Commit bd340b6f authored by Ralf Jung's avatar Ralf Jung

add missing file; also add sugar for equality test

parent 23de2030
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.
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