sem_gen_type_safety with (φ := λ _, False)=> ?.
     exists (exercise6_sem_ty Σ). split.

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