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)).
Definition Mirror_Rel (R : pRel sig F n) : pRel sig F n :=
fun t '(l, t') => R (mirror_tapes t) (l, mirror_tapes t').
Definition Mirror_T (T : tRel sig n) : tRel sig n :=
fun t k => T (mirror_tapes t) k.

Lemma Mirror_RealiseIn R (k : nat) :
pM ⊨c(k) R -> Mirror ⊨c(k) Mirror_Rel R.