// File: /Users/sergey/code/basis-gold/solidity/contracts/Oracle.sol



pragma solidity ^0.6.0;

import '@openzeppelin/contracts/math/SafeMath.sol';

import './lib/Babylonian.sol';
import './lib/FixedPoint.sol';
import './lib/UniswapV2Library.sol';
import './lib/UniswapV2OracleLibrary.sol';
import './utils/Epoch.sol';
import './interfaces/IUniswapV2Pair.sol';
import './interfaces/IUniswapV2Factory.sol';
import './interfaces/ILinkOracle.sol';

// fixed window oracle that recomputes the average price for the entire period once every period
// note that the price average is only guaranteed to be over at least 1 period, but may be over a longer period
contract Oracle is Epoch {

    event Updated(uint256 price0CumulativeLast, uint256 price1CumulativeLast);
    
    using FixedPoint for *;
    using SafeMath for uint256;

    struct PriceData {
        address token0;
        address token1;

        uint    price0CumulativeLast;
        uint    price1CumulativeLast;
        uint32  blockTimestampLast;

        uint price0Average;
        uint price1Average;
    }

    PriceData public priceData;
    IUniswapV2Pair public pair;
    ILinkOracle public linkOracle;

    constructor(
        address _factory,
        address _tokenA,
        address _tokenB,
        uint256 _period,
        uint256 _startTime,
        ILinkOracle _linkOracle
    ) public Epoch(_period, _startTime, 0) {
        pair = IUniswapV2Pair(UniswapV2Library.pairFor(_factory, _tokenA, _tokenB));
        linkOracle = _linkOracle;

        priceData.token0 = pair.token0();
        priceData.token1 = pair.token1();
        priceData.price0CumulativeLast = pair.price0CumulativeLast();
        priceData.price1CumulativeLast = pair.price1CumulativeLast();
        
        uint reserve0;
        uint reserve1;
        (reserve0, reserve1, priceData.blockTimestampLast) = pair.getReserves();
        require(reserve0 != 0 && reserve1 != 0, 'Oracle: NO_RESERVES');

        priceData.price0Average = reserve1.mul(1e18).div(reserve0);
        priceData.price1Average = reserve0.mul(1e18).div(reserve1);
    }

    /** @dev Updates 1-day EMA price from Uniswap.  */
    function update() external checkEpoch {
        priceData = priceCurrent();
        emit Updated(priceData.price0CumulativeLast, priceData.price1CumulativeLast);
    }

    function goldPriceOne() external view returns(uint256) {
        return uint256(linkOracle.latestAnswer() * 1e10);
    }

    function price0Last() public view returns (uint amountOut) {
        return priceData.price0Average;
    }

    function price1Last() public view returns (uint amountOut) {
        return priceData.price1Average;
    }

    function price0Current() public view returns (uint amountOut) {
        return priceCurrent().price0Average;
    }

    function price1Current() public view returns (uint amountOut) {
        return priceCurrent().price1Average;
    }

    function blockTimestampLast() external view returns(uint32) {
        return priceData.blockTimestampLast;
    }

    function price0CumulativeLast() external view returns(uint) {
        return priceData.price0CumulativeLast;
    }

    function price1CumulativeLast() external view returns(uint) {
        return priceData.price1CumulativeLast;
    }

    function pairFor(
        address factory,
        address tokenA,
        address tokenB
    ) external pure returns (address lpt) {
        return UniswapV2Library.pairFor(factory, tokenA, tokenB);
    }

    function priceCurrent() internal view returns(PriceData memory) {
        PriceData memory _priceData = priceData;

        (uint price0Cumulative, uint price1Cumulative, uint32 blockTimestamp) = UniswapV2OracleLibrary.currentCumulativePrices(address(pair));
        uint32 timeElapsed = blockTimestamp - _priceData.blockTimestampLast; // overflow is desired

        _priceData.price0Average = uint(
            FixedPoint.uq112x112(
                uint224((price0Cumulative - _priceData.price0CumulativeLast) / timeElapsed)
            ).mul(1e18).decode144()
        );

