Inductive InternalOrExternal: MachState -> mem -> MachState -> mem -> Prop:=
| IsInternal st st' m m':
internal_step (fst (fst st)) (snd st) m (snd st') m'  ->
InternalOrExternal st m st' m'
| IsExternal st st' m m':
external_step (fst (fst st)) (snd (fst st)) (snd st) m (fst (fst st')) (snd (fst st')) (snd st') m'  ->
InternalOrExternal st m st' m'.

Lemma safety_equivalence2: forall st m tr,
(forall U, valid (tr, st, m) U ->
safe_kstep (U, tr, st) m) <->
(forall U, valid (tr, st, m) U ->
explicit_safety U tr st m).