Module Raw.
Fixpoint of_pos (p : positive) (rest : string) : string
:= match p with
| 1 => String "1" rest
| 2 => String "2" rest
| 3 => String "3" rest
| 4 => String "4" rest
| 5 => String "5" rest
| 6 => String "6" rest
| 7 => String "7" rest
| p'~0~0~0 => of_pos p' (String "0" rest)
| p'~0~0~1 => of_pos p' (String "1" rest)
| p'~0~1~0 => of_pos p' (String "2" rest)
| p'~0~1~1 => of_pos p' (String "3" rest)
| p'~1~0~0 => of_pos p' (String "4" rest)
| p'~1~0~1 => of_pos p' (String "5" rest)
| p'~1~1~0 => of_pos p' (String "6" rest)
| p'~1~1~1 => of_pos p' (String "7" rest)
end.
Fixpoint to_N (s : string) (rest : N)
: N
:= match s with
| "" => rest
| String ch s'
=> to_N
s'
match ascii_to_digit ch with
| Some v => N.add v (N.mul 8 rest)
| None => N0
end
end.
Fixpoint to_N_of_pos (p : positive) (rest : string) (base : N) {struct p}
: to_N (of_pos p rest) base
= to_N rest match base with
| N0 => N.pos p
| Npos v => Npos (pos_oct_app v p)
end.
End Raw.
Definition of_pos (p : positive) : string
:= String "0" (String "o" (Raw.of_pos p "")).
Definition of_N (n : N) : string
:= match n with
| N0 => "0o0"
| Npos p => of_pos p
end.
Definition of_Z (z : Z) : string
:= match z with
| Zneg p => String "-" (of_pos p)
| Z0 => "0o0"
| Zpos p => of_pos p
end.
Definition of_nat (n : nat) : string
:= of_N (N.of_nat n).
Definition to_N (s : string) : N
:= match s with
| String s0 (String so s)
=> if ascii_dec s0 "0"
then if ascii_dec so "o"
then Raw.to_N s N0
else N0
else N0
| _ => N0
end.
Definition to_pos (s : string) : positive
:= match to_N s with
| N0 => 1
| Npos p => p
end.
Definition to_Z (s : string) : Z
:= let '(is_neg, n) := match s with
| String s0 s'
=> if ascii_dec s0 "-"
then (true, to_N s')
else (false, to_N s)
| EmptyString => (false, to_N s)
end in
match n with
| N0 => Z0
| Npos p => if is_neg then Zneg p else Zpos p
end.
Definition to_nat (s : string) : nat
:= N.to_nat (to_N s).

Lemma to_nat_of_nat (n : nat)
: to_nat (of_nat n)
= n.