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(** Impure monad for interface with impure code
*)
Require Export Program.
Require Export ImpConfig.
(* Theory: bind + embed => dbind
Program Definition dbind {A B} (k1: t A) (k2: forall (a:A), (mayRet k1 a) -> t B) : t B
:= bind (mk_annot k1) (fun a => k2 a _).
Lemma mayRet_dbind: forall (A B:Type) k1 k2 (b:B),
mayRet (dbind k1 k2) b -> exists a:A, exists H: (mayRet k1 a), mayRet (k2 a H) b.
Proof.
intros A B k1 k2 b H; decompose [ex and] (mayRet_bind _ _ _ _ _ H).
eapply ex_intro.
eapply ex_intro.
eauto.
Qed.
*)
Definition wlp {A:Type} (k: t A) (P: A -> Prop): Prop
:= forall a, mayRet k a -> P a.
(* Notations *)
(* Print Grammar constr. *)
Module Notations.
Bind Scope impure_scope with t.
Delimit Scope impure_scope with impure.
Notation "?? A" := (t A) (at level 0, A at level 95): impure_scope.
Notation "k '~~>' a" := (mayRet k a) (at level 75, no associativity): impure_scope.
Notation "'RET' a" := (ret a) (at level 0): impure_scope.
Notation "'DO' x '<~' k1 ';;' k2" := (bind k1 (fun x => k2))
(at level 55, k1 at level 53, x at level 99, right associativity): impure_scope.
Notation "k1 ';;' k2" := (bind k1 (fun _ => k2))
(at level 55, right associativity): impure_scope.
Notation "'WHEN' k '~>' a 'THEN' R" := (wlp k (fun a => R))
(at level 73, R at level 100, right associativity): impure_scope.
Notation "'ASSERT' P" := (ret (A:=P) _) (at level 0, only parsing): impure_scope.
End Notations.
Import Notations.
Local Open Scope impure.
Goal ((?? list nat * ??nat -> nat) = ((?? ((list nat) * ?? nat) -> nat)))%type.
Proof.
apply refl_equal.
Qed.
(* wlp lemmas for tactics *)
Lemma wlp_unfold A (k:??A)(P: A -> Prop):
(forall a, k ~~> a -> P a)
-> wlp k P.
Proof.
auto.
Qed.
Lemma wlp_monotone A (k:?? A) (P1 P2: A -> Prop):
wlp k P1
-> (forall a, k ~~> a -> P1 a -> P2 a)
-> wlp k P2.
Proof.
unfold wlp; eauto.
Qed.
Lemma wlp_forall A B (k:?? A) (P: B -> A -> Prop):
(forall x, wlp k (P x))
-> wlp k (fun a => forall x, P x a).
Proof.
unfold wlp; auto.
Qed.
Lemma wlp_ret A (P: A -> Prop) a:
P a -> wlp (ret a) P.
Proof.
unfold wlp.
intros H b H0.
rewrite <- (mayRet_ret _ a b H0).
auto.
Qed.
Lemma wlp_bind A B (k1:??A) (k2: A -> ??B) (P: B -> Prop):
wlp k1 (fun a => wlp (k2 a) P) -> wlp (bind k1 k2) P.
Proof.
unfold wlp.
intros H a H0.
case (mayRet_bind _ _ _ _ _ H0); clear H0.
intuition eauto.
Qed.
Lemma wlp_ifbool A (cond: bool) (k1 k2: ?? A) (P: A -> Prop):
(cond=true -> wlp k1 P) -> (cond=false -> wlp k2 P) -> wlp (if cond then k1 else k2) P.
Proof.
destruct cond; auto.
Qed.
Lemma wlp_letprod (A B C: Type) (p: A*B) (k: A -> B -> ??C) (P: C -> Prop):
(wlp (k (fst p) (snd p)) P)
-> (wlp (let (x,y):=p in (k x y)) P).
Proof.
destruct p; simpl; auto.
Qed.
Lemma wlp_sum (A B C: Type) (x: A+B) (k1: A -> ??C) (k2: B -> ??C) (P: C -> Prop):
(forall a, x=inl a -> wlp (k1 a) P) ->
(forall b, x=inr b -> wlp (k2 b) P) ->
(wlp (match x with inl a => k1 a | inr b => k2 b end) P).
Proof.
destruct x; simpl; auto.
Qed.
Lemma wlp_sumbool (A B:Prop) (C: Type) (x: {A}+{B}) (k1: A -> ??C) (k2: B -> ??C) (P: C -> Prop):
(forall a, x=left a -> wlp (k1 a) P) ->
(forall b, x=right b -> wlp (k2 b) P) ->
(wlp (match x with left a => k1 a | right b => k2 b end) P).
Proof.
destruct x; simpl; auto.
Qed.
Lemma wlp_option (A B: Type) (x: option A) (k1: A -> ??B) (k2: ??B) (P: B -> Prop):
(forall a, x=Some a -> wlp (k1 a) P) ->
(x=None -> wlp k2 P) ->
(wlp (match x with Some a => k1 a | None => k2 end) P).
Proof.
destruct x; simpl; auto.
Qed.
(* Tactics
MAIN tactics:
- xtsimplify "base": simplification using from hints in "base" database (in particular "wlp" lemmas).
- xtstep "base": only one step of simplification.
For good performance, it is recommanded to have several databases.
*)
Ltac introcomp :=
let a:= fresh "exta" in
let H:= fresh "Hexta" in
intros a H.
(* decompose the current wlp goal using "introduction" rules *)
Ltac wlp_decompose :=
apply wlp_ret
|| apply wlp_bind
|| apply wlp_ifbool
|| apply wlp_letprod
|| apply wlp_sum
|| apply wlp_sumbool
|| apply wlp_option
.
(* this tactic simplifies the current "wlp" goal using any hint found via tactic "hint". *)
Ltac apply_wlp_hint hint :=
eapply wlp_monotone;
[ hint; fail | idtac ] ;
simpl; introcomp.
(* one step of wlp_xsimplify
*)
Ltac wlp_step hint :=
match goal with
| |- (wlp _ _) =>
wlp_decompose
|| apply_wlp_hint hint
|| (apply wlp_unfold; introcomp)
end.
(* main general tactic
WARNING: for the good behavior of "wlp_xsimplify", "hint" must at least perform a "eauto".
Example of use:
wlp_xsimplify (intuition eauto with base).
*)
Ltac wlp_xsimplify hint :=
repeat (intros; subst; wlp_step hint; simpl; (tauto || hint)).
Create HintDb wlp discriminated.
Ltac wlp_simplify := wlp_xsimplify ltac:(intuition eauto with wlp).
|
