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// SPDX-License-Identifier: BUSL-1.1

pragma solidity 0.8.18;

import { PRBMathSD59x18 } from "@prb-math/contracts/PRBMathSD59x18.sol";
import { PRBMathUD60x18 } from "@prb-math/contracts/PRBMathUD60x18.sol";

import { IERC20 }    from "@openzeppelin/contracts/token/ERC20/IERC20.sol";
import { SafeERC20 } from "@openzeppelin/contracts/token/ERC20/utils/SafeERC20.sol";


import { 
    DepositsState, 
    EmaState, 
    InflatorState,
    InterestState, 
    PoolBalancesState, 
    PoolState 
} from '../../interfaces/pool/commons/IPoolState.sol';
import { IERC3156FlashBorrower }                             from '../../interfaces/pool/IERC3156FlashBorrower.sol';

import { 
    _dwatp,
    _htp,
    _indexOf,
    MAX_FENWICK_INDEX,
    MIN_PRICE, MAX_PRICE
} from '../helpers/PoolHelper.sol';

import { Deposits } from '../internal/Deposits.sol';
import { Buckets }  from '../internal/Buckets.sol';
import { Loans }    from '../internal/Loans.sol';
import { Maths }    from '../internal/Maths.sol';

/**
    @title  PoolCommons library
    @notice External library containing logic for common pool functionality:
            - interest rate accrual and interest rate params update
            - pool utilization
 */
library PoolCommons {
    using SafeERC20 for IERC20;


    /*****************/
    /*** Constants ***/
    /*****************/

    uint256 internal constant CUBIC_ROOT_1000000 = 100 * 1e18;
    uint256 internal constant ONE_THIRD          = 0.333333333333333334 * 1e18;

    uint256 internal constant INCREASE_COEFFICIENT = 1.1 * 1e18;
    uint256 internal constant DECREASE_COEFFICIENT = 0.9 * 1e18;
    int256  internal constant PERCENT_102          = 1.02 * 1e18;
    int256  internal constant NEG_H_MAU_HOURS      = -0.057762265046662105 * 1e18; // -ln(2)/12
    int256  internal constant NEG_H_TU_HOURS       = -0.008251752149523158 * 1e18; // -ln(2)/84

    /**************/
    /*** Events ***/
    /**************/

    // See `IPoolEvents` for descriptions
    event Flashloan(address indexed receiver, address indexed token, uint256 amount);
    event ResetInterestRate(uint256 oldRate, uint256 newRate);
    event UpdateInterestRate(uint256 oldRate, uint256 newRate);

    /**************/
    /*** Errors ***/
    /**************/

    // See `IPoolErrors` for descriptions
    error FlashloanCallbackFailed();
    error FlashloanIncorrectBalance();

    /*************************/
    /*** Local Var Structs ***/
    /*************************/

    /// @dev Struct used for `updateInterestState` function local vars.
    struct UpdateInterestLocalVars {
        uint256 debtEma;
        uint256 depositEma;
        uint256 debtColEma;
        uint256 lupt0DebtEma;
        uint256 t0Debt2ToCollateral;
        uint256 newMeaningfulDeposit;
        uint256 newDebt;
        uint256 newDebtCol;
        uint256 newLupt0Debt;
        uint256 lastEmaUpdate;
        int256 elapsed;
        int256 weightMau;
        int256 weightTu;
        uint256 newInterestRate;
        uint256 nonAuctionedT0Debt;
    }

    /**************************/
    /*** External Functions ***/
    /**************************/

    /**
     *  @notice Calculates EMAs, caches values required for calculating interest rate, and saves new values in storage.
     *  @notice Calculates new pool interest rate (Never called more than once every 12 hours) and saves new values in storage.
     *  @dev    === Write state ===
     *  @dev    `EMA`s state
     *  @dev    interest rate accumulator and `interestRateUpdate` state
     *  @dev    === Emit events ===
     *  @dev    - `UpdateInterestRate` / `ResetInterestRate`
     */
    function updateInterestState(
        InterestState storage interestParams_,
        EmaState      storage emaParams_,
        DepositsState storage deposits_,
        PoolState memory poolState_,
        uint256 lup_
    ) external {
        UpdateInterestLocalVars memory vars;
        // load existing EMA values
        vars.debtEma       = emaParams_.debtEma;
        vars.depositEma    = emaParams_.depositEma;
        vars.debtColEma    = emaParams_.debtColEma;
        vars.lupt0DebtEma  = emaParams_.lupt0DebtEma;
        vars.lastEmaUpdate = emaParams_.emaUpdate;

        vars.t0Debt2ToCollateral = interestParams_.t0Debt2ToCollateral;

        // calculate new interest params
        vars.nonAuctionedT0Debt = poolState_.t0Debt - poolState_.t0DebtInAuction;
        vars.newDebt = Maths.wmul(vars.nonAuctionedT0Debt, poolState_.inflator);
        // new meaningful deposit cannot be less than pool's debt
        vars.newMeaningfulDeposit = Maths.max(
            _meaningfulDeposit(
                deposits_,
                poolState_.t0DebtInAuction,
                vars.nonAuctionedT0Debt,
                poolState_.inflator,
                vars.t0Debt2ToCollateral
            ),
            vars.newDebt
        );
        vars.newDebtCol   = Maths.wmul(poolState_.inflator, vars.t0Debt2ToCollateral);
        vars.newLupt0Debt = Maths.wmul(lup_, vars.nonAuctionedT0Debt);

        // update EMAs only once per block
        if (vars.lastEmaUpdate != block.timestamp) {

            // first time EMAs are updated, initialize EMAs
            if (vars.lastEmaUpdate == 0) {
                vars.debtEma      = vars.newDebt;
                vars.depositEma   = vars.newMeaningfulDeposit;
                vars.debtColEma   = vars.newDebtCol;
                vars.lupt0DebtEma = vars.newLupt0Debt;
            } else {
                vars.elapsed   = int256(Maths.wdiv(block.timestamp - vars.lastEmaUpdate, 1 hours));
                vars.weightMau = PRBMathSD59x18.exp(PRBMathSD59x18.mul(NEG_H_MAU_HOURS, vars.elapsed));
                vars.weightTu  = PRBMathSD59x18.exp(PRBMathSD59x18.mul(NEG_H_TU_HOURS,  vars.elapsed));

                // calculate the t0 debt EMA, used for MAU
                vars.debtEma = uint256(
                    PRBMathSD59x18.mul(vars.weightMau, int256(vars.debtEma)) +
                    PRBMathSD59x18.mul(1e18 - vars.weightMau, int256(interestParams_.debt))
                );

                // update the meaningful deposit EMA, used for MAU
                vars.depositEma = uint256(
                    PRBMathSD59x18.mul(vars.weightMau, int256(vars.depositEma)) +
                    PRBMathSD59x18.mul(1e18 - vars.weightMau, int256(interestParams_.meaningfulDeposit))
                );

                // calculate the debt squared to collateral EMA, used for TU
                vars.debtColEma = uint256(
                    PRBMathSD59x18.mul(vars.weightTu, int256(vars.debtColEma)) +
                    PRBMathSD59x18.mul(1e18 - vars.weightTu, int256(interestParams_.debtCol))
                );

                // calculate the EMA of LUP * t0 debt
                vars.lupt0DebtEma = uint256(
                    PRBMathSD59x18.mul(vars.weightTu, int256(vars.lupt0DebtEma)) +
                    PRBMathSD59x18.mul(1e18 - vars.weightTu, int256(interestParams_.lupt0Debt))
                );
            }

            // save EMAs in storage
            emaParams_.debtEma      = vars.debtEma;
            emaParams_.depositEma   = vars.depositEma;
            emaParams_.debtColEma   = vars.debtColEma;
            emaParams_.lupt0DebtEma = vars.lupt0DebtEma;

            // save last EMA update time
            emaParams_.emaUpdate = block.timestamp;
        }

        // reset interest rate if pool rate > 10% and debtEma < 5% of depositEma
        if (
            poolState_.rate > 0.1 * 1e18
            &&
            vars.debtEma < Maths.wmul(vars.depositEma, 0.05 * 1e18)
        ) {
            interestParams_.interestRate       = uint208(0.1 * 1e18);
            interestParams_.interestRateUpdate = uint48(block.timestamp);

            emit ResetInterestRate(
                poolState_.rate,
                0.1 * 1e18
            );
        }
        // otherwise calculate and update interest rate if it has been more than 12 hours since the last update
        else if (block.timestamp - interestParams_.interestRateUpdate > 12 hours) {
            vars.newInterestRate = _calculateInterestRate(
                poolState_,
                vars.debtEma,
                vars.depositEma,
                vars.debtColEma,
                vars.lupt0DebtEma
            );

            if (poolState_.rate != vars.newInterestRate) {
                interestParams_.interestRate       = uint208(vars.newInterestRate);
                interestParams_.interestRateUpdate = uint48(block.timestamp);

                emit UpdateInterestRate(
                    poolState_.rate,
                    vars.newInterestRate
                );
            }
        }

        // save new interest rate params to storage
        interestParams_.debt              = vars.newDebt;
        interestParams_.meaningfulDeposit = vars.newMeaningfulDeposit;
        interestParams_.debtCol           = vars.newDebtCol;
        interestParams_.lupt0Debt         = vars.newLupt0Debt;
    }

    /**
     *  @notice Calculates new pool interest and scale the fenwick tree to update amount of debt owed to lenders (saved in storage).
     *  @dev    === Write state ===
     *  @dev    - `Deposits.mult` (scale `Fenwick` tree with new interest accrued):
     *  @dev      update `scaling` array state
     *  @param  emaParams_             Struct for pool `EMA`s state.
     *  @param  deposits_              Struct for pool deposits state.
     *  @param  poolState_             Current state of the pool.
     *  @param  maxT0DebtToCollateral_ Max t0 debt to collateral in Pool.
     *  @param  elapsed_               Time elapsed since last inflator update.
     *  @return newInflator_           The new value of pool inflator.
     *  @return newInterest_           The new interest accrued.
     */
    function accrueInterest(
        EmaState      storage emaParams_,
        DepositsState storage deposits_,
        PoolState calldata poolState_,
        uint256 maxT0DebtToCollateral_,
        uint256 elapsed_
    ) external returns (uint256 newInflator_, uint256 newInterest_) {
        // Scale the borrower inflator to update amount of interest owed by borrowers
        uint256 pendingFactor = PRBMathUD60x18.exp((poolState_.rate * elapsed_) / 365 days);

        // calculate the highest threshold price
        newInflator_ = Maths.wmul(poolState_.inflator, pendingFactor);
        uint256 htp  = _htp(maxT0DebtToCollateral_, poolState_.inflator);

        uint256 accrualIndex;
        if (htp > MAX_PRICE)      accrualIndex = 1;                 // if HTP is over the highest price bucket then no buckets earn interest
        else if (htp < MIN_PRICE) accrualIndex = MAX_FENWICK_INDEX; // if HTP is under the lowest price bucket then all buckets earn interest
        else                      accrualIndex = _indexOf(htp);     // else HTP bucket earn interest

        uint256 lupIndex = Deposits.findIndexOfSum(deposits_, poolState_.debt);
        // accrual price is less of lup and htp, and prices decrease as index increases
        if (lupIndex > accrualIndex) accrualIndex = lupIndex;

        uint256 interestEarningDeposit = Deposits.prefixSum(deposits_, accrualIndex);

        if (interestEarningDeposit != 0) {
            newInterest_ = Maths.wmul(
                _lenderInterestMargin(_utilization(emaParams_.debtEma, emaParams_.depositEma)),
                Maths.wmul(pendingFactor - Maths.WAD, poolState_.debt)
            );

            // lender factor computation, capped at 10x the interest factor for borrowers
            uint256 lenderFactor = Maths.min(
                Maths.floorWdiv(newInterest_, interestEarningDeposit),
                Maths.wmul(pendingFactor - Maths.WAD, Maths.wad(10))
            ) + Maths.WAD;

            // Scale the fenwick tree to update amount of debt owed to lenders
            Deposits.mult(deposits_, accrualIndex, lenderFactor);
        }
    }

    /**
     *  @notice Executes a flashloan from current pool.
     *  @dev    === Reverts on ===
     *  @dev    - `FlashloanCallbackFailed()` if receiver is not an `ERC3156FlashBorrower`
     *  @dev    - `FlashloanIncorrectBalance()` if pool balance after flashloan is different than initial balance
     *  @param  receiver_ Address of the contract which implements the appropriate interface to receive tokens.
     *  @param  token_    Address of the `ERC20` token caller wants to borrow.
     *  @param  amount_   The denormalized amount (dependent upon token precision) of tokens to borrow.
     *  @param  data_     User-defined calldata passed to the receiver.
     */
    function flashLoan(
        IERC3156FlashBorrower receiver_,
        address token_, 
        uint256 amount_,
        bytes calldata data_
    ) external {
        IERC20 tokenContract = IERC20(token_);

        uint256 initialBalance = tokenContract.balanceOf(address(this));

        tokenContract.safeTransfer(
            address(receiver_),
            amount_
        );

        if (receiver_.onFlashLoan(msg.sender, token_, amount_, 0, data_) != 
            keccak256("ERC3156FlashBorrower.onFlashLoan")) revert FlashloanCallbackFailed();

        tokenContract.safeTransferFrom(
            address(receiver_),
            address(this),
            amount_
        );

        if (tokenContract.balanceOf(address(this)) != initialBalance) revert FlashloanIncorrectBalance();

        emit Flashloan(address(receiver_), token_, amount_);
    }

    /**************************/
    /*** Internal Functions ***/
    /**************************/

    /**
     *  @notice Calculates new pool interest rate.
     */
    function _calculateInterestRate(
        PoolState memory poolState_,
        uint256 debtEma_,
        uint256 depositEma_,
        uint256 debtColEma_,
        uint256 lupt0DebtEma_
    ) internal pure returns (uint256 newInterestRate_)  {
        // meaningful actual utilization
        int256 mau;
        // meaningful actual utilization * 1.02
        int256 mau102;

        if (poolState_.debt != 0) {
            // calculate meaningful actual utilization for interest rate update
            mau    = int256(_utilization(debtEma_, depositEma_));
            mau102 = (mau * PERCENT_102) / 1e18;
        }

        // calculate target utilization
        int256 tu = (lupt0DebtEma_ != 0) ? 
            int256(Maths.wdiv(debtColEma_, lupt0DebtEma_)) : int(Maths.WAD);

        newInterestRate_ = poolState_.rate;

        // raise rates if 4*(tu-1.02*mau) < (tu+1.02*mau-1)^2-1
        if (4 * (tu - mau102) < (((tu + mau102 - 1e18) / 1e9) ** 2) - 1e18) {
            newInterestRate_ = Maths.wmul(poolState_.rate, INCREASE_COEFFICIENT);
        // decrease rates if 4*(tu-mau) > 1-(tu+mau-1)^2
        } else if (4 * (tu - mau) > 1e18 - ((tu + mau - 1e18) / 1e9) ** 2) {
            newInterestRate_ = Maths.wmul(poolState_.rate, DECREASE_COEFFICIENT);
        }

        // bound rates between 10 bps and 400%
        newInterestRate_ = Maths.min(4 * 1e18, Maths.max(0.001 * 1e18, newInterestRate_));
    }

    /**
     *  @notice Calculates pool meaningful actual utilization.
     *  @param  debtEma_     `EMA` of pool debt.
     *  @param  depositEma_  `EMA` of meaningful pool deposit.
     *  @return utilization_ Pool meaningful actual utilization value.
     */
    function _utilization(
        uint256 debtEma_,
        uint256 depositEma_
    ) internal pure returns (uint256 utilization_) {
        if (depositEma_ != 0) utilization_ = Maths.wdiv(debtEma_, depositEma_);
    }

    /**
     *  @notice Calculates lender interest margin.
     *  @param  mau_ Meaningful actual utilization.
     *  @return The lender interest margin value.
     */
    function _lenderInterestMargin(
        uint256 mau_
    ) internal pure returns (uint256) {
        // Net Interest Margin = ((1 - MAU1)^(1/3) * 0.15)
        // Where MAU1 is MAU capped at 100% (min(MAU,1))
        // Lender Interest Margin = 1 - Net Interest Margin

        // PRBMath library forbids raising a number < 1e18 to a power.  Using the product and quotient rules of 
        // exponents, rewrite the equation with a coefficient s which provides sufficient precision:
        // Net Interest Margin = ((1 - MAU1) * s)^(1/3) / s^(1/3) * 0.15

        uint256 base = 1_000_000 * 1e18 - Maths.min(mau_, 1e18) * 1_000_000;
        // If unutilized deposit is infinitessimal, lenders get 100% of interest.
        if (base < 1e18) {
            return 1e18;
        } else {
            // cubic root of the percentage of meaningful unutilized deposit
            uint256 crpud = PRBMathUD60x18.pow(base, ONE_THIRD);
            // finish calculating Net Interest Margin, and then convert to Lender Interest Margin
            return 1e18 - Maths.wdiv(Maths.wmul(crpud, 0.15 * 1e18), CUBIC_ROOT_1000000);
        }
    }

    /**
     *  @notice Calculates pool's meaningful deposit.
     *  @param  deposits_            Struct for pool deposits state.
     *  @param  t0DebtInAuction_     Value of pool's t0 debt currently in auction.
     *  @param  nonAuctionedT0Debt_  Value of pool's t0 debt that is not in auction.
     *  @param  inflator_            Pool's current inflator.
     *  @param  t0Debt2ToCollateral_ `t0Debt2ToCollateral` ratio.
     *  @return meaningfulDeposit_   Pool's meaningful deposit.
     */
    function _meaningfulDeposit(
        DepositsState storage deposits_,
        uint256 t0DebtInAuction_,
        uint256 nonAuctionedT0Debt_,
        uint256 inflator_,
        uint256 t0Debt2ToCollateral_
    ) internal view returns (uint256 meaningfulDeposit_) {
        uint256 dwatp = _dwatp(nonAuctionedT0Debt_, inflator_, t0Debt2ToCollateral_);
        if (dwatp == 0) {
            meaningfulDeposit_ = Deposits.treeSum(deposits_);
        } else {
            if      (dwatp >= MAX_PRICE) meaningfulDeposit_ = 0;
            else if (dwatp >= MIN_PRICE) meaningfulDeposit_ = Deposits.prefixSum(deposits_, _indexOf(dwatp));
            else                         meaningfulDeposit_ = Deposits.treeSum(deposits_);
        }
        meaningfulDeposit_ -= Maths.min(
            meaningfulDeposit_,
            Maths.wmul(t0DebtInAuction_, inflator_)
        );
    }

    /**********************/
    /*** View Functions ***/
    /**********************/

    /**
     *  @notice Calculates pool related debt values.
     *  @param poolBalances_  Pool debt
     *  @param inflatorState_ Interest inflator and last update time
     *  @param interestState_ Interest rate and t0Debt2ToCollateral accumulator
     *  @return Current amount of debt owed by borrowers in pool.
     *  @return Debt owed by borrowers based on last inflator snapshot.
     *  @return Total amount of debt in auction.
     *  @return t0debt accross all borrowers divided by their collateral, used in determining a collateralization weighted debt.  
     */
    function debtInfo(
        PoolBalancesState memory poolBalances_,
        InflatorState     memory inflatorState_,
        InterestState     memory interestState_
    ) external view returns (uint256, uint256, uint256, uint256) {
        uint256 t0Debt   = poolBalances_.t0Debt;
        uint256 inflator = inflatorState_.inflator;

        return (
            Maths.ceilWmul(
                t0Debt,
                pendingInflator(inflator, inflatorState_.inflatorUpdate, interestState_.interestRate)
            ),
            Maths.ceilWmul(t0Debt, inflator),
            Maths.ceilWmul(poolBalances_.t0DebtInAuction, inflator),
            interestState_.t0Debt2ToCollateral
        );
    }

    /**
     *  @notice Calculates pool interest factor for a given interest rate and time elapsed since last inflator update.
     *  @param  interestRate_   Current pool interest rate.
     *  @param  elapsed_        Time elapsed since last inflator update.
     *  @return The value of pool interest factor.
     */
    function pendingInterestFactor(
        uint256 interestRate_,
        uint256 elapsed_
    ) external pure returns (uint256) {
        return PRBMathUD60x18.exp((interestRate_ * elapsed_) / 365 days);
    }

    /**
     *  @notice Calculates pool pending inflator given the current inflator, time of last update and current interest rate.
     *  @param  inflator_      Current pool inflator.
     *  @param  inflatorUpdate Timestamp when inflator was updated.
     *  @param  interestRate_  The interest rate of the pool.
     *  @return The pending value of pool inflator.
     */
    function pendingInflator(
        uint256 inflator_,
        uint256 inflatorUpdate,
        uint256 interestRate_
    ) public view returns (uint256) {
        return Maths.wmul(
            inflator_,
            PRBMathUD60x18.exp((interestRate_ * (block.timestamp - inflatorUpdate)) / 365 days)
        );
    }

    /**
     *  @notice Calculates lender interest margin for a given meaningful actual utilization.
     *  @dev Wrapper of the internal function.
     */
    function lenderInterestMargin(
        uint256 mau_
    ) external pure returns (uint256) {
        return _lenderInterestMargin(mau_);
    }

    /**
     *  @notice Calculates pool meaningful actual utilization.
     *  @dev Wrapper of the internal function.
     */
    function utilization(
        EmaState storage emaParams_
    ) external view returns (uint256 utilization_) {
        return _utilization(emaParams_.debtEma, emaParams_.depositEma);
    }
}

// SPDX-License-Identifier: Unlicense
pragma solidity >=0.8.4;

import "./PRBMath.sol";

/// @title PRBMathSD59x18
/// @author Paul Razvan Berg
/// @notice Smart contract library for advanced fixed-point math that works with int256 numbers considered to have 18
/// trailing decimals. We call this number representation signed 59.18-decimal fixed-point, since the numbers can have
/// a sign and there can be up to 59 digits in the integer part and up to 18 decimals in the fractional part. The numbers
/// are bound by the minimum and the maximum values permitted by the Solidity type int256.
library PRBMathSD59x18 {
    /// @dev log2(e) as a signed 59.18-decimal fixed-point number.
    int256 internal constant LOG2_E = 1_442695040888963407;

    /// @dev Half the SCALE number.
    int256 internal constant HALF_SCALE = 5e17;

    /// @dev The maximum value a signed 59.18-decimal fixed-point number can have.
    int256 internal constant MAX_SD59x18 =
        57896044618658097711785492504343953926634992332820282019728_792003956564819967;

    /// @dev The maximum whole value a signed 59.18-decimal fixed-point number can have.
    int256 internal constant MAX_WHOLE_SD59x18 =
        57896044618658097711785492504343953926634992332820282019728_000000000000000000;

    /// @dev The minimum value a signed 59.18-decimal fixed-point number can have.
    int256 internal constant MIN_SD59x18 =
        -57896044618658097711785492504343953926634992332820282019728_792003956564819968;

    /// @dev The minimum whole value a signed 59.18-decimal fixed-point number can have.
    int256 internal constant MIN_WHOLE_SD59x18 =
        -57896044618658097711785492504343953926634992332820282019728_000000000000000000;

    /// @dev How many trailing decimals can be represented.
    int256 internal constant SCALE = 1e18;

    /// INTERNAL FUNCTIONS ///

    /// @notice Calculate the absolute value of x.
    ///
    /// @dev Requirements:
    /// - x must be greater than MIN_SD59x18.
    ///
    /// @param x The number to calculate the absolute value for.
    /// @param result The absolute value of x.
    function abs(int256 x) internal pure returns (int256 result) {
        unchecked {
            if (x == MIN_SD59x18) {
                revert PRBMathSD59x18__AbsInputTooSmall();
            }
            result = x < 0 ? -x : x;
        }
    }

    /// @notice Calculates the arithmetic average of x and y, rounding down.
    /// @param x The first operand as a signed 59.18-decimal fixed-point number.
    /// @param y The second operand as a signed 59.18-decimal fixed-point number.
    /// @return result The arithmetic average as a signed 59.18-decimal fixed-point number.
    function avg(int256 x, int256 y) internal pure returns (int256 result) {
        // The operations can never overflow.
        unchecked {
            int256 sum = (x >> 1) + (y >> 1);
            if (sum < 0) {
                // If at least one of x and y is odd, we add 1 to the result. This is because shifting negative numbers to the
                // right rounds down to infinity.
                assembly {
                    result := add(sum, and(or(x, y), 1))
                }
            } else {
                // If both x and y are odd, we add 1 to the result. This is because if both numbers are odd, the 0.5
                // remainder gets truncated twice.
                result = sum + (x & y & 1);
            }
        }
    }

    /// @notice Yields the least greatest signed 59.18 decimal fixed-point number greater than or equal to x.
    ///
    /// @dev Optimized for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts.
    /// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions.
    ///
    /// Requirements:
    /// - x must be less than or equal to MAX_WHOLE_SD59x18.
    ///
    /// @param x The signed 59.18-decimal fixed-point number to ceil.
    /// @param result The least integer greater than or equal to x, as a signed 58.18-decimal fixed-point number.
    function ceil(int256 x) internal pure returns (int256 result) {
        if (x > MAX_WHOLE_SD59x18) {
            revert PRBMathSD59x18__CeilOverflow(x);
        }
        unchecked {
            int256 remainder = x % SCALE;
            if (remainder == 0) {
                result = x;
            } else {
                // Solidity uses C fmod style, which returns a modulus with the same sign as x.
                result = x - remainder;
                if (x > 0) {
                    result += SCALE;
                }
            }
        }
    }

    /// @notice Divides two signed 59.18-decimal fixed-point numbers, returning a new signed 59.18-decimal fixed-point number.
    ///
    /// @dev Variant of "mulDiv" that works with signed numbers. Works by computing the signs and the absolute values separately.
    ///
    /// Requirements:
    /// - All from "PRBMath.mulDiv".
    /// - None of the inputs can be MIN_SD59x18.
    /// - The denominator cannot be zero.
    /// - The result must fit within int256.
    ///
    /// Caveats:
    /// - All from "PRBMath.mulDiv".
    ///
    /// @param x The numerator as a signed 59.18-decimal fixed-point number.
    /// @param y The denominator as a signed 59.18-decimal fixed-point number.
    /// @param result The quotient as a signed 59.18-decimal fixed-point number.
    function div(int256 x, int256 y) internal pure returns (int256 result) {
        if (x == MIN_SD59x18 || y == MIN_SD59x18) {
            revert PRBMathSD59x18__DivInputTooSmall();
        }

        // Get hold of the absolute values of x and y.
        uint256 ax;
        uint256 ay;
        unchecked {
            ax = x < 0 ? uint256(-x) : uint256(x);
            ay = y < 0 ? uint256(-y) : uint256(y);
        }

        // Compute the absolute value of (x*SCALE)÷y. The result must fit within int256.
        uint256 rAbs = PRBMath.mulDiv(ax, uint256(SCALE), ay);
        if (rAbs > uint256(MAX_SD59x18)) {
            revert PRBMathSD59x18__DivOverflow(rAbs);
        }

        // Get the signs of x and y.
        uint256 sx;
        uint256 sy;
        assembly {
            sx := sgt(x, sub(0, 1))
            sy := sgt(y, sub(0, 1))
        }

        // XOR over sx and sy. This is basically checking whether the inputs have the same sign. If yes, the result
        // should be positive. Otherwise, it should be negative.
        result = sx ^ sy == 1 ? -int256(rAbs) : int256(rAbs);
    }

    /// @notice Returns Euler's number as a signed 59.18-decimal fixed-point number.
    /// @dev See https://en.wikipedia.org/wiki/E_(mathematical_constant).
    function e() internal pure returns (int256 result) {
        result = 2_718281828459045235;
    }

    /// @notice Calculates the natural exponent of x.
    ///
    /// @dev Based on the insight that e^x = 2^(x * log2(e)).
    ///
    /// Requirements:
    /// - All from "log2".
    /// - x must be less than 133.084258667509499441.
    ///
    /// Caveats:
    /// - All from "exp2".
    /// - For any x less than -41.446531673892822322, the result is zero.
    ///
    /// @param x The exponent as a signed 59.18-decimal fixed-point number.
    /// @return result The result as a signed 59.18-decimal fixed-point number.
    function exp(int256 x) internal pure returns (int256 result) {
        // Without this check, the value passed to "exp2" would be less than -59.794705707972522261.
        if (x < -41_446531673892822322) {
            return 0;
        }

        // Without this check, the value passed to "exp2" would be greater than 192.
        if (x >= 133_084258667509499441) {
            revert PRBMathSD59x18__ExpInputTooBig(x);
        }

        // Do the fixed-point multiplication inline to save gas.
        unchecked {
            int256 doubleScaleProduct = x * LOG2_E;
            result = exp2((doubleScaleProduct + HALF_SCALE) / SCALE);
        }
    }

    /// @notice Calculates the binary exponent of x using the binary fraction method.
    ///
    /// @dev See https://ethereum.stackexchange.com/q/79903/24693.
    ///
    /// Requirements:
    /// - x must be 192 or less.
    /// - The result must fit within MAX_SD59x18.
    ///
    /// Caveats:
    /// - For any x less than -59.794705707972522261, the result is zero.
    ///
    /// @param x The exponent as a signed 59.18-decimal fixed-point number.
    /// @return result The result as a signed 59.18-decimal fixed-point number.
    function exp2(int256 x) internal pure returns (int256 result) {
        // This works because 2^(-x) = 1/2^x.
        if (x < 0) {
            // 2^59.794705707972522262 is the maximum number whose inverse does not truncate down to zero.
            if (x < -59_794705707972522261) {
                return 0;
            }

            // Do the fixed-point inversion inline to save gas. The numerator is SCALE * SCALE.
            unchecked {
                result = 1e36 / exp2(-x);
            }
        } else {
            // 2^192 doesn't fit within the 192.64-bit format used internally in this function.
            if (x >= 192e18) {
                revert PRBMathSD59x18__Exp2InputTooBig(x);
            }

            unchecked {
                // Convert x to the 192.64-bit fixed-point format.
                uint256 x192x64 = (uint256(x) << 64) / uint256(SCALE);

