add missing file; also add sugar for equality test

barrier/sugar.v

+Require Export barrier.heap_lang barrier.lifting.
+Import uPred.
+
+(** Define some syntactic sugar. LitTrue and LitFalse are defined in heap_lang.v. *)
+Definition Lam (e : {bind expr}) := Rec e.[ren(+1)].
+Definition Let (e1 : expr) (e2: {bind expr}) := App (Lam e2) e1.
+Definition Seq (e1 e2 : expr) := Let e1 e2.[ren(+1)].
+Definition If (e0 e1 e2 : expr) := Case e0 e1.[ren(+1)] e2.[ren(+1)].
+Definition Lt e1 e2 := Le (Plus e1 $ LitNat 1) e2.
+Definition Eq e1 e2 :=
+  Let e1 (Let e2.[ren(+1)]
+         (If (Le (Var 0) (Var 1)) (Le (Var 1) (Var 0)) LitFalse)).
+
+Definition LamV (e : {bind expr}) := RecV e.[ren(+1)].
+
+Definition LetCtx (K1 : ectx) (e2 : {bind expr}) := AppRCtx (LamV e2) K1.
+Definition SeqCtx (K1 : ectx) (e2 : expr) := LetCtx K1 (e2.[ren(+1)]).
+
+(** Proof rules for the sugar *)
+Lemma wp_lam E ef e v Q :
+  e2v e = Some v →
+  ▷wp (Σ:=Σ) E ef.[e/] Q ⊑ wp (Σ:=Σ) E (App (Lam ef) e) Q.
+Proof.
+  intros Hv. rewrite -wp_rec; last eassumption.
+  (* RJ: This pulls in functional extensionality. If that bothers us, we have
+     to talk to the Autosubst guys. *)
+  by asimpl.
+Qed.
+
+Lemma wp_let e1 e2 E Q :
+  wp (Σ:=Σ) E e1 (λ v, ▷wp (Σ:=Σ) E (e2.[v2e v/]) Q) ⊑ wp (Σ:=Σ) E (Let e1 e2) Q.
+Proof.
+  rewrite -(wp_bind (LetCtx EmptyCtx e2)). apply wp_mono=>v.
+  rewrite -wp_lam //. by rewrite v2v.
+Qed.
+
+Lemma wp_if_true e1 e2 E Q :
+  ▷wp (Σ:=Σ) E e1 Q ⊑ wp (Σ:=Σ) E (If LitTrue e1 e2) Q.
+Proof.
+  rewrite -wp_case_inl //. by asimpl.
+Qed.
+
+Lemma wp_if_false e1 e2 E Q :
+  ▷wp (Σ:=Σ) E e2 Q ⊑ wp (Σ:=Σ) E (If LitFalse e1 e2) Q.
+Proof.
+  rewrite -wp_case_inr //. by asimpl.
+Qed.
+
+Lemma wp_lt n1 n2 E P Q :
+  (n1 < n2 → P ⊑ ▷Q LitTrueV) →
+  (n1 ≥ n2 → P ⊑ ▷Q LitFalseV) →
+  P ⊑ wp (Σ:=Σ) E (Lt (LitNat n1) (LitNat n2)) Q.
+Proof.
+  intros HPlt HPge.
+  rewrite -(wp_bind (LeLCtx EmptyCtx _)) -wp_plus -later_intro. simpl.
+  apply wp_le; intros; [apply HPlt|apply HPge]; omega.
+Qed.
+
+Lemma wp_eq n1 n2 E P Q :
+  (n1 = n2 → P ⊑ ▷Q LitTrueV) →
+  (n1 ≠ n2 → P ⊑ ▷Q LitFalseV) →
+  P ⊑ wp (Σ:=Σ) E (Eq (LitNat n1) (LitNat n2)) Q.
+Proof.
+  intros HPeq HPne.
+  rewrite -wp_let -wp_value' // -later_intro. asimpl.
+  rewrite -wp_rec //. asimpl.
+  rewrite -(wp_bind (CaseCtx EmptyCtx _ _)) -later_intro.
+  apply wp_le; intros Hn12.
+  - asimpl. rewrite -wp_case_inl // -!later_intro. apply wp_le; intros Hn12'.
+    + apply HPeq; omega.
+    + apply HPne; omega.
+  - asimpl. rewrite -wp_case_inr // -later_intro -wp_value' //.
+    apply HPne; omega.
+Qed.