Contract Source Code:

// SPDX-License-Identifier: GPL-v3
pragma solidity =0.8.21;

import './interfaces/IDeltaSwapERC20.sol';

contract DeltaSwapERC20 is IDeltaSwapERC20 {
    string public constant override name = 'DeltaSwap V1';
    string public constant override symbol = 'DS-V1';
    uint8 public constant override decimals = 18;
    uint256  public override totalSupply;
    mapping(address => uint256) public override balanceOf;
    mapping(address => mapping(address => uint256)) public override allowance;

    bytes32 public override DOMAIN_SEPARATOR;
    // keccak256("Permit(address owner,address spender,uint256 value,uint256 nonce,uint256 deadline)");
    bytes32 public constant override PERMIT_TYPEHASH = 0x6e71edae12b1b97f4d1f60370fef10105fa2faae0126114a169c64845d6126c9;
    mapping(address => uint256) public override nonces;

    constructor() {
        uint256 chainId;
        assembly {
            chainId := chainid()
        }
        DOMAIN_SEPARATOR = keccak256(
            abi.encode(
                keccak256('EIP712Domain(string name,string version,uint256 chainId,address verifyingContract)'),
                keccak256(bytes(name)),
                keccak256(bytes('1')),
                chainId,
                address(this)
            )
        );
    }

    function _mint(address to, uint256 value) internal {
        totalSupply = totalSupply + value;
        balanceOf[to] = balanceOf[to] + value;
        emit Transfer(address(0), to, value);
    }

    function _burn(address from, uint256 value) internal {
        balanceOf[from] = balanceOf[from]- value;
        totalSupply = totalSupply - value;
        emit Transfer(from, address(0), value);
    }

    function _approve(address owner, address spender, uint256 value) private {
        allowance[owner][spender] = value;
        emit Approval(owner, spender, value);
    }

    function _transfer(address from, address to, uint256 value) private {
        balanceOf[from] = balanceOf[from] - value;
        balanceOf[to] = balanceOf[to] + value;
        emit Transfer(from, to, value);
    }

    function approve(address spender, uint256 value) external override returns (bool) {
        _approve(msg.sender, spender, value);
        return true;
    }

    function transfer(address to, uint256 value) external override returns (bool) {
        _transfer(msg.sender, to, value);
        return true;
    }

    function transferFrom(address from, address to, uint256 value) external override returns (bool) {
        if (allowance[from][msg.sender] != type(uint256).max) {
            allowance[from][msg.sender] -= value;
        }
        _transfer(from, to, value);
        return true;
    }

    function permit(address owner, address spender, uint256 value, uint256 deadline, uint8 v, bytes32 r, bytes32 s) external override {
        require(deadline >= block.timestamp, 'DeltaSwap: EXPIRED');
        bytes32 digest = keccak256(
            abi.encodePacked(
                '\x19\x01',
                DOMAIN_SEPARATOR,
                keccak256(abi.encode(PERMIT_TYPEHASH, owner, spender, value, nonces[owner]++, deadline))
            )
        );
        address recoveredAddress = ecrecover(digest, v, r, s);
        require(recoveredAddress != address(0) && recoveredAddress == owner, 'DeltaSwap: INVALID_SIGNATURE');
        _approve(owner, spender, value);
    }
}

// SPDX-License-Identifier: GPL-v3
pragma solidity =0.8.21;

import './libraries/DSMath.sol';
import './libraries/UQ112x112.sol';
import './interfaces/IERC20.sol';
import './interfaces/IDeltaSwapPair.sol';
import './interfaces/IDeltaSwapFactory.sol';
import './interfaces/IDeltaSwapCallee.sol';
import './DeltaSwapERC20.sol';

contract DeltaSwapPair is DeltaSwapERC20, IDeltaSwapPair {
    using UQ112x112 for uint224;

    uint256 public constant override MINIMUM_LIQUIDITY = 10**3;
    bytes4 private constant SELECTOR = bytes4(keccak256(bytes('transfer(address,uint256)')));

    address public override factory;
    address public override token0;
    address public override token1;

    address public override gammaPool;

    uint112 private liquidityEMA;
    uint32 private lastLiquidityBlockNumber;