                // Safe to convert the result to int256 directly because the maximum input allowed is 192.
                result = int256(PRBMath.exp2(x192x64));
            }
        }
    }

    /// @notice Yields the greatest signed 59.18 decimal fixed-point number less than or equal to x.
    ///
    /// @dev Optimized for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts.
    /// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions.
    ///
    /// Requirements:
    /// - x must be greater than or equal to MIN_WHOLE_SD59x18.
    ///
    /// @param x The signed 59.18-decimal fixed-point number to floor.
    /// @param result The greatest integer less than or equal to x, as a signed 58.18-decimal fixed-point number.
    function floor(int256 x) internal pure returns (int256 result) {
        if (x < MIN_WHOLE_SD59x18) {
            revert PRBMathSD59x18__FloorUnderflow(x);
        }
        unchecked {
            int256 remainder = x % SCALE;
            if (remainder == 0) {
                result = x;
            } else {
                // Solidity uses C fmod style, which returns a modulus with the same sign as x.
                result = x - remainder;
                if (x < 0) {
                    result -= SCALE;
                }
            }
        }
    }

    /// @notice Yields the excess beyond the floor of x for positive numbers and the part of the number to the right
    /// of the radix point for negative numbers.
    /// @dev Based on the odd function definition. https://en.wikipedia.org/wiki/Fractional_part
    /// @param x The signed 59.18-decimal fixed-point number to get the fractional part of.
    /// @param result The fractional part of x as a signed 59.18-decimal fixed-point number.
    function frac(int256 x) internal pure returns (int256 result) {
        unchecked {
            result = x % SCALE;
        }
    }

    /// @notice Converts a number from basic integer form to signed 59.18-decimal fixed-point representation.
    ///
    /// @dev Requirements:
    /// - x must be greater than or equal to MIN_SD59x18 divided by SCALE.
    /// - x must be less than or equal to MAX_SD59x18 divided by SCALE.
    ///
    /// @param x The basic integer to convert.
    /// @param result The same number in signed 59.18-decimal fixed-point representation.
    function fromInt(int256 x) internal pure returns (int256 result) {
        unchecked {
            if (x < MIN_SD59x18 / SCALE) {
                revert PRBMathSD59x18__FromIntUnderflow(x);
            }
            if (x > MAX_SD59x18 / SCALE) {
                revert PRBMathSD59x18__FromIntOverflow(x);
            }
            result = x * SCALE;
        }
    }

    /// @notice Calculates geometric mean of x and y, i.e. sqrt(x * y), rounding down.
    ///
    /// @dev Requirements:
    /// - x * y must fit within MAX_SD59x18, lest it overflows.
    /// - x * y cannot be negative.
    ///
    /// @param x The first operand as a signed 59.18-decimal fixed-point number.
    /// @param y The second operand as a signed 59.18-decimal fixed-point number.
    /// @return result The result as a signed 59.18-decimal fixed-point number.
    function gm(int256 x, int256 y) internal pure returns (int256 result) {
        if (x == 0) {
            return 0;
        }

        unchecked {
            // Checking for overflow this way is faster than letting Solidity do it.
            int256 xy = x * y;
            if (xy / x != y) {
                revert PRBMathSD59x18__GmOverflow(x, y);
            }

            // The product cannot be negative.
            if (xy < 0) {
                revert PRBMathSD59x18__GmNegativeProduct(x, y);
            }

            // We don't need to multiply by the SCALE here because the x*y product had already picked up a factor of SCALE
            // during multiplication. See the comments within the "sqrt" function.
            result = int256(PRBMath.sqrt(uint256(xy)));
        }
    }

    /// @notice Calculates 1 / x, rounding toward zero.
    ///
    /// @dev Requirements:
    /// - x cannot be zero.
    ///
    /// @param x The signed 59.18-decimal fixed-point number for which to calculate the inverse.
    /// @return result The inverse as a signed 59.18-decimal fixed-point number.
    function inv(int256 x) internal pure returns (int256 result) {
        unchecked {
            // 1e36 is SCALE * SCALE.
            result = 1e36 / x;
        }
    }

    /// @notice Calculates the natural logarithm of x.
    ///
    /// @dev Based on the insight that ln(x) = log2(x) / log2(e).
    ///
    /// Requirements:
    /// - All from "log2".
    ///
    /// Caveats:
    /// - All from "log2".
    /// - This doesn't return exactly 1 for 2718281828459045235, for that we would need more fine-grained precision.
    ///
    /// @param x The signed 59.18-decimal fixed-point number for which to calculate the natural logarithm.
    /// @return result The natural logarithm as a signed 59.18-decimal fixed-point number.
    function ln(int256 x) internal pure returns (int256 result) {
        // Do the fixed-point multiplication inline to save gas. This is overflow-safe because the maximum value that log2(x)
        // can return is 195205294292027477728.
        unchecked {
            result = (log2(x) * SCALE) / LOG2_E;
        }
    }

    /// @notice Calculates the common logarithm of x.
    ///
    /// @dev First checks if x is an exact power of ten and it stops if yes. If it's not, calculates the common
    /// logarithm based on the insight that log10(x) = log2(x) / log2(10).
    ///
    /// Requirements:
    /// - All from "log2".
    ///
    /// Caveats:
    /// - All from "log2".
    ///
    /// @param x The signed 59.18-decimal fixed-point number for which to calculate the common logarithm.
    /// @return result The common logarithm as a signed 59.18-decimal fixed-point number.
    function log10(int256 x) internal pure returns (int256 result) {
        if (x <= 0) {
            revert PRBMathSD59x18__LogInputTooSmall(x);
        }

        // Note that the "mul" in this block is the assembly mul operation, not the "mul" function defined in this contract.
        // prettier-ignore
        assembly {
            switch x
            case 1 { result := mul(SCALE, sub(0, 18)) }
            case 10 { result := mul(SCALE, sub(1, 18)) }
            case 100 { result := mul(SCALE, sub(2, 18)) }
            case 1000 { result := mul(SCALE, sub(3, 18)) }
            case 10000 { result := mul(SCALE, sub(4, 18)) }
            case 100000 { result := mul(SCALE, sub(5, 18)) }
            case 1000000 { result := mul(SCALE, sub(6, 18)) }
            case 10000000 { result := mul(SCALE, sub(7, 18)) }
            case 100000000 { result := mul(SCALE, sub(8, 18)) }
            case 1000000000 { result := mul(SCALE, sub(9, 18)) }
            case 10000000000 { result := mul(SCALE, sub(10, 18)) }
            case 100000000000 { result := mul(SCALE, sub(11, 18)) }
            case 1000000000000 { result := mul(SCALE, sub(12, 18)) }
            case 10000000000000 { result := mul(SCALE, sub(13, 18)) }
            case 100000000000000 { result := mul(SCALE, sub(14, 18)) }
            case 1000000000000000 { result := mul(SCALE, sub(15, 18)) }
            case 10000000000000000 { result := mul(SCALE, sub(16, 18)) }
            case 100000000000000000 { result := mul(SCALE, sub(17, 18)) }
            case 1000000000000000000 { result := 0 }
            case 10000000000000000000 { result := SCALE }
            case 100000000000000000000 { result := mul(SCALE, 2) }
            case 1000000000000000000000 { result := mul(SCALE, 3) }
            case 10000000000000000000000 { result := mul(SCALE, 4) }
            case 100000000000000000000000 { result := mul(SCALE, 5) }
            case 1000000000000000000000000 { result := mul(SCALE, 6) }
            case 10000000000000000000000000 { result := mul(SCALE, 7) }
            case 100000000000000000000000000 { result := mul(SCALE, 8) }
            case 1000000000000000000000000000 { result := mul(SCALE, 9) }
            case 10000000000000000000000000000 { result := mul(SCALE, 10) }
            case 100000000000000000000000000000 { result := mul(SCALE, 11) }
            case 1000000000000000000000000000000 { result := mul(SCALE, 12) }
            case 10000000000000000000000000000000 { result := mul(SCALE, 13) }
            case 100000000000000000000000000000000 { result := mul(SCALE, 14) }
            case 1000000000000000000000000000000000 { result := mul(SCALE, 15) }
            case 10000000000000000000000000000000000 { result := mul(SCALE, 16) }
            case 100000000000000000000000000000000000 { result := mul(SCALE, 17) }
            case 1000000000000000000000000000000000000 { result := mul(SCALE, 18) }
            case 10000000000000000000000000000000000000 { result := mul(SCALE, 19) }
            case 100000000000000000000000000000000000000 { result := mul(SCALE, 20) }
            case 1000000000000000000000000000000000000000 { result := mul(SCALE, 21) }
            case 10000000000000000000000000000000000000000 { result := mul(SCALE, 22) }
            case 100000000000000000000000000000000000000000 { result := mul(SCALE, 23) }
            case 1000000000000000000000000000000000000000000 { result := mul(SCALE, 24) }
            case 10000000000000000000000000000000000000000000 { result := mul(SCALE, 25) }
            case 100000000000000000000000000000000000000000000 { result := mul(SCALE, 26) }
            case 1000000000000000000000000000000000000000000000 { result := mul(SCALE, 27) }
            case 10000000000000000000000000000000000000000000000 { result := mul(SCALE, 28) }
            case 100000000000000000000000000000000000000000000000 { result := mul(SCALE, 29) }
            case 1000000000000000000000000000000000000000000000000 { result := mul(SCALE, 30) }
            case 10000000000000000000000000000000000000000000000000 { result := mul(SCALE, 31) }
            case 100000000000000000000000000000000000000000000000000 { result := mul(SCALE, 32) }
            case 1000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 33) }
            case 10000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 34) }
            case 100000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 35) }
            case 1000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 36) }
            case 10000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 37) }
            case 100000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 38) }
            case 1000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 39) }
            case 10000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 40) }
            case 100000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 41) }
            case 1000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 42) }
            case 10000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 43) }
            case 100000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 44) }
            case 1000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 45) }
            case 10000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 46) }
            case 100000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 47) }
            case 1000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 48) }
            case 10000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 49) }
            case 100000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 50) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 51) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 52) }
            case 100000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 53) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 54) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 55) }
            case 100000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 56) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 57) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 58) }
            default {
                result := MAX_SD59x18
            }
        }

        if (result == MAX_SD59x18) {
            // Do the fixed-point division inline to save gas. The denominator is log2(10).
            unchecked {
                result = (log2(x) * SCALE) / 3_321928094887362347;
            }
        }
    }

    /// @notice Calculates the binary logarithm of x.
    ///
    /// @dev Based on the iterative approximation algorithm.
    /// https://en.wikipedia.org/wiki/Binary_logarithm#Iterative_approximation
    ///
    /// Requirements:
    /// - x must be greater than zero.
    ///
    /// Caveats:
    /// - The results are not perfectly accurate to the last decimal, due to the lossy precision of the iterative approximation.
    ///
    /// @param x The signed 59.18-decimal fixed-point number for which to calculate the binary logarithm.
    /// @return result The binary logarithm as a signed 59.18-decimal fixed-point number.
    function log2(int256 x) internal pure returns (int256 result) {
        if (x <= 0) {
            revert PRBMathSD59x18__LogInputTooSmall(x);
        }
        unchecked {
            // This works because log2(x) = -log2(1/x).
            int256 sign;
            if (x >= SCALE) {
                sign = 1;
            } else {
                sign = -1;
                // Do the fixed-point inversion inline to save gas. The numerator is SCALE * SCALE.
                assembly {
                    x := div(1000000000000000000000000000000000000, x)
                }
            }

            // Calculate the integer part of the logarithm and add it to the result and finally calculate y = x * 2^(-n).
            uint256 n = PRBMath.mostSignificantBit(uint256(x / SCALE));

            // The integer part of the logarithm as a signed 59.18-decimal fixed-point number. The operation can't overflow
            // because n is maximum 255, SCALE is 1e18 and sign is either 1 or -1.
            result = int256(n) * SCALE;

            // This is y = x * 2^(-n).
            int256 y = x >> n;

            // If y = 1, the fractional part is zero.
            if (y == SCALE) {
                return result * sign;
            }

            // Calculate the fractional part via the iterative approximation.
            // The "delta >>= 1" part is equivalent to "delta /= 2", but shifting bits is faster.
            for (int256 delta = int256(HALF_SCALE); delta > 0; delta >>= 1) {
                y = (y * y) / SCALE;

                // Is y^2 > 2 and so in the range [2,4)?
                if (y >= 2 * SCALE) {
                    // Add the 2^(-m) factor to the logarithm.
                    result += delta;

                    // Corresponds to z/2 on Wikipedia.
                    y >>= 1;
                }
            }
            result *= sign;
        }
    }

    /// @notice Multiplies two signed 59.18-decimal fixed-point numbers together, returning a new signed 59.18-decimal
    /// fixed-point number.
    ///
    /// @dev Variant of "mulDiv" that works with signed numbers and employs constant folding, i.e. the denominator is
    /// always 1e18.
    ///
    /// Requirements:
    /// - All from "PRBMath.mulDivFixedPoint".
    /// - None of the inputs can be MIN_SD59x18
    /// - The result must fit within MAX_SD59x18.
    ///
    /// Caveats:
    /// - The body is purposely left uncommented; see the NatSpec comments in "PRBMath.mulDiv" to understand how this works.
    ///
    /// @param x The multiplicand as a signed 59.18-decimal fixed-point number.
    /// @param y The multiplier as a signed 59.18-decimal fixed-point number.
    /// @return result The product as a signed 59.18-decimal fixed-point number.
    function mul(int256 x, int256 y) internal pure returns (int256 result) {
        if (x == MIN_SD59x18 || y == MIN_SD59x18) {
            revert PRBMathSD59x18__MulInputTooSmall();
        }

        unchecked {
            uint256 ax;
            uint256 ay;
            ax = x < 0 ? uint256(-x) : uint256(x);
            ay = y < 0 ? uint256(-y) : uint256(y);

            uint256 rAbs = PRBMath.mulDivFixedPoint(ax, ay);
            if (rAbs > uint256(MAX_SD59x18)) {
                revert PRBMathSD59x18__MulOverflow(rAbs);
            }

            uint256 sx;
            uint256 sy;
            assembly {
                sx := sgt(x, sub(0, 1))
                sy := sgt(y, sub(0, 1))
            }
            result = sx ^ sy == 1 ? -int256(rAbs) : int256(rAbs);
        }
    }

    /// @notice Returns PI as a signed 59.18-decimal fixed-point number.
    function pi() internal pure returns (int256 result) {
        result = 3_141592653589793238;
    }

    /// @notice Raises x to the power of y.
    ///
    /// @dev Based on the insight that x^y = 2^(log2(x) * y).
    ///
    /// Requirements:
    /// - All from "exp2", "log2" and "mul".
    /// - z cannot be zero.
    ///
    /// Caveats:
    /// - All from "exp2", "log2" and "mul".
    /// - Assumes 0^0 is 1.
    ///
    /// @param x Number to raise to given power y, as a signed 59.18-decimal fixed-point number.
    /// @param y Exponent to raise x to, as a signed 59.18-decimal fixed-point number.
    /// @return result x raised to power y, as a signed 59.18-decimal fixed-point number.
    function pow(int256 x, int256 y) internal pure returns (int256 result) {
        if (x == 0) {
            result = y == 0 ? SCALE : int256(0);
        } else {
            result = exp2(mul(log2(x), y));
        }
    }

    /// @notice Raises x (signed 59.18-decimal fixed-point number) to the power of y (basic unsigned integer) using the
    /// famous algorithm "exponentiation by squaring".
    ///
    /// @dev See https://en.wikipedia.org/wiki/Exponentiation_by_squaring
    ///
    /// Requirements:
    /// - All from "abs" and "PRBMath.mulDivFixedPoint".
    /// - The result must fit within MAX_SD59x18.
    ///
    /// Caveats:
    /// - All from "PRBMath.mulDivFixedPoint".
    /// - Assumes 0^0 is 1.
    ///
    /// @param x The base as a signed 59.18-decimal fixed-point number.
    /// @param y The exponent as an uint256.
    /// @return result The result as a signed 59.18-decimal fixed-point number.
    function powu(int256 x, uint256 y) internal pure returns (int256 result) {
        uint256 xAbs = uint256(abs(x));

        // Calculate the first iteration of the loop in advance.
        uint256 rAbs = y & 1 > 0 ? xAbs : uint256(SCALE);

        // Equivalent to "for(y /= 2; y > 0; y /= 2)" but faster.
        uint256 yAux = y;
        for (yAux >>= 1; yAux > 0; yAux >>= 1) {
            xAbs = PRBMath.mulDivFixedPoint(xAbs, xAbs);

            // Equivalent to "y % 2 == 1" but faster.
            if (yAux & 1 > 0) {
                rAbs = PRBMath.mulDivFixedPoint(rAbs, xAbs);
            }
        }

        // The result must fit within the 59.18-decimal fixed-point representation.
        if (rAbs > uint256(MAX_SD59x18)) {
            revert PRBMathSD59x18__PowuOverflow(rAbs);
        }

        // Is the base negative and the exponent an odd number?
        bool isNegative = x < 0 && y & 1 == 1;
        result = isNegative ? -int256(rAbs) : int256(rAbs);
    }

    /// @notice Returns 1 as a signed 59.18-decimal fixed-point number.
    function scale() internal pure returns (int256 result) {
        result = SCALE;
    }

    /// @notice Calculates the square root of x, rounding down.
    /// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method.
    ///
    /// Requirements:
    /// - x cannot be negative.
    /// - x must be less than MAX_SD59x18 / SCALE.
    ///
    /// @param x The signed 59.18-decimal fixed-point number for which to calculate the square root.
    /// @return result The result as a signed 59.18-decimal fixed-point .
    function sqrt(int256 x) internal pure returns (int256 result) {
        unchecked {
            if (x < 0) {
                revert PRBMathSD59x18__SqrtNegativeInput(x);
            }
            if (x > MAX_SD59x18 / SCALE) {
                revert PRBMathSD59x18__SqrtOverflow(x);
            }
            // Multiply x by the SCALE to account for the factor of SCALE that is picked up when multiplying two signed
            // 59.18-decimal fixed-point numbers together (in this case, those two numbers are both the square root).
            result = int256(PRBMath.sqrt(uint256(x * SCALE)));
        }
    }

    /// @notice Converts a signed 59.18-decimal fixed-point number to basic integer form, rounding down in the process.
    /// @param x The signed 59.18-decimal fixed-point number to convert.
    /// @return result The same number in basic integer form.
    function toInt(int256 x) internal pure returns (int256 result) {
        unchecked {
            result = x / SCALE;
        }
    }
}

// SPDX-License-Identifier: Unlicense
pragma solidity >=0.8.4;

import "./PRBMath.sol";

/// @title PRBMathUD60x18
/// @author Paul Razvan Berg
/// @notice Smart contract library for advanced fixed-point math that works with uint256 numbers considered to have 18
/// trailing decimals. We call this number representation unsigned 60.18-decimal fixed-point, since there can be up to 60
/// digits in the integer part and up to 18 decimals in the fractional part. The numbers are bound by the minimum and the
/// maximum values permitted by the Solidity type uint256.
library PRBMathUD60x18 {
    /// @dev Half the SCALE number.
    uint256 internal constant HALF_SCALE = 5e17;

    /// @dev log2(e) as an unsigned 60.18-decimal fixed-point number.
    uint256 internal constant LOG2_E = 1_442695040888963407;

    /// @dev The maximum value an unsigned 60.18-decimal fixed-point number can have.
    uint256 internal constant MAX_UD60x18 =
        115792089237316195423570985008687907853269984665640564039457_584007913129639935;

    /// @dev The maximum whole value an unsigned 60.18-decimal fixed-point number can have.
    uint256 internal constant MAX_WHOLE_UD60x18 =
        115792089237316195423570985008687907853269984665640564039457_000000000000000000;

    /// @dev How many trailing decimals can be represented.
    uint256 internal constant SCALE = 1e18;

    /// @notice Calculates the arithmetic average of x and y, rounding down.
    /// @param x The first operand as an unsigned 60.18-decimal fixed-point number.
    /// @param y The second operand as an unsigned 60.18-decimal fixed-point number.
    /// @return result The arithmetic average as an unsigned 60.18-decimal fixed-point number.
    function avg(uint256 x, uint256 y) internal pure returns (uint256 result) {
        // The operations can never overflow.
        unchecked {
            // The last operand checks if both x and y are odd and if that is the case, we add 1 to the result. We need
            // to do this because if both numbers are odd, the 0.5 remainder gets truncated twice.
            result = (x >> 1) + (y >> 1) + (x & y & 1);
        }
    }

    /// @notice Yields the least unsigned 60.18 decimal fixed-point number greater than or equal to x.
    ///
    /// @dev Optimized for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts.
    /// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions.
    ///
    /// Requirements:
    /// - x must be less than or equal to MAX_WHOLE_UD60x18.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number to ceil.
    /// @param result The least integer greater than or equal to x, as an unsigned 60.18-decimal fixed-point number.
    function ceil(uint256 x) internal pure returns (uint256 result) {
        if (x > MAX_WHOLE_UD60x18) {
            revert PRBMathUD60x18__CeilOverflow(x);
        }
        assembly {
            // Equivalent to "x % SCALE" but faster.
            let remainder := mod(x, SCALE)

            // Equivalent to "SCALE - remainder" but faster.
            let delta := sub(SCALE, remainder)

            // Equivalent to "x + delta * (remainder > 0 ? 1 : 0)" but faster.
            result := add(x, mul(delta, gt(remainder, 0)))
        }
    }

    /// @notice Divides two unsigned 60.18-decimal fixed-point numbers, returning a new unsigned 60.18-decimal fixed-point number.
    ///
    /// @dev Uses mulDiv to enable overflow-safe multiplication and division.
    ///
    /// Requirements:
    /// - The denominator cannot be zero.
    ///
    /// @param x The numerator as an unsigned 60.18-decimal fixed-point number.
    /// @param y The denominator as an unsigned 60.18-decimal fixed-point number.
    /// @param result The quotient as an unsigned 60.18-decimal fixed-point number.
    function div(uint256 x, uint256 y) internal pure returns (uint256 result) {
        result = PRBMath.mulDiv(x, SCALE, y);
    }

    /// @notice Returns Euler's number as an unsigned 60.18-decimal fixed-point number.
    /// @dev See https://en.wikipedia.org/wiki/E_(mathematical_constant).
    function e() internal pure returns (uint256 result) {
        result = 2_718281828459045235;
    }

    /// @notice Calculates the natural exponent of x.
    ///
    /// @dev Based on the insight that e^x = 2^(x * log2(e)).
    ///
    /// Requirements:
    /// - All from "log2".
    /// - x must be less than 133.084258667509499441.
    ///
    /// @param x The exponent as an unsigned 60.18-decimal fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function exp(uint256 x) internal pure returns (uint256 result) {
        // Without this check, the value passed to "exp2" would be greater than 192.
        if (x >= 133_084258667509499441) {
            revert PRBMathUD60x18__ExpInputTooBig(x);
        }

        // Do the fixed-point multiplication inline to save gas.
        unchecked {
            uint256 doubleScaleProduct = x * LOG2_E;
            result = exp2((doubleScaleProduct + HALF_SCALE) / SCALE);
        }
    }

    /// @notice Calculates the binary exponent of x using the binary fraction method.
    ///
    /// @dev See https://ethereum.stackexchange.com/q/79903/24693.
    ///
    /// Requirements:
    /// - x must be 192 or less.
    /// - The result must fit within MAX_UD60x18.
    ///
    /// @param x The exponent as an unsigned 60.18-decimal fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function exp2(uint256 x) internal pure returns (uint256 result) {
        // 2^192 doesn't fit within the 192.64-bit format used internally in this function.
        if (x >= 192e18) {
            revert PRBMathUD60x18__Exp2InputTooBig(x);
        }

        unchecked {
            // Convert x to the 192.64-bit fixed-point format.
            uint256 x192x64 = (x << 64) / SCALE;

            // Pass x to the PRBMath.exp2 function, which uses the 192.64-bit fixed-point number representation.
            result = PRBMath.exp2(x192x64);
        }
    }

    /// @notice Yields the greatest unsigned 60.18 decimal fixed-point number less than or equal to x.
    /// @dev Optimized for fractional value inputs, because for every whole value there are (1e18 - 1) fractional counterparts.
    /// See https://en.wikipedia.org/wiki/Floor_and_ceiling_functions.
    /// @param x The unsigned 60.18-decimal fixed-point number to floor.
    /// @param result The greatest integer less than or equal to x, as an unsigned 60.18-decimal fixed-point number.
    function floor(uint256 x) internal pure returns (uint256 result) {
        assembly {
            // Equivalent to "x % SCALE" but faster.
            let remainder := mod(x, SCALE)

            // Equivalent to "x - remainder * (remainder > 0 ? 1 : 0)" but faster.
            result := sub(x, mul(remainder, gt(remainder, 0)))
        }
    }

    /// @notice Yields the excess beyond the floor of x.
    /// @dev Based on the odd function definition https://en.wikipedia.org/wiki/Fractional_part.
    /// @param x The unsigned 60.18-decimal fixed-point number to get the fractional part of.
    /// @param result The fractional part of x as an unsigned 60.18-decimal fixed-point number.
    function frac(uint256 x) internal pure returns (uint256 result) {
        assembly {
            result := mod(x, SCALE)
        }
    }

    /// @notice Converts a number from basic integer form to unsigned 60.18-decimal fixed-point representation.
    ///
    /// @dev Requirements:
    /// - x must be less than or equal to MAX_UD60x18 divided by SCALE.
    ///
    /// @param x The basic integer to convert.
    /// @param result The same number in unsigned 60.18-decimal fixed-point representation.
    function fromUint(uint256 x) internal pure returns (uint256 result) {
        unchecked {
            if (x > MAX_UD60x18 / SCALE) {
                revert PRBMathUD60x18__FromUintOverflow(x);
            }
            result = x * SCALE;
        }
    }

    /// @notice Calculates geometric mean of x and y, i.e. sqrt(x * y), rounding down.
    ///
    /// @dev Requirements:
    /// - x * y must fit within MAX_UD60x18, lest it overflows.
    ///
    /// @param x The first operand as an unsigned 60.18-decimal fixed-point number.
    /// @param y The second operand as an unsigned 60.18-decimal fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function gm(uint256 x, uint256 y) internal pure returns (uint256 result) {
        if (x == 0) {
            return 0;
        }

        unchecked {
            // Checking for overflow this way is faster than letting Solidity do it.
            uint256 xy = x * y;
            if (xy / x != y) {
                revert PRBMathUD60x18__GmOverflow(x, y);
            }

            // We don't need to multiply by the SCALE here because the x*y product had already picked up a factor of SCALE
            // during multiplication. See the comments within the "sqrt" function.
            result = PRBMath.sqrt(xy);
        }
    }

    /// @notice Calculates 1 / x, rounding toward zero.
    ///
    /// @dev Requirements:
    /// - x cannot be zero.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the inverse.
    /// @return result The inverse as an unsigned 60.18-decimal fixed-point number.
    function inv(uint256 x) internal pure returns (uint256 result) {
        unchecked {
            // 1e36 is SCALE * SCALE.
            result = 1e36 / x;
        }
    }

    /// @notice Calculates the natural logarithm of x.
    ///
    /// @dev Based on the insight that ln(x) = log2(x) / log2(e).
    ///
    /// Requirements:
    /// - All from "log2".
    ///
    /// Caveats:
    /// - All from "log2".
    /// - This doesn't return exactly 1 for 2.718281828459045235, for that we would need more fine-grained precision.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the natural logarithm.
    /// @return result The natural logarithm as an unsigned 60.18-decimal fixed-point number.
    function ln(uint256 x) internal pure returns (uint256 result) {
        // Do the fixed-point multiplication inline to save gas. This is overflow-safe because the maximum value that log2(x)
        // can return is 196205294292027477728.
        unchecked {
            result = (log2(x) * SCALE) / LOG2_E;
        }
    }

    /// @notice Calculates the common logarithm of x.
    ///
    /// @dev First checks if x is an exact power of ten and it stops if yes. If it's not, calculates the common
    /// logarithm based on the insight that log10(x) = log2(x) / log2(10).
    ///
    /// Requirements:
    /// - All from "log2".
    ///
    /// Caveats:
    /// - All from "log2".
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the common logarithm.
    /// @return result The common logarithm as an unsigned 60.18-decimal fixed-point number.
    function log10(uint256 x) internal pure returns (uint256 result) {
        if (x < SCALE) {
            revert PRBMathUD60x18__LogInputTooSmall(x);
        }

