(** * Generation of Hoare proof obligations in partial correctness  This file is part of the "Tutorial on Hoare Logic". For an introduction to this Coq library, see README #or <a href=index.html>index.html</a>#.  This file gives a syntactic definition of the weakest liberal precondition [wlp] introduced in #<a href=hoarelogicsemantics.html>#[hoarelogicsemantics]#</a>#. *)
Global Set Asymmetric Patterns.
Set Implicit Arguments.
Require Export hoarelogicsemantics.
Module PartialHoareLogic (HD: HoareLogicDefs).
Export HD.
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.

(** This property is also trivially satisfied by [wlp]. We need it here to prove the soundness. *)
Lemma synt_wp_monotonic:
forall (p: ImpProg) (post1 post2: Pred),
(post1 |= post2) -> (synt_wp p post1) |= (synt_wp p post2).