(* * 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.
Definition atHSB (t : tape sigPos^+) (p : positive) :=
exists (ls : list sigPos^+),
t = midtape (inl START :: ls) (inr sigPos_xH) (map inr (tl (encode p)) ++ [inl STOP]).

Lemma atBit_moveLeft_atHSB (t : tape sigPos^+) (b : bool) (rs : list bool) :
atBit t 1 b rs ->
atHSB (tape_move t Lmove) (bits_to_pos (b :: rs)).