Module HLD:=HD.
Definition sem_wp := wlp.
(** * Syntactic definition of the weakest liberal precondition.  In the following, we show that this definition is logically equivalent to [wlp]. *)
Fixpoint synt_wp (prog: ImpProg) : Pred -> Pred
:= fun post e =>
match prog with
| Iskip => post e
| (Iset A x expr) => post (E.upd x (E.eval expr e) e)
| (Iif cond p1 p2) =>
((E.eval cond e)=true -> (synt_wp p1 post e))
/\ ((E.eval cond e)=false -> (synt_wp p2 post e))
| (Iseq p1 p2) => synt_wp p1 (synt_wp p2 post) e
| (Iwhile cond p) =>
exists inv:Pred,
(inv e)
/\ (forall e', (inv e')
-> (E.eval cond e')=false -> (post e'))
/\ (forall e', (inv e')
-> (E.eval cond e')=true -> (synt_wp p inv e'))
end.
Hint Resolve synt_wp_monotonic: hoare.
(** * Combining the previous results with transitivity of [ |= ] *)
Hint Resolve wp_complete wp_sound: hoare.

Theorem soundness: forall pre p post, pre |= (synt_wp p post) -> pre |= p {=post=}.