(* 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_pre_bupd {CS: compspecs} {Espec: OracleKind} :
forall P' Delta P c R,
(forall rho,  !!(typecheck_environ Delta rho) &&  P rho |-- bupd (P' rho) )%pred ->
semax Espec Delta P' c R  -> semax Espec Delta P c R.