Section Mirror.
Variable (n : nat) (sig : finType).
Definition mirror_act : (option sig * move) -> (option sig * move) :=
map_snd mirror_move.
Definition mirror_acts : Vector.t (option sig * move) n -> Vector.t (option sig * move) n :=
Vector.map mirror_act.
Variable F : finType.
Variable pM : pTM sig F n.
Definition Mirror_trans :
state (projT1 pM) * Vector.t (option sig) n ->
state (projT1 pM) *
Vector.t (option sig * move) n :=
fun qsym =>
let (q', act) := trans qsym in
(q', mirror_acts act).
Definition MirrorTM : TM sig n :=
{|
trans := Mirror_trans;
start := start (projT1 pM);
halt := halt (m := projT1 pM);
|}.
Definition Mirror : pTM sig F n :=
(MirrorTM; projT2 pM).
Definition mirrorConf : mconfig sig (state (projT1 pM)) n -> mconfig sig (state (projT1 pM)) n :=
fun c => mk_mconfig (cstate c) (mirror_tapes (ctapes c)).

Lemma mirror_lift k c1 c2 :
loopM (M := projT1 Mirror)             c1  k = Some             c2 ->
loopM (M := projT1 pM    ) (mirrorConf c1) k = Some (mirrorConf c2).