    uint112 private tradeLiquidityEMA;     // uses single storage slot
    uint112 private lastTradeLiquiditySum; // uses single storage slot
    uint32 private lastTradeBlockNumber;   // uses single storage slot

    uint112 private reserve0;           // uses single storage slot, accessible via getReserves
    uint112 private reserve1;           // uses single storage slot, accessible via getReserves
    uint32  private blockTimestampLast; // uses single storage slot, accessible via getReserves

    uint256 public override price0CumulativeLast;
    uint256 public override price1CumulativeLast;
    uint256 public override kLast; // reserve0 * reserve1, as of immediately after the most recent liquidity event

    uint256 private unlocked = 1;
    modifier lock() {
        require(unlocked == 1, 'DeltaSwap: LOCKED');
        unlocked = 0;
        _;
        unlocked = 1;
    }

    function getReserves() public override view returns (uint112 _reserve0, uint112 _reserve1, uint32 _blockTimestampLast) {
        _reserve0 = reserve0;
        _reserve1 = reserve1;
        _blockTimestampLast = blockTimestampLast;
    }

    function _safeTransfer(address token, address to, uint256 value) private {
        (bool success, bytes memory data) = token.call(abi.encodeWithSelector(SELECTOR, to, value));
        require(success && (data.length == 0 || abi.decode(data, (bool))), 'DeltaSwap: TRANSFER_FAILED');
    }

    constructor() {
        factory = msg.sender;
    }

    // called once by the factory at time of deployment
    function initialize(address _token0, address _token1) external override {
        require(msg.sender == factory, 'DeltaSwap: FORBIDDEN'); // sufficient check
        token0 = _token0;
        token1 = _token1;
    }

    // called by the factory after deployment
    function setGammaPool(address pool) external override {
        require(msg.sender == factory, 'DeltaSwap: FORBIDDEN'); // sufficient check
        gammaPool = pool;
    }

    // update reserves and, on the first call per block, price accumulators
    function _update(uint256 balance0, uint256 balance1, uint112 _reserve0, uint112 _reserve1) private {
        require(balance0 <= type(uint112).max && balance1 <= type(uint112).max, 'DeltaSwap: OVERFLOW');
        uint32 blockTimestamp = uint32(block.timestamp % 2**32);
        uint32 timeElapsed;
        unchecked {
            timeElapsed = blockTimestamp - blockTimestampLast; // overflow is desired
        }
        if (timeElapsed > 0 && _reserve0 != 0 && _reserve1 != 0) {
            // * never overflows, and + overflow is desired
            unchecked{
                price0CumulativeLast += uint256(UQ112x112.encode(_reserve1).uqdiv(_reserve0)) * timeElapsed;
                price1CumulativeLast += uint256(UQ112x112.encode(_reserve0).uqdiv(_reserve1)) * timeElapsed;
            }
        }
        (uint112 _liquidityEMA, uint32 _lastLiquidityBlockNumber) = getLiquidityEMA(); // saves gas
        if(block.number != _lastLiquidityBlockNumber) {
            liquidityEMA = uint112(DSMath.calcEMA(DSMath.sqrt(balance0 * balance1), _liquidityEMA, DSMath.max(block.number - _lastLiquidityBlockNumber, 10)));
            lastLiquidityBlockNumber = uint32(block.number);
        }
        reserve0 = uint112(balance0);
        reserve1 = uint112(balance1);
        blockTimestampLast = blockTimestamp;
        emit Sync(reserve0, reserve1);
    }

    function _updateLiquidityTradedEMA(uint256 tradeLiquidity) internal virtual returns(uint256 _tradeLiquidityEMA) {
        require(tradeLiquidity > 0, "DeltaSwap: ZERO_TRADE_LIQUIDITY");
        uint256 blockNum = block.number;
        uint256 tradeLiquiditySum;
        (_tradeLiquidityEMA,,tradeLiquiditySum) = _getTradeLiquidityEMA(tradeLiquidity, blockNum);
        lastTradeLiquiditySum = uint112(tradeLiquiditySum);
        tradeLiquidityEMA = uint112(_tradeLiquidityEMA);
        if(lastTradeBlockNumber != blockNum) {
            lastTradeBlockNumber = uint32(blockNum);
        }
    }

