Section Fix_Sigma.
Variable sig : Type.
Notation tape := (tape sig).
(* ** Definition of the tape *)
(* A tape is essentially a triple 〈left,current,right〉 where, however, the current symbol could be missing. This may happen for three different reasons: both tapes are empty, we are on the left extremity of a non-empty tape (left overflow), or we are on the right extremity of a non-empty tape (right overflow). *)
(* Note that the alphabet has type [Type], not [finType]. *)
(* Print tape. *)
Definition tapes n := Vector.t tape n.
Definition tapeToList (t : tape) : list sig :=
match t with
| niltape => []
| leftof s r => s :: r
| rightof s l => List.rev (s :: l)
| midtape l c r => (List.rev l) ++ [c] ++ r
end.
Definition sizeOfTape t := |tapeToList t|.
Definition sizeOfmTapes n (v : tapes n) :=
Vector.fold_left max 0 (Vector.map sizeOfTape v).
Definition left :=
fun (t : tape) =>
match t with
| niltape => []
| leftof _ _ => []
| rightof s l => s :: l
| midtape l _ _ => l
end.
Definition right :=
fun (t : tape) =>
match t with
| niltape => []
| leftof s r => s :: r
| rightof _ _ => []
| midtape _ _ r => r
end.

(* Lemmas for [midtape] *)
Lemma tape_midtape_current_right t rs s :
current t = Some s ->
right t = rs ->
t = midtape (left t) s rs.