        // Note that the "mul" in this block is the assembly multiplication operation, not the "mul" function defined
        // in this contract.
        // prettier-ignore
        assembly {
            switch x
            case 1 { result := mul(SCALE, sub(0, 18)) }
            case 10 { result := mul(SCALE, sub(1, 18)) }
            case 100 { result := mul(SCALE, sub(2, 18)) }
            case 1000 { result := mul(SCALE, sub(3, 18)) }
            case 10000 { result := mul(SCALE, sub(4, 18)) }
            case 100000 { result := mul(SCALE, sub(5, 18)) }
            case 1000000 { result := mul(SCALE, sub(6, 18)) }
            case 10000000 { result := mul(SCALE, sub(7, 18)) }
            case 100000000 { result := mul(SCALE, sub(8, 18)) }
            case 1000000000 { result := mul(SCALE, sub(9, 18)) }
            case 10000000000 { result := mul(SCALE, sub(10, 18)) }
            case 100000000000 { result := mul(SCALE, sub(11, 18)) }
            case 1000000000000 { result := mul(SCALE, sub(12, 18)) }
            case 10000000000000 { result := mul(SCALE, sub(13, 18)) }
            case 100000000000000 { result := mul(SCALE, sub(14, 18)) }
            case 1000000000000000 { result := mul(SCALE, sub(15, 18)) }
            case 10000000000000000 { result := mul(SCALE, sub(16, 18)) }
            case 100000000000000000 { result := mul(SCALE, sub(17, 18)) }
            case 1000000000000000000 { result := 0 }
            case 10000000000000000000 { result := SCALE }
            case 100000000000000000000 { result := mul(SCALE, 2) }
            case 1000000000000000000000 { result := mul(SCALE, 3) }
            case 10000000000000000000000 { result := mul(SCALE, 4) }
            case 100000000000000000000000 { result := mul(SCALE, 5) }
            case 1000000000000000000000000 { result := mul(SCALE, 6) }
            case 10000000000000000000000000 { result := mul(SCALE, 7) }
            case 100000000000000000000000000 { result := mul(SCALE, 8) }
            case 1000000000000000000000000000 { result := mul(SCALE, 9) }
            case 10000000000000000000000000000 { result := mul(SCALE, 10) }
            case 100000000000000000000000000000 { result := mul(SCALE, 11) }
            case 1000000000000000000000000000000 { result := mul(SCALE, 12) }
            case 10000000000000000000000000000000 { result := mul(SCALE, 13) }
            case 100000000000000000000000000000000 { result := mul(SCALE, 14) }
            case 1000000000000000000000000000000000 { result := mul(SCALE, 15) }
            case 10000000000000000000000000000000000 { result := mul(SCALE, 16) }
            case 100000000000000000000000000000000000 { result := mul(SCALE, 17) }
            case 1000000000000000000000000000000000000 { result := mul(SCALE, 18) }
            case 10000000000000000000000000000000000000 { result := mul(SCALE, 19) }
            case 100000000000000000000000000000000000000 { result := mul(SCALE, 20) }
            case 1000000000000000000000000000000000000000 { result := mul(SCALE, 21) }
            case 10000000000000000000000000000000000000000 { result := mul(SCALE, 22) }
            case 100000000000000000000000000000000000000000 { result := mul(SCALE, 23) }
            case 1000000000000000000000000000000000000000000 { result := mul(SCALE, 24) }
            case 10000000000000000000000000000000000000000000 { result := mul(SCALE, 25) }
            case 100000000000000000000000000000000000000000000 { result := mul(SCALE, 26) }
            case 1000000000000000000000000000000000000000000000 { result := mul(SCALE, 27) }
            case 10000000000000000000000000000000000000000000000 { result := mul(SCALE, 28) }
            case 100000000000000000000000000000000000000000000000 { result := mul(SCALE, 29) }
            case 1000000000000000000000000000000000000000000000000 { result := mul(SCALE, 30) }
            case 10000000000000000000000000000000000000000000000000 { result := mul(SCALE, 31) }
            case 100000000000000000000000000000000000000000000000000 { result := mul(SCALE, 32) }
            case 1000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 33) }
            case 10000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 34) }
            case 100000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 35) }
            case 1000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 36) }
            case 10000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 37) }
            case 100000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 38) }
            case 1000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 39) }
            case 10000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 40) }
            case 100000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 41) }
            case 1000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 42) }
            case 10000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 43) }
            case 100000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 44) }
            case 1000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 45) }
            case 10000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 46) }
            case 100000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 47) }
            case 1000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 48) }
            case 10000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 49) }
            case 100000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 50) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 51) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 52) }
            case 100000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 53) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 54) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 55) }
            case 100000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 56) }
            case 1000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 57) }
            case 10000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 58) }
            case 100000000000000000000000000000000000000000000000000000000000000000000000000000 { result := mul(SCALE, 59) }
            default {
                result := MAX_UD60x18
            }
        }

        if (result == MAX_UD60x18) {
            // Do the fixed-point division inline to save gas. The denominator is log2(10).
            unchecked {
                result = (log2(x) * SCALE) / 3_321928094887362347;
            }
        }
    }

    /// @notice Calculates the binary logarithm of x.
    ///
    /// @dev Based on the iterative approximation algorithm.
    /// https://en.wikipedia.org/wiki/Binary_logarithm#Iterative_approximation
    ///
    /// Requirements:
    /// - x must be greater than or equal to SCALE, otherwise the result would be negative.
    ///
    /// Caveats:
    /// - The results are nor perfectly accurate to the last decimal, due to the lossy precision of the iterative approximation.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the binary logarithm.
    /// @return result The binary logarithm as an unsigned 60.18-decimal fixed-point number.
    function log2(uint256 x) internal pure returns (uint256 result) {
        if (x < SCALE) {
            revert PRBMathUD60x18__LogInputTooSmall(x);
        }
        unchecked {
            // Calculate the integer part of the logarithm and add it to the result and finally calculate y = x * 2^(-n).
            uint256 n = PRBMath.mostSignificantBit(x / SCALE);

            // The integer part of the logarithm as an unsigned 60.18-decimal fixed-point number. The operation can't overflow
            // because n is maximum 255 and SCALE is 1e18.
            result = n * SCALE;

            // This is y = x * 2^(-n).
            uint256 y = x >> n;

            // If y = 1, the fractional part is zero.
            if (y == SCALE) {
                return result;
            }

            // Calculate the fractional part via the iterative approximation.
            // The "delta >>= 1" part is equivalent to "delta /= 2", but shifting bits is faster.
            for (uint256 delta = HALF_SCALE; delta > 0; delta >>= 1) {
                y = (y * y) / SCALE;

                // Is y^2 > 2 and so in the range [2,4)?
                if (y >= 2 * SCALE) {
                    // Add the 2^(-m) factor to the logarithm.
                    result += delta;

                    // Corresponds to z/2 on Wikipedia.
                    y >>= 1;
                }
            }
        }
    }

    /// @notice Multiplies two unsigned 60.18-decimal fixed-point numbers together, returning a new unsigned 60.18-decimal
    /// fixed-point number.
    /// @dev See the documentation for the "PRBMath.mulDivFixedPoint" function.
    /// @param x The multiplicand as an unsigned 60.18-decimal fixed-point number.
    /// @param y The multiplier as an unsigned 60.18-decimal fixed-point number.
    /// @return result The product as an unsigned 60.18-decimal fixed-point number.
    function mul(uint256 x, uint256 y) internal pure returns (uint256 result) {
        result = PRBMath.mulDivFixedPoint(x, y);
    }

    /// @notice Returns PI as an unsigned 60.18-decimal fixed-point number.
    function pi() internal pure returns (uint256 result) {
        result = 3_141592653589793238;
    }

    /// @notice Raises x to the power of y.
    ///
    /// @dev Based on the insight that x^y = 2^(log2(x) * y).
    ///
    /// Requirements:
    /// - All from "exp2", "log2" and "mul".
    ///
    /// Caveats:
    /// - All from "exp2", "log2" and "mul".
    /// - Assumes 0^0 is 1.
    ///
    /// @param x Number to raise to given power y, as an unsigned 60.18-decimal fixed-point number.
    /// @param y Exponent to raise x to, as an unsigned 60.18-decimal fixed-point number.
    /// @return result x raised to power y, as an unsigned 60.18-decimal fixed-point number.
    function pow(uint256 x, uint256 y) internal pure returns (uint256 result) {
        if (x == 0) {
            result = y == 0 ? SCALE : uint256(0);
        } else {
            result = exp2(mul(log2(x), y));
        }
    }

    /// @notice Raises x (unsigned 60.18-decimal fixed-point number) to the power of y (basic unsigned integer) using the
    /// famous algorithm "exponentiation by squaring".
    ///
    /// @dev See https://en.wikipedia.org/wiki/Exponentiation_by_squaring
    ///
    /// Requirements:
    /// - The result must fit within MAX_UD60x18.
    ///
    /// Caveats:
    /// - All from "mul".
    /// - Assumes 0^0 is 1.
    ///
    /// @param x The base as an unsigned 60.18-decimal fixed-point number.
    /// @param y The exponent as an uint256.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function powu(uint256 x, uint256 y) internal pure returns (uint256 result) {
        // Calculate the first iteration of the loop in advance.
        result = y & 1 > 0 ? x : SCALE;

        // Equivalent to "for(y /= 2; y > 0; y /= 2)" but faster.
        for (y >>= 1; y > 0; y >>= 1) {
            x = PRBMath.mulDivFixedPoint(x, x);

            // Equivalent to "y % 2 == 1" but faster.
            if (y & 1 > 0) {
                result = PRBMath.mulDivFixedPoint(result, x);
            }
        }
    }

    /// @notice Returns 1 as an unsigned 60.18-decimal fixed-point number.
    function scale() internal pure returns (uint256 result) {
        result = SCALE;
    }

    /// @notice Calculates the square root of x, rounding down.
    /// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method.
    ///
    /// Requirements:
    /// - x must be less than MAX_UD60x18 / SCALE.
    ///
    /// @param x The unsigned 60.18-decimal fixed-point number for which to calculate the square root.
    /// @return result The result as an unsigned 60.18-decimal fixed-point .
    function sqrt(uint256 x) internal pure returns (uint256 result) {
        unchecked {
            if (x > MAX_UD60x18 / SCALE) {
                revert PRBMathUD60x18__SqrtOverflow(x);
            }
            // Multiply x by the SCALE to account for the factor of SCALE that is picked up when multiplying two unsigned
            // 60.18-decimal fixed-point numbers together (in this case, those two numbers are both the square root).
            result = PRBMath.sqrt(x * SCALE);
        }
    }

    /// @notice Converts a unsigned 60.18-decimal fixed-point number to basic integer form, rounding down in the process.
    /// @param x The unsigned 60.18-decimal fixed-point number to convert.
    /// @return result The same number in basic integer form.
    function toUint(uint256 x) internal pure returns (uint256 result) {
        unchecked {
            result = x / SCALE;
        }
    }
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.6.0) (token/ERC20/IERC20.sol)

pragma solidity ^0.8.0;

/**
 * @dev Interface of the ERC20 standard as defined in the EIP.
 */
interface IERC20 {
    /**
     * @dev Emitted when `value` tokens are moved from one account (`from`) to
     * another (`to`).
     *
     * Note that `value` may be zero.
     */
    event Transfer(address indexed from, address indexed to, uint256 value);

    /**
     * @dev Emitted when the allowance of a `spender` for an `owner` is set by
     * a call to {approve}. `value` is the new allowance.
     */
    event Approval(address indexed owner, address indexed spender, uint256 value);

    /**
     * @dev Returns the amount of tokens in existence.
     */
    function totalSupply() external view returns (uint256);

    /**
     * @dev Returns the amount of tokens owned by `account`.
     */
    function balanceOf(address account) external view returns (uint256);

    /**
     * @dev Moves `amount` tokens from the caller's account to `to`.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * Emits a {Transfer} event.
     */
    function transfer(address to, uint256 amount) external returns (bool);

    /**
     * @dev Returns the remaining number of tokens that `spender` will be
     * allowed to spend on behalf of `owner` through {transferFrom}. This is
     * zero by default.
     *
     * This value changes when {approve} or {transferFrom} are called.
     */
    function allowance(address owner, address spender) external view returns (uint256);

    /**
     * @dev Sets `amount` as the allowance of `spender` over the caller's tokens.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * IMPORTANT: Beware that changing an allowance with this method brings the risk
     * that someone may use both the old and the new allowance by unfortunate
     * transaction ordering. One possible solution to mitigate this race
     * condition is to first reduce the spender's allowance to 0 and set the
     * desired value afterwards:
     * https://github.com/ethereum/EIPs/issues/20#issuecomment-263524729
     *
     * Emits an {Approval} event.
     */
    function approve(address spender, uint256 amount) external returns (bool);

    /**
     * @dev Moves `amount` tokens from `from` to `to` using the
     * allowance mechanism. `amount` is then deducted from the caller's
     * allowance.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * Emits a {Transfer} event.
     */
    function transferFrom(
        address from,
        address to,
        uint256 amount
    ) external returns (bool);
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.8.0) (token/ERC20/utils/SafeERC20.sol)

pragma solidity ^0.8.0;

import "../IERC20.sol";
import "../extensions/draft-IERC20Permit.sol";
import "../../../utils/Address.sol";

/**
 * @title SafeERC20
 * @dev Wrappers around ERC20 operations that throw on failure (when the token
 * contract returns false). Tokens that return no value (and instead revert or
 * throw on failure) are also supported, non-reverting calls are assumed to be
 * successful.
 * To use this library you can add a `using SafeERC20 for IERC20;` statement to your contract,
 * which allows you to call the safe operations as `token.safeTransfer(...)`, etc.
 */
library SafeERC20 {
    using Address for address;

    function safeTransfer(
        IERC20 token,
        address to,
        uint256 value
    ) internal {
        _callOptionalReturn(token, abi.encodeWithSelector(token.transfer.selector, to, value));
    }

    function safeTransferFrom(
        IERC20 token,
        address from,
        address to,
        uint256 value
    ) internal {
        _callOptionalReturn(token, abi.encodeWithSelector(token.transferFrom.selector, from, to, value));
    }

    /**
     * @dev Deprecated. This function has issues similar to the ones found in
     * {IERC20-approve}, and its usage is discouraged.
     *
     * Whenever possible, use {safeIncreaseAllowance} and
     * {safeDecreaseAllowance} instead.
     */
    function safeApprove(
        IERC20 token,
        address spender,
        uint256 value
    ) internal {
        // safeApprove should only be called when setting an initial allowance,
        // or when resetting it to zero. To increase and decrease it, use
        // 'safeIncreaseAllowance' and 'safeDecreaseAllowance'
        require(
            (value == 0) || (token.allowance(address(this), spender) == 0),
            "SafeERC20: approve from non-zero to non-zero allowance"
        );
        _callOptionalReturn(token, abi.encodeWithSelector(token.approve.selector, spender, value));
    }

    function safeIncreaseAllowance(
        IERC20 token,
        address spender,
        uint256 value
    ) internal {
        uint256 newAllowance = token.allowance(address(this), spender) + value;
        _callOptionalReturn(token, abi.encodeWithSelector(token.approve.selector, spender, newAllowance));
    }

    function safeDecreaseAllowance(
        IERC20 token,
        address spender,
        uint256 value
    ) internal {
        unchecked {
            uint256 oldAllowance = token.allowance(address(this), spender);
            require(oldAllowance >= value, "SafeERC20: decreased allowance below zero");
            uint256 newAllowance = oldAllowance - value;
            _callOptionalReturn(token, abi.encodeWithSelector(token.approve.selector, spender, newAllowance));
        }
    }

    function safePermit(
        IERC20Permit token,
        address owner,
        address spender,
        uint256 value,
        uint256 deadline,
        uint8 v,
        bytes32 r,
        bytes32 s
    ) internal {
        uint256 nonceBefore = token.nonces(owner);
        token.permit(owner, spender, value, deadline, v, r, s);
        uint256 nonceAfter = token.nonces(owner);
        require(nonceAfter == nonceBefore + 1, "SafeERC20: permit did not succeed");
    }

    /**
     * @dev Imitates a Solidity high-level call (i.e. a regular function call to a contract), relaxing the requirement
     * on the return value: the return value is optional (but if data is returned, it must not be false).
     * @param token The token targeted by the call.
     * @param data The call data (encoded using abi.encode or one of its variants).
     */
    function _callOptionalReturn(IERC20 token, bytes memory data) private {
        // We need to perform a low level call here, to bypass Solidity's return data size checking mechanism, since
        // we're implementing it ourselves. We use {Address-functionCall} to perform this call, which verifies that
        // the target address contains contract code and also asserts for success in the low-level call.

        bytes memory returndata = address(token).functionCall(data, "SafeERC20: low-level call failed");
        if (returndata.length > 0) {
            // Return data is optional
            require(abi.decode(returndata, (bool)), "SafeERC20: ERC20 operation did not succeed");
        }
    }
}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

/**
 * @title Pool State
 */
interface IPoolState {

    /**
     *  @notice Returns details of an auction for a given borrower address.
     *  @param  borrower_         Address of the borrower that is liquidated.
     *  @return kicker_           Address of the kicker that is kicking the auction.
     *  @return bondFactor_       The factor used for calculating bond size.
     *  @return bondSize_         The bond amount in quote token terms.
     *  @return kickTime_         Time the liquidation was initiated.
     *  @return referencePrice_   Price used to determine auction start price.
     *  @return neutralPrice_     `Neutral Price` of auction.
     *  @return debtToCollateral_ Borrower debt to collateral, which is used in BPF for kicker's reward calculation.
     *  @return head_             Address of the head auction.
     *  @return next_             Address of the next auction in queue.
     *  @return prev_             Address of the prev auction in queue.
     */
    function auctionInfo(address borrower_)
        external
        view
        returns (
            address kicker_,
            uint256 bondFactor_,
            uint256 bondSize_,
            uint256 kickTime_,
            uint256 referencePrice_,
            uint256 neutralPrice_,
            uint256 debtToCollateral_,
            address head_,
            address next_,
            address prev_
        );

    /**
     *  @notice Returns pool related debt values.
     *  @return debt_                Current amount of debt owed by borrowers in pool.
     *  @return accruedDebt_         Debt owed by borrowers based on last inflator snapshot.
     *  @return debtInAuction_       Total amount of debt in auction.
     *  @return t0Debt2ToCollateral_ t0debt accross all borrowers divided by their collateral, used in determining a collateralization weighted debt.
     */
    function debtInfo()
        external
        view
        returns (
            uint256 debt_,
            uint256 accruedDebt_,
            uint256 debtInAuction_,
            uint256 t0Debt2ToCollateral_
        );

    /**
     *  @notice Mapping of borrower addresses to `Borrower` structs.
     *  @dev    NOTE: Cannot use appended underscore syntax for return params since struct is used.
     *  @param  borrower_   Address of the borrower.
     *  @return t0Debt_     Amount of debt borrower would have had if their loan was the first debt drawn from the pool.
     *  @return collateral_ Amount of collateral that the borrower has deposited, in collateral token.
     *  @return npTpRatio_  Np to Tp ratio of borrower at the time of last borrow or pull collateral.
     */
    function borrowerInfo(address borrower_)
        external
        view
        returns (
            uint256 t0Debt_,
            uint256 collateral_,
            uint256 npTpRatio_
        );

    /**
     *  @notice Mapping of buckets indexes to `Bucket` structs.
     *  @dev    NOTE: Cannot use appended underscore syntax for return params since struct is used.
     *  @param  index_               Bucket index.
     *  @return lpAccumulator_       Amount of `LP` accumulated in current bucket.
     *  @return availableCollateral_ Amount of collateral available in current bucket.
     *  @return bankruptcyTime_      Timestamp when bucket become insolvent, `0` if healthy.
     *  @return bucketDeposit_       Amount of quote tokens in bucket.
     *  @return bucketScale_         Bucket multiplier.
     */
    function bucketInfo(uint256 index_)
        external
        view
        returns (
            uint256 lpAccumulator_,
            uint256 availableCollateral_,
            uint256 bankruptcyTime_,
            uint256 bucketDeposit_,
            uint256 bucketScale_
        );

    /**
     *  @notice Mapping of burnEventEpoch to `BurnEvent` structs.
     *  @dev    Reserve auctions correspond to burn events.
     *  @param  burnEventEpoch_  Id of the current reserve auction.
     *  @return burnBlock_       Block in which a reserve auction started.
     *  @return totalInterest_   Total interest as of the reserve auction.
     *  @return totalBurned_     Total ajna tokens burned as of the reserve auction.
     */
    function burnInfo(uint256 burnEventEpoch_) external view returns (uint256, uint256, uint256);

    /**
     *  @notice Returns the latest `burnEventEpoch` of reserve auctions.
     *  @dev    If a reserve auction is active, it refers to the current reserve auction. If no reserve auction is active, it refers to the last reserve auction.
     *  @return Current `burnEventEpoch`.
     */
    function currentBurnEpoch() external view returns (uint256);

    /**
     *  @notice Returns information about the pool `EMA (Exponential Moving Average)` variables.
     *  @return debtColEma_   Debt squared to collateral Exponential, numerator to `TU` calculation.
     *  @return lupt0DebtEma_ Exponential of `LUP * t0 debt`, denominator to `TU` calculation
     *  @return debtEma_      Exponential debt moving average.
     *  @return depositEma_   sample of meaningful deposit Exponential, denominator to `MAU` calculation.
     */
    function emasInfo()
        external
        view
        returns (
            uint256 debtColEma_,
            uint256 lupt0DebtEma_,
            uint256 debtEma_,
            uint256 depositEma_
    );

    /**
     *  @notice Returns information about pool inflator.
     *  @return inflator_   Pool inflator value.
     *  @return lastUpdate_ The timestamp of the last `inflator` update.
     */
    function inflatorInfo()
        external
        view
        returns (
            uint256 inflator_,
            uint256 lastUpdate_
    );

    /**
     *  @notice Returns information about pool interest rate.
     *  @return interestRate_       Current interest rate in pool.
     *  @return interestRateUpdate_ The timestamp of the last interest rate update.
     */
    function interestRateInfo()
        external
        view
        returns (
            uint256 interestRate_,
            uint256 interestRateUpdate_
        );


    /**
     *  @notice Returns details about kicker balances.
     *  @param  kicker_    The address of the kicker to retrieved info for.
     *  @return claimable_ Amount of quote token kicker can claim / withdraw from pool at any time.
     *  @return locked_    Amount of quote token kicker locked in auctions (as bonds).
     */
    function kickerInfo(address kicker_)
        external
        view
        returns (
            uint256 claimable_,
            uint256 locked_
        );

    /**
     *  @notice Mapping of buckets indexes and owner addresses to `Lender` structs.
     *  @param  index_       Bucket index.
     *  @param  lender_      Address of the liquidity provider.
     *  @return lpBalance_   Amount of `LP` owner has in current bucket.
     *  @return depositTime_ Time the user last deposited quote token.
     */
    function lenderInfo(
        uint256 index_,
        address lender_
    )
        external
        view
        returns (
            uint256 lpBalance_,
            uint256 depositTime_
    );

    /**
     *  @notice Return the `LP` allowance a `LP` owner provided to a spender.
     *  @param  index_     Bucket index.
     *  @param  spender_   Address of the `LP` spender.
     *  @param  owner_     The initial owner of the `LP`.
     *  @return allowance_ Amount of `LP` spender can utilize.
     */
    function lpAllowance(
        uint256 index_,
        address spender_,
        address owner_
    ) external view returns (uint256 allowance_);

    /**
     *  @notice Returns information about a loan in the pool.
     *  @param  loanId_             Loan's id within loan heap. Max loan is position `1`.
     *  @return borrower_           Borrower address at the given position.
     *  @return t0DebtToCollateral_ Borrower t0 debt to collateral.
     */
    function loanInfo(
        uint256 loanId_
    )
        external
        view
        returns (
            address borrower_,
            uint256 t0DebtToCollateral_
    );

    /**
     *  @notice Returns information about pool loans.
     *  @return maxBorrower_           Borrower address with highest t0 debt to collateral.
     *  @return maxT0DebtToCollateral_ Highest t0 debt to collateral in pool.
     *  @return noOfLoans_             Total number of loans.
     */
    function loansInfo()
        external
        view
        returns (
            address maxBorrower_,
            uint256 maxT0DebtToCollateral_,
            uint256 noOfLoans_
    );

    /**
     *  @notice Returns information about pool reserves.
     *  @return liquidationBondEscrowed_ Amount of liquidation bond across all liquidators.
     *  @return reserveAuctionUnclaimed_ Amount of claimable reserves which has not been taken in the `Claimable Reserve Auction`.
     *  @return reserveAuctionKicked_    Time a `Claimable Reserve Auction` was last kicked.
     *  @return lastKickedReserves_      Amount of reserves upon last kick, used to calculate price.
     *  @return totalInterestEarned_     Total interest earned by all lenders in the pool
     */
    function reservesInfo()
        external
        view
        returns (
            uint256 liquidationBondEscrowed_,
            uint256 reserveAuctionUnclaimed_,
            uint256 reserveAuctionKicked_,
            uint256 lastKickedReserves_,
            uint256 totalInterestEarned_
    );

    /**
     *  @notice Returns the `pledgedCollateral` state variable.
     *  @return The total pledged collateral in the system, in WAD units.
     */
    function pledgedCollateral() external view returns (uint256);

    /**
     *  @notice Returns the total number of active auctions in pool.
     *  @return totalAuctions_ Number of active auctions.
     */
    function totalAuctionsInPool() external view returns (uint256);

     /**
     *  @notice Returns the `t0Debt` state variable.
     *  @dev    This value should be multiplied by inflator in order to calculate current debt of the pool.
     *  @return The total `t0Debt` in the system, in `WAD` units.
     */
    function totalT0Debt() external view returns (uint256);

    /**
     *  @notice Returns the `t0DebtInAuction` state variable.
     *  @dev    This value should be multiplied by inflator in order to calculate current debt in auction of the pool.
     *  @return The total `t0DebtInAuction` in the system, in `WAD` units.
     */
    function totalT0DebtInAuction() external view returns (uint256);

    /**
     *  @notice Mapping of addresses that can transfer `LP` to a given lender.
     *  @param  lender_     Lender that receives `LP`.
     *  @param  transferor_ Transferor that transfers `LP`.
     *  @return True if the transferor is approved by lender.
     */
    function approvedTransferors(
        address lender_,
        address transferor_
    ) external view returns (bool);

}

/*********************/
/*** State Structs ***/
/*********************/

/******************/
/*** Pool State ***/
/******************/

/// @dev Struct holding inflator state.
struct InflatorState {
    uint208 inflator;       // [WAD] pool's inflator
    uint48  inflatorUpdate; // [SEC] last time pool's inflator was updated
}

/// @dev Struct holding pool interest state.
struct InterestState {
    uint208 interestRate;        // [WAD] pool's interest rate
    uint48  interestRateUpdate;  // [SEC] last time pool's interest rate was updated (not before 12 hours passed)
    uint256 debt;                // [WAD] previous update's debt
    uint256 meaningfulDeposit;   // [WAD] previous update's meaningfulDeposit
    uint256 t0Debt2ToCollateral; // [WAD] utilization weight accumulator, tracks debt and collateral relationship accross borrowers 
    uint256 debtCol;             // [WAD] previous debt squared to collateral
    uint256 lupt0Debt;           // [WAD] previous LUP * t0 debt
}

/// @dev Struct holding pool EMAs state.
struct EmaState {
    uint256 debtEma;             // [WAD] sample of debt EMA, numerator to MAU calculation
    uint256 depositEma;          // [WAD] sample of meaningful deposit EMA, denominator to MAU calculation
    uint256 debtColEma;          // [WAD] debt squared to collateral EMA, numerator to TU calculation
    uint256 lupt0DebtEma;        // [WAD] EMA of LUP * t0 debt, denominator to TU calculation
    uint256 emaUpdate;           // [SEC] last time pool's EMAs were updated
}

/// @dev Struct holding pool balances state.
struct PoolBalancesState {
    uint256 pledgedCollateral; // [WAD] total collateral pledged in pool
    uint256 t0DebtInAuction;   // [WAD] Total debt in auction used to restrict LPB holder from withdrawing
    uint256 t0Debt;            // [WAD] Pool debt as if the whole amount was incurred upon the first loan
}

/// @dev Struct holding pool params (in memory only).
struct PoolState {
    uint8   poolType;             // pool type, can be ERC20 or ERC721
    uint256 t0Debt;               // [WAD] t0 debt in pool
    uint256 t0DebtInAuction;      // [WAD] t0 debt in auction within pool
    uint256 debt;                 // [WAD] total debt in pool, accrued in current block
    uint256 collateral;           // [WAD] total collateral pledged in pool
    uint256 inflator;             // [WAD] current pool inflator
    bool    isNewInterestAccrued; // true if new interest already accrued in current block
    uint256 rate;                 // [WAD] pool's current interest rate
    uint256 quoteTokenScale;      // [WAD] quote token scale of the pool. Same as quote token dust.
}

/*********************/
/*** Buckets State ***/
/*********************/

/// @dev Struct holding lender state.
struct Lender {
    uint256 lps;         // [WAD] Lender LP accumulator
    uint256 depositTime; // timestamp of last deposit
}

/// @dev Struct holding bucket state.
struct Bucket {
    uint256 lps;                        // [WAD] Bucket LP accumulator
    uint256 collateral;                 // [WAD] Available collateral tokens deposited in the bucket
    uint256 bankruptcyTime;             // Timestamp when bucket become insolvent, 0 if healthy
    mapping(address => Lender) lenders; // lender address to Lender struct mapping
}

/**********************/
/*** Deposits State ***/
/**********************/

/// @dev Struct holding deposits (Fenwick) values and scaling.
struct DepositsState {
    uint256[8193] values;  // Array of values in the FenwickTree.
    uint256[8193] scaling; // Array of values which scale (multiply) the FenwickTree accross indexes.
}

/*******************/
/*** Loans State ***/
/*******************/

/// @dev Struct holding loans state.
struct LoansState {
    Loan[] loans;
    mapping (address => uint)     indices;   // borrower address => loan index mapping
    mapping (address => Borrower) borrowers; // borrower address => Borrower struct mapping
}

/// @dev Struct holding loan state.
struct Loan {
    address borrower;           // borrower address
    uint96  t0DebtToCollateral; // [WAD] Borrower t0 debt to collateral.
}

/// @dev Struct holding borrower state.
struct Borrower {
    uint256 t0Debt;                    // [WAD] Borrower debt time-adjusted as if it was incurred upon first loan of pool.
    uint256 collateral;                // [WAD] Collateral deposited by borrower.
    uint256 npTpRatio;                 // [WAD] Np to Tp ratio at the time of last borrow or pull collateral.
}

/**********************/
/*** Auctions State ***/
/**********************/

/// @dev Struct holding pool auctions state.
struct AuctionsState {
    uint96  noOfAuctions;                         // total number of auctions in pool
    address head;                                 // first address in auction queue
    address tail;                                 // last address in auction queue
    uint256 totalBondEscrowed;                    // [WAD] total amount of quote token posted as auction kick bonds
    mapping(address => Liquidation) liquidations; // mapping of borrower address and auction details
    mapping(address => Kicker)      kickers;      // mapping of kicker address and kicker balances
}