    function estimateTradingFee(uint256 tradeLiquidity) external virtual override view returns(uint256 fee) {
        (uint256 _tradeLiquidityEMA,,) = _getTradeLiquidityEMA(tradeLiquidity, block.number);
        fee = calcTradingFee(tradeLiquidity, _tradeLiquidityEMA, liquidityEMA);
    }

    function calcTradingFee(uint256 tradeLiquidity, uint256 lastLiquidityTradedEMA, uint256 lastLiquidityEMA) public virtual override view returns(uint256) {
        (uint8 dsFee, uint8 dsFeeThreshold) = IDeltaSwapFactory(factory).dsFeeInfo();
        if(DSMath.max(tradeLiquidity, lastLiquidityTradedEMA) >= lastLiquidityEMA * dsFeeThreshold / 1000) { // if trade >= threshold, charge fee
            return dsFee;
        }
        return 0;
    }

    function getLiquidityEMA() public virtual override view returns(uint112 _liquidityEMA, uint32 _lastLiquidityBlockNumber) {
        _liquidityEMA = liquidityEMA;
        _lastLiquidityBlockNumber = lastLiquidityBlockNumber;
    }

    function getTradeLiquidityEMAParams() external virtual override view returns(uint112 _tradeLiquidityEMA, uint112 _lastTradeLiquiditySum, uint32 _lastTradeBlockNumber) {
        _tradeLiquidityEMA = tradeLiquidityEMA;
        _lastTradeLiquiditySum = lastTradeLiquiditySum;
        _lastTradeBlockNumber = lastTradeBlockNumber;
    }

    function getTradeLiquidityEMA(uint256 tradeLiquidity) external virtual override view
        returns(uint256 _tradeLiquidityEMA, uint256 lastTradeLiquidityEMA, uint256 tradeLiquiditySum) {
        return _getTradeLiquidityEMA(tradeLiquidity, block.number);
    }

    function _getTradeLiquidityEMA(uint256 tradeLiquidity, uint256 blockNumber) internal virtual view
        returns(uint256 _tradeLiquidityEMA, uint256 lastTradeLiquidityEMA, uint256 tradeLiquiditySum) {
        uint256 blockDiff = blockNumber - lastTradeBlockNumber;
        tradeLiquiditySum = _getLastTradeLiquiditySum(tradeLiquidity, blockDiff);
        lastTradeLiquidityEMA = _getLastTradeLiquidityEMA(blockDiff);
        _tradeLiquidityEMA = tradeLiquidity > 0 ? DSMath.calcEMA(tradeLiquiditySum, lastTradeLiquidityEMA, 20) : lastTradeLiquidityEMA;
    }

    function getLastTradeLiquiditySum(uint256 tradeLiquidity) external virtual override view returns(uint112 _tradeLiquiditySum, uint32 _lastTradeBlockNumber) {
        _lastTradeBlockNumber = lastTradeBlockNumber;
        _tradeLiquiditySum = uint112(_getLastTradeLiquiditySum(tradeLiquidity, block.number - _lastTradeBlockNumber));
    }

    function getLastTradeLiquidityEMA() external virtual override view returns(uint256) {
        return _getLastTradeLiquidityEMA(block.number - lastLiquidityBlockNumber);
    }

    function _getLastTradeLiquiditySum(uint256 tradeLiquidity, uint256 blockDiff) internal virtual view returns(uint256) {
        if(blockDiff > 0) {
            return tradeLiquidity;
        } else {
            return lastTradeLiquiditySum + tradeLiquidity;
        }
    }

    function _getLastTradeLiquidityEMA(uint256 blockDiff) internal virtual view returns(uint256) {
        if(blockDiff > 50) {
            // if no trade in 50 blocks (~10 minutes), then reset
            return 0;
        }
        return tradeLiquidityEMA;
    }

