Definition semax vars (G: funspecs) (P: assert) (k: control) : Prop :=
typecheck vars k = true /\ forall n,  semax' (SemaxArg vars (fun s => P s && funassert G) k) n.
Definition semax_func (G: funspecs) (p: program) (G': funspecs) :=
match_specs p G' /\
forall n, believe_all semax' G p G' n.

Lemma semax_prop:
forall (R: Prop) vars G P c,
typecheck vars c = true ->
(R -> semax vars G P c) ->
semax vars G (fun s => !! R && P s) c.