Inductive return_inner_gen (S: list mpred): option val -> (environ -> mpred) -> (environ -> mpred) -> Prop :=
| return_inner_gen_main: forall ov_gen P ts u,
return_inner_gen S ov_gen (main_post P ts u) (PROPx nil (LOCALx nil (SEPx (TT :: S))))
| return_inner_gen_canon_nil':
forall ov_gen P R,
return_inner_gen S ov_gen
(PROPx P (LOCALx nil (SEPx R)))
(PROPx P (LOCALx nil (SEPx (R ++ S))))
| return_inner_gen_canon_Some':
forall P v R v_gen,
return_inner_gen S (Some v_gen)
(PROPx P (LOCALx (temp ret_temp v :: nil) (SEPx R)))
(PROPx (P ++ (v_gen = v) :: nil) (LOCALx nil (SEPx (R ++ S))))
| return_inner_gen_EX':
forall ov_gen (A: Type) (post1 post2: A -> environ -> mpred),
(forall a: A, return_inner_gen S ov_gen (post1 a) (post2 a)) ->
return_inner_gen S ov_gen (exp post1) (exp post2).

Lemma remove_PROP_LOCAL_left':
forall P Q R S, `R |-- S ->
PROPx P (LOCALx Q (SEPx (R::nil))) |-- S.