/// @dev Struct holding liquidation state.
struct Liquidation {
    address kicker;                // address that initiated liquidation
    uint96  bondFactor;            // [WAD] bond factor used to start liquidation
    uint96  kickTime;              // timestamp when liquidation was started
    address prev;                  // previous liquidated borrower in auctions queue
    uint96  referencePrice;        // [WAD] used to calculate auction start price
    address next;                  // next liquidated borrower in auctions queue
    uint160 bondSize;              // [WAD] liquidation bond size
    uint96  neutralPrice;          // [WAD] Neutral Price when liquidation was started
    uint256 debtToCollateral;      // [WAD] Borrower debt to collateral, which is used in BPF for kicker's reward calculation
    uint256 t0ReserveSettleAmount; // [WAD] Amount of t0Debt that could be settled via reserves in this auction
}

/// @dev Struct holding kicker state.
struct Kicker {
    uint256 claimable; // [WAD] kicker's claimable balance
    uint256 locked;    // [WAD] kicker's balance of tokens locked in auction bonds
}

/******************************/
/*** Reserve Auctions State ***/
/******************************/

/// @dev Struct holding reserve auction state.
struct ReserveAuctionState {
    uint256 kicked;                            // Time a Claimable Reserve Auction was last kicked.
    uint256 lastKickedReserves;                // [WAD] Amount of reserves upon last kick, used to calculate price.
    uint256 unclaimed;                         // [WAD] Amount of claimable reserves which has not been taken in the Claimable Reserve Auction.
    uint256 latestBurnEventEpoch;              // Latest burn event epoch.
    uint256 totalAjnaBurned;                   // [WAD] Total ajna burned in the pool.
    uint256 totalInterestEarned;               // [WAD] Total interest earned by all lenders in the pool.
    mapping (uint256 => BurnEvent) burnEvents; // Mapping burnEventEpoch => BurnEvent.
}

/// @dev Struct holding burn event state.
struct BurnEvent {
    uint256 timestamp;     // time at which the burn event occured
    uint256 totalInterest; // [WAD] current pool interest accumulator `PoolCommons.accrueInterest().newInterest`
    uint256 totalBurned;   // [WAD] burn amount accumulator
}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

interface IERC3156FlashBorrower {

    /**
     * @dev    Receive a flash loan.
     * @param  initiator The initiator of the loan.
     * @param  token     The loan currency.
     * @param  amount    The amount of tokens lent (token precision).
     * @param  fee       The additional amount of tokens to repay.
     * @param  data      Arbitrary data structure, intended to contain user-defined parameters.
     * @return The `keccak256` hash of `ERC3156FlashBorrower.onFlashLoan`
     */
    function onFlashLoan(
        address initiator,
        address token,
        uint256 amount,
        uint256 fee,
        bytes   calldata data
    ) external returns (bytes32);
}

// SPDX-License-Identifier: BUSL-1.1

pragma solidity 0.8.18;

import { PRBMathSD59x18 } from "@prb-math/contracts/PRBMathSD59x18.sol";
import { Math }           from '@openzeppelin/contracts/utils/math/Math.sol';
import { SafeCast }       from "@openzeppelin/contracts/utils/math/SafeCast.sol";

import { PoolType }                 from '../../interfaces/pool/IPool.sol';
import { InflatorState, PoolState } from '../../interfaces/pool/commons/IPoolState.sol';

import { Buckets } from '../internal/Buckets.sol';
import { Maths }   from '../internal/Maths.sol';

    error BucketIndexOutOfBounds();
    error BucketPriceOutOfBounds();

    /*************************/
    /*** Price Conversions ***/
    /*************************/

    /// @dev constant price indices defining the min and max of the potential price range
    int256  constant MAX_BUCKET_INDEX  =  4_156;
    int256  constant MIN_BUCKET_INDEX  = -3_232;
    uint256 constant MAX_FENWICK_INDEX =  7_388;

    uint256 constant MIN_PRICE = 99_836_282_890;
    uint256 constant MAX_PRICE = 1_004_968_987.606512354182109771 * 1e18;

    uint256 constant MAX_INFLATED_PRICE = 50_248_449_380.325617709105488550 * 1e18; // 50 * MAX_PRICE

    /// @dev deposit buffer (extra margin) used for calculating reserves
    uint256 constant DEPOSIT_BUFFER = 1.000000001 * 1e18;

    /// @dev step amounts in basis points. This is a constant across pools at `0.005`, achieved by dividing `WAD` by `10,000`
    int256 constant FLOAT_STEP_INT = 1.005 * 1e18;

    /// @dev collateralization factor used to calculate borrrower HTP/TP/collateralization.
    uint256 constant COLLATERALIZATION_FACTOR = 1.04 * 1e18;

    /**
     *  @notice Calculates the price (`WAD` precision) for a given `Fenwick` index.
     *  @dev    Reverts with `BucketIndexOutOfBounds` if index exceeds maximum constant.
     *  @dev    Uses fixed-point math to get around lack of floating point numbers in `EVM`.
     *  @dev    Fenwick index is converted to bucket index.
     *  @dev    Fenwick index to bucket index conversion:
     *  @dev      `1.00`      : bucket index `0`,     fenwick index `4156`: `7388-4156-3232=0`.
     *  @dev      `MAX_PRICE` : bucket index `4156`,  fenwick index `0`:    `7388-0-3232=4156`.
     *  @dev      `MIN_PRICE` : bucket index - `3232`, fenwick index `7388`: `7388-7388-3232=-3232`.
     *  @dev    `V1`: `price = MIN_PRICE + (FLOAT_STEP * index)`
     *  @dev    `V2`: `price = MAX_PRICE * (FLOAT_STEP ** (abs(int256(index - MAX_PRICE_INDEX))));`
     *  @dev    `V3 (final)`: `x^y = 2^(y*log_2(x))`
     */
    function _priceAt(
        uint256 index_
    ) pure returns (uint256) {
        // Lowest Fenwick index is highest price, so invert the index and offset by highest bucket index.
        int256 bucketIndex = MAX_BUCKET_INDEX - int256(index_);
        if (bucketIndex < MIN_BUCKET_INDEX || bucketIndex > MAX_BUCKET_INDEX) revert BucketIndexOutOfBounds();

        return uint256(
            PRBMathSD59x18.exp2(
                PRBMathSD59x18.mul(
                    PRBMathSD59x18.fromInt(bucketIndex),
                    PRBMathSD59x18.log2(FLOAT_STEP_INT)
                )
            )
        );
    }

    /**
     *  @notice Calculates the  Fenwick  index for a given price.
     *  @dev    Reverts with `BucketPriceOutOfBounds` if price exceeds maximum constant.
     *  @dev    Price expected to be inputted as a `WAD` (`18` decimal).
     *  @dev    `V1`: `bucket index = (price - MIN_PRICE) / FLOAT_STEP`
     *  @dev    `V2`: `bucket index = (log(FLOAT_STEP) * price) /  MAX_PRICE`
     *  @dev    `V3 (final)`: `bucket index =  log_2(price) / log_2(FLOAT_STEP)`
     *  @dev    `Fenwick index = 7388 - bucket index + 3232`
     */
    function _indexOf(
        uint256 price_
    ) pure returns (uint256) {
        if (price_ < MIN_PRICE || price_ > MAX_PRICE) revert BucketPriceOutOfBounds();

        int256 index = PRBMathSD59x18.div(
            PRBMathSD59x18.log2(int256(price_)),
            PRBMathSD59x18.log2(FLOAT_STEP_INT)
        );

        int256 ceilIndex = PRBMathSD59x18.ceil(index);
        if (index < 0 && ceilIndex - index > 0.5 * 1e18) {
            return uint256(4157 - PRBMathSD59x18.toInt(ceilIndex));
        }
        return uint256(4156 - PRBMathSD59x18.toInt(ceilIndex));
    }

    /**********************/
    /*** Pool Utilities ***/
    /**********************/

    /**
     *  @notice Calculates the minimum debt amount that can be borrowed or can remain in a loan in pool.
     *  @param  debt_          The debt amount to calculate minimum debt amount for.
     *  @param  loansCount_    The number of loans in pool.
     *  @return minDebtAmount_ Minimum debt amount value of the pool.
     */
    function _minDebtAmount(
        uint256 debt_,
        uint256 loansCount_
    ) pure returns (uint256 minDebtAmount_) {
        if (loansCount_ != 0) {
            minDebtAmount_ = Maths.wdiv(Maths.wdiv(debt_, Maths.wad(loansCount_)), 10**19);
        }
    }

    /**
     *  @notice Calculates origination fee for a given interest rate.
     *  @notice Calculated as greater of the current annualized interest rate divided by `52` (one week of interest) or `5` bps.
     *  @param  interestRate_ The current interest rate.
     *  @return Fee rate based upon the given interest rate.
     */
    function _borrowFeeRate(
        uint256 interestRate_
    ) pure returns (uint256) {
        // greater of the current annualized interest rate divided by 52 (one week of interest) or 5 bps
        return Maths.max(Maths.wdiv(interestRate_, 52 * 1e18), 0.0005 * 1e18);
    }

    /**
     * @notice Calculates the unutilized deposit fee, charged to lenders who deposit below the `LUP`.
     * @param  interestRate_ The current interest rate.
     * @return Fee rate based upon the given interest rate
     */
    function _depositFeeRate(
        uint256 interestRate_
    ) pure returns (uint256) {
        // current annualized rate divided by 365 * 3 (8 hours of interest)
        return Maths.wdiv(interestRate_, 365 * 3e18);
    }

    /**
     * @notice Determines how the inflator state should be updated
     * @param  poolState_     State of the pool after updateInterestState was called.
     * @param  inflatorState_ Old inflator state.
     * @return newInflator_     New inflator value.
     * @return updateTimestamp_ `True` if timestamp of last update should be updated.
     */
    function _determineInflatorState(
        PoolState memory poolState_,
        InflatorState memory inflatorState_
    ) view returns (uint208 newInflator_, bool updateTimestamp_) {
        newInflator_ = inflatorState_.inflator;

        // update pool inflator
        if (poolState_.isNewInterestAccrued) {
            newInflator_     = SafeCast.toUint208(poolState_.inflator);
            updateTimestamp_ = true;
        // if the debt in the current pool state is 0, also update the inflator and inflatorUpdate fields in inflatorState
        // slither-disable-next-line incorrect-equality
        } else if (poolState_.debt == 0) {
            newInflator_     = SafeCast.toUint208(Maths.WAD);
            updateTimestamp_ = true;
        // if the first loan has just been drawn, update the inflator timestamp
        // slither-disable-next-line incorrect-equality
        } else if (inflatorState_.inflator == Maths.WAD && inflatorState_.inflatorUpdate != block.timestamp){
            updateTimestamp_ = true;
        }
    }

    /**
     *  @notice Calculates `HTP` price.
     *  @param  maxT0DebtToCollateral_ Max t0 debt to collateral in pool.
     *  @param  inflator_              Pool's inflator.
     */
    function _htp(
        uint256 maxT0DebtToCollateral_,
        uint256 inflator_
    ) pure returns (uint256) {
        return Maths.wmul(
            Maths.wmul(maxT0DebtToCollateral_, inflator_),
            COLLATERALIZATION_FACTOR
        );
    }

    /**
     *  @notice Calculates debt-weighted average threshold price.
     *  @param  t0Debt_              Pool debt owed by borrowers in `t0` terms.
     *  @param  inflator_            Pool's borrower inflator.
     *  @param  t0Debt2ToCollateral_ `t0-debt-squared-to-collateral` accumulator. 
     */
    function _dwatp(
        uint256 t0Debt_,
        uint256 inflator_,
        uint256 t0Debt2ToCollateral_
    ) pure returns (uint256) {
        return t0Debt_ == 0 ? 0 : Maths.wdiv(
            Maths.wmul(
                Maths.wmul(inflator_, t0Debt2ToCollateral_),
                COLLATERALIZATION_FACTOR
            ),
            t0Debt_
        );
    }

    /**
     *  @notice Collateralization calculation.
     *  @param debt_       Debt to calculate collateralization for.
     *  @param collateral_ Collateral to calculate collateralization for.
     *  @param price_      Price to calculate collateralization for.
     *  @param type_       Type of the pool.
     *  @return `True` if value of collateral exceeds or equals debt.
     */
    function _isCollateralized(
        uint256 debt_,
        uint256 collateral_,
        uint256 price_,
        uint8 type_
    ) pure returns (bool) {
        // `False` if LUP = MIN_PRICE unless there is no debt
        if (price_ == MIN_PRICE && debt_ != 0) return false;

        // Use collateral floor for NFT pools
        if (type_ == uint8(PoolType.ERC721)) {
            //slither-disable-next-line divide-before-multiply
            collateral_ = (collateral_ / Maths.WAD) * Maths.WAD; // use collateral floor
        }
        
        return Maths.wmul(collateral_, price_) >= Maths.wmul(COLLATERALIZATION_FACTOR, debt_);
    }

    /**
     *  @notice Price precision adjustment used in calculating collateral dust for a bucket.
     *          To ensure the accuracy of the exchange rate calculation, buckets with smaller prices require
     *          larger minimum amounts of collateral.  This formula imposes a lower bound independent of token scale.
     *  @param  bucketIndex_              Index of the bucket, or `0` for encumbered collateral with no bucket affinity.
     *  @return pricePrecisionAdjustment_ Unscaled integer of the minimum number of decimal places the dust limit requires.
     */
    function _getCollateralDustPricePrecisionAdjustment(
        uint256 bucketIndex_
    ) pure returns (uint256 pricePrecisionAdjustment_) {
        // conditional is a gas optimization
        if (bucketIndex_ > 3900) {
            int256 bucketOffset = int256(bucketIndex_ - 3900);
            int256 result = PRBMathSD59x18.sqrt(PRBMathSD59x18.div(bucketOffset * 1e18, int256(36 * 1e18)));
            pricePrecisionAdjustment_ = uint256(result / 1e18);
        }
    }

    /**
     *  @notice Returns the amount of collateral calculated for the given amount of `LP`.
     *  @dev    The value returned is capped at collateral amount available in bucket.
     *  @param  bucketCollateral_ Amount of collateral in bucket.
     *  @param  bucketLP_         Amount of `LP` in bucket.
     *  @param  deposit_          Current bucket deposit (quote tokens). Used to calculate bucket's exchange rate / `LP`.
     *  @param  lenderLPBalance_  The amount of `LP` to calculate collateral for.
     *  @param  bucketPrice_      Bucket's price.
     *  @return collateralAmount_ Amount of collateral calculated for the given `LP `amount.
     */
    function _lpToCollateral(
        uint256 bucketCollateral_,
        uint256 bucketLP_,
        uint256 deposit_,
        uint256 lenderLPBalance_,
        uint256 bucketPrice_
    ) pure returns (uint256 collateralAmount_) {
        collateralAmount_ = Buckets.lpToCollateral(
            bucketCollateral_,
            bucketLP_,
            deposit_,
            lenderLPBalance_,
            bucketPrice_,
            Math.Rounding.Down
        );

        if (collateralAmount_ > bucketCollateral_) {
            // user is owed more collateral than is available in the bucket
            collateralAmount_ = bucketCollateral_;
        }
    }

    /**
     *  @notice Returns the amount of quote tokens calculated for the given amount of `LP`.
     *  @dev    The value returned is capped at available bucket deposit.
     *  @param  bucketLP_         Amount of `LP` in bucket.
     *  @param  bucketCollateral_ Amount of collateral in bucket.
     *  @param  deposit_          Current bucket deposit (quote tokens). Used to calculate bucket's exchange rate / `LP`.
     *  @param  lenderLPBalance_  The amount of `LP` to calculate quote token amount for.
     *  @param  bucketPrice_      Bucket's price.
     *  @return quoteTokenAmount_ Amount of quote tokens calculated for the given `LP` amount, capped at available bucket deposit.
     */
    function _lpToQuoteToken(
        uint256 bucketLP_,
        uint256 bucketCollateral_,
        uint256 deposit_,
        uint256 lenderLPBalance_,
        uint256 bucketPrice_
    ) pure returns (uint256 quoteTokenAmount_) {
        quoteTokenAmount_ = Buckets.lpToQuoteTokens(
            bucketCollateral_,
            bucketLP_,
            deposit_,
            lenderLPBalance_,
            bucketPrice_,
            Math.Rounding.Down
        );

        if (quoteTokenAmount_ > deposit_) quoteTokenAmount_ = deposit_;
    }

    /**
     *  @notice Rounds a token amount down to the minimum amount permissible by the token scale.
     *  @param  amount_       Value to be rounded.
     *  @param  tokenScale_   Scale of the token, presented as a power of `10`.
     *  @return scaledAmount_ Rounded value.
     */
    function _roundToScale(
        uint256 amount_,
        uint256 tokenScale_
    ) pure returns (uint256 scaledAmount_) {
        scaledAmount_ = (amount_ / tokenScale_) * tokenScale_;
    }

    /**
     *  @notice Rounds a token amount up to the next amount permissible by the token scale.
     *  @param  amount_       Value to be rounded.
     *  @param  tokenScale_   Scale of the token, presented as a power of `10`.
     *  @return scaledAmount_ Rounded value.
     */
    function _roundUpToScale(
        uint256 amount_,
        uint256 tokenScale_
    ) pure returns (uint256 scaledAmount_) {
        if (amount_ % tokenScale_ == 0)
            scaledAmount_ = amount_;
        else
            scaledAmount_ = _roundToScale(amount_, tokenScale_) + tokenScale_;
    }

    /*********************************/
    /*** Reserve Auction Utilities ***/
    /*********************************/

    uint256 constant MINUTE_HALF_LIFE    = 0.988514020352896135_356867505 * 1e27;  // 0.5^(1/60)

    /**
     *  @notice Calculates claimable reserves within the pool.
     *  @dev    Claimable reserve auctions and escrowed auction bonds are guaranteed by the pool.
     *  @param  debt_                    Pool's debt.
     *  @param  poolSize_                Pool's deposit size.
     *  @param  totalBondEscrowed_       Total bond escrowed.
     *  @param  reserveAuctionUnclaimed_ Pool's unclaimed reserve auction.
     *  @param  quoteTokenBalance_       Pool's quote token balance.
     *  @return claimable_               Calculated pool reserves.
     */  
    function _claimableReserves(
        uint256 debt_,
        uint256 poolSize_,
        uint256 totalBondEscrowed_,
        uint256 reserveAuctionUnclaimed_,
        uint256 quoteTokenBalance_
    ) pure returns (uint256 claimable_) {
        uint256 guaranteedFunds = totalBondEscrowed_ + reserveAuctionUnclaimed_;

        // calculate claimable reserves if there's quote token excess
        if (quoteTokenBalance_ > guaranteedFunds) {
            claimable_ = debt_ + quoteTokenBalance_;

            claimable_ -= Maths.min(
                claimable_,
                // require 1.0 + 1e-9 deposit buffer (extra margin) for deposits
                Maths.wmul(DEPOSIT_BUFFER, poolSize_) + guaranteedFunds
            );

            // incremental claimable reserve should not exceed excess quote in pool
            claimable_ = Maths.min(
                claimable_,
                quoteTokenBalance_ - guaranteedFunds
            );
        }
    }

    /**
     *  @notice Calculates reserves auction price.
     *  @param  reserveAuctionKicked_ Time when reserve auction was started (kicked).
     *  @param  lastKickedReserves_   Reserves to be auctioned when started (kicked).
     *  @return price_                Calculated auction price.
     */     
    function _reserveAuctionPrice(
        uint256 reserveAuctionKicked_,
        uint256 lastKickedReserves_
    ) view returns (uint256 price_) {
        if (reserveAuctionKicked_ != 0) {
            uint256 secondsElapsed   = block.timestamp - reserveAuctionKicked_;
            uint256 hoursComponent   = 1e27 >> secondsElapsed / 3600;
            uint256 minutesComponent = Maths.rpow(MINUTE_HALF_LIFE, secondsElapsed % 3600 / 60);
            uint256 initialPrice     = lastKickedReserves_ == 0 ? 0 : Maths.wdiv(1_000_000_000 * 1e18, lastKickedReserves_);

            price_ = initialPrice * Maths.rmul(hoursComponent, minutesComponent) / 1e27;
        }
    }

    /*************************/
    /*** Auction Utilities ***/
    /*************************/

    /// @dev min bond factor.
    uint256 constant MIN_BOND_FACTOR = 0.005 * 1e18;
    /// @dev max bond factor.
    uint256 constant MAX_BOND_FACTOR = 0.03 * 1e18;

    /**
     *  @notice Calculates auction price.
     *  @param  referencePrice_ Recorded at kick, used to calculate start price.
     *  @param  kickTime_       Time when auction was kicked.
     *  @return price_          Calculated auction price.
     */
    function _auctionPrice(
        uint256 referencePrice_,
        uint256 kickTime_
    ) view returns (uint256 price_) {
        uint256 elapsedMinutes = Maths.wdiv((block.timestamp - kickTime_) * 1e18, 1 minutes * 1e18);

        int256 timeAdjustment;
        if (elapsedMinutes < 120 * 1e18) {
            timeAdjustment = PRBMathSD59x18.mul(-1 * 1e18, int256(elapsedMinutes / 20));
            price_ = 256 * Maths.wmul(referencePrice_, uint256(PRBMathSD59x18.exp2(timeAdjustment)));
        } else if (elapsedMinutes < 840 * 1e18) {
            timeAdjustment = PRBMathSD59x18.mul(-1 * 1e18, int256((elapsedMinutes - 120 * 1e18) / 120));
            price_ = 4 * Maths.wmul(referencePrice_, uint256(PRBMathSD59x18.exp2(timeAdjustment)));
        } else {
            timeAdjustment = PRBMathSD59x18.mul(-1 * 1e18, int256((elapsedMinutes - 840 * 1e18) / 60));
            price_ = Maths.wmul(referencePrice_, uint256(PRBMathSD59x18.exp2(timeAdjustment))) / 16;
        }
    }

    /**
     *  @notice Calculates bond penalty factor.
     *  @dev    Called in kick and take.
     *  @param debtToCollateral_ Borrower debt to collateral at time of kick.
     *  @param neutralPrice_     `NP` of auction.
     *  @param bondFactor_       Factor used to determine bondSize.
     *  @param auctionPrice_     Auction price at the time of call or, for bucket takes, bucket price.
     *  @return bpf_             Factor used in determining bond `reward` (positive) or `penalty` (negative).
     */
    function _bpf(
        uint256 debtToCollateral_,
        uint256 neutralPrice_,
        uint256 bondFactor_,
        uint256 auctionPrice_
    ) pure returns (int256) {
        int256 sign;
        if (debtToCollateral_ < neutralPrice_) {
            // BPF = BondFactor * min(1, max(-1, (neutralPrice - price) / (neutralPrice - debtToCollateral)))
            sign = Maths.minInt(
                1e18,
                Maths.maxInt(
                    -1 * 1e18,
                    PRBMathSD59x18.div(
                        int256(neutralPrice_) - int256(auctionPrice_),
                        int256(neutralPrice_) - int256(debtToCollateral_)
                    )
                )
            );
        } else {
            int256 val = int256(neutralPrice_) - int256(auctionPrice_);
            if (val < 0 )      sign = -1e18;
            else if (val != 0) sign = 1e18;
        }

        return PRBMathSD59x18.mul(int256(bondFactor_), sign);
    }

    /**
     *  @notice Calculates bond parameters of an auction.
     *  @param  borrowerDebt_   Borrower's debt before entering in liquidation.
     *  @param  npTpRatio_      Borrower's Np to Tp ratio
     */
    function _bondParams(
        uint256 borrowerDebt_,
        uint256 npTpRatio_
    ) pure returns (uint256 bondFactor_, uint256 bondSize_) {
        // bondFactor = max(min(MAX_BOND_FACTOR, (NP/TP_ratio - 1) / 10), MIN_BOND_FACTOR)
        bondFactor_ = Maths.max(
            Maths.min(
                MAX_BOND_FACTOR,
                (npTpRatio_ - 1e18) / 10
            ),
            MIN_BOND_FACTOR
        );

        bondSize_ = Maths.wmul(bondFactor_,  borrowerDebt_);
    }

// SPDX-License-Identifier: BUSL-1.1

pragma solidity 0.8.18;

import { Math } from '@openzeppelin/contracts/utils/math/Math.sol';

import { DepositsState } from '../../interfaces/pool/commons/IPoolState.sol';

import { _priceAt, MAX_FENWICK_INDEX } from '../helpers/PoolHelper.sol';

import { Maths } from './Maths.sol';

/**
    @title  Deposits library
    @notice Internal library containing common logic for deposits management.
    @dev    Implemented as `Fenwick Tree` data structure.
 */
library Deposits {

    /// @dev Max index supported in the `Fenwick` tree
    uint256 internal constant SIZE = 8192;

    /**************/
    /*** Errors ***/
    /**************/

    // See `IPoolErrors` for descriptions
    error InvalidAmount();

    /**
     *  @notice Increase a value in the FenwickTree at an index.
     *  @dev    Starts at leaf/target and moved up towards root
     *  @dev    === Reverts on ===
     *  @dev    unscaled amount to add is 0 `InvalidAmount()`
     *  @param  deposits_          Deposits state struct.
     *  @param  index_             The deposit index.
     *  @param  unscaledAddAmount_ The unscaled amount to increase deposit by.
     */
    function unscaledAdd(
        DepositsState storage deposits_,
        uint256 index_,
        uint256 unscaledAddAmount_
    ) internal {

        // revert if 0 amount is added.
        if (unscaledAddAmount_ == 0) revert InvalidAmount();

        // price buckets are indexed starting at 0, Fenwick bit logic is more elegant starting at 1
        ++index_;

        // unscaledAddAmount_ is the raw amount to add directly to the value at index_, unaffected by the scale array
        // For example, to denote an amount of deposit added to the array, we would need to call unscaledAdd with
        // (deposit amount) / scale(index).  There are two reasons for this:
        // 1- scale(index) is often already known in the context of where unscaledAdd(..) is called, and we want to avoid
        //    redundant iterations through the Fenwick tree.
        // 2- We often need to precisely change the value in the tree, avoiding the rounding that dividing by scale(index).
        //    This is more relevant to unscaledRemove(...), where we need to ensure the value is precisely set to 0, but we
        //    also prefer it here for consistency.

        uint256 value;
        uint256 scaling;
        uint256 newValue;

        while (index_ <= SIZE) {
            value    = deposits_.values[index_];
            scaling  = deposits_.scaling[index_];

            // Compute the new value to be put in location index_
            newValue = value + unscaledAddAmount_;

            // Update unscaledAddAmount to propogate up the Fenwick tree
            // Note: we can't just multiply addAmount_ by scaling[i_] due to rounding
            // We need to track the precice change in values[i_] in order to ensure
            // obliterated indices remain zero after subsequent adding to related indices
            // if scaling==0, the actual scale value is 1, otherwise it is scaling
            if (scaling != 0) unscaledAddAmount_ = Maths.wmul(newValue, scaling) - Maths.wmul(value, scaling);

            deposits_.values[index_] = newValue;

            // traverse upwards through tree via "update" route
            index_ += lsb(index_);
        }
    }

    /**
     *  @notice Finds index and sum of first bucket that EXCEEDS the given sum
     *  @dev    Used in `LUP` calculation
     *  @param  deposits_      Struct for deposits state.
     *  @param  targetSum_     The sum to find index for.
     *  @return sumIndex_      Smallest index where prefixsum greater than the sum.
     *  @return sumIndexSum_   Sum at index PRECEDING `sumIndex_`.
     *  @return sumIndexScale_ Scale of bucket PRECEDING `sumIndex_`.
     */
    function findIndexAndSumOfSum(
        DepositsState storage deposits_,
        uint256 targetSum_
    ) internal view returns (uint256 sumIndex_, uint256 sumIndexSum_, uint256 sumIndexScale_) {
        // i iterates over bits from MSB to LSB.  We check at each stage if the target sum is to the left or right of sumIndex_+i
        uint256 i  = 4096; // 1 << (_numBits - 1) = 1 << (13 - 1) = 4096
        uint256 runningScale = Maths.WAD;

        // We construct the target sumIndex_ bit by bit, from MSB to LSB.  lowerIndexSum_ always maintains the sum
        // up to the current value of sumIndex_
        uint256 lowerIndexSum;
        uint256 curIndex;
        uint256 value;
        uint256 scaling;
        uint256 scaledValue;

        while (i > 0) {
            // Consider if the target index is less than or greater than sumIndex_ + i
            curIndex = sumIndex_ + i;
            value    = deposits_.values[curIndex];
            scaling  = deposits_.scaling[curIndex];

            // Compute sum up to sumIndex_ + i
            scaledValue =
                lowerIndexSum +
                (
                    scaling != 0 ? Math.mulDiv(
                        runningScale * scaling,
                        value,
                        1e36
                    ) : Maths.wmul(runningScale, value)
                );

            if (scaledValue  < targetSum_) {
                // Target value is too small, need to consider increasing sumIndex_ still
                if (curIndex <= MAX_FENWICK_INDEX) {
                    // sumIndex_+i is in range of Fenwick prices.  Target index has this bit set to 1.  
                    sumIndex_ = curIndex;
                    lowerIndexSum = scaledValue;
                }
            } else {
                // Target index has this bit set to 0
                // scaling == 0 means scale factor == 1, otherwise scale factor == scaling
                if (scaling != 0) runningScale = Maths.floorWmul(runningScale, scaling);

                // Current scaledValue is <= targetSum_, it's a candidate value for sumIndexSum_
                sumIndexSum_   = scaledValue;
                sumIndexScale_ = runningScale;
            }
            // Shift i to next less significant bit
            i = i >> 1;
        }
    }

    /**
     *  @notice Finds index of passed sum. Helper function for `findIndexAndSumOfSum`.
     *  @dev    Used in `LUP` calculation
     *  @param  deposits_ Deposits state struct.
     *  @param  sum_      The sum to find index for.
     *  @return sumIndex_ Smallest index where prefixsum greater than the sum.
     */
    function findIndexOfSum(
        DepositsState storage deposits_,
        uint256 sum_
    ) internal view returns (uint256 sumIndex_) {
        (sumIndex_,,) = findIndexAndSumOfSum(deposits_, sum_);
    }

    /**
     *  @notice Get least significant bit (`LSB`) of integer `i_`.
     *  @dev    Used primarily to decrement the binary index in loops, iterating over range parents.
     *  @param  i_  The integer with which to return the `LSB`.
     */
    function lsb(
        uint256 i_
    ) internal pure returns (uint256 lsb_) {
        if (i_ != 0) {
            // "i & (-i)"
            lsb_ = i_ & ((i_ ^ 0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff) + 1);
        }
    }