        _priceData.price1Average = uint(
            FixedPoint.uq112x112(
                uint224((price1Cumulative - _priceData.price1CumulativeLast) / timeElapsed)
            ).mul(1e18).decode144()
        );

        _priceData.price0CumulativeLast = price0Cumulative;
        _priceData.price1CumulativeLast = price1Cumulative;
        _priceData.blockTimestampLast = blockTimestamp;

        return _priceData;
    }
}

// File: /Users/sergey/code/basis-gold/solidity/contracts/interfaces/ILinkOracle.sol



pragma solidity ^0.6.0;

interface ILinkOracle {

  function latestAnswer() external view returns (int256);
}

// File: /Users/sergey/code/basis-gold/solidity/contracts/interfaces/IUniswapV2Factory.sol



pragma solidity ^0.6.0;

interface IUniswapV2Factory {
    event PairCreated(
        address indexed token0,
        address indexed token1,
        address pair,
        uint256
    );

    function getPair(address tokenA, address tokenB)
        external
        view
        returns (address pair);

    function allPairs(uint256) external view returns (address pair);

    function allPairsLength() external view returns (uint256);

    function feeTo() external view returns (address);

    function feeToSetter() external view returns (address);

    function createPair(address tokenA, address tokenB)
        external
        returns (address pair);
}

// File: /Users/sergey/code/basis-gold/solidity/contracts/interfaces/IUniswapV2Pair.sol



pragma solidity ^0.6.0;

interface IUniswapV2Pair {
    event Approval(
        address indexed owner,
        address indexed spender,
        uint256 value
    );
    event Transfer(address indexed from, address indexed to, uint256 value);

    function name() external pure returns (string memory);

    function symbol() external pure returns (string memory);

    function decimals() external pure returns (uint8);

    function totalSupply() external view returns (uint256);

    function balanceOf(address owner) external view returns (uint256);

    function allowance(address owner, address spender)
        external
        view
        returns (uint256);

    function approve(address spender, uint256 value) external returns (bool);

    function transfer(address to, uint256 value) external returns (bool);

    function transferFrom(
        address from,
        address to,
        uint256 value
    ) external returns (bool);

    function DOMAIN_SEPARATOR() external view returns (bytes32);

    function PERMIT_TYPEHASH() external pure returns (bytes32);

    function nonces(address owner) external view returns (uint256);

    function permit(
        address owner,
        address spender,
        uint256 value,
        uint256 deadline,
        uint8 v,
        bytes32 r,
        bytes32 s
    ) external;

    event Mint(address indexed sender, uint256 amount0, uint256 amount1);
    event Burn(
        address indexed sender,
        uint256 amount0,
        uint256 amount1,
        address indexed to
    );
    event Swap(
        address indexed sender,
        uint256 amount0In,
        uint256 amount1In,
        uint256 amount0Out,
        uint256 amount1Out,
        address indexed to
    );
    event Sync(uint112 reserve0, uint112 reserve1);

    function MINIMUM_LIQUIDITY() external pure returns (uint256);

    function factory() external view returns (address);

    function token0() external view returns (address);

    function token1() external view returns (address);

    function getReserves()
        external
        view
        returns (
            uint112 reserve0,
            uint112 reserve1,
            uint32 blockTimestampLast
        );

    function price0CumulativeLast() external view returns (uint256);

    function price1CumulativeLast() external view returns (uint256);

    function kLast() external view returns (uint256);

    function mint(address to) external returns (uint256 liquidity);

    function burn(address to)
        external
        returns (uint256 amount0, uint256 amount1);

    function swap(
        uint256 amount0Out,
        uint256 amount1Out,
        address to,
        bytes calldata data
    ) external;

    function skim(address to) external;

    function sync() external;

    function initialize(address, address) external;
}

// File: /Users/sergey/code/basis-gold/solidity/contracts/lib/Babylonian.sol



pragma solidity ^0.6.0;

library Babylonian {
    function sqrt(uint256 y) internal pure returns (uint256 z) {
        if (y > 3) {
            z = y;
            uint256 x = y / 2 + 1;
            while (x < z) {
                z = x;
                x = (y / x + x) / 2;
            }
        } else if (y != 0) {
            z = 1;
        }
        // else z = 0
    }
}

