Module Raw.
Fixpoint of_pos (p : positive) (rest : string) : string
:= match p with
| 1 => String "1" rest
| p'~0 => of_pos p' (String "0" rest)
| p'~1 => of_pos p' (String "1" 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.double 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_bin_app v p)
end.
End Raw.
Definition of_pos (p : positive) : string
:= String "0" (String "b" (Raw.of_pos p "")).
Definition of_N (n : N) : string
:= match n with
| N0 => "0b0"
| Npos p => of_pos p
end.
Definition of_Z (z : Z) : string
:= match z with
| Zneg p => String "-" (of_pos p)
| Z0 => "0b0"
| 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 sb s)
=> if ascii_dec s0 "0"
then if ascii_dec sb "b"
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.