    /**
     *  @notice Scale values in the tree from the index provided, upwards.
     *  @dev    Starts at passed in node and increments through range parent nodes, and ends at `8192`.
     *  @param  deposits_ Deposits state struct.
     *  @param  index_    The index to start scaling from.
     *  @param  factor_   The factor to scale the values by.
     */
    function mult(
        DepositsState storage deposits_,
        uint256 index_,
        uint256 factor_
    ) internal {
        // price buckets are indexed starting at 0, Fenwick bit logic is more elegant starting at 1
        ++index_;

        uint256 sum;
        uint256 value;
        uint256 scaling;
        uint256 bit = lsb(index_);

        // Starting with the LSB of index, we iteratively move up towards the MSB of SIZE
        // Case 1:     the bit of index_ is set to 1.  In this case, the entire subtree below index_
        //             is scaled.  So, we include factor_ into scaling[index_], and remember in sum how much
        //             we increased the subtree by, so that we can use it in case we encounter 0 bits (below).
        // Case 2:     The bit of index_ is set to 0.  In this case, consider the subtree below the node
        //             index_+bit. The subtree below that is not entirely scaled, but it does contain the
        //             subtree what was scaled earlier.  Therefore: we need to increment it's stored value
        //             (in sum) which was set in a prior interation in case 1.
        while (bit <= SIZE) {
            if ((bit & index_) != 0) {
                // Case 1 as described above
                value   = deposits_.values[index_];
                scaling = deposits_.scaling[index_];

                // Calc sum, will only be stored in range parents of starting node, index_
                if (scaling != 0) {
                    // Note: we can't just multiply by factor_ - 1 in the following line, as rounding will
                    // cause obliterated indices to have nonzero values.  Need to track the actual
                    // precise delta in the value array
                    uint256 scaledFactor = Maths.wmul(factor_, scaling);

                    sum += Maths.wmul(scaledFactor, value) - Maths.wmul(scaling, value);

                    // Apply scaling to all range parents less then starting node, index_
                    deposits_.scaling[index_] = scaledFactor;
                } else {
                    // this node's scale factor is 1
                    sum += Maths.wmul(factor_, value) - value;
                    deposits_.scaling[index_] = factor_;
                }
                // Unset the bit in index to continue traversing up the Fenwick tree
                index_ -= bit;
            } else {
                // Case 2 above.  superRangeIndex is the index of the node to consider that
                //                contains the sub range that was already scaled in prior iteration
                uint256 superRangeIndex = index_ + bit;

                value   = (deposits_.values[superRangeIndex] += sum);
                scaling = deposits_.scaling[superRangeIndex];

                // Need to be careful due to rounding to propagate actual changes upwards in tree.
                // sum is always equal to the actual value we changed deposits_.values[] by
                if (scaling != 0) sum = Maths.wmul(value, scaling) - Maths.wmul(value - sum, scaling);
            }
            // consider next most significant bit
            bit = bit << 1;
        }
    }

    /**
     *  @notice Get prefix sum of all indexes from provided index downwards.
     *  @dev    Starts at tree root and decrements through range parent nodes summing from index `sumIndex_`'s range to index `0`.
     *  @param  deposits_  Deposits state struct.
     *  @param  sumIndex_  The index to receive the prefix sum.
     *  @param  sum_       The prefix sum from current index downwards.
     */
    function prefixSum(
        DepositsState storage deposits_,
        uint256 sumIndex_
    ) internal view returns (uint256 sum_) {
        // price buckets are indexed starting at 0, Fenwick bit logic is more elegant starting at 1
        ++sumIndex_;

        uint256 runningScale = Maths.WAD; // Tracks scale(index_) as we move down Fenwick tree
        uint256 j            = SIZE;      // bit that iterates from MSB to LSB
        uint256 index        = 0;         // build up sumIndex bit by bit

        // Used to terminate loop.  We don't need to consider final 0 bits of sumIndex_
        uint256 indexLSB = lsb(sumIndex_);
        uint256 curIndex;

        while (j >= indexLSB) {
            curIndex = index + j;

            // Skip considering indices outside bounds of Fenwick tree
            if (curIndex > SIZE) continue;

            // We are considering whether to include node index + j in the sum or not.  Either way, we need to scaling[index + j],
            // either to increment sum_ or to accumulate in runningScale
            uint256 scaled = deposits_.scaling[curIndex];

            if (sumIndex_ & j != 0) {
                // node index + j of tree is included in sum
                uint256 value = deposits_.values[curIndex];

                // Accumulate in sum_, recall that scaled==0 means that the scale factor is actually 1
                sum_  += scaled != 0 ? Math.mulDiv(
                    runningScale * scaled,
                    value,
                    1e36
                ) : Maths.wmul(runningScale, value);

                // Build up index bit by bit
                index = curIndex;

                // terminate if we've already matched sumIndex_
                if (index == sumIndex_) break;
            } else {
                // node is not included in sum, but its scale needs to be included for subsequent sums
                if (scaled != 0) runningScale = Maths.floorWmul(runningScale, scaled);
            }
            // shift j to consider next less signficant bit
            j = j >> 1;
        }
    }

    /**
     *  @notice Decrease a node in the `FenwickTree` at an index.
     *  @dev    Starts at leaf/target and moved up towards root.
     *  @dev    === Reverts on ===
     *  @dev    unscaled amount to remove is 0 `InvalidAmount()`
     *  @param  deposits_             Deposits state struct.
     *  @param  index_                The deposit index.
     *  @param  unscaledRemoveAmount_ Unscaled amount to decrease deposit by.
     */
    function unscaledRemove(
        DepositsState storage deposits_,
        uint256 index_,
        uint256 unscaledRemoveAmount_
    ) internal {

        // revert if 0 amount is removed.
        if (unscaledRemoveAmount_ == 0) revert InvalidAmount();

        // price buckets are indexed starting at 0, Fenwick bit logic is more elegant starting at 1
        ++index_;

        // We operate with unscaledRemoveAmount_ here instead of a scaled quantity to avoid duplicate computation of scale factor
        // (thus redundant iterations through the Fenwick tree), and ALSO so that we can set the value of a given deposit exactly
        // to 0.
        
        while (index_ <= SIZE) {
            // Decrement deposits_ at index_ for removeAmount, storing new value in value
            uint256 value   = (deposits_.values[index_] -= unscaledRemoveAmount_);
            uint256 scaling = deposits_.scaling[index_];

            // If scale factor != 1, we need to adjust unscaledRemoveAmount by scale factor to adjust values further up in tree
            // On the line below, it would be tempting to replace this with:
            // unscaledRemoveAmount_ = Maths.wmul(unscaledRemoveAmount, scaling).  This will introduce nonzero values up
            // the tree due to rounding.  It's important to compute the actual change in deposits_.values[index_]
            // and propogate that upwards.
            if (scaling != 0) unscaledRemoveAmount_ = Maths.wmul(value + unscaledRemoveAmount_, scaling) - Maths.wmul(value,  scaling);

            // Traverse upward through the "update" path of the Fenwick tree
            index_ += lsb(index_);
        }
    }

    /**
     *  @notice Scale tree starting from given index.
     *  @dev    Starts at leaf/target and moved up towards root.
     *  @param  deposits_ Deposits state struct.
     *  @param  index_    The deposit index.
     *  @return scaled_   Scaled value.
     */
    function scale(
        DepositsState storage deposits_,
        uint256 index_
    ) internal view returns (uint256 scaled_) {
        // price buckets are indexed starting at 0, Fenwick bit logic is more elegant starting at 1
        ++index_;

        // start with scaled_1 = 1
        scaled_ = Maths.WAD;
        while (index_ <= SIZE) {
            // Traverse up through Fenwick tree via "update" path, accumulating scale factors as we go
            uint256 scaling = deposits_.scaling[index_];
            // scaling==0 means actual scale factor is 1
            if (scaling != 0) scaled_ = Maths.wmul(scaled_, scaling);
            index_ += lsb(index_);
        }
    }

    /**
     *  @notice Returns sum of all deposits.
     *  @param  deposits_ Deposits state struct.
     *  @return Sum of all deposits in tree.
     */
    function treeSum(
        DepositsState storage deposits_
    ) internal view returns (uint256) {
        // In a scaled Fenwick tree, sum is at the root node and never scaled
        return deposits_.values[SIZE];
    }

    /**
     *  @notice Returns deposit value for a given deposit index.
     *  @param  deposits_     Deposits state struct.
     *  @param  index_        The deposit index.
     *  @return depositValue_ Value of the deposit.
     */
    function valueAt(
        DepositsState storage deposits_,
        uint256 index_
    ) internal view returns (uint256 depositValue_) {
        // Get unscaled value at index and multiply by scale
        depositValue_ = Maths.wmul(unscaledValueAt(deposits_, index_), scale(deposits_,index_));
    }

    /**
     *  @notice Returns unscaled (deposit without interest) deposit value for a given deposit index.
     *  @param  deposits_             Deposits state struct.
     *  @param  index_                The deposit index.
     *  @return unscaledDepositValue_ Value of unscaled deposit.
     */
    function unscaledValueAt(
        DepositsState storage deposits_,
        uint256 index_
    ) internal view returns (uint256 unscaledDepositValue_) {
        // In a scaled Fenwick tree, sum is at the root node, but needs to be scaled
        ++index_;

        uint256 j = 1;

        // Returns the unscaled value at the node.  We consider the unscaled value for two reasons:
        // 1- If we want to zero out deposit in bucket, we need to subtract the exact unscaled value
        // 2- We may already have computed the scale factor, so we can avoid duplicate traversal

        unscaledDepositValue_ = deposits_.values[index_];
        uint256 curIndex;
        uint256 value;
        uint256 scaling;

        while (j & index_ == 0) {
            curIndex = index_ - j;

            value   = deposits_.values[curIndex];
            scaling = deposits_.scaling[curIndex];

            unscaledDepositValue_ -= scaling != 0 ? Maths.wmul(scaling, value) : value;
            j = j << 1;
        }
    }

    /**
     *  @notice Returns `LUP` for a given debt value (capped at min bucket price).
     *  @param  deposits_ Deposits state struct.
     *  @param  debt_     The debt amount to calculate `LUP` for.
     *  @return `LUP` for given debt.
     */
    function getLup(
        DepositsState storage deposits_,
        uint256 debt_
    ) internal view returns (uint256) {
        return _priceAt(findIndexOfSum(deposits_, debt_));
    }
}

// SPDX-License-Identifier: BUSL-1.1

pragma solidity 0.8.18;

import { Math } from '@openzeppelin/contracts/utils/math/Math.sol';

import { Bucket, Lender } from '../../interfaces/pool/commons/IPoolState.sol';

import { Maths } from './Maths.sol';

/**
    @title  Buckets library
    @notice Internal library containing common logic for buckets management.
 */
library Buckets {

    /**************/
    /*** Events ***/
    /**************/

    // See `IPoolError` for descriptions
    error BucketBankruptcyBlock();

    /***********************************/
    /*** Bucket Management Functions ***/
    /***********************************/

    /**
     *  @notice Add collateral to a bucket and updates `LP` for bucket and lender with the amount coresponding to collateral amount added.
     *  @dev    Increment `bucket.collateral` and `bucket.lps` accumulator
     *  @dev    - `addLenderLP`:
     *  @dev    increment `lender.lps` accumulator and `lender.depositTime` state
     *  @param  lender_                Address of the lender.
     *  @param  deposit_               Current bucket deposit (quote tokens). Used to calculate bucket's exchange rate / `LP`.
     *  @param  collateralAmountToAdd_ Additional collateral amount to add to bucket.
     *  @param  bucketPrice_           Bucket price.
     *  @return addedLP_               Amount of bucket `LP` for the collateral amount added.
     */
    function addCollateral(
        Bucket storage bucket_,
        address lender_,
        uint256 deposit_,
        uint256 collateralAmountToAdd_,
        uint256 bucketPrice_
    ) internal returns (uint256 addedLP_) {
        // cannot deposit in the same block when bucket becomes insolvent
        uint256 bankruptcyTime = bucket_.bankruptcyTime;
        if (bankruptcyTime == block.timestamp) revert BucketBankruptcyBlock();

        // calculate amount of LP to be added for the amount of collateral added to bucket
        addedLP_ = collateralToLP(
            bucket_.collateral,
            bucket_.lps,
            deposit_,
            collateralAmountToAdd_,
            bucketPrice_,
            Math.Rounding.Down
        );
        // update bucket LP balance and collateral

        // update bucket collateral
        bucket_.collateral += collateralAmountToAdd_;
        // update bucket and lender LP balance and deposit timestamp
        bucket_.lps += addedLP_;

        addLenderLP(bucket_, bankruptcyTime, lender_, addedLP_);
    }

    /**
     *  @notice Add amount of `LP` for a given lender in a given bucket.
     *  @dev    Increments lender lps accumulator and updates the deposit time.
     *  @param  bucket_         Bucket to record lender `LP`.
     *  @param  bankruptcyTime_ Time when bucket become insolvent.
     *  @param  lender_         Lender address to add `LP` for in the given bucket.
     *  @param  lpAmount_       Amount of `LP` to be recorded for the given lender.
     */
    function addLenderLP(
        Bucket storage bucket_,
        uint256 bankruptcyTime_,
        address lender_,
        uint256 lpAmount_
    ) internal {
        if (lpAmount_ != 0) {
            Lender storage lender = bucket_.lenders[lender_];

            if (bankruptcyTime_ >= lender.depositTime) lender.lps = lpAmount_;
            else lender.lps += lpAmount_;

            lender.depositTime = block.timestamp;
        }
    }

    /**********************/
    /*** View Functions ***/
    /**********************/

    /****************************/
    /*** Assets to LP helpers ***/
    /****************************/

    /**
     *  @notice Returns the amount of bucket `LP` calculated for the given amount of collateral.
     *  @param  bucketCollateral_ Amount of collateral in bucket.
     *  @param  bucketLP_         Amount of `LP` in bucket.
     *  @param  deposit_          Current bucket deposit (quote tokens). Used to calculate bucket's exchange rate / `LP`.
     *  @param  collateral_       The amount of collateral to calculate bucket LP for.
     *  @param  bucketPrice_      Bucket's price.
     *  @param  rounding_         The direction of rounding when calculating LP (down when adding, up when removing collateral from pool).
     *  @return Amount of `LP` calculated for the amount of collateral.
     */
    function collateralToLP(
        uint256 bucketCollateral_,
        uint256 bucketLP_,
        uint256 deposit_,
        uint256 collateral_,
        uint256 bucketPrice_,
        Math.Rounding rounding_
    ) internal pure returns (uint256) {
        // case when there's no deposit nor collateral in bucket
        if (deposit_ == 0 && bucketCollateral_ == 0) return Maths.wmul(collateral_, bucketPrice_);

        // case when there's deposit or collateral in bucket but no LP to cover
        if (bucketLP_ == 0) return Maths.wmul(collateral_, bucketPrice_);

        // case when there's deposit or collateral and bucket has LP balance
        return Math.mulDiv(
            bucketLP_,
            collateral_ * bucketPrice_,
            deposit_ * Maths.WAD + bucketCollateral_ * bucketPrice_,
            rounding_
        );
    }

    /**
     *  @notice Returns the amount of `LP` calculated for the given amount of quote tokens.
     *  @param  bucketCollateral_ Amount of collateral in bucket.
     *  @param  bucketLP_         Amount of `LP` in bucket.
     *  @param  deposit_          Current bucket deposit (quote tokens). Used to calculate bucket's exchange rate / `LP`.
     *  @param  quoteTokens_      The amount of quote tokens to calculate `LP` amount for.
     *  @param  bucketPrice_      Bucket's price.
     *  @param  rounding_         The direction of rounding when calculating LP (down when adding, up when removing quote tokens from pool).
     *  @return The amount of `LP` coresponding to the given quote tokens in current bucket.
     */
    function quoteTokensToLP(
        uint256 bucketCollateral_,
        uint256 bucketLP_,
        uint256 deposit_,
        uint256 quoteTokens_,
        uint256 bucketPrice_,
        Math.Rounding rounding_
    ) internal pure returns (uint256) {
        // case when there's no deposit nor collateral in bucket
        if (deposit_ == 0 && bucketCollateral_ == 0) return quoteTokens_;

        // case when there's deposit or collateral in bucket but no LP to cover
        if (bucketLP_ == 0) return quoteTokens_;

        // case when there's deposit or collateral and bucket has LP balance
        return Math.mulDiv(
            bucketLP_,
            quoteTokens_ * Maths.WAD,
            deposit_ * Maths.WAD + bucketCollateral_ * bucketPrice_,
            rounding_
        );
    }

    /****************************/
    /*** LP to Assets helpers ***/
    /****************************/

    /**
     *  @notice Returns the amount of collateral calculated for the given amount of lp
     *  @dev    The value returned is not capped at collateral amount available in bucket.
     *  @param  bucketCollateral_ Amount of collateral in bucket.
     *  @param  bucketLP_         Amount of `LP` in bucket.
     *  @param  deposit_          Current bucket deposit (quote tokens). Used to calculate bucket's exchange rate / `LP`.
     *  @param  lp_               The amount of LP to calculate collateral amount for.
     *  @param  bucketPrice_      Bucket's price.
     *  @return The amount of collateral coresponding to the given `LP` in current bucket.
     */
    function lpToCollateral(
        uint256 bucketCollateral_,
        uint256 bucketLP_,
        uint256 deposit_,
        uint256 lp_,
        uint256 bucketPrice_,
        Math.Rounding rounding_
    ) internal pure returns (uint256) {
        // case when there's no deposit nor collateral in bucket
        if (deposit_ == 0 && bucketCollateral_ == 0) return Maths.wdiv(lp_, bucketPrice_);

        // case when there's deposit or collateral in bucket but no LP to cover
        if (bucketLP_ == 0) return Maths.wdiv(lp_, bucketPrice_);

        // case when there's deposit or collateral and bucket has LP balance
        return Math.mulDiv(
            deposit_ * Maths.WAD + bucketCollateral_ * bucketPrice_,
            lp_,
            bucketLP_ * bucketPrice_,
            rounding_
        );
    }

    /**
     *  @notice Returns the amount of quote token (in value) calculated for the given amount of `LP`.
     *  @dev    The value returned is not capped at available bucket deposit.
     *  @param  bucketCollateral_ Amount of collateral in bucket.
     *  @param  bucketLP_         Amount of `LP` in bucket.
     *  @param  deposit_          Current bucket deposit (quote tokens). Used to calculate bucket's exchange rate / `LP`.
     *  @param  lp_               The amount of LP to calculate quote tokens amount for.
     *  @param  bucketPrice_      Bucket's price.
     *  @return The amount coresponding to the given quote tokens in current bucket.
     */
    function lpToQuoteTokens(
        uint256 bucketCollateral_,
        uint256 bucketLP_,
        uint256 deposit_,
        uint256 lp_,
        uint256 bucketPrice_,
        Math.Rounding rounding_
    ) internal pure returns (uint256) {
        // case when there's no deposit nor collateral in bucket
        if (deposit_ == 0 && bucketCollateral_ == 0) return lp_;

        // case when there's deposit or collateral in bucket but no LP to cover
        if (bucketLP_ == 0) return lp_;

        // case when there's deposit or collateral and bucket has LP balance
        return Math.mulDiv(
            deposit_ * Maths.WAD + bucketCollateral_ * bucketPrice_,
            lp_,
            bucketLP_ * Maths.WAD,
            rounding_
        );
    }

    /****************************/
    /*** Exchange Rate helper ***/
    /****************************/

    /**
     *  @notice Returns the exchange rate for a given bucket (conversion of 1 lp to quote token).
     *  @param  bucketCollateral_ Amount of collateral in bucket.
     *  @param  bucketLP_         Amount of `LP` in bucket.
     *  @param  bucketDeposit_    The amount of quote tokens deposited in the given bucket.
     *  @param  bucketPrice_      Bucket's price.
     */
    function getExchangeRate(
        uint256 bucketCollateral_,
        uint256 bucketLP_,
        uint256 bucketDeposit_,
        uint256 bucketPrice_
    ) internal pure returns (uint256) {
        return lpToQuoteTokens(
            bucketCollateral_,
            bucketLP_,
            bucketDeposit_,
            Maths.WAD,
            bucketPrice_,
            Math.Rounding.Up
        );
    }
}

// SPDX-License-Identifier: BUSL-1.1

pragma solidity 0.8.18;

import { SafeCast }       from "@openzeppelin/contracts/utils/math/SafeCast.sol";
import { PRBMathSD59x18 } from "@prb-math/contracts/PRBMathSD59x18.sol";

import {
    AuctionsState,
    Borrower,
    DepositsState,
    Loan,
    LoansState
} from '../../interfaces/pool/commons/IPoolState.sol';

import { _priceAt } from '../helpers/PoolHelper.sol';

import { Deposits } from './Deposits.sol';
import { Maths }    from './Maths.sol';

/**
    @title  Loans library
    @notice Internal library containing common logic for loans management.
    @dev    The `Loans` heap is a `Max Heap` data structure (complete binary tree), the root node is the loan with the highest t0 threshold price (`TP`)
            at a given time. The heap is represented as an array, where the first element is a dummy element (`Loan(address(0), 0)`) and the first
            value of the heap starts at index `1`, `ROOT_INDEX`. The t0 threshold price of a loan's parent is always greater than or equal to the
            t0 threshold price of the loan.
    @dev    This code was modified from the following source: https://github.com/zmitton/eth-heap/blob/master/contracts/Heap.sol
 */
library Loans {

    uint256 constant ROOT_INDEX = 1;

    /**************/
    /*** Errors ***/
    /**************/

    // See `IPoolErrors` for descriptions
    error ZeroDebtToCollateral();

    /***********************/
    /***  Initialization ***/
    /***********************/

    /**
     *  @notice Initializes Loans Max Heap.
     *  @dev    Organizes loans so `Highest t0 threshold price` can be retrieved easily.
     *  @param  loans_ Holds Loan heap data.
     */
    function init(LoansState storage loans_) internal {
        loans_.loans.push(Loan(address(0), 0));
    }

    /***********************************/
    /***  Loans Management Functions ***/
    /***********************************/

    /**
     *  @notice Updates a loan: updates heap (`upsert` if `TP` not `0`, `remove` otherwise) and borrower balance.
     *  @dev    === Write state ===
     *  @dev    - `_upsert`:
     *  @dev      insert or update loan in `loans` array
     *  @dev    - `remove`:
     *  @dev      remove loan from `loans` array
     *  @dev    - update borrower in `address => borrower` mapping
     *  @param loans_           Holds loans heap data.
     *  @param borrower_        Borrower struct with borrower details.
     *  @param borrowerAddress_ Borrower's address to update.
     *  @param poolRate_        Pool's current rate.
     *  @param inAuction_       Whether the loan is in auction or not.
     *  @param npTpRatioUpdate_ Whether the Np to Tp ratio of borrower should be updated or not.
     */
    function update(
        LoansState storage loans_,
        Borrower memory borrower_,
        address borrowerAddress_,
        uint256 poolRate_,
        bool inAuction_,
        bool npTpRatioUpdate_
    ) internal {

        bool activeBorrower = borrower_.t0Debt != 0 && borrower_.collateral != 0;

        uint256 t0DebtToCollateral = activeBorrower ? Maths.wdiv(borrower_.t0Debt, borrower_.collateral) : 0;

        // loan not in auction, update t0 threshold price and position in heap
        if (!inAuction_ ) {
            // get the loan id inside the heap
            uint256 loanId = loans_.indices[borrowerAddress_];
            if (activeBorrower) {
                // revert if t0 threshold price is zero
                if (t0DebtToCollateral == 0) revert ZeroDebtToCollateral();

                // update heap, insert if a new loan, update loan if already in heap
                _upsert(loans_, borrowerAddress_, loanId, SafeCast.toUint96(t0DebtToCollateral));

            // if loan is in heap and borrwer is no longer active (no debt, no collateral) then remove loan from heap
            } else if (loanId != 0) {
                remove(loans_, borrowerAddress_, loanId);
            }
        }

        // update Np to Tp ratio of borrower
        if (npTpRatioUpdate_) {
            borrower_.npTpRatio = 1e18 + uint256(PRBMathSD59x18.sqrt(int256(poolRate_))) / 2;
        }

        // save borrower state
        loans_.borrowers[borrowerAddress_] = borrower_;
    }

    /**************************************/
    /***  Loans Heap Internal Functions ***/
    /**************************************/

    /**
     *  @notice Moves a `Loan` up the heap.
     *  @param loans_ Holds loans heap data.
     *  @param loan_  `Loan` to be moved.
     *  @param index_ Index of `Loan` to be moved to.
     */
    function _bubbleUp(LoansState storage loans_, Loan memory loan_, uint index_) private {
        uint256 count = loans_.loans.length;
        if (index_ == ROOT_INDEX || loan_.t0DebtToCollateral <= loans_.loans[index_ / 2].t0DebtToCollateral){
          _insert(loans_, loan_, index_, count);
        } else {
          _insert(loans_, loans_.loans[index_ / 2], index_, count);
          _bubbleUp(loans_, loan_, index_ / 2);
        }
    }

    /**
     *  @notice Moves a `Loan` down the heap.
     *  @param loans_ Holds loans heap data.
     *  @param loan_  `Loan` to be moved.
     *  @param index_ Index of `Loan` to be moved to.
     */
    function _bubbleDown(LoansState storage loans_, Loan memory loan_, uint index_) private {
        // Left child index.
        uint cIndex = index_ * 2;

        uint256 count = loans_.loans.length;
        if (count <= cIndex) {
            _insert(loans_, loan_, index_, count);
        } else {
            Loan memory largestChild = loans_.loans[cIndex];

            if (count > cIndex + 1 && loans_.loans[cIndex + 1].t0DebtToCollateral > largestChild.t0DebtToCollateral) {
                largestChild = loans_.loans[++cIndex];
            }

            if (largestChild.t0DebtToCollateral <= loan_.t0DebtToCollateral) {
              _insert(loans_, loan_, index_, count);
            } else {
              _insert(loans_, largestChild, index_, count);
              _bubbleDown(loans_, loan_, cIndex);
            }
        }
    }

    /**
     *  @notice Inserts a `Loan` in the heap.
     *  @param loans_ Holds loans heap data.
     *  @param loan_  `Loan` to be inserted.
     *  @param index_ Index of `Loan` to be inserted at.
     */
    function _insert(LoansState storage loans_, Loan memory loan_, uint index_, uint256 count_) private {
        if (index_ == count_) loans_.loans.push(loan_);
        else loans_.loans[index_] = loan_;

        loans_.indices[loan_.borrower] = index_;
    }

    /**
     *  @notice Removes `Loan` from heap given borrower address.
     *  @param loans_    Holds loans heap data.
     *  @param borrower_ Borrower address whose `Loan` is being updated or inserted.
     *  @param index_    Index of `Loan` to be removed.
     */
    function remove(LoansState storage loans_, address borrower_, uint256 index_) internal {
        delete loans_.indices[borrower_];
        uint256 tailIndex = loans_.loans.length - 1;
        if (index_ == tailIndex) loans_.loans.pop(); // we're removing the tail, pop without sorting
        else {
            Loan memory tail = loans_.loans[tailIndex];
            loans_.loans.pop();            // remove tail loan
            _bubbleUp(loans_, tail, index_);
            _bubbleDown(loans_, loans_.loans[index_], index_);
        }
    }

    /**
     *  @notice Performs an insert or an update dependent on borrowers existance.
     *  @param loans_              Holds loans heap data.
     *  @param borrower_           Borrower address that is being updated or inserted.
     *  @param index_              Index of `Loan` to be upserted.
     *  @param t0DebtToCollateral_ Borrower t0 debt to collateral that is updated or inserted.
     */
    function _upsert(
        LoansState storage loans_,
        address borrower_,
        uint256 index_,
        uint96 t0DebtToCollateral_
    ) internal {
        // Loan exists, update in place.
        if (index_ != 0) {
            Loan memory currentLoan = loans_.loans[index_];
            if (currentLoan.t0DebtToCollateral > t0DebtToCollateral_) {
                currentLoan.t0DebtToCollateral = t0DebtToCollateral_;
                _bubbleDown(loans_, currentLoan, index_);
            } else {
                currentLoan.t0DebtToCollateral = t0DebtToCollateral_;
                _bubbleUp(loans_, currentLoan, index_);
            }

        // New loan, insert it
        } else {
            _bubbleUp(loans_, Loan(borrower_, t0DebtToCollateral_), loans_.loans.length);
        }
    }


    /**********************/
    /*** View Functions ***/
    /**********************/

    /**
     *  @notice Retreives `Loan` by index, `index_`.
     *  @param loans_ Holds loans heap data.
     *  @param index_ Index to retrieve `Loan`.
     *  @return `Loan` struct retrieved by index.
     */
    function getByIndex(LoansState storage loans_, uint256 index_) internal view returns(Loan memory) {
        return loans_.loans.length > index_ ? loans_.loans[index_] : Loan(address(0), 0);
    }

    /**
     *  @notice Retreives `Loan` with the highest t0 threshold price value.
     *  @param loans_ Holds loans heap data.
     *  @return `Max Loan` in the heap.
     */
    function getMax(LoansState storage loans_) internal view returns(Loan memory) {
        return getByIndex(loans_, ROOT_INDEX);
    }

    /**
     *  @notice Returns number of loans in pool.
     *  @param loans_ Holds loans heap data.
     *  @return Number of loans in pool.
     */
    function noOfLoans(LoansState storage loans_) internal view returns (uint256) {
        return loans_.loans.length - 1;
    }
}

// SPDX-License-Identifier: GPL-3.0-or-later

pragma solidity 0.8.18;