// File: /Users/sergey/code/basis-gold/solidity/contracts/lib/FixedPoint.sol



pragma solidity ^0.6.0;

import './Babylonian.sol';

// a library for handling binary fixed point numbers (https://en.wikipedia.org/wiki/Q_(number_format))
library FixedPoint {
    // range: [0, 2**112 - 1]
    // resolution: 1 / 2**112
    struct uq112x112 {
        uint224 _x;
    }

    // range: [0, 2**144 - 1]
    // resolution: 1 / 2**112
    struct uq144x112 {
        uint256 _x;
    }

    uint8 private constant RESOLUTION = 112;
    uint256 private constant Q112 = uint256(1) << RESOLUTION;
    uint256 private constant Q224 = Q112 << RESOLUTION;

    // encode a uint112 as a UQ112x112
    function encode(uint112 x) internal pure returns (uq112x112 memory) {
        return uq112x112(uint224(x) << RESOLUTION);
    }

    // encodes a uint144 as a UQ144x112
    function encode144(uint144 x) internal pure returns (uq144x112 memory) {
        return uq144x112(uint256(x) << RESOLUTION);
    }

    // divide a UQ112x112 by a uint112, returning a UQ112x112
    function div(uq112x112 memory self, uint112 x)
        internal
        pure
        returns (uq112x112 memory)
    {
        require(x != 0, 'FixedPoint: DIV_BY_ZERO');
        return uq112x112(self._x / uint224(x));
    }

    // multiply a UQ112x112 by a uint, returning a UQ144x112
    // reverts on overflow
    function mul(uq112x112 memory self, uint256 y)
        internal
        pure
        returns (uq144x112 memory)
    {
        uint256 z;
        require(
            y == 0 || (z = uint256(self._x) * y) / y == uint256(self._x),
            'FixedPoint: MULTIPLICATION_OVERFLOW'
        );
        return uq144x112(z);
    }

    // returns a UQ112x112 which represents the ratio of the numerator to the denominator
    // equivalent to encode(numerator).div(denominator)
    function fraction(uint112 numerator, uint112 denominator)
        internal
        pure
        returns (uq112x112 memory)
    {
        require(denominator > 0, 'FixedPoint: DIV_BY_ZERO');
        return uq112x112((uint224(numerator) << RESOLUTION) / denominator);
    }

    // decode a UQ112x112 into a uint112 by truncating after the radix point
    function decode(uq112x112 memory self) internal pure returns (uint112) {
        return uint112(self._x >> RESOLUTION);
    }

    // decode a UQ144x112 into a uint144 by truncating after the radix point
    function decode144(uq144x112 memory self) internal pure returns (uint144) {
        return uint144(self._x >> RESOLUTION);
    }

    // take the reciprocal of a UQ112x112
    function reciprocal(uq112x112 memory self)
        internal
        pure
        returns (uq112x112 memory)
    {
        require(self._x != 0, 'FixedPoint: ZERO_RECIPROCAL');
        return uq112x112(uint224(Q224 / self._x));
    }

    // square root of a UQ112x112
    function sqrt(uq112x112 memory self)
        internal
        pure
        returns (uq112x112 memory)
    {
        return uq112x112(uint224(Babylonian.sqrt(uint256(self._x)) << 56));
    }
}

// File: /Users/sergey/code/basis-gold/solidity/contracts/lib/UniswapV2Library.sol



pragma solidity ^0.6.0;

import '@openzeppelin/contracts/math/SafeMath.sol';
import '../interfaces/IUniswapV2Pair.sol';

library UniswapV2Library {
    using SafeMath for uint256;

    // returns sorted token addresses, used to handle return values from pairs sorted in this order
    function sortTokens(address tokenA, address tokenB)
        internal
        pure
        returns (address token0, address token1)
    {
        require(tokenA != tokenB, 'UniswapV2Library: IDENTICAL_ADDRESSES');
        (token0, token1) = tokenA < tokenB
            ? (tokenA, tokenB)
            : (tokenB, tokenA);
        require(token0 != address(0), 'UniswapV2Library: ZERO_ADDRESS');
    }

