library SafeMath {

 /**
 * @dev Multiplies two numbers, throws on overflow.
 */
 function mul(uint256 _a, uint256 _b) internal pure returns (uint256 c) {
  // Gas optimization: this is cheaper than asserting 'a' not being zero, but the
  // benefit is lost if 'b' is also tested.
  // See: https://github.com/OpenZeppelin/openzeppelin-solidity/pull/522
  if (_a == 0) {
   return 0;
  }

  c = _a * _b;
  assert(c / _a == _b);
  return c;
 }

 /**
 * @dev Integer division of two numbers, truncating the quotient.
 */
 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).
 */
 function sub(uint256 _a, uint256 _b) internal pure returns (uint256) {
  assert(_b <= _a);
  return _a - _b;
 }

 /**
 * @dev Adds two numbers, throws on overflow.
 */
 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 add(uint256 _a, uint256 _b) internal pure returns (uint256 c) {
 c = _a + _b;
 assert(c >= _a);
 return c;
}

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