(* * Pointer bookkeeping for machines using (positive) binary numbers *)
Local Open Scope positive_scope.
Definition atBit (t : tape sigPos^+) (lp : positive) (mb : bool) (rs : list bool) :=
exists (ls : list sigPos^+),
t = midtape (map inr (rev (encode lp)) ++ inl START :: ls) (bitToSigPos' mb) (map bitToSigPos' rs ++ [inl STOP]).
Definition atLSB (t : tape sigPos^+) (lp : positive) (mb : bool) := atBit t lp mb nil.

Lemma atBit_moveRight_cons (t : tape sigPos^+) (lp : positive) (b b' : bool) (rs : list bool) :
atBit t lp b (b' :: rs) ->
atBit (tape_move t Rmove) (lp ~~ b) b' rs.