(* The following two lemmas is not now used. but after deep embedded hoare logic (SL_as_Logic) is ported, the frame does not need to be quantified in the semantic definition of semax. Then, these two lemmas can replace the other two afterwards. *)

Lemma semax_conseq {CS: compspecs} {Espec: OracleKind}:
forall Delta P' (R': ret_assert) P c (R: ret_assert) ,
(forall rho,  !!(typecheck_environ Delta rho) && (allp_fun_id Delta rho && P rho)
|-- bupd (|> FF || P' rho) )%pred ->
(forall rho,  !!(typecheck_environ Delta rho) && (allp_fun_id Delta rho && RA_normal R' rho)
|-- bupd (|> FF || RA_normal R rho)) ->
(forall rho, !! (typecheck_environ Delta rho) && (allp_fun_id Delta rho && RA_break R' rho)
|-- bupd (|> FF || RA_break R rho)) ->
(forall rho, !! (typecheck_environ Delta rho) && (allp_fun_id Delta rho && RA_continue R' rho)
|-- bupd (|> FF || RA_continue R rho)) ->
(forall vl rho, !! (typecheck_environ Delta rho) && (allp_fun_id Delta rho && RA_return R' vl rho)
|-- bupd (|> FF || RA_return R vl rho)) ->
semax Espec Delta P' c R' ->  semax Espec Delta P c R.