Notations for LitTrue and LitFalse.

heap_lang/heap_lang.v

@@ -7,6 +7,8 @@ Definition loc := positive. (* Really, any countable type. *)
 
 Inductive base_lit : Set :=
   | LitNat (n : nat) | LitBool (b : bool) | LitUnit.
+Notation LitTrue := (LitBool true).
+Notation LitFalse := (LitBool false).
 Inductive un_op : Set :=
   | NegOp.
 Inductive bin_op : Set :=
@@ -177,9 +179,9 @@ Inductive head_step : expr -> state -> expr -> state -> option expr -> Prop :=
      bin_op_eval op l1 l2 = Some l' → 
      head_step (BinOp op (Lit l1) (Lit l2)) σ (Lit l') σ None
   | IfTrueS e1 e2 σ :
-     head_step (If (Lit (LitBool true)) e1 e2) σ e1 σ None
+     head_step (If (Lit LitTrue) e1 e2) σ e1 σ None
   | IfFalseS e1 e2 σ :
-     head_step (If (Lit (LitBool false)) e1 e2) σ e2 σ None
+     head_step (If (Lit LitFalse) e1 e2) σ e2 σ None
   | FstS e1 v1 e2 v2 σ :
      to_val e1 = Some v1 → to_val e2 = Some v2 →
      head_step (Fst (Pair e1 e2)) σ e1 σ None
@@ -206,11 +208,11 @@ Inductive head_step : expr -> state -> expr -> state -> option expr -> Prop :=
   | CasFailS l e1 v1 e2 v2 vl σ :
      to_val e1 = Some v1 → to_val e2 = Some v2 →
      σ !! l = Some vl → vl ≠ v1 →
-     head_step (Cas (Loc l) e1 e2) σ (Lit (LitBool false)) σ None
+     head_step (Cas (Loc l) e1 e2) σ (Lit LitFalse) σ None
   | CasSucS l e1 v1 e2 v2 σ :
      to_val e1 = Some v1 → to_val e2 = Some v2 →
      σ !! l = Some v1 →
-     head_step (Cas (Loc l) e1 e2) σ (Lit (LitBool true)) (<[l:=v2]>σ) None.
+     head_step (Cas (Loc l) e1 e2) σ (Lit LitTrue) (<[l:=v2]>σ) None.
 
 (** Atomic expressions *)
 Definition atomic (e: expr) : Prop :=

heap_lang/lifting.v

@@ -56,23 +56,24 @@ Qed.
 
 Lemma wp_cas_fail_pst E σ l e1 v1 e2 v2 v' Q :
   to_val e1 = Some v1 → to_val e2 = Some v2 → σ !! l = Some v' → v' ≠ v1 →
-  (ownP σ ★ ▷ (ownP σ -★ Q $ LitV $ LitBool false)) ⊑ wp E (Cas (Loc l) e1 e2) Q.
+  (ownP σ ★ ▷ (ownP σ -★ Q (LitV LitFalse))) ⊑ wp E (Cas (Loc l) e1 e2) Q.
 Proof.
-  intros. rewrite -(wp_lift_atomic_det_step σ (LitV $ LitBool false) σ None) ?right_id //;
-    last by intros; inv_step; eauto.
+  intros. rewrite -(wp_lift_atomic_det_step σ (LitV LitFalse) σ None)
+    ?right_id //; last by intros; inv_step; eauto.
 Qed.
 
 Lemma wp_cas_suc_pst E σ l e1 v1 e2 v2 Q :
   to_val e1 = Some v1 → to_val e2 = Some v2 → σ !! l = Some v1 →
-  (ownP σ ★ ▷ (ownP (<[l:=v2]>σ) -★ Q $ LitV $ LitBool true)) ⊑ wp E (Cas (Loc l) e1 e2) Q.
+  (ownP σ ★ ▷ (ownP (<[l:=v2]>σ) -★ Q (LitV LitTrue)))
+  ⊑ wp E (Cas (Loc l) e1 e2) Q.
 Proof.
-  intros. rewrite -(wp_lift_atomic_det_step σ (LitV $ LitBool true) (<[l:=v2]>σ) None)
+  intros. rewrite -(wp_lift_atomic_det_step σ (LitV LitTrue) (<[l:=v2]>σ) None)
     ?right_id //; last by intros; inv_step; eauto.
 Qed.
 
 (** Base axioms for core primitives of the language: Stateless reductions *)
 Lemma wp_fork E e :
-  ▷ wp (Σ:=Σ) coPset_all e (λ _, True) ⊑ wp E (Fork e) (λ v, ■(v = LitV $ LitUnit)).
+  ▷ wp (Σ:=Σ) coPset_all e (λ _, True) ⊑ wp E (Fork e) (λ v, ■(v = LitV LitUnit)).
 Proof.
   rewrite -(wp_lift_pure_det_step (Fork e) (Lit LitUnit) (Some e)) //=;
     last by intros; inv_step; eauto.
@@ -107,15 +108,13 @@ Proof.
   by rewrite -wp_value'.
 Qed.
 
-Lemma wp_if_true E e1 e2 Q :
-  ▷ wp E e1 Q ⊑ wp E (If (Lit $ LitBool true) e1 e2) Q.
+Lemma wp_if_true E e1 e2 Q : ▷ wp E e1 Q ⊑ wp E (If (Lit LitTrue) e1 e2) Q.
 Proof.
   rewrite -(wp_lift_pure_det_step (If _ _ _) e1 None)
     ?right_id //; last by intros; inv_step; eauto.
 Qed.
 
-Lemma wp_if_false E e1 e2 Q :
-  ▷ wp E e2 Q ⊑ wp E (If (Lit $ LitBool false) e1 e2) Q.
+Lemma wp_if_false E e1 e2 Q : ▷ wp E e2 Q ⊑ wp E (If (Lit LitFalse) e1 e2) Q.
 Proof.
   rewrite -(wp_lift_pure_det_step (If _ _ _) e2 None)
     ?right_id //; last by intros; inv_step; eauto.
@@ -123,7 +122,7 @@ Qed.
 
 Lemma wp_fst E e1 v1 e2 v2 Q :
   to_val e1 = Some v1 → to_val e2 = Some v2 →
-  ▷Q v1 ⊑ wp E (Fst (Pair e1 e2)) Q.
+  ▷ Q v1 ⊑ wp E (Fst (Pair e1 e2)) Q.
 Proof.
   intros. rewrite -(wp_lift_pure_det_step (Fst _) e1 None) ?right_id //;
     last by intros; inv_step; eauto.