Next Obligation.
destruct f; simpl; unfold funspecOracle2pre, pureat; simpl; destruct f; simpl;
destruct f; simpl; intros e t0 ge_s typs args z.
if_tac [e0|e0].
* destruct e; try discriminate; injection e0 as E; subst i sg; intros a a' Hage.
intros [phi0 [phi1 [Hjoin [Hx Hy]]]].
apply age1_juicy_mem_unpack in Hage.
destruct Hage as [Hage Hdry].
destruct (age1_join2 phi0 Hjoin Hage) as [x' [y' [Hjoin' [Hage' H]]]].
exists x', y'; split; auto.
destruct m. split. eapply h; eauto.
apply (necR_trans (fst t0) phi1 y'); auto.
unfold necR. constructor; auto.
* intros ? ?; auto.
destruct Espec; simpl; apply JE_pre_hered.
Qed.
Next Obligation.
destruct f; simpl; unfold funspecOracle2post, pureat; simpl; destruct f; simpl;
destruct f; simpl. intros e t0 ge_s tret rv z.
if_tac [e0|e0].
* destruct e; try discriminate; injection e0 as E; subst i sg; intros a a' Hage; destruct m0; simpl.
intros [phi0 [phi1 [Hjoin [Hx Hy]]]].
apply age1_juicy_mem_unpack in Hage.
destruct Hage as [Hage Hdry].
destruct (age1_join2 phi0 Hjoin Hage) as [x' [y' [Hjoin' [Hage' H]]]].
exists x', y'; split; auto.
split; [solve[eapply h; eauto]|].
apply (necR_trans (fst t0) phi1 y'); auto.
unfold necR. constructor; auto.
* intros ? ?; auto.
destruct Espec; simpl; apply JE_post_hered.
Qed.
Next Obligation.
intros ? ? ? ?; destruct f; destruct f; destruct f; simpl.
intros a' Hage; auto.
Qed.
End funspecsOracle2jspec.
Fixpoint add_funspecsOracle_rec (ext_link: string -> ident) Z (Espec : juicy_ext_spec Z) (fs : list (ident * funspecOracle Z)) :=
match fs with
| nil => Espec
| cons (i,f) fs' => funspecOracle2jspec Z (add_funspecsOracle_rec ext_link Z Espec fs') ext_link (i,f)
end.

(* Specifications: *)
Lemma add_funspecs_pre  (ext_link: Strings.String.string -> ident)
{Z fs id sig cc A P Q x args m} Espec tys ge_s phi0 phi1 z :
let ef := EF_external id (funsig2signature sig cc) in
list_norepet (map fst fs) ->
In (ext_link id, (mk_funspecOracle Z sig cc A P Q)) fs ->
join phi0 phi1 (m_phi m) ->
P x z (make_ext_args (filter_genv (symb2genv ge_s)) (fst (split (fst sig))) args) phi0 ->
exists x' : ext_spec_type (JE_spec _ (add_funspecsOracle_rec ext_link Z Espec fs)) ef,
JMeq (phi1, x) x'
/\ ext_spec_pre (add_funspecsOracle_rec ext_link Z Espec fs) ef x' ge_s tys args z m.
Lemma add_funspecs_post (ext_link: Strings.String.string -> ident)
{Z Espec tret fs id sig cc A}
{P Q : A -> Z -> environ -> mpred} {x ret z m ge_s} :
let ef := EF_external id (funsig2signature sig cc) in
list_norepet (map fst fs) ->
In (ext_link id, (mk_funspecOracle Z sig cc A P Q)) fs ->
In (ext_link id) (map fst fs) ->
ext_spec_post (add_funspecsOracle_rec ext_link Z Espec fs) ef x ge_s tret ret z m ->
exists (phi0 phi1 phi1' : rmap) (x' : A),
join phi0 phi1 (m_phi m)
/\ necR phi1' phi1
/\ JMeq x (phi1', x')
/\ Q x' z (make_ext_rval (filter_genv (symb2genv ge_s)) ret) phi0.