    // if fee is on, mint liquidity equivalent to 1/6th of the growth in sqrt(k)
    function _mintFee(uint112 _reserve0, uint112 _reserve1) private returns (bool feeOn) {
        (address feeTo, uint256 feeNum) = IDeltaSwapFactory(factory).feeInfo();
        feeOn = feeTo != address(0);
        uint256 _kLast = kLast; // gas savings
        if (feeOn) {
            if (_kLast != 0) {
                uint256 rootK = DSMath.sqrt(uint256(_reserve0) * _reserve1);
                uint256 rootKLast = DSMath.sqrt(_kLast);
                if (rootK > rootKLast) {
                    uint256 numerator = totalSupply * (rootK - rootKLast);
                    uint256 denominator = rootK * feeNum / 1000 + rootKLast;
                    uint256 liquidity = numerator / denominator;
                    if (liquidity > 0) _mint(feeTo, liquidity);
                }
            }
        } else if (_kLast != 0) {
            kLast = 0;
        }
    }

    // this low-level function should be called from a contract which performs important safety checks
    function mint(address to) external override lock returns (uint256 liquidity) {
        (uint112 _reserve0, uint112 _reserve1,) = getReserves(); // gas savings
        uint256 balance0 = IERC20(token0).balanceOf(address(this));
        uint256 balance1 = IERC20(token1).balanceOf(address(this));
        uint256 amount0 = balance0 - _reserve0;
        uint256 amount1 = balance1 - _reserve1;

        bool feeOn = _mintFee(_reserve0, _reserve1);
        uint256 _totalSupply = totalSupply; // gas savings, must be defined here since totalSupply can update in _mintFee
        if (_totalSupply == 0) {
            liquidity = DSMath.sqrt(amount0 * amount1) - MINIMUM_LIQUIDITY;
            _mint(address(0), MINIMUM_LIQUIDITY); // permanently lock the first MINIMUM_LIQUIDITY tokens
        } else {
            liquidity = DSMath.min(amount0 * _totalSupply / _reserve0, amount1 * _totalSupply / _reserve1);
        }
        require(liquidity > 0, 'DeltaSwap: INSUFFICIENT_LIQUIDITY_MINTED');
        _mint(to, liquidity);

        _update(balance0, balance1, _reserve0, _reserve1);
        if (feeOn) kLast = uint256(reserve0) * reserve1; // reserve0 and reserve1 are up-to-date
        emit Mint(msg.sender, amount0, amount1);
    }

    // this low-level function should be called from a contract which performs important safety checks
    function burn(address to) external override lock returns (uint256 amount0, uint256 amount1) {
        (uint112 _reserve0, uint112 _reserve1,) = getReserves(); // gas savings
        address _token0 = token0;                                // gas savings
        address _token1 = token1;                                // gas savings
        uint256 balance0 = IERC20(_token0).balanceOf(address(this));
        uint256 balance1 = IERC20(_token1).balanceOf(address(this));
        uint256 liquidity = balanceOf[address(this)];

        bool feeOn = _mintFee(_reserve0, _reserve1);
        uint256 _totalSupply = totalSupply; // gas savings, must be defined here since totalSupply can update in _mintFee
        amount0 = liquidity * balance0 / _totalSupply; // using balances ensures pro-rata distribution
        amount1 = liquidity * balance1 / _totalSupply; // using balances ensures pro-rata distribution
        require(amount0 > 0 && amount1 > 0, 'DeltaSwap: INSUFFICIENT_LIQUIDITY_BURNED');
        _burn(address(this), liquidity);
        _safeTransfer(_token0, to, amount0);
        _safeTransfer(_token1, to, amount1);
        balance0 = IERC20(_token0).balanceOf(address(this));
        balance1 = IERC20(_token1).balanceOf(address(this));

        _update(balance0, balance1, _reserve0, _reserve1);
        if (feeOn) kLast = uint256(reserve0) * reserve1; // reserve0 and reserve1 are up-to-date
        emit Burn(msg.sender, amount0, amount1, to);
    }