    // calculates the CREATE2 address for a pair without making any external calls
    function pairFor(
        address factory,
        address tokenA,
        address tokenB
    ) internal pure returns (address pair) {
        (address token0, address token1) = sortTokens(tokenA, tokenB);
        pair = address(
            uint256(
                keccak256(
                    abi.encodePacked(
                        hex'ff',
                        factory,
                        keccak256(abi.encodePacked(token0, token1)),
                        hex'96e8ac4277198ff8b6f785478aa9a39f403cb768dd02cbee326c3e7da348845f' // init code hash
                    )
                )
            )
        );
    }

    // fetches and sorts the reserves for a pair
    function getReserves(
        address factory,
        address tokenA,
        address tokenB
    ) internal view returns (uint256 reserveA, uint256 reserveB) {
        (address token0, ) = sortTokens(tokenA, tokenB);
        (uint256 reserve0, uint256 reserve1, ) = IUniswapV2Pair(
            pairFor(factory, tokenA, tokenB)
        )
            .getReserves();
        (reserveA, reserveB) = tokenA == token0
            ? (reserve0, reserve1)
            : (reserve1, reserve0);
    }

    // given some amount of an asset and pair reserves, returns an equivalent amount of the other asset
    function quote(
        uint256 amountA,
        uint256 reserveA,
        uint256 reserveB
    ) internal pure returns (uint256 amountB) {
        require(amountA > 0, 'UniswapV2Library: INSUFFICIENT_AMOUNT');
        require(
            reserveA > 0 && reserveB > 0,
            'UniswapV2Library: INSUFFICIENT_LIQUIDITY'
        );
        amountB = amountA.mul(reserveB) / reserveA;
    }

    // given an input amount of an asset and pair reserves, returns the maximum output amount of the other asset
    function getAmountOut(
        uint256 amountIn,
        uint256 reserveIn,
        uint256 reserveOut
    ) internal pure returns (uint256 amountOut) {
        require(amountIn > 0, 'UniswapV2Library: INSUFFICIENT_INPUT_AMOUNT');
        require(
            reserveIn > 0 && reserveOut > 0,
            'UniswapV2Library: INSUFFICIENT_LIQUIDITY'
        );
        uint256 amountInWithFee = amountIn.mul(997);
        uint256 numerator = amountInWithFee.mul(reserveOut);
        uint256 denominator = reserveIn.mul(1000).add(amountInWithFee);
        amountOut = numerator / denominator;
    }

    // given an output amount of an asset and pair reserves, returns a required input amount of the other asset
    function getAmountIn(
        uint256 amountOut,
        uint256 reserveIn,
        uint256 reserveOut
    ) internal pure returns (uint256 amountIn) {
        require(amountOut > 0, 'UniswapV2Library: INSUFFICIENT_OUTPUT_AMOUNT');
        require(
            reserveIn > 0 && reserveOut > 0,
            'UniswapV2Library: INSUFFICIENT_LIQUIDITY'
        );
        uint256 numerator = reserveIn.mul(amountOut).mul(1000);
        uint256 denominator = reserveOut.sub(amountOut).mul(997);
        amountIn = (numerator / denominator).add(1);
    }

    // performs chained getAmountOut calculations on any number of pairs
    function getAmountsOut(
        address factory,
        uint256 amountIn,
        address[] memory path
    ) internal view returns (uint256[] memory amounts) {
        require(path.length >= 2, 'UniswapV2Library: INVALID_PATH');
        amounts = new uint256[](path.length);
        amounts[0] = amountIn;
        for (uint256 i; i < path.length - 1; i++) {
            (uint256 reserveIn, uint256 reserveOut) = getReserves(
                factory,
                path[i],
                path[i + 1]
            );
            amounts[i + 1] = getAmountOut(amounts[i], reserveIn, reserveOut);
        }
    }