/**
    @title  Maths library
    @notice Internal library containing common maths.
 */
library Maths {

    uint256 internal constant WAD = 1e18;
    uint256 internal constant RAY = 1e27;

    function wmul(uint256 x, uint256 y) internal pure returns (uint256) {
        return (x * y + WAD / 2) / WAD;
    }

    function floorWmul(uint256 x, uint256 y) internal pure returns (uint256) {
        return (x * y) / WAD;
    }

    function ceilWmul(uint256 x, uint256 y) internal pure returns (uint256) {
        return (x * y + WAD - 1) / WAD;
    }

    function wdiv(uint256 x, uint256 y) internal pure returns (uint256) {
        return (x * WAD + y / 2) / y;
    }

    function floorWdiv(uint256 x, uint256 y) internal pure returns (uint256) {
        return (x * WAD) / y;
    }

    function ceilWdiv(uint256 x, uint256 y) internal pure returns (uint256) {
        return (x * WAD + y - 1) / y;
    }

    function ceilDiv(uint256 x, uint256 y) internal pure returns (uint256) {
        return (x + y - 1) / y;
    }

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

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

    function wad(uint256 x) internal pure returns (uint256) {
        return x * WAD;
    }

    function rmul(uint256 x, uint256 y) internal pure returns (uint256) {
        return (x * y + RAY / 2) / RAY;
    }

    function rpow(uint256 x, uint256 n) internal pure returns (uint256 z) {
        z = n % 2 != 0 ? x : RAY;

        for (n /= 2; n != 0; n /= 2) {
            x = rmul(x, x);

            if (n % 2 != 0) {
                z = rmul(z, x);
            }
        }
    }

    /*************************/
    /*** Integer Functions ***/
    /*************************/

    function maxInt(int256 x, int256 y) internal pure returns (int256) {
        return x >= y ? x : y;
    }

    function minInt(int256 x, int256 y) internal pure returns (int256) {
        return x <= y ? x : y;
    }

}

// SPDX-License-Identifier: Unlicense
pragma solidity >=0.8.4;

/// @notice Emitted when the result overflows uint256.
error PRBMath__MulDivFixedPointOverflow(uint256 prod1);

/// @notice Emitted when the result overflows uint256.
error PRBMath__MulDivOverflow(uint256 prod1, uint256 denominator);

/// @notice Emitted when one of the inputs is type(int256).min.
error PRBMath__MulDivSignedInputTooSmall();

/// @notice Emitted when the intermediary absolute result overflows int256.
error PRBMath__MulDivSignedOverflow(uint256 rAbs);

/// @notice Emitted when the input is MIN_SD59x18.
error PRBMathSD59x18__AbsInputTooSmall();

/// @notice Emitted when ceiling a number overflows SD59x18.
error PRBMathSD59x18__CeilOverflow(int256 x);

/// @notice Emitted when one of the inputs is MIN_SD59x18.
error PRBMathSD59x18__DivInputTooSmall();

/// @notice Emitted when one of the intermediary unsigned results overflows SD59x18.
error PRBMathSD59x18__DivOverflow(uint256 rAbs);

/// @notice Emitted when the input is greater than 133.084258667509499441.
error PRBMathSD59x18__ExpInputTooBig(int256 x);

/// @notice Emitted when the input is greater than 192.
error PRBMathSD59x18__Exp2InputTooBig(int256 x);

/// @notice Emitted when flooring a number underflows SD59x18.
error PRBMathSD59x18__FloorUnderflow(int256 x);

/// @notice Emitted when converting a basic integer to the fixed-point format overflows SD59x18.
error PRBMathSD59x18__FromIntOverflow(int256 x);

/// @notice Emitted when converting a basic integer to the fixed-point format underflows SD59x18.
error PRBMathSD59x18__FromIntUnderflow(int256 x);

/// @notice Emitted when the product of the inputs is negative.
error PRBMathSD59x18__GmNegativeProduct(int256 x, int256 y);

/// @notice Emitted when multiplying the inputs overflows SD59x18.
error PRBMathSD59x18__GmOverflow(int256 x, int256 y);

/// @notice Emitted when the input is less than or equal to zero.
error PRBMathSD59x18__LogInputTooSmall(int256 x);

/// @notice Emitted when one of the inputs is MIN_SD59x18.
error PRBMathSD59x18__MulInputTooSmall();

/// @notice Emitted when the intermediary absolute result overflows SD59x18.
error PRBMathSD59x18__MulOverflow(uint256 rAbs);

/// @notice Emitted when the intermediary absolute result overflows SD59x18.
error PRBMathSD59x18__PowuOverflow(uint256 rAbs);

/// @notice Emitted when the input is negative.
error PRBMathSD59x18__SqrtNegativeInput(int256 x);

/// @notice Emitted when the calculating the square root overflows SD59x18.
error PRBMathSD59x18__SqrtOverflow(int256 x);

/// @notice Emitted when addition overflows UD60x18.
error PRBMathUD60x18__AddOverflow(uint256 x, uint256 y);

/// @notice Emitted when ceiling a number overflows UD60x18.
error PRBMathUD60x18__CeilOverflow(uint256 x);

/// @notice Emitted when the input is greater than 133.084258667509499441.
error PRBMathUD60x18__ExpInputTooBig(uint256 x);

/// @notice Emitted when the input is greater than 192.
error PRBMathUD60x18__Exp2InputTooBig(uint256 x);

/// @notice Emitted when converting a basic integer to the fixed-point format format overflows UD60x18.
error PRBMathUD60x18__FromUintOverflow(uint256 x);

/// @notice Emitted when multiplying the inputs overflows UD60x18.
error PRBMathUD60x18__GmOverflow(uint256 x, uint256 y);

/// @notice Emitted when the input is less than 1.
error PRBMathUD60x18__LogInputTooSmall(uint256 x);

/// @notice Emitted when the calculating the square root overflows UD60x18.
error PRBMathUD60x18__SqrtOverflow(uint256 x);

/// @notice Emitted when subtraction underflows UD60x18.
error PRBMathUD60x18__SubUnderflow(uint256 x, uint256 y);

/// @dev Common mathematical functions used in both PRBMathSD59x18 and PRBMathUD60x18. Note that this shared library
/// does not always assume the signed 59.18-decimal fixed-point or the unsigned 60.18-decimal fixed-point
/// representation. When it does not, it is explicitly mentioned in the NatSpec documentation.
library PRBMath {
    /// STRUCTS ///

    struct SD59x18 {
        int256 value;
    }

    struct UD60x18 {
        uint256 value;
    }

    /// STORAGE ///

    /// @dev How many trailing decimals can be represented.
    uint256 internal constant SCALE = 1e18;

    /// @dev Largest power of two divisor of SCALE.
    uint256 internal constant SCALE_LPOTD = 262144;

    /// @dev SCALE inverted mod 2^256.
    uint256 internal constant SCALE_INVERSE =
        78156646155174841979727994598816262306175212592076161876661_508869554232690281;

    /// FUNCTIONS ///

    /// @notice Calculates the binary exponent of x using the binary fraction method.
    /// @dev Has to use 192.64-bit fixed-point numbers.
    /// See https://ethereum.stackexchange.com/a/96594/24693.
    /// @param x The exponent as an unsigned 192.64-bit fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function exp2(uint256 x) internal pure returns (uint256 result) {
        unchecked {
            // Start from 0.5 in the 192.64-bit fixed-point format.
            result = 0x800000000000000000000000000000000000000000000000;

            // Multiply the result by root(2, 2^-i) when the bit at position i is 1. None of the intermediary results overflows
            // because the initial result is 2^191 and all magic factors are less than 2^65.
            if (x & 0x8000000000000000 > 0) {
                result = (result * 0x16A09E667F3BCC909) >> 64;
            }
            if (x & 0x4000000000000000 > 0) {
                result = (result * 0x1306FE0A31B7152DF) >> 64;
            }
            if (x & 0x2000000000000000 > 0) {
                result = (result * 0x1172B83C7D517ADCE) >> 64;
            }
            if (x & 0x1000000000000000 > 0) {
                result = (result * 0x10B5586CF9890F62A) >> 64;
            }
            if (x & 0x800000000000000 > 0) {
                result = (result * 0x1059B0D31585743AE) >> 64;
            }
            if (x & 0x400000000000000 > 0) {
                result = (result * 0x102C9A3E778060EE7) >> 64;
            }
            if (x & 0x200000000000000 > 0) {
                result = (result * 0x10163DA9FB33356D8) >> 64;
            }
            if (x & 0x100000000000000 > 0) {
                result = (result * 0x100B1AFA5ABCBED61) >> 64;
            }
            if (x & 0x80000000000000 > 0) {
                result = (result * 0x10058C86DA1C09EA2) >> 64;
            }
            if (x & 0x40000000000000 > 0) {
                result = (result * 0x1002C605E2E8CEC50) >> 64;
            }
            if (x & 0x20000000000000 > 0) {
                result = (result * 0x100162F3904051FA1) >> 64;
            }
            if (x & 0x10000000000000 > 0) {
                result = (result * 0x1000B175EFFDC76BA) >> 64;
            }
            if (x & 0x8000000000000 > 0) {
                result = (result * 0x100058BA01FB9F96D) >> 64;
            }
            if (x & 0x4000000000000 > 0) {
                result = (result * 0x10002C5CC37DA9492) >> 64;
            }
            if (x & 0x2000000000000 > 0) {
                result = (result * 0x1000162E525EE0547) >> 64;
            }
            if (x & 0x1000000000000 > 0) {
                result = (result * 0x10000B17255775C04) >> 64;
            }
            if (x & 0x800000000000 > 0) {
                result = (result * 0x1000058B91B5BC9AE) >> 64;
            }
            if (x & 0x400000000000 > 0) {
                result = (result * 0x100002C5C89D5EC6D) >> 64;
            }
            if (x & 0x200000000000 > 0) {
                result = (result * 0x10000162E43F4F831) >> 64;
            }
            if (x & 0x100000000000 > 0) {
                result = (result * 0x100000B1721BCFC9A) >> 64;
            }
            if (x & 0x80000000000 > 0) {
                result = (result * 0x10000058B90CF1E6E) >> 64;
            }
            if (x & 0x40000000000 > 0) {
                result = (result * 0x1000002C5C863B73F) >> 64;
            }
            if (x & 0x20000000000 > 0) {
                result = (result * 0x100000162E430E5A2) >> 64;
            }
            if (x & 0x10000000000 > 0) {
                result = (result * 0x1000000B172183551) >> 64;
            }
            if (x & 0x8000000000 > 0) {
                result = (result * 0x100000058B90C0B49) >> 64;
            }
            if (x & 0x4000000000 > 0) {
                result = (result * 0x10000002C5C8601CC) >> 64;
            }
            if (x & 0x2000000000 > 0) {
                result = (result * 0x1000000162E42FFF0) >> 64;
            }
            if (x & 0x1000000000 > 0) {
                result = (result * 0x10000000B17217FBB) >> 64;
            }
            if (x & 0x800000000 > 0) {
                result = (result * 0x1000000058B90BFCE) >> 64;
            }
            if (x & 0x400000000 > 0) {
                result = (result * 0x100000002C5C85FE3) >> 64;
            }
            if (x & 0x200000000 > 0) {
                result = (result * 0x10000000162E42FF1) >> 64;
            }
            if (x & 0x100000000 > 0) {
                result = (result * 0x100000000B17217F8) >> 64;
            }
            if (x & 0x80000000 > 0) {
                result = (result * 0x10000000058B90BFC) >> 64;
            }
            if (x & 0x40000000 > 0) {
                result = (result * 0x1000000002C5C85FE) >> 64;
            }
            if (x & 0x20000000 > 0) {
                result = (result * 0x100000000162E42FF) >> 64;
            }
            if (x & 0x10000000 > 0) {
                result = (result * 0x1000000000B17217F) >> 64;
            }
            if (x & 0x8000000 > 0) {
                result = (result * 0x100000000058B90C0) >> 64;
            }
            if (x & 0x4000000 > 0) {
                result = (result * 0x10000000002C5C860) >> 64;
            }
            if (x & 0x2000000 > 0) {
                result = (result * 0x1000000000162E430) >> 64;
            }
            if (x & 0x1000000 > 0) {
                result = (result * 0x10000000000B17218) >> 64;
            }
            if (x & 0x800000 > 0) {
                result = (result * 0x1000000000058B90C) >> 64;
            }
            if (x & 0x400000 > 0) {
                result = (result * 0x100000000002C5C86) >> 64;
            }
            if (x & 0x200000 > 0) {
                result = (result * 0x10000000000162E43) >> 64;
            }
            if (x & 0x100000 > 0) {
                result = (result * 0x100000000000B1721) >> 64;
            }
            if (x & 0x80000 > 0) {
                result = (result * 0x10000000000058B91) >> 64;
            }
            if (x & 0x40000 > 0) {
                result = (result * 0x1000000000002C5C8) >> 64;
            }
            if (x & 0x20000 > 0) {
                result = (result * 0x100000000000162E4) >> 64;
            }
            if (x & 0x10000 > 0) {
                result = (result * 0x1000000000000B172) >> 64;
            }
            if (x & 0x8000 > 0) {
                result = (result * 0x100000000000058B9) >> 64;
            }
            if (x & 0x4000 > 0) {
                result = (result * 0x10000000000002C5D) >> 64;
            }
            if (x & 0x2000 > 0) {
                result = (result * 0x1000000000000162E) >> 64;
            }
            if (x & 0x1000 > 0) {
                result = (result * 0x10000000000000B17) >> 64;
            }
            if (x & 0x800 > 0) {
                result = (result * 0x1000000000000058C) >> 64;
            }
            if (x & 0x400 > 0) {
                result = (result * 0x100000000000002C6) >> 64;
            }
            if (x & 0x200 > 0) {
                result = (result * 0x10000000000000163) >> 64;
            }
            if (x & 0x100 > 0) {
                result = (result * 0x100000000000000B1) >> 64;
            }
            if (x & 0x80 > 0) {
                result = (result * 0x10000000000000059) >> 64;
            }
            if (x & 0x40 > 0) {
                result = (result * 0x1000000000000002C) >> 64;
            }
            if (x & 0x20 > 0) {
                result = (result * 0x10000000000000016) >> 64;
            }
            if (x & 0x10 > 0) {
                result = (result * 0x1000000000000000B) >> 64;
            }
            if (x & 0x8 > 0) {
                result = (result * 0x10000000000000006) >> 64;
            }
            if (x & 0x4 > 0) {
                result = (result * 0x10000000000000003) >> 64;
            }
            if (x & 0x2 > 0) {
                result = (result * 0x10000000000000001) >> 64;
            }
            if (x & 0x1 > 0) {
                result = (result * 0x10000000000000001) >> 64;
            }

            // We're doing two things at the same time:
            //
            //   1. Multiply the result by 2^n + 1, where "2^n" is the integer part and the one is added to account for
            //      the fact that we initially set the result to 0.5. This is accomplished by subtracting from 191
            //      rather than 192.
            //   2. Convert the result to the unsigned 60.18-decimal fixed-point format.
            //
            // This works because 2^(191-ip) = 2^ip / 2^191, where "ip" is the integer part "2^n".
            result *= SCALE;
            result >>= (191 - (x >> 64));
        }
    }

    /// @notice Finds the zero-based index of the first one in the binary representation of x.
    /// @dev See the note on msb in the "Find First Set" Wikipedia article https://en.wikipedia.org/wiki/Find_first_set
    /// @param x The uint256 number for which to find the index of the most significant bit.
    /// @return msb The index of the most significant bit as an uint256.
    function mostSignificantBit(uint256 x) internal pure returns (uint256 msb) {
        if (x >= 2**128) {
            x >>= 128;
            msb += 128;
        }
        if (x >= 2**64) {
            x >>= 64;
            msb += 64;
        }
        if (x >= 2**32) {
            x >>= 32;
            msb += 32;
        }
        if (x >= 2**16) {
            x >>= 16;
            msb += 16;
        }
        if (x >= 2**8) {
            x >>= 8;
            msb += 8;
        }
        if (x >= 2**4) {
            x >>= 4;
            msb += 4;
        }
        if (x >= 2**2) {
            x >>= 2;
            msb += 2;
        }
        if (x >= 2**1) {
            // No need to shift x any more.
            msb += 1;
        }
    }

    /// @notice Calculates floor(x*y÷denominator) with full precision.
    ///
    /// @dev Credit to Remco Bloemen under MIT license https://xn--2-umb.com/21/muldiv.
    ///
    /// Requirements:
    /// - The denominator cannot be zero.
    /// - The result must fit within uint256.
    ///
    /// Caveats:
    /// - This function does not work with fixed-point numbers.
    ///
    /// @param x The multiplicand as an uint256.
    /// @param y The multiplier as an uint256.
    /// @param denominator The divisor as an uint256.
    /// @return result The result as an uint256.
    function mulDiv(
        uint256 x,
        uint256 y,
        uint256 denominator
    ) internal pure returns (uint256 result) {
        // 512-bit multiply [prod1 prod0] = x * y. Compute the product mod 2^256 and mod 2^256 - 1, then use
        // use the Chinese Remainder Theorem to reconstruct the 512 bit result. The result is stored in two 256
        // variables such that product = prod1 * 2^256 + prod0.
        uint256 prod0; // Least significant 256 bits of the product
        uint256 prod1; // Most significant 256 bits of the product
        assembly {
            let mm := mulmod(x, y, not(0))
            prod0 := mul(x, y)
            prod1 := sub(sub(mm, prod0), lt(mm, prod0))
        }

        // Handle non-overflow cases, 256 by 256 division.
        if (prod1 == 0) {
            unchecked {
                result = prod0 / denominator;
            }
            return result;
        }

        // Make sure the result is less than 2^256. Also prevents denominator == 0.
        if (prod1 >= denominator) {
            revert PRBMath__MulDivOverflow(prod1, denominator);
        }

        ///////////////////////////////////////////////
        // 512 by 256 division.
        ///////////////////////////////////////////////

        // Make division exact by subtracting the remainder from [prod1 prod0].
        uint256 remainder;
        assembly {
            // Compute remainder using mulmod.
            remainder := mulmod(x, y, denominator)

            // Subtract 256 bit number from 512 bit number.
            prod1 := sub(prod1, gt(remainder, prod0))
            prod0 := sub(prod0, remainder)
        }

        // Factor powers of two out of denominator and compute largest power of two divisor of denominator. Always >= 1.
        // See https://cs.stackexchange.com/q/138556/92363.
        unchecked {
            // Does not overflow because the denominator cannot be zero at this stage in the function.
            uint256 lpotdod = denominator & (~denominator + 1);
            assembly {
                // Divide denominator by lpotdod.
                denominator := div(denominator, lpotdod)

                // Divide [prod1 prod0] by lpotdod.
                prod0 := div(prod0, lpotdod)

                // Flip lpotdod such that it is 2^256 / lpotdod. If lpotdod is zero, then it becomes one.
                lpotdod := add(div(sub(0, lpotdod), lpotdod), 1)
            }

            // Shift in bits from prod1 into prod0.
            prod0 |= prod1 * lpotdod;

            // Invert denominator mod 2^256. Now that denominator is an odd number, it has an inverse modulo 2^256 such
            // that denominator * inv = 1 mod 2^256. Compute the inverse by starting with a seed that is correct for
            // four bits. That is, denominator * inv = 1 mod 2^4.
            uint256 inverse = (3 * denominator) ^ 2;

            // Use the Newton-Raphson iteration to improve the precision. Thanks to Hensel's lifting lemma, this also works
            // in modular arithmetic, doubling the correct bits in each step.
            inverse *= 2 - denominator * inverse; // inverse mod 2^8
            inverse *= 2 - denominator * inverse; // inverse mod 2^16
            inverse *= 2 - denominator * inverse; // inverse mod 2^32
            inverse *= 2 - denominator * inverse; // inverse mod 2^64
            inverse *= 2 - denominator * inverse; // inverse mod 2^128
            inverse *= 2 - denominator * inverse; // inverse mod 2^256

            // Because the division is now exact we can divide by multiplying with the modular inverse of denominator.
            // This will give us the correct result modulo 2^256. Since the preconditions guarantee that the outcome is
            // less than 2^256, this is the final result. We don't need to compute the high bits of the result and prod1
            // is no longer required.
            result = prod0 * inverse;
            return result;
        }
    }

    /// @notice Calculates floor(x*y÷1e18) with full precision.
    ///
    /// @dev Variant of "mulDiv" with constant folding, i.e. in which the denominator is always 1e18. Before returning the
    /// final result, we add 1 if (x * y) % SCALE >= HALF_SCALE. Without this, 6.6e-19 would be truncated to 0 instead of
    /// being rounded to 1e-18.  See "Listing 6" and text above it at https://accu.org/index.php/journals/1717.
    ///
    /// Requirements:
    /// - The result must fit within uint256.
    ///
    /// Caveats:
    /// - The body is purposely left uncommented; see the NatSpec comments in "PRBMath.mulDiv" to understand how this works.
    /// - It is assumed that the result can never be type(uint256).max when x and y solve the following two equations:
    ///     1. x * y = type(uint256).max * SCALE
    ///     2. (x * y) % SCALE >= SCALE / 2
    ///
    /// @param x The multiplicand as an unsigned 60.18-decimal fixed-point number.
    /// @param y The multiplier as an unsigned 60.18-decimal fixed-point number.
    /// @return result The result as an unsigned 60.18-decimal fixed-point number.
    function mulDivFixedPoint(uint256 x, uint256 y) internal pure returns (uint256 result) {
        uint256 prod0;
        uint256 prod1;
        assembly {
            let mm := mulmod(x, y, not(0))
            prod0 := mul(x, y)
            prod1 := sub(sub(mm, prod0), lt(mm, prod0))
        }

        if (prod1 >= SCALE) {
            revert PRBMath__MulDivFixedPointOverflow(prod1);
        }

        uint256 remainder;
        uint256 roundUpUnit;
        assembly {
            remainder := mulmod(x, y, SCALE)
            roundUpUnit := gt(remainder, 499999999999999999)
        }

        if (prod1 == 0) {
            unchecked {
                result = (prod0 / SCALE) + roundUpUnit;
                return result;
            }
        }

        assembly {
            result := add(
                mul(
                    or(
                        div(sub(prod0, remainder), SCALE_LPOTD),
                        mul(sub(prod1, gt(remainder, prod0)), add(div(sub(0, SCALE_LPOTD), SCALE_LPOTD), 1))
                    ),
                    SCALE_INVERSE
                ),
                roundUpUnit
            )
        }
    }

    /// @notice Calculates floor(x*y÷denominator) with full precision.
    ///
    /// @dev An extension of "mulDiv" for signed numbers. Works by computing the signs and the absolute values separately.
    ///
    /// Requirements:
    /// - None of the inputs can be type(int256).min.
    /// - The result must fit within int256.
    ///
    /// @param x The multiplicand as an int256.
    /// @param y The multiplier as an int256.
    /// @param denominator The divisor as an int256.
    /// @return result The result as an int256.
    function mulDivSigned(
        int256 x,
        int256 y,
        int256 denominator
    ) internal pure returns (int256 result) {
        if (x == type(int256).min || y == type(int256).min || denominator == type(int256).min) {
            revert PRBMath__MulDivSignedInputTooSmall();
        }

        // Get hold of the absolute values of x, y and the denominator.
        uint256 ax;
        uint256 ay;
        uint256 ad;
        unchecked {
            ax = x < 0 ? uint256(-x) : uint256(x);
            ay = y < 0 ? uint256(-y) : uint256(y);
            ad = denominator < 0 ? uint256(-denominator) : uint256(denominator);
        }

        // Compute the absolute value of (x*y)÷denominator. The result must fit within int256.
        uint256 rAbs = mulDiv(ax, ay, ad);
        if (rAbs > uint256(type(int256).max)) {
            revert PRBMath__MulDivSignedOverflow(rAbs);
        }

        // Get the signs of x, y and the denominator.
        uint256 sx;
        uint256 sy;
        uint256 sd;
        assembly {
            sx := sgt(x, sub(0, 1))
            sy := sgt(y, sub(0, 1))
            sd := sgt(denominator, sub(0, 1))
        }

        // XOR over sx, sy and sd. This is checking whether there are one or three negative signs in the inputs.
        // If yes, the result should be negative.
        result = sx ^ sy ^ sd == 0 ? -int256(rAbs) : int256(rAbs);
    }

    /// @notice Calculates the square root of x, rounding down.
    /// @dev Uses the Babylonian method https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method.
    ///
    /// Caveats:
    /// - This function does not work with fixed-point numbers.
    ///
    /// @param x The uint256 number for which to calculate the square root.
    /// @return result The result as an uint256.
    function sqrt(uint256 x) internal pure returns (uint256 result) {
        if (x == 0) {
            return 0;
        }

        // Set the initial guess to the least power of two that is greater than or equal to sqrt(x).
        uint256 xAux = uint256(x);
        result = 1;
        if (xAux >= 0x100000000000000000000000000000000) {
            xAux >>= 128;
            result <<= 64;
        }
        if (xAux >= 0x10000000000000000) {
            xAux >>= 64;
            result <<= 32;
        }
        if (xAux >= 0x100000000) {
            xAux >>= 32;
            result <<= 16;
        }
        if (xAux >= 0x10000) {
            xAux >>= 16;
            result <<= 8;
        }
        if (xAux >= 0x100) {
            xAux >>= 8;
            result <<= 4;
        }
        if (xAux >= 0x10) {
            xAux >>= 4;
            result <<= 2;
        }
        if (xAux >= 0x8) {
            result <<= 1;
        }

        // The operations can never overflow because the result is max 2^127 when it enters this block.
        unchecked {
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1;
            result = (result + x / result) >> 1; // Seven iterations should be enough
            uint256 roundedDownResult = x / result;
            return result >= roundedDownResult ? roundedDownResult : result;
        }
    }
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts v4.4.1 (token/ERC20/extensions/draft-IERC20Permit.sol)

pragma solidity ^0.8.0;

/**
 * @dev Interface of the ERC20 Permit extension allowing approvals to be made via signatures, as defined in
 * https://eips.ethereum.org/EIPS/eip-2612[EIP-2612].
 *
 * Adds the {permit} method, which can be used to change an account's ERC20 allowance (see {IERC20-allowance}) by
 * presenting a message signed by the account. By not relying on {IERC20-approve}, the token holder account doesn't
 * need to send a transaction, and thus is not required to hold Ether at all.
 */
interface IERC20Permit {
    /**
     * @dev Sets `value` as the allowance of `spender` over ``owner``'s tokens,
     * given ``owner``'s signed approval.
     *
     * IMPORTANT: The same issues {IERC20-approve} has related to transaction
     * ordering also apply here.
     *
     * Emits an {Approval} event.
     *
     * Requirements:
     *
     * - `spender` cannot be the zero address.
     * - `deadline` must be a timestamp in the future.
     * - `v`, `r` and `s` must be a valid `secp256k1` signature from `owner`
     * over the EIP712-formatted function arguments.
     * - the signature must use ``owner``'s current nonce (see {nonces}).
     *
     * For more information on the signature format, see the
     * https://eips.ethereum.org/EIPS/eip-2612#specification[relevant EIP
     * section].
     */
    function permit(
        address owner,
        address spender,
        uint256 value,
        uint256 deadline,
        uint8 v,
        bytes32 r,
        bytes32 s
    ) external;

    /**
     * @dev Returns the current nonce for `owner`. This value must be
     * included whenever a signature is generated for {permit}.
     *
     * Every successful call to {permit} increases ``owner``'s nonce by one. This
     * prevents a signature from being used multiple times.
     */
    function nonces(address owner) external view returns (uint256);

    /**
     * @dev Returns the domain separator used in the encoding of the signature for {permit}, as defined by {EIP712}.
     */
    // solhint-disable-next-line func-name-mixedcase
    function DOMAIN_SEPARATOR() external view returns (bytes32);
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.8.0) (utils/Address.sol)

pragma solidity ^0.8.1;

/**
 * @dev Collection of functions related to the address type
 */
library Address {
    /**
     * @dev Returns true if `account` is a contract.
     *
     * [IMPORTANT]
     * ====
     * It is unsafe to assume that an address for which this function returns
     * false is an externally-owned account (EOA) and not a contract.
     *
     * Among others, `isContract` will return false for the following
     * types of addresses:
     *
     *  - an externally-owned account
     *  - a contract in construction
     *  - an address where a contract will be created
     *  - an address where a contract lived, but was destroyed
     * ====
     *
     * [IMPORTANT]
     * ====
     * You shouldn't rely on `isContract` to protect against flash loan attacks!
     *
     * Preventing calls from contracts is highly discouraged. It breaks composability, breaks support for smart wallets
     * like Gnosis Safe, and does not provide security since it can be circumvented by calling from a contract
     * constructor.
     * ====
     */
    function isContract(address account) internal view returns (bool) {
        // This method relies on extcodesize/address.code.length, which returns 0
        // for contracts in construction, since the code is only stored at the end
        // of the constructor execution.

        return account.code.length > 0;
    }

    /**
     * @dev Replacement for Solidity's `transfer`: sends `amount` wei to
     * `recipient`, forwarding all available gas and reverting on errors.
     *
     * https://eips.ethereum.org/EIPS/eip-1884[EIP1884] increases the gas cost
     * of certain opcodes, possibly making contracts go over the 2300 gas limit
     * imposed by `transfer`, making them unable to receive funds via
     * `transfer`. {sendValue} removes this limitation.
     *
     * https://diligence.consensys.net/posts/2019/09/stop-using-soliditys-transfer-now/[Learn more].
     *
     * IMPORTANT: because control is transferred to `recipient`, care must be
     * taken to not create reentrancy vulnerabilities. Consider using
     * {ReentrancyGuard} or the
     * https://solidity.readthedocs.io/en/v0.5.11/security-considerations.html#use-the-checks-effects-interactions-pattern[checks-effects-interactions pattern].
     */
    function sendValue(address payable recipient, uint256 amount) internal {
        require(address(this).balance >= amount, "Address: insufficient balance");

        (bool success, ) = recipient.call{value: amount}("");
        require(success, "Address: unable to send value, recipient may have reverted");
    }

