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 sub(uint256 a, uint256 b) internal pure returns (uint256) {
assert(b <= a);
return a - b;
}
/**
 * @dev Subtracts two numbers, throws on overflow (i.e. if subtrahend is greater than minuend).
 */