Section red.
Variable M : TM.TM (finType_CS bool) 1.
Definition num_states := (projT1 (finite_n (TM.state M))).
Definition f := (projT1 (projT2 (finite_n (TM.state M)))).
Definition g := (proj1_sig (projT2 (projT2 (finite_n (TM.state M))))).
Definition Hf := (proj1 (proj2_sig (projT2 (projT2 (finite_n (TM.state M)))))).
Definition Hg := (proj2 (proj2_sig (projT2 (projT2 (finite_n (TM.state M)))))).
Definition conv_move : TM.move -> SBTM.move :=
fun m => match m with TM.Lmove => SBTM.Lmove | TM.Nmove => SBTM.Nmove | TM.Rmove => SBTM.Rmove end.
Definition conv_state : TM.state M -> Fin.t (S num_states) := fun q => (Fin.FS (f q)).
Definition trans : Fin.t (S num_states) * option bool -> option (Fin.t (S num_states) * option bool * SBTM.move) :=
fun '(q, o) => Fin.caseS' q (fun _ => _) (Some (conv_state (TM.start M), None, SBTM.Nmove))
(fun q => if TM.halt (g q) then None
else let '(q', vec) := TM.trans M (g q, [| o |]) in
let '(w, m) := vector_hd vec in
Some (conv_state q', w, conv_move m)
).
Definition conv_tape (t : Vector.t (TM.tape bool) 1) : SBTM.tape :=
let t := vector_hd t in
match TM.current t with
| Some c => (TM.Util.TM_facts.left t, Some c, TM.Util.TM_facts.right t)
| None => (TM.Util.TM_facts.left t, None, TM.Util.TM_facts.right t)
end.

Lemma red_correct2 q t q'_ t'_ :
SBTM.eval (SBTM.Build_SBTM num_states trans) (conv_state q) (conv_tape t) q'_ t'_ ->
exists q' t', q'_ = conv_state q' /\ t'_ = conv_tape t' /\
TM.eval M q t q' t'.