    // performs chained getAmountIn calculations on any number of pairs
    function getAmountsIn(
        address factory,
        uint256 amountOut,
        address[] memory path
    ) internal view returns (uint256[] memory amounts) {
        require(path.length >= 2, 'UniswapV2Library: INVALID_PATH');
        amounts = new uint256[](path.length);
        amounts[amounts.length - 1] = amountOut;
        for (uint256 i = path.length - 1; i > 0; i--) {
            (uint256 reserveIn, uint256 reserveOut) = getReserves(
                factory,
                path[i - 1],
                path[i]
            );
            amounts[i - 1] = getAmountIn(amounts[i], reserveIn, reserveOut);
        }
    }
}

// File: /Users/sergey/code/basis-gold/solidity/contracts/lib/UniswapV2OracleLibrary.sol



pragma solidity ^0.6.0;

import './FixedPoint.sol';
import '../interfaces/IUniswapV2Pair.sol';

// library with helper methods for oracles that are concerned with computing average prices
library UniswapV2OracleLibrary {
    using FixedPoint for *;

    // helper function that returns the current block timestamp within the range of uint32, i.e. [0, 2**32 - 1]
    function currentBlockTimestamp() internal view returns (uint32) {
        return uint32(block.timestamp % 2**32);
    }

    // produces the cumulative price using counterfactuals to save gas and avoid a call to sync.
    function currentCumulativePrices(address pair)
        internal
        view
        returns (
            uint256 price0Cumulative,
            uint256 price1Cumulative,
            uint32 blockTimestamp
        )
    {
        blockTimestamp = currentBlockTimestamp();
        price0Cumulative = IUniswapV2Pair(pair).price0CumulativeLast();
        price1Cumulative = IUniswapV2Pair(pair).price1CumulativeLast();

        // if time has elapsed since the last update on the pair, mock the accumulated price values
        (
            uint112 reserve0,
            uint112 reserve1,
            uint32 blockTimestampLast
        ) = IUniswapV2Pair(pair).getReserves();
        if (blockTimestampLast != blockTimestamp) {
            // subtraction overflow is desired
            uint32 timeElapsed = blockTimestamp - blockTimestampLast;
            // addition overflow is desired
            // counterfactual
            price0Cumulative +=
                uint256(FixedPoint.fraction(reserve1, reserve0)._x) *
                timeElapsed;
            // counterfactual
            price1Cumulative +=
                uint256(FixedPoint.fraction(reserve0, reserve1)._x) *
                timeElapsed;
        }
    }
}

// File: /Users/sergey/code/basis-gold/solidity/contracts/owner/Operator.sol



pragma solidity ^0.6.0;

import '@openzeppelin/contracts/GSN/Context.sol';
import '@openzeppelin/contracts/access/Ownable.sol';

contract Operator is Context, Ownable {
    address private _operator;

    event OperatorTransferred(
        address indexed previousOperator,
        address indexed newOperator
    );

    constructor() internal {
        _operator = _msgSender();
        emit OperatorTransferred(address(0), _operator);
    }

    function operator() public view returns (address) {
        return _operator;
    }

    modifier onlyOperator() {
        require(
            _operator == msg.sender,
            'operator: caller is not the operator'
        );
        _;
    }

    function isOperator() public view returns (bool) {
        return _msgSender() == _operator;
    }

    function transferOperator(address newOperator_) public onlyOwner {
        _transferOperator(newOperator_);
    }

    function _transferOperator(address newOperator_) internal {
        require(
            newOperator_ != address(0),
            'operator: zero address given for new operator'
        );
        emit OperatorTransferred(address(0), newOperator_);
        _operator = newOperator_;
    }
}

// File: /Users/sergey/code/basis-gold/solidity/contracts/utils/Epoch.sol



pragma solidity ^0.6.0;

import '@openzeppelin/contracts/math/SafeMath.sol';

import '../owner/Operator.sol';

contract Epoch is Operator {
    using SafeMath for uint256;

    uint256 private period;
    uint256 private startTime;
    uint256 private epoch;

    /* ========== CONSTRUCTOR ========== */

    constructor(
        uint256 _period,
        uint256 _startTime,
        uint256 _startEpoch
    ) public {
        period = _period;
        startTime = _startTime;
        epoch = _startEpoch;
    }

