(* 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_post {CS: compspecs} {Espec: OracleKind}:
forall (R': ret_assert) Delta (R: ret_assert) P c,
(forall rho,  !!(typecheck_environ Delta rho)
&&  RA_normal R' rho |-- RA_normal R rho) ->
(forall rho, !! (typecheck_environ Delta rho)
&& RA_break R' rho |-- RA_break R rho) ->
(forall rho, !! (typecheck_environ Delta rho)
&& RA_continue R' rho |-- RA_continue R rho) ->
(forall vl rho, !! (typecheck_environ Delta rho)
&& RA_return R' vl rho |-- RA_return R vl rho) ->
semax Espec Delta P c R' ->  semax Espec Delta P c R.