struct Fraction {
  int n;
  int d;
}

library FRMath {
  static function add(Fraction x, Fraction y) : Fraction {
    return {x.n * y.d + y.n * x.d, x.d * y.d};
  }
  static function sAdd(Fraction x, Fraction y) : Fraction {
    require(x.d > 0 && y.d > 0);
    return {x.n * y.d + y.n * x.d, x.d * y.d};
  }
  static function sub(Fraction x, Fraction y) : Fraction {
    return {x.n * y.d - y.n * x.d, x.d * y.d};
  }
  static function sSub(Fraction x, Fraction y) : Fraction {
    require(x.d > 0 && y.d > 0);
    return {x.n * y.d - y.n * x.d, x.d * y.d};
  }
  static function mul(Fraction x, Fraction y) : Fraction {
    return {x.n * y.n, x.d * y.d};
  }
  static function sMul(Fraction x, Fraction y) : Fraction {
    require(x.d > 0 && y.d > 0);
    return {x.n * y.n, x.d * y.d};
  }
  static function div(Fraction x, Fraction y) : Fraction {
    return {x.n * y.d, x.d * y.n};
  }
  static function sDiv(Fraction x, Fraction y) : Fraction {
    require(x.d > 0 && y.d > 0 && y.n != 0);
    return {x.n * y.d, x.d * y.n};
  }
  static function abs(Fraction x) : Fraction {
    return {abs(x.n), abs(x.d)};
  }
  static function sAbs(Fraction x) : Fraction {
    require(x.d > 0);
    return {abs(x.n), x.d};
  }
  static function equal(Fraction x, Fraction y) : bool {
    return FRMath.sub(x, y).n == 0;
  }
  static function sEqual(Fraction x, Fraction y) : bool {
    return FRMath.sSub(x, y).n == 0;
  }
  static function toInt(Fraction x) : int {
    return x.n / x.d;
  }
  static function fromInt(int numerator, int denominator) : Fraction {
    return {numerator, denominator};
  }
  static function scaleUp(Fraction x, int s) : int {
    return (x.n * s) / x.d;
  }
}