Some clean up.

heap_lang/lifting.v

@@ -17,15 +17,10 @@ Lemma wp_bind {E e} K Φ :
   WP e @ E {{ v, WP fill K (of_val v) @ E {{ Φ }} }} ⊢ WP fill K e @ E {{ Φ }}.
 Proof. exact: wp_ectx_bind. Qed.
 
-Lemma wp_bindi {E e} Ki Φ :
-  WP e @ E {{ v, WP fill_item Ki (of_val v) @ E {{ Φ }} }} ⊢
-     WP fill_item Ki e @ E {{ Φ }}.
-Proof. exact: weakestpre.wp_bind. Qed.
-
 (** Base axioms for core primitives of the language: Stateful reductions. *)
 Lemma wp_alloc_pst E σ e v Φ :
   to_val e = Some v →
-  (▷ ownP σ ★ ▷ (∀ l, σ !! l = None ∧ ownP (<[l:=v]>σ) -★ Φ (LitV (LitLoc l))))
+  ▷ ownP σ ★ ▷ (∀ l, σ !! l = None ∧ ownP (<[l:=v]>σ) -★ Φ (LitV (LitLoc l)))
   ⊢ WP Alloc e @ E {{ Φ }}.
 Proof.
   iIntros {?}  "[HP HΦ]".
@@ -41,7 +36,7 @@ Qed.
 
 Lemma wp_load_pst E σ l v Φ :
   σ !! l = Some v →
-  (▷ ownP σ ★ ▷ (ownP σ -★ Φ v)) ⊢ WP Load (Lit (LitLoc l)) @ E {{ Φ }}.
+  ▷ ownP σ ★ ▷ (ownP σ -★ Φ v) ⊢ WP Load (Lit (LitLoc l)) @ E {{ Φ }}.
 Proof.
   intros. rewrite -(wp_lift_atomic_det_head_step σ v σ None) ?right_id //;
     last (by intros; inv_head_step; eauto using to_of_val); simpl; by eauto.
@@ -49,7 +44,7 @@ Qed.
 
 Lemma wp_store_pst E σ l e v v' Φ :
   to_val e = Some v → σ !! l = Some v' →
-  (▷ ownP σ ★ ▷ (ownP (<[l:=v]>σ) -★ Φ (LitV LitUnit)))
+  ▷ ownP σ ★ ▷ (ownP (<[l:=v]>σ) -★ Φ (LitV LitUnit))
   ⊢ WP Store (Lit (LitLoc l)) e @ E {{ Φ }}.
 Proof.
   intros. rewrite-(wp_lift_atomic_det_head_step σ (LitV LitUnit) (<[l:=v]>σ) None)
@@ -58,7 +53,7 @@ Qed.
 
 Lemma wp_cas_fail_pst E σ l e1 v1 e2 v2 v' Φ :
   to_val e1 = Some v1 → to_val e2 = Some v2 → σ !! l = Some v' → v' ≠ v1 →
-  (▷ ownP σ ★ ▷ (ownP σ -★ Φ (LitV $ LitBool false)))
+  ▷ ownP σ ★ ▷ (ownP σ -★ Φ (LitV $ LitBool false))
   ⊢ WP CAS (Lit (LitLoc l)) e1 e2 @ E {{ Φ }}.
 Proof.
   intros. rewrite -(wp_lift_atomic_det_head_step σ (LitV $ LitBool false) σ None)
@@ -68,7 +63,7 @@ Qed.
 
 Lemma wp_cas_suc_pst E σ l e1 v1 e2 v2 Φ :
   to_val e1 = Some v1 → to_val e2 = Some v2 → σ !! l = Some v1 →
-  (▷ ownP σ ★ ▷ (ownP (<[l:=v2]>σ) -★ Φ (LitV $ LitBool true)))
+  ▷ ownP σ ★ ▷ (ownP (<[l:=v2]>σ) -★ Φ (LitV $ LitBool true))
   ⊢ WP CAS (Lit (LitLoc l)) e1 e2 @ E {{ Φ }}.
 Proof.
   intros. rewrite -(wp_lift_atomic_det_head_step σ (LitV $ LitBool true)
@@ -78,7 +73,7 @@ Qed.
 
 (** Base axioms for core primitives of the language: Stateless reductions *)
 Lemma wp_fork E e Φ :
-  (▷ Φ (LitV LitUnit) ★ ▷ WP e {{ _, True }}) ⊢ WP Fork e @ E {{ Φ }}.
+  ▷ Φ (LitV LitUnit) ★ ▷ WP e {{ _, True }} ⊢ WP Fork e @ E {{ Φ }}.
 Proof.
   rewrite -(wp_lift_pure_det_head_step (Fork e) (Lit LitUnit) (Some e)) //=;
     last by intros; inv_head_step; eauto.
@@ -88,8 +83,7 @@ Qed.
 Lemma wp_rec E f x erec e1 e2 v2 Φ :
   e1 = Rec f x erec →
   to_val e2 = Some v2 →
-  ▷ WP subst' x e2 (subst' f e1 erec) @ E {{ Φ }}
-  ⊢ WP App e1 e2 @ E {{ Φ }}.
+  ▷ WP subst' x e2 (subst' f e1 erec) @ E {{ Φ }} ⊢ WP App e1 e2 @ E {{ Φ }}.
 Proof.
   intros -> ?. rewrite -(wp_lift_pure_det_head_step (App _ _)
     (subst' x e2 (subst' f (Rec f x erec) erec)) None) //= ?right_id;