    /**
     * @dev Performs a Solidity function call using a low level `call`. A
     * plain `call` is an unsafe replacement for a function call: use this
     * function instead.
     *
     * If `target` reverts with a revert reason, it is bubbled up by this
     * function (like regular Solidity function calls).
     *
     * Returns the raw returned data. To convert to the expected return value,
     * use https://solidity.readthedocs.io/en/latest/units-and-global-variables.html?highlight=abi.decode#abi-encoding-and-decoding-functions[`abi.decode`].
     *
     * Requirements:
     *
     * - `target` must be a contract.
     * - calling `target` with `data` must not revert.
     *
     * _Available since v3.1._
     */
    function functionCall(address target, bytes memory data) internal returns (bytes memory) {
        return functionCallWithValue(target, data, 0, "Address: low-level call failed");
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`], but with
     * `errorMessage` as a fallback revert reason when `target` reverts.
     *
     * _Available since v3.1._
     */
    function functionCall(
        address target,
        bytes memory data,
        string memory errorMessage
    ) internal returns (bytes memory) {
        return functionCallWithValue(target, data, 0, errorMessage);
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but also transferring `value` wei to `target`.
     *
     * Requirements:
     *
     * - the calling contract must have an ETH balance of at least `value`.
     * - the called Solidity function must be `payable`.
     *
     * _Available since v3.1._
     */
    function functionCallWithValue(
        address target,
        bytes memory data,
        uint256 value
    ) internal returns (bytes memory) {
        return functionCallWithValue(target, data, value, "Address: low-level call with value failed");
    }

    /**
     * @dev Same as {xref-Address-functionCallWithValue-address-bytes-uint256-}[`functionCallWithValue`], but
     * with `errorMessage` as a fallback revert reason when `target` reverts.
     *
     * _Available since v3.1._
     */
    function functionCallWithValue(
        address target,
        bytes memory data,
        uint256 value,
        string memory errorMessage
    ) internal returns (bytes memory) {
        require(address(this).balance >= value, "Address: insufficient balance for call");
        (bool success, bytes memory returndata) = target.call{value: value}(data);
        return verifyCallResultFromTarget(target, success, returndata, errorMessage);
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but performing a static call.
     *
     * _Available since v3.3._
     */
    function functionStaticCall(address target, bytes memory data) internal view returns (bytes memory) {
        return functionStaticCall(target, data, "Address: low-level static call failed");
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-string-}[`functionCall`],
     * but performing a static call.
     *
     * _Available since v3.3._
     */
    function functionStaticCall(
        address target,
        bytes memory data,
        string memory errorMessage
    ) internal view returns (bytes memory) {
        (bool success, bytes memory returndata) = target.staticcall(data);
        return verifyCallResultFromTarget(target, success, returndata, errorMessage);
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but performing a delegate call.
     *
     * _Available since v3.4._
     */
    function functionDelegateCall(address target, bytes memory data) internal returns (bytes memory) {
        return functionDelegateCall(target, data, "Address: low-level delegate call failed");
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-string-}[`functionCall`],
     * but performing a delegate call.
     *
     * _Available since v3.4._
     */
    function functionDelegateCall(
        address target,
        bytes memory data,
        string memory errorMessage
    ) internal returns (bytes memory) {
        (bool success, bytes memory returndata) = target.delegatecall(data);
        return verifyCallResultFromTarget(target, success, returndata, errorMessage);
    }

    /**
     * @dev Tool to verify that a low level call to smart-contract was successful, and revert (either by bubbling
     * the revert reason or using the provided one) in case of unsuccessful call or if target was not a contract.
     *
     * _Available since v4.8._
     */
    function verifyCallResultFromTarget(
        address target,
        bool success,
        bytes memory returndata,
        string memory errorMessage
    ) internal view returns (bytes memory) {
        if (success) {
            if (returndata.length == 0) {
                // only check isContract if the call was successful and the return data is empty
                // otherwise we already know that it was a contract
                require(isContract(target), "Address: call to non-contract");
            }
            return returndata;
        } else {
            _revert(returndata, errorMessage);
        }
    }

    /**
     * @dev Tool to verify that a low level call was successful, and revert if it wasn't, either by bubbling the
     * revert reason or using the provided one.
     *
     * _Available since v4.3._
     */
    function verifyCallResult(
        bool success,
        bytes memory returndata,
        string memory errorMessage
    ) internal pure returns (bytes memory) {
        if (success) {
            return returndata;
        } else {
            _revert(returndata, errorMessage);
        }
    }

    function _revert(bytes memory returndata, string memory errorMessage) private pure {
        // Look for revert reason and bubble it up if present
        if (returndata.length > 0) {
            // The easiest way to bubble the revert reason is using memory via assembly
            /// @solidity memory-safe-assembly
            assembly {
                let returndata_size := mload(returndata)
                revert(add(32, returndata), returndata_size)
            }
        } else {
            revert(errorMessage);
        }
    }
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.8.0) (utils/math/Math.sol)

pragma solidity ^0.8.0;

/**
 * @dev Standard math utilities missing in the Solidity language.
 */
library Math {
    enum Rounding {
        Down, // Toward negative infinity
        Up, // Toward infinity
        Zero // Toward zero
    }

    /**
     * @dev Returns the largest of two numbers.
     */
    function max(uint256 a, uint256 b) internal pure returns (uint256) {
        return a > b ? a : b;
    }

    /**
     * @dev Returns the smallest of two numbers.
     */
    function min(uint256 a, uint256 b) internal pure returns (uint256) {
        return a < b ? a : b;
    }

    /**
     * @dev Returns the average of two numbers. The result is rounded towards
     * zero.
     */
    function average(uint256 a, uint256 b) internal pure returns (uint256) {
        // (a + b) / 2 can overflow.
        return (a & b) + (a ^ b) / 2;
    }

    /**
     * @dev Returns the ceiling of the division of two numbers.
     *
     * This differs from standard division with `/` in that it rounds up instead
     * of rounding down.
     */
    function ceilDiv(uint256 a, uint256 b) internal pure returns (uint256) {
        // (a + b - 1) / b can overflow on addition, so we distribute.
        return a == 0 ? 0 : (a - 1) / b + 1;
    }

    /**
     * @notice Calculates floor(x * y / denominator) with full precision. Throws if result overflows a uint256 or denominator == 0
     * @dev Original credit to Remco Bloemen under MIT license (https://xn--2-umb.com/21/muldiv)
     * with further edits by Uniswap Labs also under MIT license.
     */
    function mulDiv(
        uint256 x,
        uint256 y,
        uint256 denominator
    ) internal pure returns (uint256 result) {
        unchecked {
            // 512-bit multiply [prod1 prod0] = x * y. Compute the product mod 2^256 and mod 2^256 - 1, then use
            // use the Chinese Remainder Theorem to reconstruct the 512 bit result. The result is stored in two 256
            // variables such that product = prod1 * 2^256 + prod0.
            uint256 prod0; // Least significant 256 bits of the product
            uint256 prod1; // Most significant 256 bits of the product
            assembly {
                let mm := mulmod(x, y, not(0))
                prod0 := mul(x, y)
                prod1 := sub(sub(mm, prod0), lt(mm, prod0))
            }

            // Handle non-overflow cases, 256 by 256 division.
            if (prod1 == 0) {
                return prod0 / denominator;
            }

            // Make sure the result is less than 2^256. Also prevents denominator == 0.
            require(denominator > prod1);

            ///////////////////////////////////////////////
            // 512 by 256 division.
            ///////////////////////////////////////////////

            // Make division exact by subtracting the remainder from [prod1 prod0].
            uint256 remainder;
            assembly {
                // Compute remainder using mulmod.
                remainder := mulmod(x, y, denominator)

                // Subtract 256 bit number from 512 bit number.
                prod1 := sub(prod1, gt(remainder, prod0))
                prod0 := sub(prod0, remainder)
            }

            // Factor powers of two out of denominator and compute largest power of two divisor of denominator. Always >= 1.
            // See https://cs.stackexchange.com/q/138556/92363.

            // Does not overflow because the denominator cannot be zero at this stage in the function.
            uint256 twos = denominator & (~denominator + 1);
            assembly {
                // Divide denominator by twos.
                denominator := div(denominator, twos)

                // Divide [prod1 prod0] by twos.
                prod0 := div(prod0, twos)

                // Flip twos such that it is 2^256 / twos. If twos is zero, then it becomes one.
                twos := add(div(sub(0, twos), twos), 1)
            }

            // Shift in bits from prod1 into prod0.
            prod0 |= prod1 * twos;

            // Invert denominator mod 2^256. Now that denominator is an odd number, it has an inverse modulo 2^256 such
            // that denominator * inv = 1 mod 2^256. Compute the inverse by starting with a seed that is correct for
            // four bits. That is, denominator * inv = 1 mod 2^4.
            uint256 inverse = (3 * denominator) ^ 2;

            // Use the Newton-Raphson iteration to improve the precision. Thanks to Hensel's lifting lemma, this also works
            // in modular arithmetic, doubling the correct bits in each step.
            inverse *= 2 - denominator * inverse; // inverse mod 2^8
            inverse *= 2 - denominator * inverse; // inverse mod 2^16
            inverse *= 2 - denominator * inverse; // inverse mod 2^32
            inverse *= 2 - denominator * inverse; // inverse mod 2^64
            inverse *= 2 - denominator * inverse; // inverse mod 2^128
            inverse *= 2 - denominator * inverse; // inverse mod 2^256

            // Because the division is now exact we can divide by multiplying with the modular inverse of denominator.
            // This will give us the correct result modulo 2^256. Since the preconditions guarantee that the outcome is
            // less than 2^256, this is the final result. We don't need to compute the high bits of the result and prod1
            // is no longer required.
            result = prod0 * inverse;
            return result;
        }
    }

    /**
     * @notice Calculates x * y / denominator with full precision, following the selected rounding direction.
     */
    function mulDiv(
        uint256 x,
        uint256 y,
        uint256 denominator,
        Rounding rounding
    ) internal pure returns (uint256) {
        uint256 result = mulDiv(x, y, denominator);
        if (rounding == Rounding.Up && mulmod(x, y, denominator) > 0) {
            result += 1;
        }
        return result;
    }

    /**
     * @dev Returns the square root of a number. If the number is not a perfect square, the value is rounded down.
     *
     * Inspired by Henry S. Warren, Jr.'s "Hacker's Delight" (Chapter 11).
     */
    function sqrt(uint256 a) internal pure returns (uint256) {
        if (a == 0) {
            return 0;
        }

        // For our first guess, we get the biggest power of 2 which is smaller than the square root of the target.
        //
        // We know that the "msb" (most significant bit) of our target number `a` is a power of 2 such that we have
        // `msb(a) <= a < 2*msb(a)`. This value can be written `msb(a)=2**k` with `k=log2(a)`.
        //
        // This can be rewritten `2**log2(a) <= a < 2**(log2(a) + 1)`
        // → `sqrt(2**k) <= sqrt(a) < sqrt(2**(k+1))`
        // → `2**(k/2) <= sqrt(a) < 2**((k+1)/2) <= 2**(k/2 + 1)`
        //
        // Consequently, `2**(log2(a) / 2)` is a good first approximation of `sqrt(a)` with at least 1 correct bit.
        uint256 result = 1 << (log2(a) >> 1);

        // At this point `result` is an estimation with one bit of precision. We know the true value is a uint128,
        // since it is the square root of a uint256. Newton's method converges quadratically (precision doubles at
        // every iteration). We thus need at most 7 iteration to turn our partial result with one bit of precision
        // into the expected uint128 result.
        unchecked {
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            result = (result + a / result) >> 1;
            return min(result, a / result);
        }
    }

    /**
     * @notice Calculates sqrt(a), following the selected rounding direction.
     */
    function sqrt(uint256 a, Rounding rounding) internal pure returns (uint256) {
        unchecked {
            uint256 result = sqrt(a);
            return result + (rounding == Rounding.Up && result * result < a ? 1 : 0);
        }
    }

    /**
     * @dev Return the log in base 2, rounded down, of a positive value.
     * Returns 0 if given 0.
     */
    function log2(uint256 value) internal pure returns (uint256) {
        uint256 result = 0;
        unchecked {
            if (value >> 128 > 0) {
                value >>= 128;
                result += 128;
            }
            if (value >> 64 > 0) {
                value >>= 64;
                result += 64;
            }
            if (value >> 32 > 0) {
                value >>= 32;
                result += 32;
            }
            if (value >> 16 > 0) {
                value >>= 16;
                result += 16;
            }
            if (value >> 8 > 0) {
                value >>= 8;
                result += 8;
            }
            if (value >> 4 > 0) {
                value >>= 4;
                result += 4;
            }
            if (value >> 2 > 0) {
                value >>= 2;
                result += 2;
            }
            if (value >> 1 > 0) {
                result += 1;
            }
        }
        return result;
    }

    /**
     * @dev Return the log in base 2, following the selected rounding direction, of a positive value.
     * Returns 0 if given 0.
     */
    function log2(uint256 value, Rounding rounding) internal pure returns (uint256) {
        unchecked {
            uint256 result = log2(value);
            return result + (rounding == Rounding.Up && 1 << result < value ? 1 : 0);
        }
    }

    /**
     * @dev Return the log in base 10, rounded down, of a positive value.
     * Returns 0 if given 0.
     */
    function log10(uint256 value) internal pure returns (uint256) {
        uint256 result = 0;
        unchecked {
            if (value >= 10**64) {
                value /= 10**64;
                result += 64;
            }
            if (value >= 10**32) {
                value /= 10**32;
                result += 32;
            }
            if (value >= 10**16) {
                value /= 10**16;
                result += 16;
            }
            if (value >= 10**8) {
                value /= 10**8;
                result += 8;
            }
            if (value >= 10**4) {
                value /= 10**4;
                result += 4;
            }
            if (value >= 10**2) {
                value /= 10**2;
                result += 2;
            }
            if (value >= 10**1) {
                result += 1;
            }
        }
        return result;
    }

    /**
     * @dev Return the log in base 10, following the selected rounding direction, of a positive value.
     * Returns 0 if given 0.
     */
    function log10(uint256 value, Rounding rounding) internal pure returns (uint256) {
        unchecked {
            uint256 result = log10(value);
            return result + (rounding == Rounding.Up && 10**result < value ? 1 : 0);
        }
    }

    /**
     * @dev Return the log in base 256, rounded down, of a positive value.
     * Returns 0 if given 0.
     *
     * Adding one to the result gives the number of pairs of hex symbols needed to represent `value` as a hex string.
     */
    function log256(uint256 value) internal pure returns (uint256) {
        uint256 result = 0;
        unchecked {
            if (value >> 128 > 0) {
                value >>= 128;
                result += 16;
            }
            if (value >> 64 > 0) {
                value >>= 64;
                result += 8;
            }
            if (value >> 32 > 0) {
                value >>= 32;
                result += 4;
            }
            if (value >> 16 > 0) {
                value >>= 16;
                result += 2;
            }
            if (value >> 8 > 0) {
                result += 1;
            }
        }
        return result;
    }

    /**
     * @dev Return the log in base 10, following the selected rounding direction, of a positive value.
     * Returns 0 if given 0.
     */
    function log256(uint256 value, Rounding rounding) internal pure returns (uint256) {
        unchecked {
            uint256 result = log256(value);
            return result + (rounding == Rounding.Up && 1 << (result * 8) < value ? 1 : 0);
        }
    }
}

// SPDX-License-Identifier: MIT
// OpenZeppelin Contracts (last updated v4.8.0) (utils/math/SafeCast.sol)
// This file was procedurally generated from scripts/generate/templates/SafeCast.js.

pragma solidity ^0.8.0;

/**
 * @dev Wrappers over Solidity's uintXX/intXX casting operators with added overflow
 * checks.
 *
 * Downcasting from uint256/int256 in Solidity does not revert on overflow. This can
 * easily result in undesired exploitation or bugs, since developers usually
 * assume that overflows raise errors. `SafeCast` restores this intuition by
 * reverting the transaction when such 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.
 *
 * Can be combined with {SafeMath} and {SignedSafeMath} to extend it to smaller types, by performing
 * all math on `uint256` and `int256` and then downcasting.
 */
library SafeCast {
    /**
     * @dev Returns the downcasted uint248 from uint256, reverting on
     * overflow (when the input is greater than largest uint248).
     *
     * Counterpart to Solidity's `uint248` operator.
     *
     * Requirements:
     *
     * - input must fit into 248 bits
     *
     * _Available since v4.7._
     */
    function toUint248(uint256 value) internal pure returns (uint248) {
        require(value <= type(uint248).max, "SafeCast: value doesn't fit in 248 bits");
        return uint248(value);
    }

    /**
     * @dev Returns the downcasted uint240 from uint256, reverting on
     * overflow (when the input is greater than largest uint240).
     *
     * Counterpart to Solidity's `uint240` operator.
     *
     * Requirements:
     *
     * - input must fit into 240 bits
     *
     * _Available since v4.7._
     */
    function toUint240(uint256 value) internal pure returns (uint240) {
        require(value <= type(uint240).max, "SafeCast: value doesn't fit in 240 bits");
        return uint240(value);
    }

    /**
     * @dev Returns the downcasted uint232 from uint256, reverting on
     * overflow (when the input is greater than largest uint232).
     *
     * Counterpart to Solidity's `uint232` operator.
     *
     * Requirements:
     *
     * - input must fit into 232 bits
     *
     * _Available since v4.7._
     */
    function toUint232(uint256 value) internal pure returns (uint232) {
        require(value <= type(uint232).max, "SafeCast: value doesn't fit in 232 bits");
        return uint232(value);
    }

    /**
     * @dev Returns the downcasted uint224 from uint256, reverting on
     * overflow (when the input is greater than largest uint224).
     *
     * Counterpart to Solidity's `uint224` operator.
     *
     * Requirements:
     *
     * - input must fit into 224 bits
     *
     * _Available since v4.2._
     */
    function toUint224(uint256 value) internal pure returns (uint224) {
        require(value <= type(uint224).max, "SafeCast: value doesn't fit in 224 bits");
        return uint224(value);
    }

    /**
     * @dev Returns the downcasted uint216 from uint256, reverting on
     * overflow (when the input is greater than largest uint216).
     *
     * Counterpart to Solidity's `uint216` operator.
     *
     * Requirements:
     *
     * - input must fit into 216 bits
     *
     * _Available since v4.7._
     */
    function toUint216(uint256 value) internal pure returns (uint216) {
        require(value <= type(uint216).max, "SafeCast: value doesn't fit in 216 bits");
        return uint216(value);
    }

    /**
     * @dev Returns the downcasted uint208 from uint256, reverting on
     * overflow (when the input is greater than largest uint208).
     *
     * Counterpart to Solidity's `uint208` operator.
     *
     * Requirements:
     *
     * - input must fit into 208 bits
     *
     * _Available since v4.7._
     */
    function toUint208(uint256 value) internal pure returns (uint208) {
        require(value <= type(uint208).max, "SafeCast: value doesn't fit in 208 bits");
        return uint208(value);
    }

    /**
     * @dev Returns the downcasted uint200 from uint256, reverting on
     * overflow (when the input is greater than largest uint200).
     *
     * Counterpart to Solidity's `uint200` operator.
     *
     * Requirements:
     *
     * - input must fit into 200 bits
     *
     * _Available since v4.7._
     */
    function toUint200(uint256 value) internal pure returns (uint200) {
        require(value <= type(uint200).max, "SafeCast: value doesn't fit in 200 bits");
        return uint200(value);
    }

    /**
     * @dev Returns the downcasted uint192 from uint256, reverting on
     * overflow (when the input is greater than largest uint192).
     *
     * Counterpart to Solidity's `uint192` operator.
     *
     * Requirements:
     *
     * - input must fit into 192 bits
     *
     * _Available since v4.7._
     */
    function toUint192(uint256 value) internal pure returns (uint192) {
        require(value <= type(uint192).max, "SafeCast: value doesn't fit in 192 bits");
        return uint192(value);
    }

    /**
     * @dev Returns the downcasted uint184 from uint256, reverting on
     * overflow (when the input is greater than largest uint184).
     *
     * Counterpart to Solidity's `uint184` operator.
     *
     * Requirements:
     *
     * - input must fit into 184 bits
     *
     * _Available since v4.7._
     */
    function toUint184(uint256 value) internal pure returns (uint184) {
        require(value <= type(uint184).max, "SafeCast: value doesn't fit in 184 bits");
        return uint184(value);
    }

    /**
     * @dev Returns the downcasted uint176 from uint256, reverting on
     * overflow (when the input is greater than largest uint176).
     *
     * Counterpart to Solidity's `uint176` operator.
     *
     * Requirements:
     *
     * - input must fit into 176 bits
     *
     * _Available since v4.7._
     */
    function toUint176(uint256 value) internal pure returns (uint176) {
        require(value <= type(uint176).max, "SafeCast: value doesn't fit in 176 bits");
        return uint176(value);
    }

    /**
     * @dev Returns the downcasted uint168 from uint256, reverting on
     * overflow (when the input is greater than largest uint168).
     *
     * Counterpart to Solidity's `uint168` operator.
     *
     * Requirements:
     *
     * - input must fit into 168 bits
     *
     * _Available since v4.7._
     */
    function toUint168(uint256 value) internal pure returns (uint168) {
        require(value <= type(uint168).max, "SafeCast: value doesn't fit in 168 bits");
        return uint168(value);
    }

    /**
     * @dev Returns the downcasted uint160 from uint256, reverting on
     * overflow (when the input is greater than largest uint160).
     *
     * Counterpart to Solidity's `uint160` operator.
     *
     * Requirements:
     *
     * - input must fit into 160 bits
     *
     * _Available since v4.7._
     */
    function toUint160(uint256 value) internal pure returns (uint160) {
        require(value <= type(uint160).max, "SafeCast: value doesn't fit in 160 bits");
        return uint160(value);
    }

    /**
     * @dev Returns the downcasted uint152 from uint256, reverting on
     * overflow (when the input is greater than largest uint152).
     *
     * Counterpart to Solidity's `uint152` operator.
     *
     * Requirements:
     *
     * - input must fit into 152 bits
     *
     * _Available since v4.7._
     */
    function toUint152(uint256 value) internal pure returns (uint152) {
        require(value <= type(uint152).max, "SafeCast: value doesn't fit in 152 bits");
        return uint152(value);
    }

    /**
     * @dev Returns the downcasted uint144 from uint256, reverting on
     * overflow (when the input is greater than largest uint144).
     *
     * Counterpart to Solidity's `uint144` operator.
     *
     * Requirements:
     *
     * - input must fit into 144 bits
     *
     * _Available since v4.7._
     */
    function toUint144(uint256 value) internal pure returns (uint144) {
        require(value <= type(uint144).max, "SafeCast: value doesn't fit in 144 bits");
        return uint144(value);
    }

    /**
     * @dev Returns the downcasted uint136 from uint256, reverting on
     * overflow (when the input is greater than largest uint136).
     *
     * Counterpart to Solidity's `uint136` operator.
     *
     * Requirements:
     *
     * - input must fit into 136 bits
     *
     * _Available since v4.7._
     */
    function toUint136(uint256 value) internal pure returns (uint136) {
        require(value <= type(uint136).max, "SafeCast: value doesn't fit in 136 bits");
        return uint136(value);
    }

    /**
     * @dev Returns the downcasted uint128 from uint256, reverting on
     * overflow (when the input is greater than largest uint128).
     *
     * Counterpart to Solidity's `uint128` operator.
     *
     * Requirements:
     *
     * - input must fit into 128 bits
     *
     * _Available since v2.5._
     */
    function toUint128(uint256 value) internal pure returns (uint128) {
        require(value <= type(uint128).max, "SafeCast: value doesn't fit in 128 bits");
        return uint128(value);
    }

    /**
     * @dev Returns the downcasted uint120 from uint256, reverting on
     * overflow (when the input is greater than largest uint120).
     *
     * Counterpart to Solidity's `uint120` operator.
     *
     * Requirements:
     *
     * - input must fit into 120 bits
     *
     * _Available since v4.7._
     */
    function toUint120(uint256 value) internal pure returns (uint120) {
        require(value <= type(uint120).max, "SafeCast: value doesn't fit in 120 bits");
        return uint120(value);
    }

    /**
     * @dev Returns the downcasted uint112 from uint256, reverting on
     * overflow (when the input is greater than largest uint112).
     *
     * Counterpart to Solidity's `uint112` operator.
     *
     * Requirements:
     *
     * - input must fit into 112 bits
     *
     * _Available since v4.7._
     */
    function toUint112(uint256 value) internal pure returns (uint112) {
        require(value <= type(uint112).max, "SafeCast: value doesn't fit in 112 bits");
        return uint112(value);
    }

    /**
     * @dev Returns the downcasted uint104 from uint256, reverting on
     * overflow (when the input is greater than largest uint104).
     *
     * Counterpart to Solidity's `uint104` operator.
     *
     * Requirements:
     *
     * - input must fit into 104 bits
     *
     * _Available since v4.7._
     */
    function toUint104(uint256 value) internal pure returns (uint104) {
        require(value <= type(uint104).max, "SafeCast: value doesn't fit in 104 bits");
        return uint104(value);
    }

    /**
     * @dev Returns the downcasted uint96 from uint256, reverting on
     * overflow (when the input is greater than largest uint96).
     *
     * Counterpart to Solidity's `uint96` operator.
     *
     * Requirements:
     *
     * - input must fit into 96 bits
     *
     * _Available since v4.2._
     */
    function toUint96(uint256 value) internal pure returns (uint96) {
        require(value <= type(uint96).max, "SafeCast: value doesn't fit in 96 bits");
        return uint96(value);
    }

    /**
     * @dev Returns the downcasted uint88 from uint256, reverting on
     * overflow (when the input is greater than largest uint88).
     *
     * Counterpart to Solidity's `uint88` operator.
     *
     * Requirements:
     *
     * - input must fit into 88 bits
     *
     * _Available since v4.7._
     */
    function toUint88(uint256 value) internal pure returns (uint88) {
        require(value <= type(uint88).max, "SafeCast: value doesn't fit in 88 bits");
        return uint88(value);
    }

    /**
     * @dev Returns the downcasted uint80 from uint256, reverting on
     * overflow (when the input is greater than largest uint80).
     *
     * Counterpart to Solidity's `uint80` operator.
     *
     * Requirements:
     *
     * - input must fit into 80 bits
     *
     * _Available since v4.7._
     */
    function toUint80(uint256 value) internal pure returns (uint80) {
        require(value <= type(uint80).max, "SafeCast: value doesn't fit in 80 bits");
        return uint80(value);
    }

    /**
     * @dev Returns the downcasted uint72 from uint256, reverting on
     * overflow (when the input is greater than largest uint72).
     *
     * Counterpart to Solidity's `uint72` operator.
     *
     * Requirements:
     *
     * - input must fit into 72 bits
     *
     * _Available since v4.7._
     */
    function toUint72(uint256 value) internal pure returns (uint72) {
        require(value <= type(uint72).max, "SafeCast: value doesn't fit in 72 bits");
        return uint72(value);
    }

    /**
     * @dev Returns the downcasted uint64 from uint256, reverting on
     * overflow (when the input is greater than largest uint64).
     *
     * Counterpart to Solidity's `uint64` operator.
     *
     * Requirements:
     *
     * - input must fit into 64 bits
     *
     * _Available since v2.5._
     */
    function toUint64(uint256 value) internal pure returns (uint64) {
        require(value <= type(uint64).max, "SafeCast: value doesn't fit in 64 bits");
        return uint64(value);
    }

    /**
     * @dev Returns the downcasted uint56 from uint256, reverting on
     * overflow (when the input is greater than largest uint56).
     *
     * Counterpart to Solidity's `uint56` operator.
     *
     * Requirements:
     *
     * - input must fit into 56 bits
     *
     * _Available since v4.7._
     */
    function toUint56(uint256 value) internal pure returns (uint56) {
        require(value <= type(uint56).max, "SafeCast: value doesn't fit in 56 bits");
        return uint56(value);
    }

    /**
     * @dev Returns the downcasted uint48 from uint256, reverting on
     * overflow (when the input is greater than largest uint48).
     *
     * Counterpart to Solidity's `uint48` operator.
     *
     * Requirements:
     *
     * - input must fit into 48 bits
     *
     * _Available since v4.7._
     */
    function toUint48(uint256 value) internal pure returns (uint48) {
        require(value <= type(uint48).max, "SafeCast: value doesn't fit in 48 bits");
        return uint48(value);
    }

    /**
     * @dev Returns the downcasted uint40 from uint256, reverting on
     * overflow (when the input is greater than largest uint40).
     *
     * Counterpart to Solidity's `uint40` operator.
     *
     * Requirements:
     *
     * - input must fit into 40 bits
     *
     * _Available since v4.7._
     */
    function toUint40(uint256 value) internal pure returns (uint40) {
        require(value <= type(uint40).max, "SafeCast: value doesn't fit in 40 bits");
        return uint40(value);
    }

    /**
     * @dev Returns the downcasted uint32 from uint256, reverting on
     * overflow (when the input is greater than largest uint32).
     *
     * Counterpart to Solidity's `uint32` operator.
     *
     * Requirements:
     *
     * - input must fit into 32 bits
     *
     * _Available since v2.5._
     */
    function toUint32(uint256 value) internal pure returns (uint32) {
        require(value <= type(uint32).max, "SafeCast: value doesn't fit in 32 bits");
        return uint32(value);
    }

    /**
     * @dev Returns the downcasted uint24 from uint256, reverting on
     * overflow (when the input is greater than largest uint24).
     *
     * Counterpart to Solidity's `uint24` operator.
     *
     * Requirements:
     *
     * - input must fit into 24 bits
     *
     * _Available since v4.7._
     */
    function toUint24(uint256 value) internal pure returns (uint24) {
        require(value <= type(uint24).max, "SafeCast: value doesn't fit in 24 bits");
        return uint24(value);
    }

    /**
     * @dev Returns the downcasted uint16 from uint256, reverting on
     * overflow (when the input is greater than largest uint16).
     *
     * Counterpart to Solidity's `uint16` operator.
     *
     * Requirements:
     *
     * - input must fit into 16 bits
     *
     * _Available since v2.5._
     */
    function toUint16(uint256 value) internal pure returns (uint16) {
        require(value <= type(uint16).max, "SafeCast: value doesn't fit in 16 bits");
        return uint16(value);
    }

