(* The following four 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 assert_safe_bupd:
forall {Espec: OracleKind} gx vx tx rho (F P: environ -> pred rmap) Delta f k,
let PP1 := !! guard_environ Delta f rho in
let PP2 := funassert Delta rho in
PP1 && (F rho * P rho) && PP2 >=>
assert_safe Espec gx vx tx k rho =
PP1 && (F rho * bupd (P rho)) && PP2 >=>
assert_safe Espec gx vx tx k rho.