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_Steps n st0 st1:
Steps Σ tm n (F_ES st0) st1->
exists st1',
Steps Σ' tm' n st0 st1' /\
F_ES st1' = st1.