Nearly prove Th2.3, assuming 1.6 and one small subgoal
This commit is contained in:
parent
82eb5f2189
commit
3ed5ef8ea4
1 changed files with 88 additions and 6 deletions
94
wp.v
94
wp.v
|
@ -112,7 +112,7 @@ Fixpoint interp (inst: Instr) (mem: MemCpo) : MemCpo :=
|
|||
Fixpoint nth_iterate (instr: Instr) (n: nat) : Instr :=
|
||||
match n with
|
||||
| 0 => skip
|
||||
| S m => seq instr (nth_iterate instr m)
|
||||
| S m => seq (nth_iterate instr m) instr
|
||||
end.
|
||||
|
||||
Definition satisfies_expr (mem: MemCpo) (expr: Expr) : Prop := match mem with
|
||||
|
@ -127,14 +127,35 @@ Definition expr_neg (expr: Expr) : Expr :=
|
|||
| _ => 0%Z
|
||||
end.
|
||||
|
||||
Lemma certain_termination_exists body guard mem :
|
||||
interp (while guard body) mem <> MemError ->
|
||||
exists n: nat,
|
||||
interp (nth_iterate (ifonly guard body) n) mem |=e expr_neg guard.
|
||||
Proof.
|
||||
intros noError.
|
||||
Admitted.
|
||||
|
||||
Lemma certain_termination_exists_minimal body guard mem :
|
||||
interp (while guard body) mem <> MemError ->
|
||||
exists n: nat,
|
||||
(forall p: nat, p < n ->
|
||||
interp (nth_iterate (ifonly guard body) p) mem |=e guard)
|
||||
/\ interp (nth_iterate (ifonly guard body) n) mem |=e expr_neg guard.
|
||||
Proof.
|
||||
intros not_error.
|
||||
Admitted.
|
||||
|
||||
Lemma certain_termination body guard mem :
|
||||
interp (while guard body) mem <> MemError ->
|
||||
exists n: nat,
|
||||
(interp (nth_iterate body n) mem) |=e (expr_neg guard)
|
||||
/\ forall p, p < n -> (interp (nth_iterate body p) mem) |=e guard
|
||||
/\ (forall p, p < n -> (interp (nth_iterate body p) mem) |=e guard)
|
||||
/\ interp (while guard body) mem = interp (nth_iterate body n) mem.
|
||||
Proof.
|
||||
intros notError.
|
||||
intros notError.
|
||||
elim (certain_termination_exists_minimal body guard mem).
|
||||
intros n. intros [notBeforeN atN]. exists n.
|
||||
split.
|
||||
Admitted.
|
||||
|
||||
(***** Validite, prouvabilite pour Hoare *************************************)
|
||||
|
@ -256,6 +277,40 @@ Proof.
|
|||
unfold MemError. destruct s; cbv; trivial.
|
||||
Qed.
|
||||
|
||||
Lemma conseq_or_bottom_is_conseq (y: Assert) (m: Mem) :
|
||||
conseq_or_bottom y (MemElem m) -> y m.
|
||||
Proof.
|
||||
intros src; unfold conseq_or_bottom; simpl; trivial.
|
||||
Qed.
|
||||
|
||||
Lemma unfold_one_iter (s: Instr) (m: Mem) (n: nat):
|
||||
interp (nth_iterate s (S n)) (MemElem m)
|
||||
= interp s (interp (nth_iterate s n) (MemElem m)).
|
||||
Proof.
|
||||
simpl; congruence.
|
||||
Qed.
|
||||
|
||||
Lemma error_leads_to_no_success s:
|
||||
forall m, interp s MemError <> MemElem m.
|
||||
Proof.
|
||||
intros mem. unfold interp; destruct s; simpl ;
|
||||
unfold MemElem; unfold MemError; congruence.
|
||||
Qed.
|
||||
|
||||
Lemma Sn_noerror_n_noerror (n: nat) (s: Instr) (sMem: Mem) (m: Mem):
|
||||
interp (nth_iterate s (S n)) (MemElem sMem) = MemElem m
|
||||
-> exists m0, interp (nth_iterate s n) (MemElem sMem) = MemElem m0
|
||||
/\ interp s (MemElem m0) = MemElem m.
|
||||
Proof.
|
||||
intro HSn.
|
||||
destruct (interp (nth_iterate s n) (MemElem sMem)) eqn:nRel.
|
||||
- rewrite (unfold_one_iter s sMem n) in HSn.
|
||||
rewrite nRel in HSn. apply error_leads_to_no_success in HSn.
|
||||
elimtype False. apply HSn.
|
||||
- exists m0; unfold MemElem. split.
|
||||
* trivial.
|
||||
* rewrite <- nRel; simpl. apply HSn.
|
||||
Qed.
|
||||
|
||||
Theorem hoare_provability_implies_consequence :
|
||||
forall (pre: Assert), forall (s: Instr), forall (post: Assert),
|
||||
|
@ -286,9 +341,36 @@ Proof.
|
|||
- unfold conseq_or_bottom.
|
||||
destruct (interp (while expr sBody) (MemElem mem)) eqn:interpRel.
|
||||
* trivial.
|
||||
* unfold assertAnd. split.
|
||||
+ specialize (IHdeduction m) as IHdeduction_m.
|
||||
unfold conseq_or_bottom in IHdeduction_m.
|
||||
* elim (certain_termination sBody expr (MemElem mem)).
|
||||
intros n. intros [lastIter [notLastIter isWhile] ].
|
||||
rewrite isWhile in interpRel.
|
||||
split.
|
||||
+
|
||||
apply conseq_or_bottom_is_conseq. unfold MemElem.
|
||||
rewrite <- interpRel.
|
||||
induction n; simpl.
|
||||
{ assumption. }
|
||||
{ elim (Sn_noerror_n_noerror n sBody mem m interpRel).
|
||||
intros memN [relMemN stepMemN].
|
||||
specialize (IHdeduction memN) as IHmem.
|
||||
rewrite stepMemN in IHmem.
|
||||
rewrite relMemN.
|
||||
rewrite relMemN in IHn.
|
||||
rewrite stepMemN.
|
||||
apply IHmem.
|
||||
unfold assertAnd; split.
|
||||
{ admit. }
|
||||
{ unfold assertOfExpr.
|
||||
specialize (notLastIter n).
|
||||
rewrite relMemN in notLastIter.
|
||||
unfold satisfies_expr in notLastIter; simpl in notLastIter.
|
||||
apply notLastIter. unfold Peano.lt. trivial. }
|
||||
}
|
||||
+ unfold assertNot; unfold assertOfExpr. rewrite interpRel in lastIter.
|
||||
unfold satisfies_expr in lastIter.
|
||||
unfold expr_neg in lastIter.
|
||||
destruct (expr m); simpl; congruence.
|
||||
+ rewrite interpRel; unfold MemError; congruence.
|
||||
Admitted.
|
||||
|
||||
(***** Weakest precondition **************************************************)
|
||||
|
|
Loading…
Reference in a new issue