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static class Rat {
  BigInteger a, b;
  
  *() {} // for persistence
  *(BigInteger *a, BigInteger *b) { simplify(); }
  *(BigInteger *a) { b = bigint(1); }
  *(long a) { this(bigint(a)); }
  *(long a, long b) { this(bigint(a), bigint(b)); }
  
  Rat multiply(BigInteger i) { ret multiply(new Rat(i)); }
  Rat divide(BigInteger i) { ret divide(new Rat(i)); }
  
  Rat multiply(Rat r) {
    ret new Rat(a.multiply(r.a), b.multiply(r.b));
  }
  
  Rat divide(Rat r) {
    ret new Rat(a.multiply(r.b), b.multiply(r.a));
  }
  
  void simplify() {
    // TODO: does this work for negative fractions (especially negative b)?
    BigInteger gcd = a.gcd(b);
    if (!eq(gcd, 1)) {
      a = a.divide(gcd);
      b = b.divide(gcd);
    }
  }
  
  public S toString() {
    if (eq(b, 1))
      ret a.toString();
    else
      ret a + "/" + b;
  }
  
  // toString, return as mixed number if applicable
  // todo: negatives
  public S mixed() {
    if (a.compareTo(b) > 0 && neq(b, 1)) {
      BigInteger rest = a.mod(b);
      BigInteger whole = a.divide(b);
      ret whole + "+" + rest + "/" + b;
    }
    ret toString();
  }
  
  // TODO (maybe): compare with int etc.
  public boolean equals(O o) {
    if (!(o instanceof Rat)) ret false;
    Rat r = cast o;
    ret eq(a, r.a) && eq(b, r.b);
  }
}