    // this low-level function should be called from a contract which performs important safety checks
    function swap(uint256 amount0Out, uint256 amount1Out, address to, bytes calldata data) external override lock {
        require(amount0Out > 0 || amount1Out > 0, 'DeltaSwap: INSUFFICIENT_OUTPUT_AMOUNT');
        (uint112 _reserve0, uint112 _reserve1,) = getReserves(); // gas savings
        require(amount0Out < _reserve0 && amount1Out < _reserve1, 'DeltaSwap: INSUFFICIENT_LIQUIDITY');

        uint256 balance0;
        uint256 balance1;
        { // scope for _token{0,1}, avoids stack too deep errors
            address _token0 = token0;
            address _token1 = token1;
            require(to != _token0 && to != _token1, 'DeltaSwap: INVALID_TO');
            if (amount0Out > 0) _safeTransfer(_token0, to, amount0Out); // optimistically transfer tokens
            if (amount1Out > 0) _safeTransfer(_token1, to, amount1Out); // optimistically transfer tokens
            if (data.length > 0) IDeltaSwapCallee(to).deltaSwapCall(msg.sender, amount0Out, amount1Out, data);
            balance0 = IERC20(_token0).balanceOf(address(this));
            balance1 = IERC20(_token1).balanceOf(address(this));
        }
        uint256 amount0In = balance0 > _reserve0 - amount0Out ? balance0 - (_reserve0 - amount0Out) : 0;
        uint256 amount1In = balance1 > _reserve1 - amount1Out ? balance1 - (_reserve1 - amount1Out) : 0;
        require(amount0In > 0 || amount1In > 0, 'DeltaSwap: INSUFFICIENT_INPUT_AMOUNT');
        { // scope for reserve{0,1}Adjusted, avoids stack too deep errors
            uint256 fee;
            if(msg.sender != gammaPool) {
                uint256 tradeLiquidity = DSMath.calcTradeLiquidity(amount0In, amount1In, _reserve0, _reserve1);
                fee = calcTradingFee(tradeLiquidity, _updateLiquidityTradedEMA(tradeLiquidity), liquidityEMA);
            } else {
                fee = IDeltaSwapFactory(factory).gsFee();
            }
            uint256 balance0Adjusted = balance0 * 1000 - amount0In * fee;
            uint256 balance1Adjusted = balance1 * 1000 - amount1In * fee;
            require(balance0Adjusted * balance1Adjusted >= uint256(_reserve0) * _reserve1 * (1000**2), 'DeltaSwap: K');
        }

        _update(balance0, balance1, _reserve0, _reserve1);
        emit Swap(msg.sender, amount0In, amount1In, amount0Out, amount1Out, to);
    }

    // force balances to match reserves
    function skim(address to) external override lock {
        address _token0 = token0; // gas savings
        address _token1 = token1; // gas savings
        _safeTransfer(_token0, to, IERC20(_token0).balanceOf(address(this)) - reserve0);
        _safeTransfer(_token1, to, IERC20(_token1).balanceOf(address(this)) - reserve1);
    }

    // force reserves to match balances
    function sync() external override lock {
        _update(IERC20(token0).balanceOf(address(this)), IERC20(token1).balanceOf(address(this)), reserve0, reserve1);
    }
}

// SPDX-License-Identifier: GPL-v3
pragma solidity >=0.5.0;

interface IDeltaSwapCallee {
    function deltaSwapCall(address sender, uint256 amount0, uint256 amount1, bytes calldata data) external;
}

// SPDX-License-Identifier: GPL-v3
pragma solidity >=0.5.0;

interface IDeltaSwapERC20 {
    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;
}

// SPDX-License-Identifier: GPL-v3
pragma solidity >=0.5.0;

interface IDeltaSwapFactory {
    event PairCreated(address indexed token0, address indexed token1, address pair, uint256);
    event GammaPoolSet(address indexed pair, address gammaPool);

    function feeTo() external view returns (address);
    function feeNum() external view returns (uint16);
    function feeToSetter() external view returns (address);
    function gammaPoolSetter() external view returns (address);
    function gsFactory() external view returns(address);
    function gsProtocolId() external view returns(uint16);
    function gsFee() external view returns(uint8);
    function dsFee() external view returns(uint8);
    function dsFeeThreshold() external view returns(uint8);

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

    function setFeeNum(uint16) external;
    function setFeeTo(address) external;
    function setFeeToSetter(address) external;

    function setGSFactory(address factory) external;
    function setGSProtocolId(uint16 protocolId) external;
    function setGSFee(uint8 fee) external;
    function setDSFee(uint8 fee) external;
    function setDSFeeThreshold(uint8 feeThreshold) external;

    function feeInfo() external view returns (address,uint16);
    function dsFeeInfo() external view returns (uint8,uint8);

    function setGammaPoolSetter(address) external;
    function updateGammaPool(address tokenA, address tokenB) external;
}

