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'.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'.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).