Definition ascii_to_digit (ch : ascii) : option N
  := (if ascii_dec ch "0" then Some 0
      else if ascii_dec ch "1" then Some 1
      else if ascii_dec ch "2" then Some 2
      else if ascii_dec ch "3" then Some 3
      else if ascii_dec ch "4" then Some 4
      else if ascii_dec ch "5" then Some 5
      else if ascii_dec ch "6" then Some 6
      else if ascii_dec ch "7" then Some 7
      else None)%N.Fixpoint pos_oct_app (p q:positive) : positive :=
  match q with
  | 1 => p~0~0~1
  | 2 => p~0~1~0
  | 3 => p~0~1~1
  | 4 => p~1~0~0
  | 5 => p~1~0~1
  | 6 => p~1~1~0
  | 7 => p~1~1~1
  | q~0~0~0 => (pos_oct_app p q)~0~0~0
  | q~0~0~1 => (pos_oct_app p q)~0~0~1
  | q~0~1~0 => (pos_oct_app p q)~0~1~0
  | q~0~1~1 => (pos_oct_app p q)~0~1~1
  | q~1~0~0 => (pos_oct_app p q)~1~0~0
  | q~1~0~1 => (pos_oct_app p q)~1~0~1
  | q~1~1~0 => (pos_oct_app p q)~1~1~0
  | q~1~1~1 => (pos_oct_app p q)~1~1~1
  end.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.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.