Section TMwithExtraInfo.
Hypothesis Σ:Set.
Hypothesis Σ':Set.
Hypothesis Σ0':Σ'.
Hypothesis F:Σ'->Σ.
Hypothesis tm:TM Σ.
Hypothesis tm':TM Σ'.
Hypothesis HF:
forall (s:St)(i:Σ'),
match tm' s i with
| Some tr =>
let (s',d,o) := tr in
tm s (F i) = Some {| nxt:=s'; dir:=d; out:= F o |}
| None => tm s (F i) = None
end.
Definition F_ES: (ExecState Σ')->(ExecState Σ) :=
fun st =>
let (s,t):=st in
(s,fun x => F (t x)).

Lemma F_step st0 st1:
step Σ tm (F_ES st0) = Some st1->
exists st1',
step Σ' tm' st0 = Some st1' /\
F_ES st1' = st1.