    /* ========== Modifier ========== */

    modifier checkStartTime {
        require(now >= startTime, 'Epoch: not started yet');

        _;
    }

    modifier checkEpoch {
        require(now >= nextEpochPoint(), 'Epoch: not allowed');

        _;

        epoch = epoch.add(1);
    }

    /* ========== VIEW FUNCTIONS ========== */

    function getCurrentEpoch() public view returns (uint256) {
        return epoch;
    }

    function getPeriod() public view returns (uint256) {
        return period;
    }

    function getStartTime() public view returns (uint256) {
        return startTime;
    }

    function nextEpochPoint() public view returns (uint256) {
        return startTime.add(epoch.mul(period));
    }

    /* ========== GOVERNANCE ========== */

    function setPeriod(uint256 _period) external onlyOperator {
        period = _period;
    }
}

// File: @openzeppelin/contracts/GSN/Context.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.6.0;

/*
 * @dev Provides information about the current execution context, including the
 * sender of the transaction and its data. While these are generally available
 * via msg.sender and msg.data, they should not be accessed in such a direct
 * manner, since when dealing with GSN meta-transactions the account sending and
 * paying for execution may not be the actual sender (as far as an application
 * is concerned).
 *
 * This contract is only required for intermediate, library-like contracts.
 */
abstract contract Context {
    function _msgSender() internal view virtual returns (address payable) {
        return msg.sender;
    }

    function _msgData() internal view virtual returns (bytes memory) {
        this; // silence state mutability warning without generating bytecode - see https://github.com/ethereum/solidity/issues/2691
        return msg.data;
    }
}

// File: @openzeppelin/contracts/access/Ownable.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.6.0;

import "../GSN/Context.sol";
/**
 * @dev Contract module which provides a basic access control mechanism, where
 * there is an account (an owner) that can be granted exclusive access to
 * specific functions.
 *
 * By default, the owner account will be the one that deploys the contract. This
 * can later be changed with {transferOwnership}.
 *
 * This module is used through inheritance. It will make available the modifier
 * `onlyOwner`, which can be applied to your functions to restrict their use to
 * the owner.
 */
contract Ownable is Context {
    address private _owner;

    event OwnershipTransferred(address indexed previousOwner, address indexed newOwner);

    /**
     * @dev Initializes the contract setting the deployer as the initial owner.
     */
    constructor () internal {
        address msgSender = _msgSender();
        _owner = msgSender;
        emit OwnershipTransferred(address(0), msgSender);
    }

    /**
     * @dev Returns the address of the current owner.
     */
    function owner() public view returns (address) {
        return _owner;
    }

    /**
     * @dev Throws if called by any account other than the owner.
     */
    modifier onlyOwner() {
        require(_owner == _msgSender(), "Ownable: caller is not the owner");
        _;
    }

    /**
     * @dev Leaves the contract without owner. It will not be possible to call
     * `onlyOwner` functions anymore. Can only be called by the current owner.
     *
     * NOTE: Renouncing ownership will leave the contract without an owner,
     * thereby removing any functionality that is only available to the owner.
     */
    function renounceOwnership() public virtual onlyOwner {
        emit OwnershipTransferred(_owner, address(0));
        _owner = address(0);
    }

    /**
     * @dev Transfers ownership of the contract to a new account (`newOwner`).
     * Can only be called by the current owner.
     */
    function transferOwnership(address newOwner) public virtual onlyOwner {
        require(newOwner != address(0), "Ownable: new owner is the zero address");
        emit OwnershipTransferred(_owner, newOwner);
        _owner = newOwner;
    }
}

// File: @openzeppelin/contracts/math/SafeMath.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.6.0;

