library SafeMath {

  /**
  * @dev Multiplies two numbers, throws on overflow.
  */

  /*@CTK SafeMath_mul
    @tag spec
    @post __reverted == __has_assertion_failure
    @post __has_assertion_failure == __has_overflow
    @post __reverted == false -> c == a * b
    @post msg == msg__post
   */
  /* CertiK Smart Labelling, for more details visit: https://certik.org */
  function mul(uint256 a, uint256 b) internal pure returns (uint256 c) {
    if (a == 0) {
      return 0;
    }
    c = a * b;
    assert(c / a == b);
    return c;
  }

  /**
  * @dev Integer division of two numbers, truncating the quotient.
  */
  /*@CTK SafeMath_div
    @tag spec
    @pre b != 0
    @post __reverted == __has_assertion_failure
    @post __has_overflow == true -> __has_assertion_failure == true
    @post __reverted == false -> __return == a / b
    @post msg == msg__post
   */
  /* CertiK Smart Labelling, for more details visit: https://certik.org */
  function div(uint256 a, uint256 b) internal pure returns (uint256) {
    // assert(b > 0); // Solidity automatically throws when dividing by 0
    // uint256 c = a / b;
    // assert(a == b * c + a % b); // There is no case in which this doesn't hold
    return a / b;
  }

  /**
  * @dev Subtracts two numbers, throws on overflow (i.e. if subtrahend is greater than minuend).
  */
  /*@CTK SafeMath_sub
    @tag spec
    @post __reverted == __has_assertion_failure
    @post __has_overflow == true -> __has_assertion_failure == true
    @post __reverted == false -> __return == a - b
    @post msg == msg__post
   */
  /* CertiK Smart Labelling, for more details visit: https://certik.org */
  function sub(uint256 a, uint256 b) internal pure returns (uint256) {
    assert(b <= a);
    return a - b;
  }

  /**
  * @dev Adds two numbers, throws on overflow.
  */
  /*@CTK SafeMath_add
    @tag spec
    @post __reverted == __has_assertion_failure
    @post __has_assertion_failure == __has_overflow
    @post __reverted == false -> c == a + b
    @post msg == msg__post
   */
  /* CertiK Smart Labelling, for more details visit: https://certik.org */
  function add(uint256 a, uint256 b) internal pure returns (uint256 c) {
    c = a + b;
    assert(c >= a);
    return c;
  }
}
/**
 * @title SafeMath
 * @dev Math operations with safety checks that throw on error
 */
function div(uint256 a, uint256 b) internal pure returns (uint256) {
    // assert(b > 0); // Solidity automatically throws when dividing by 0
    // uint256 c = a / b;
    // assert(a == b * c + a % b); // There is no case in which this doesn't hold
    return a / b;
  }
/* CertiK Smart Labelling, for more details visit: https://certik.org */