// SPDX-License-Identifier: GPL-v3
pragma solidity >=0.5.0;

import "./IDeltaSwapERC20.sol";

interface IDeltaSwapPair is IDeltaSwapERC20 {

    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;

    function gammaPool() external view returns (address);
    function setGammaPool(address gammaPool) external;

    function getLiquidityEMA() external view returns(uint112 liquidityEMA, uint32 lastLiquidityEMABlockNumber);
    function getLastTradeLiquidityEMA() external view returns(uint256);
    function getTradeLiquidityEMA(uint256 tradeLiquidity) external view returns(uint256 tradeLiquidityEMA, uint256 lastTradeLiquidityEMA, uint256 tradeLiquiditySum);
    function getTradeLiquidityEMAParams() external view returns(uint112 _tradeLiquidityEMA, uint112 _lastTradeLiquiditySum, uint32 _lastTradeBlockNumber);
    function getLastTradeLiquiditySum(uint256 tradeLiquidity) external view returns(uint112 _tradeLiquiditySum, uint32 _lastTradeBlockNum);
    function estimateTradingFee(uint256 tradeLiquidity) external view returns(uint256 fee);
    function calcTradingFee(uint256 tradeLiquidity, uint256 tradeLiquidityEMA, uint256 liquidityEMA) external view returns(uint256 fee);
}

// SPDX-License-Identifier: GPL-v3
pragma solidity >=0.5.0;

interface IERC20 {
    event Approval(address indexed owner, address indexed spender, uint256 value);
    event Transfer(address indexed from, address indexed to, uint256 value);

    function name() external view returns (string memory);
    function symbol() external view returns (string memory);
    function decimals() external view 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);
}

// SPDX-License-Identifier: GPL-v3
pragma solidity >=0.5.16;

// a library for performing various math operations

library DSMath {

    function max(uint256 x, uint256 y) internal pure returns (uint256 z) {
        z = x > y ? x : y;
    }

    function min(uint256 x, uint256 y) internal pure returns (uint256 z) {
        z = x < y ? x : y;
    }

    // babylonian method (https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method)
    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;
        }
    }

    /// @dev Update pool invariant, LP tokens borrowed plus interest, interest rate index, and last block update
    /// @param last - last value added to ema calculation
    /// @param emaLast - last calculated ema
    /// @param emaWeight - weight given to last value in ema calculation compared to last ema value
    /// @return ema - result of ema calculation
    function calcEMA(uint256 last, uint256 emaLast, uint256 emaWeight) internal pure returns(uint256) {
        if(emaLast == 0) {
            return last;
        } else {
            emaWeight = min(100, emaWeight);
            // EMA_1 = last * weight + EMA_0 * (1 - weight)
            return last * emaWeight / 100 + emaLast * (100 - emaWeight) / 100;
        }
    }

    function calcSingleSideLiquidity(uint256 amount, uint256 reserve0, uint256 reserve1) internal pure returns(uint256) {
        uint256 amount0 = amount / 2;
        uint256 amount1 = amount0 * reserve1 / reserve0;
        return DSMath.sqrt(amount0 * amount1);
    }

    function calcTradeLiquidity(uint256 amount0, uint256 amount1, uint256 reserve0, uint256 reserve1) internal pure returns(uint256) {
        return max(
            calcSingleSideLiquidity(amount0, reserve0, reserve1),
                calcSingleSideLiquidity(amount1, reserve1, reserve0));
    }
}

// SPDX-License-Identifier: GPL-v3
pragma solidity >=0.5.16;

// a library for handling binary fixed point numbers (https://en.wikipedia.org/wiki/Q_(number_format))

// range: [0, 2**112 - 1]
// resolution: 1 / 2**112

library UQ112x112 {
    uint224 constant Q112 = 2**112;

    // encode a uint112 as a UQ112x112
    function encode(uint112 y) internal pure returns (uint224 z) {
        z = uint224(y) * Q112; // never overflows
    }

    // divide a UQ112x112 by a uint112, returning a UQ112x112
    function uqdiv(uint224 x, uint112 y) internal pure returns (uint224 z) {
        z = x / uint224(y);
    }
}