/**
 * @dev Wrappers over Solidity's arithmetic operations with added overflow
 * checks.
 *
 * Arithmetic operations in Solidity wrap on overflow. This can easily result
 * in bugs, because programmers usually assume that an overflow raises an
 * error, which is the standard behavior in high level programming languages.
 * `SafeMath` restores this intuition by reverting the transaction when an
 * operation overflows.
 *
 * Using this library instead of the unchecked operations eliminates an entire
 * class of bugs, so it's recommended to use it always.
 */
library SafeMath {
    /**
     * @dev Returns the addition of two unsigned integers, reverting on
     * overflow.
     *
     * Counterpart to Solidity's `+` operator.
     *
     * Requirements:
     *
     * - Addition cannot overflow.
     */
    function add(uint256 a, uint256 b) internal pure returns (uint256) {
        uint256 c = a + b;
        require(c >= a, "SafeMath: addition overflow");

        return c;
    }

    /**
     * @dev Returns the subtraction of two unsigned integers, reverting on
     * overflow (when the result is negative).
     *
     * Counterpart to Solidity's `-` operator.
     *
     * Requirements:
     *
     * - Subtraction cannot overflow.
     */
    function sub(uint256 a, uint256 b) internal pure returns (uint256) {
        return sub(a, b, "SafeMath: subtraction overflow");
    }

    /**
     * @dev Returns the subtraction of two unsigned integers, reverting with custom message on
     * overflow (when the result is negative).
     *
     * Counterpart to Solidity's `-` operator.
     *
     * Requirements:
     *
     * - Subtraction cannot overflow.
     */
    function sub(uint256 a, uint256 b, string memory errorMessage) internal pure returns (uint256) {
        require(b <= a, errorMessage);
        uint256 c = a - b;

        return c;
    }

    /**
     * @dev Returns the multiplication of two unsigned integers, reverting on
     * overflow.
     *
     * Counterpart to Solidity's `*` operator.
     *
     * Requirements:
     *
     * - Multiplication cannot overflow.
     */
    function mul(uint256 a, uint256 b) internal pure returns (uint256) {
        // Gas optimization: this is cheaper than requiring 'a' not being zero, but the
        // benefit is lost if 'b' is also tested.
        // See: https://github.com/OpenZeppelin/openzeppelin-contracts/pull/522
        if (a == 0) {
            return 0;
        }

        uint256 c = a * b;
        require(c / a == b, "SafeMath: multiplication overflow");

        return c;
    }

    /**
     * @dev Returns the integer division of two unsigned integers. Reverts on
     * division by zero. The result is rounded towards zero.
     *
     * Counterpart to Solidity's `/` operator. Note: this function uses a
     * `revert` opcode (which leaves remaining gas untouched) while Solidity
     * uses an invalid opcode to revert (consuming all remaining gas).
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function div(uint256 a, uint256 b) internal pure returns (uint256) {
        return div(a, b, "SafeMath: division by zero");
    }

    /**
     * @dev Returns the integer division of two unsigned integers. Reverts with custom message on
     * division by zero. The result is rounded towards zero.
     *
     * Counterpart to Solidity's `/` operator. Note: this function uses a
     * `revert` opcode (which leaves remaining gas untouched) while Solidity
     * uses an invalid opcode to revert (consuming all remaining gas).
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function div(uint256 a, uint256 b, string memory errorMessage) internal pure returns (uint256) {
        require(b > 0, errorMessage);
        uint256 c = a / b;
        // assert(a == b * c + a % b); // There is no case in which this doesn't hold

        return c;
    }

    /**
     * @dev Returns the remainder of dividing two unsigned integers. (unsigned integer modulo),
     * Reverts when dividing by zero.
     *
     * Counterpart to Solidity's `%` operator. This function uses a `revert`
     * opcode (which leaves remaining gas untouched) while Solidity uses an
     * invalid opcode to revert (consuming all remaining gas).
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function mod(uint256 a, uint256 b) internal pure returns (uint256) {
        return mod(a, b, "SafeMath: modulo by zero");
    }

    /**
     * @dev Returns the remainder of dividing two unsigned integers. (unsigned integer modulo),
     * Reverts with custom message when dividing by zero.
     *
     * Counterpart to Solidity's `%` operator. This function uses a `revert`
     * opcode (which leaves remaining gas untouched) while Solidity uses an
     * invalid opcode to revert (consuming all remaining gas).
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function mod(uint256 a, uint256 b, string memory errorMessage) internal pure returns (uint256) {
        require(b != 0, errorMessage);
        return a % b;
    }
}