    /**
     * @dev Returns the downcasted uint8 from uint256, reverting on
     * overflow (when the input is greater than largest uint8).
     *
     * Counterpart to Solidity's `uint8` operator.
     *
     * Requirements:
     *
     * - input must fit into 8 bits
     *
     * _Available since v2.5._
     */
    function toUint8(uint256 value) internal pure returns (uint8) {
        require(value <= type(uint8).max, "SafeCast: value doesn't fit in 8 bits");
        return uint8(value);
    }

    /**
     * @dev Converts a signed int256 into an unsigned uint256.
     *
     * Requirements:
     *
     * - input must be greater than or equal to 0.
     *
     * _Available since v3.0._
     */
    function toUint256(int256 value) internal pure returns (uint256) {
        require(value >= 0, "SafeCast: value must be positive");
        return uint256(value);
    }

    /**
     * @dev Returns the downcasted int248 from int256, reverting on
     * overflow (when the input is less than smallest int248 or
     * greater than largest int248).
     *
     * Counterpart to Solidity's `int248` operator.
     *
     * Requirements:
     *
     * - input must fit into 248 bits
     *
     * _Available since v4.7._
     */
    function toInt248(int256 value) internal pure returns (int248 downcasted) {
        downcasted = int248(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 248 bits");
    }

    /**
     * @dev Returns the downcasted int240 from int256, reverting on
     * overflow (when the input is less than smallest int240 or
     * greater than largest int240).
     *
     * Counterpart to Solidity's `int240` operator.
     *
     * Requirements:
     *
     * - input must fit into 240 bits
     *
     * _Available since v4.7._
     */
    function toInt240(int256 value) internal pure returns (int240 downcasted) {
        downcasted = int240(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 240 bits");
    }

    /**
     * @dev Returns the downcasted int232 from int256, reverting on
     * overflow (when the input is less than smallest int232 or
     * greater than largest int232).
     *
     * Counterpart to Solidity's `int232` operator.
     *
     * Requirements:
     *
     * - input must fit into 232 bits
     *
     * _Available since v4.7._
     */
    function toInt232(int256 value) internal pure returns (int232 downcasted) {
        downcasted = int232(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 232 bits");
    }

    /**
     * @dev Returns the downcasted int224 from int256, reverting on
     * overflow (when the input is less than smallest int224 or
     * greater than largest int224).
     *
     * Counterpart to Solidity's `int224` operator.
     *
     * Requirements:
     *
     * - input must fit into 224 bits
     *
     * _Available since v4.7._
     */
    function toInt224(int256 value) internal pure returns (int224 downcasted) {
        downcasted = int224(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 224 bits");
    }

    /**
     * @dev Returns the downcasted int216 from int256, reverting on
     * overflow (when the input is less than smallest int216 or
     * greater than largest int216).
     *
     * Counterpart to Solidity's `int216` operator.
     *
     * Requirements:
     *
     * - input must fit into 216 bits
     *
     * _Available since v4.7._
     */
    function toInt216(int256 value) internal pure returns (int216 downcasted) {
        downcasted = int216(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 216 bits");
    }

    /**
     * @dev Returns the downcasted int208 from int256, reverting on
     * overflow (when the input is less than smallest int208 or
     * greater than largest int208).
     *
     * Counterpart to Solidity's `int208` operator.
     *
     * Requirements:
     *
     * - input must fit into 208 bits
     *
     * _Available since v4.7._
     */
    function toInt208(int256 value) internal pure returns (int208 downcasted) {
        downcasted = int208(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 208 bits");
    }

    /**
     * @dev Returns the downcasted int200 from int256, reverting on
     * overflow (when the input is less than smallest int200 or
     * greater than largest int200).
     *
     * Counterpart to Solidity's `int200` operator.
     *
     * Requirements:
     *
     * - input must fit into 200 bits
     *
     * _Available since v4.7._
     */
    function toInt200(int256 value) internal pure returns (int200 downcasted) {
        downcasted = int200(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 200 bits");
    }

    /**
     * @dev Returns the downcasted int192 from int256, reverting on
     * overflow (when the input is less than smallest int192 or
     * greater than largest int192).
     *
     * Counterpart to Solidity's `int192` operator.
     *
     * Requirements:
     *
     * - input must fit into 192 bits
     *
     * _Available since v4.7._
     */
    function toInt192(int256 value) internal pure returns (int192 downcasted) {
        downcasted = int192(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 192 bits");
    }

    /**
     * @dev Returns the downcasted int184 from int256, reverting on
     * overflow (when the input is less than smallest int184 or
     * greater than largest int184).
     *
     * Counterpart to Solidity's `int184` operator.
     *
     * Requirements:
     *
     * - input must fit into 184 bits
     *
     * _Available since v4.7._
     */
    function toInt184(int256 value) internal pure returns (int184 downcasted) {
        downcasted = int184(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 184 bits");
    }

    /**
     * @dev Returns the downcasted int176 from int256, reverting on
     * overflow (when the input is less than smallest int176 or
     * greater than largest int176).
     *
     * Counterpart to Solidity's `int176` operator.
     *
     * Requirements:
     *
     * - input must fit into 176 bits
     *
     * _Available since v4.7._
     */
    function toInt176(int256 value) internal pure returns (int176 downcasted) {
        downcasted = int176(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 176 bits");
    }

    /**
     * @dev Returns the downcasted int168 from int256, reverting on
     * overflow (when the input is less than smallest int168 or
     * greater than largest int168).
     *
     * Counterpart to Solidity's `int168` operator.
     *
     * Requirements:
     *
     * - input must fit into 168 bits
     *
     * _Available since v4.7._
     */
    function toInt168(int256 value) internal pure returns (int168 downcasted) {
        downcasted = int168(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 168 bits");
    }

    /**
     * @dev Returns the downcasted int160 from int256, reverting on
     * overflow (when the input is less than smallest int160 or
     * greater than largest int160).
     *
     * Counterpart to Solidity's `int160` operator.
     *
     * Requirements:
     *
     * - input must fit into 160 bits
     *
     * _Available since v4.7._
     */
    function toInt160(int256 value) internal pure returns (int160 downcasted) {
        downcasted = int160(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 160 bits");
    }

    /**
     * @dev Returns the downcasted int152 from int256, reverting on
     * overflow (when the input is less than smallest int152 or
     * greater than largest int152).
     *
     * Counterpart to Solidity's `int152` operator.
     *
     * Requirements:
     *
     * - input must fit into 152 bits
     *
     * _Available since v4.7._
     */
    function toInt152(int256 value) internal pure returns (int152 downcasted) {
        downcasted = int152(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 152 bits");
    }

    /**
     * @dev Returns the downcasted int144 from int256, reverting on
     * overflow (when the input is less than smallest int144 or
     * greater than largest int144).
     *
     * Counterpart to Solidity's `int144` operator.
     *
     * Requirements:
     *
     * - input must fit into 144 bits
     *
     * _Available since v4.7._
     */
    function toInt144(int256 value) internal pure returns (int144 downcasted) {
        downcasted = int144(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 144 bits");
    }

    /**
     * @dev Returns the downcasted int136 from int256, reverting on
     * overflow (when the input is less than smallest int136 or
     * greater than largest int136).
     *
     * Counterpart to Solidity's `int136` operator.
     *
     * Requirements:
     *
     * - input must fit into 136 bits
     *
     * _Available since v4.7._
     */
    function toInt136(int256 value) internal pure returns (int136 downcasted) {
        downcasted = int136(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 136 bits");
    }

    /**
     * @dev Returns the downcasted int128 from int256, reverting on
     * overflow (when the input is less than smallest int128 or
     * greater than largest int128).
     *
     * Counterpart to Solidity's `int128` operator.
     *
     * Requirements:
     *
     * - input must fit into 128 bits
     *
     * _Available since v3.1._
     */
    function toInt128(int256 value) internal pure returns (int128 downcasted) {
        downcasted = int128(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 128 bits");
    }

    /**
     * @dev Returns the downcasted int120 from int256, reverting on
     * overflow (when the input is less than smallest int120 or
     * greater than largest int120).
     *
     * Counterpart to Solidity's `int120` operator.
     *
     * Requirements:
     *
     * - input must fit into 120 bits
     *
     * _Available since v4.7._
     */
    function toInt120(int256 value) internal pure returns (int120 downcasted) {
        downcasted = int120(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 120 bits");
    }

    /**
     * @dev Returns the downcasted int112 from int256, reverting on
     * overflow (when the input is less than smallest int112 or
     * greater than largest int112).
     *
     * Counterpart to Solidity's `int112` operator.
     *
     * Requirements:
     *
     * - input must fit into 112 bits
     *
     * _Available since v4.7._
     */
    function toInt112(int256 value) internal pure returns (int112 downcasted) {
        downcasted = int112(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 112 bits");
    }

    /**
     * @dev Returns the downcasted int104 from int256, reverting on
     * overflow (when the input is less than smallest int104 or
     * greater than largest int104).
     *
     * Counterpart to Solidity's `int104` operator.
     *
     * Requirements:
     *
     * - input must fit into 104 bits
     *
     * _Available since v4.7._
     */
    function toInt104(int256 value) internal pure returns (int104 downcasted) {
        downcasted = int104(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 104 bits");
    }

    /**
     * @dev Returns the downcasted int96 from int256, reverting on
     * overflow (when the input is less than smallest int96 or
     * greater than largest int96).
     *
     * Counterpart to Solidity's `int96` operator.
     *
     * Requirements:
     *
     * - input must fit into 96 bits
     *
     * _Available since v4.7._
     */
    function toInt96(int256 value) internal pure returns (int96 downcasted) {
        downcasted = int96(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 96 bits");
    }

    /**
     * @dev Returns the downcasted int88 from int256, reverting on
     * overflow (when the input is less than smallest int88 or
     * greater than largest int88).
     *
     * Counterpart to Solidity's `int88` operator.
     *
     * Requirements:
     *
     * - input must fit into 88 bits
     *
     * _Available since v4.7._
     */
    function toInt88(int256 value) internal pure returns (int88 downcasted) {
        downcasted = int88(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 88 bits");
    }

    /**
     * @dev Returns the downcasted int80 from int256, reverting on
     * overflow (when the input is less than smallest int80 or
     * greater than largest int80).
     *
     * Counterpart to Solidity's `int80` operator.
     *
     * Requirements:
     *
     * - input must fit into 80 bits
     *
     * _Available since v4.7._
     */
    function toInt80(int256 value) internal pure returns (int80 downcasted) {
        downcasted = int80(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 80 bits");
    }

    /**
     * @dev Returns the downcasted int72 from int256, reverting on
     * overflow (when the input is less than smallest int72 or
     * greater than largest int72).
     *
     * Counterpart to Solidity's `int72` operator.
     *
     * Requirements:
     *
     * - input must fit into 72 bits
     *
     * _Available since v4.7._
     */
    function toInt72(int256 value) internal pure returns (int72 downcasted) {
        downcasted = int72(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 72 bits");
    }

    /**
     * @dev Returns the downcasted int64 from int256, reverting on
     * overflow (when the input is less than smallest int64 or
     * greater than largest int64).
     *
     * Counterpart to Solidity's `int64` operator.
     *
     * Requirements:
     *
     * - input must fit into 64 bits
     *
     * _Available since v3.1._
     */
    function toInt64(int256 value) internal pure returns (int64 downcasted) {
        downcasted = int64(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 64 bits");
    }

    /**
     * @dev Returns the downcasted int56 from int256, reverting on
     * overflow (when the input is less than smallest int56 or
     * greater than largest int56).
     *
     * Counterpart to Solidity's `int56` operator.
     *
     * Requirements:
     *
     * - input must fit into 56 bits
     *
     * _Available since v4.7._
     */
    function toInt56(int256 value) internal pure returns (int56 downcasted) {
        downcasted = int56(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 56 bits");
    }

    /**
     * @dev Returns the downcasted int48 from int256, reverting on
     * overflow (when the input is less than smallest int48 or
     * greater than largest int48).
     *
     * Counterpart to Solidity's `int48` operator.
     *
     * Requirements:
     *
     * - input must fit into 48 bits
     *
     * _Available since v4.7._
     */
    function toInt48(int256 value) internal pure returns (int48 downcasted) {
        downcasted = int48(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 48 bits");
    }

    /**
     * @dev Returns the downcasted int40 from int256, reverting on
     * overflow (when the input is less than smallest int40 or
     * greater than largest int40).
     *
     * Counterpart to Solidity's `int40` operator.
     *
     * Requirements:
     *
     * - input must fit into 40 bits
     *
     * _Available since v4.7._
     */
    function toInt40(int256 value) internal pure returns (int40 downcasted) {
        downcasted = int40(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 40 bits");
    }

    /**
     * @dev Returns the downcasted int32 from int256, reverting on
     * overflow (when the input is less than smallest int32 or
     * greater than largest int32).
     *
     * Counterpart to Solidity's `int32` operator.
     *
     * Requirements:
     *
     * - input must fit into 32 bits
     *
     * _Available since v3.1._
     */
    function toInt32(int256 value) internal pure returns (int32 downcasted) {
        downcasted = int32(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 32 bits");
    }

    /**
     * @dev Returns the downcasted int24 from int256, reverting on
     * overflow (when the input is less than smallest int24 or
     * greater than largest int24).
     *
     * Counterpart to Solidity's `int24` operator.
     *
     * Requirements:
     *
     * - input must fit into 24 bits
     *
     * _Available since v4.7._
     */
    function toInt24(int256 value) internal pure returns (int24 downcasted) {
        downcasted = int24(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 24 bits");
    }

    /**
     * @dev Returns the downcasted int16 from int256, reverting on
     * overflow (when the input is less than smallest int16 or
     * greater than largest int16).
     *
     * Counterpart to Solidity's `int16` operator.
     *
     * Requirements:
     *
     * - input must fit into 16 bits
     *
     * _Available since v3.1._
     */
    function toInt16(int256 value) internal pure returns (int16 downcasted) {
        downcasted = int16(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 16 bits");
    }

    /**
     * @dev Returns the downcasted int8 from int256, reverting on
     * overflow (when the input is less than smallest int8 or
     * greater than largest int8).
     *
     * Counterpart to Solidity's `int8` operator.
     *
     * Requirements:
     *
     * - input must fit into 8 bits
     *
     * _Available since v3.1._
     */
    function toInt8(int256 value) internal pure returns (int8 downcasted) {
        downcasted = int8(value);
        require(downcasted == value, "SafeCast: value doesn't fit in 8 bits");
    }

    /**
     * @dev Converts an unsigned uint256 into a signed int256.
     *
     * Requirements:
     *
     * - input must be less than or equal to maxInt256.
     *
     * _Available since v3.0._
     */
    function toInt256(uint256 value) internal pure returns (int256) {
        // Note: Unsafe cast below is okay because `type(int256).max` is guaranteed to be positive
        require(value <= uint256(type(int256).max), "SafeCast: value doesn't fit in an int256");
        return int256(value);
    }
}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

import { IPoolBorrowerActions } from './commons/IPoolBorrowerActions.sol';
import { IPoolLPActions }       from './commons/IPoolLPActions.sol';
import { IPoolLenderActions }   from './commons/IPoolLenderActions.sol';
import { IPoolKickerActions }   from './commons/IPoolKickerActions.sol';
import { IPoolTakerActions }    from './commons/IPoolTakerActions.sol';
import { IPoolSettlerActions }  from './commons/IPoolSettlerActions.sol';

import { IPoolImmutables }      from './commons/IPoolImmutables.sol';
import { IPoolState }           from './commons/IPoolState.sol';
import { IPoolDerivedState }    from './commons/IPoolDerivedState.sol';
import { IPoolEvents }          from './commons/IPoolEvents.sol';
import { IPoolErrors }          from './commons/IPoolErrors.sol';
import { IERC3156FlashLender }  from './IERC3156FlashLender.sol';

/**
 * @title Base Pool Interface
 */
interface IPool is
    IPoolBorrowerActions,
    IPoolLPActions,
    IPoolLenderActions,
    IPoolKickerActions,
    IPoolTakerActions,
    IPoolSettlerActions,
    IPoolImmutables,
    IPoolState,
    IPoolDerivedState,
    IPoolEvents,
    IPoolErrors,
    IERC3156FlashLender
{

}

/// @dev Pool type enum - `ERC20` and `ERC721`
enum PoolType { ERC20, ERC721 }

/// @dev `ERC20` token interface.
interface IERC20Token {
    function balanceOf(address account) external view returns (uint256);
    function burn(uint256 amount) external;
    function decimals() external view returns (uint8);
}

/// @dev `ERC721` token interface.
interface IERC721Token {
    function transferFrom(
        address from,
        address to,
        uint256 tokenId
    ) external;
}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

/**
 * @title Pool Borrower Actions
 */
interface IPoolBorrowerActions {

    /**
     *  @notice Called by fully collateralized borrowers to restamp the `Np to Tp ratio` of the loan (only if loan is fully collateralized and not in auction).
     *          The reason for stamping the `Np to Tp ratio` on the loan is to provide some certainty to the borrower as to at what price they can expect to be liquidated.
     *          This action can restamp only the loan of `msg.sender`.
     */
    function stampLoan() external;

}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

/**
 * @title Pool `LP` Actions
 */
interface IPoolLPActions {

    /**
     *  @notice Called by `LP` owners to approve transfer of an amount of `LP` to a new owner.
     *  @dev    Intended for use by the `PositionManager` contract.
     *  @param  spender_ The new owner of the `LP`.
     *  @param  indexes_ Bucket indexes from where `LP` are transferred.
     *  @param  amounts_ The amounts of `LP` approved to transfer (`WAD` precision).
     */
    function increaseLPAllowance(
        address spender_,
        uint256[] calldata indexes_,
        uint256[] calldata amounts_
    ) external;

    /**
     *  @notice Called by `LP` owners to decrease the amount of `LP` that can be spend by a new owner.
     *  @dev    Intended for use by the `PositionManager` contract.
     *  @param  spender_ The new owner of the `LP`.
     *  @param  indexes_ Bucket indexes from where `LP` are transferred.
     *  @param  amounts_ The amounts of `LP` disapproved to transfer (`WAD` precision).
     */
    function decreaseLPAllowance(
        address spender_,
        uint256[] calldata indexes_,
        uint256[] calldata amounts_
    ) external;

    /**
     *  @notice Called by `LP` owners to decrease the amount of `LP` that can be spend by a new owner.
     *  @param  spender_ Address that is having it's allowance revoked.
     *  @param  indexes_ List of bucket index to remove the allowance from.
     */
    function revokeLPAllowance(
        address spender_,
        uint256[] calldata indexes_
    ) external;

    /**
     *  @notice Called by `LP` owners to allow addresses that can transfer LP.
     *  @dev    Intended for use by the `PositionManager` contract.
     *  @param  transferors_ Addresses that are allowed to transfer `LP` to new owner.
     */
    function approveLPTransferors(
        address[] calldata transferors_
    ) external;

    /**
     *  @notice Called by `LP` owners to revoke addresses that can transfer `LP`.
     *  @dev    Intended for use by the `PositionManager` contract.
     *  @param  transferors_ Addresses that are revoked to transfer `LP` to new owner.
     */
    function revokeLPTransferors(
        address[] calldata transferors_
    ) external;

    /**
     *  @notice Called by `LP` owners to transfers their `LP` to a different address. `approveLpOwnership` needs to be run first.
     *  @dev    Used by `PositionManager.memorializePositions()`.
     *  @param  owner_    The original owner address of the position.
     *  @param  newOwner_ The new owner address of the position.
     *  @param  indexes_  Array of price buckets index at which `LP` were moved.
     */
    function transferLP(
        address owner_,
        address newOwner_,
        uint256[] calldata indexes_
    ) external;
}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

/**
 * @title Pool Lender Actions
 */
interface IPoolLenderActions {

    /*********************************************/
    /*** Quote/collateral management functions ***/
    /*********************************************/

    /**
     *  @notice Called by lenders to add an amount of credit at a specified price bucket.
     *  @param  amount_           The amount of quote token to be added by a lender (`WAD` precision).
     *  @param  index_            The index of the bucket to which the quote tokens will be added.
     *  @param  expiry_           Timestamp after which this transaction will revert, preventing inclusion in a block with unfavorable price.
     *  @return bucketLP_         The amount of `LP` changed for the added quote tokens (`WAD` precision).
     *  @return addedAmount_      The amount of quote token added (`WAD` precision).
     */
    function addQuoteToken(
        uint256 amount_,
        uint256 index_,
        uint256 expiry_
    ) external returns (uint256 bucketLP_, uint256 addedAmount_);

    /**
     *  @notice Called by lenders to move an amount of credit from a specified price bucket to another specified price bucket.
     *  @param  maxAmount_        The maximum amount of quote token to be moved by a lender (`WAD` precision).
     *  @param  fromIndex_        The bucket index from which the quote tokens will be removed.
     *  @param  toIndex_          The bucket index to which the quote tokens will be added.
     *  @param  expiry_           Timestamp after which this transaction will revert, preventing inclusion in a block with unfavorable price.
     *  @return fromBucketLP_     The amount of `LP` moved out from bucket (`WAD` precision).
     *  @return toBucketLP_       The amount of `LP` moved to destination bucket (`WAD` precision).
     *  @return movedAmount_      The amount of quote token moved (`WAD` precision).
     */
    function moveQuoteToken(
        uint256 maxAmount_,
        uint256 fromIndex_,
        uint256 toIndex_,
        uint256 expiry_
    ) external returns (uint256 fromBucketLP_, uint256 toBucketLP_, uint256 movedAmount_);

    /**
     *  @notice Called by lenders to claim collateral from a price bucket.
     *  @param  maxAmount_     The amount of collateral (`WAD` precision for `ERC20` pools, number of `NFT` tokens for `ERC721` pools) to claim.
     *  @param  index_         The bucket index from which collateral will be removed.
     *  @return removedAmount_ The amount of collateral removed (`WAD` precision).
     *  @return redeemedLP_    The amount of `LP` used for removing collateral amount (`WAD` precision).
     */
    function removeCollateral(
        uint256 maxAmount_,
        uint256 index_
    ) external returns (uint256 removedAmount_, uint256 redeemedLP_);

    /**
     *  @notice Called by lenders to remove an amount of credit at a specified price bucket.
     *  @param  maxAmount_     The max amount of quote token to be removed by a lender (`WAD` precision).
     *  @param  index_         The bucket index from which quote tokens will be removed.
     *  @return removedAmount_ The amount of quote token removed (`WAD` precision).
     *  @return redeemedLP_    The amount of `LP` used for removing quote tokens amount (`WAD` precision).
     */
    function removeQuoteToken(
        uint256 maxAmount_,
        uint256 index_
    ) external returns (uint256 removedAmount_, uint256 redeemedLP_);

    /********************************/
    /*** Interest update function ***/
    /********************************/

    /**
     *  @notice Called by actors to update pool interest rate (can be updated only once in a `12` hours period of time).
     */
    function updateInterest() external;

}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

/**
 * @title Pool Kicker Actions
 */
interface IPoolKickerActions {

    /********************/
    /*** Liquidations ***/
    /********************/

    /**
     *  @notice Called by actors to initiate a liquidation.
     *  @param  borrower_     Identifies the loan to liquidate.
     *  @param  npLimitIndex_ Index of the lower bound of `NP` tolerated when kicking the auction.
     */
    function kick(
        address borrower_,
        uint256 npLimitIndex_
    ) external;

    /**
     *  @notice Called by lenders to liquidate the top loan.
     *  @param  index_        The deposit index to use for kicking the top loan.
     *  @param  npLimitIndex_ Index of the lower bound of `NP` tolerated when kicking the auction.
     */
    function lenderKick(
        uint256 index_,
        uint256 npLimitIndex_
    ) external;

    /**
     *  @notice Called by kickers to withdraw their auction bonds (the amount of quote tokens that are not locked in active auctions).
     *  @param  recipient_ Address to receive claimed bonds amount.
     *  @param  maxAmount_ The max amount to withdraw from auction bonds (`WAD` precision). Constrained by claimable amounts and liquidity.
     *  @return withdrawnAmount_ The amount withdrawn (`WAD` precision).
     */
    function withdrawBonds(
        address recipient_,
        uint256 maxAmount_
    ) external returns (uint256 withdrawnAmount_);

    /***********************/
    /*** Reserve Auction ***/
    /***********************/

    /**
     *  @notice Called by actor to start a `Claimable Reserve Auction` (`CRA`).
     */
    function kickReserveAuction() external;
}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

/**
 * @title Pool Taker Actions
 */
interface IPoolTakerActions {

    /**
     *  @notice Called by actors to use quote token to arb higher-priced deposit off the book.
     *  @param  borrowerAddress_  Address of the borower take is being called upon.
     *  @param  depositTake_      If `true` then the take will happen at an auction price equal with bucket price. Auction price is used otherwise.
     *  @param  index_            Index of a bucket, likely the `HPB`, in which collateral will be deposited.
     */
    function bucketTake(
        address borrowerAddress_,
        bool    depositTake_,
        uint256 index_
    ) external;

    /**
     *  @notice Called by actors to purchase collateral from the auction in exchange for quote token.
     *  @param  borrowerAddress_  Address of the borower take is being called upon.
     *  @param  maxAmount_        Max amount of collateral that will be taken from the auction (`WAD` precision for `ERC20` pools, max number of `NFT`s for `ERC721` pools).
     *  @param  callee_           Identifies where collateral should be sent and where quote token should be obtained.
     *  @param  data_             If provided, take will assume the callee implements `IERC*Taker`.  Take will send collateral to 
     *                            callee before passing this data to `IERC*Taker.atomicSwapCallback`.  If not provided, 
     *                            the callback function will not be invoked.
     *  @return collateralTaken_  Amount of collateral taken from the auction (`WAD` precision for `ERC20` pools, max number of `NFT`s for `ERC721` pools).
     */
    function take(
        address        borrowerAddress_,
        uint256        maxAmount_,
        address        callee_,
        bytes calldata data_
    ) external returns (uint256 collateralTaken_);

    /***********************/
    /*** Reserve Auction ***/
    /***********************/

    /**
     *  @notice Purchases claimable reserves during a `CRA` using `Ajna` token.
     *  @param  maxAmount_ Maximum amount of quote token to purchase at the current auction price (`WAD` precision).
     *  @return amount_    Actual amount of reserves taken (`WAD` precision).
     */
    function takeReserves(
        uint256 maxAmount_
    ) external returns (uint256 amount_);

}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

/**
 * @title Pool Settler Actions
 */
interface IPoolSettlerActions {

    /**
     *  @notice Called by actors to settle an amount of debt in a completed liquidation.
     *  @param  borrowerAddress_   Address of the auctioned borrower.
     *  @param  maxDepth_          Measured from `HPB`, maximum number of buckets deep to settle debt.
     *  @return collateralSettled_ Amount of collateral settled.
     *  @return isBorrowerSettled_ True if all borrower's debt is settled.
     *  @dev    `maxDepth_` is used to prevent unbounded iteration clearing large liquidations.
     */
    function settle(
        address borrowerAddress_,
        uint256 maxDepth_
    ) external returns (uint256 collateralSettled_, bool isBorrowerSettled_);

}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

/**
 * @title Pool Immutables
 */
interface IPoolImmutables {

    /**
     *  @notice Returns the type of the pool (`0` for `ERC20`, `1` for `ERC721`).
     */
    function poolType() external pure returns (uint8);

    /**
     *  @notice Returns the address of the pool's collateral token.
     */
    function collateralAddress() external pure returns (address);

    /**
     *  @notice Returns the address of the pool's quote token.
     */
    function quoteTokenAddress() external pure returns (address);

    /**
     *  @notice Returns the `quoteTokenScale` state variable.
     *  @notice Token scale is also the minimum amount a lender may have in a bucket (dust amount).
     *  @return The precision of the quote `ERC20` token based on decimals.
     */
    function quoteTokenScale() external pure returns (uint256);
}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

/**
 * @title Pool Derived State
 */
interface IPoolDerivedState {

    /**
     *  @notice Returns the exchange rate for a given bucket index.
     *  @param  index_        The bucket index.
     *  @return exchangeRate_ Exchange rate of the bucket (`WAD` precision).
     */
    function bucketExchangeRate(
        uint256 index_
    ) external view returns (uint256 exchangeRate_);

    /**
     *  @notice Returns the prefix sum of a given bucket.
     *  @param  index_   The bucket index.
     *  @return The deposit up to given index (`WAD` precision).
     */
    function depositUpToIndex(
        uint256 index_
    ) external view returns (uint256);

    /**
     *  @notice Returns the bucket index for a given debt amount.
     *  @param  debt_  The debt amount to calculate bucket index for (`WAD` precision).
     *  @return Bucket index.
     */
    function depositIndex(
        uint256 debt_
    ) external view returns (uint256);

    /**
     *  @notice Returns the total amount of quote tokens deposited in pool.
     *  @return Total amount of deposited quote tokens (`WAD` precision).
     */
    function depositSize() external view returns (uint256);

    /**
     *  @notice Returns the meaningful actual utilization of the pool.
     *  @return Deposit utilization (`WAD` precision).
     */
    function depositUtilization() external view returns (uint256);

    /**
     *  @notice Returns the scaling value of deposit at given index.
     *  @param  index_  Deposit index.
     *  @return Deposit scaling (`WAD` precision).
     */
    function depositScale(
        uint256 index_
    ) external view returns (uint256);

}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

/**
 * @title Pool Events
 */
interface IPoolEvents {

    /*********************/
    /*** Lender events ***/
    /*********************/

    /**
     *  @notice Emitted when lender adds quote token to the pool.
     *  @param  lender    Recipient that added quote tokens.
     *  @param  index     Index at which quote tokens were added.
     *  @param  amount    Amount of quote tokens added to the pool (`WAD` precision).
     *  @param  lpAwarded Amount of `LP` awarded for the deposit (`WAD` precision).
     *  @param  lup       `LUP` calculated after deposit.
     */
    event AddQuoteToken(
        address indexed lender,
        uint256 indexed index,
        uint256 amount,
        uint256 lpAwarded,
        uint256 lup
    );

    /**
     *  @notice Emitted when lender moves quote token from a bucket price to another.
     *  @param  lender         Recipient that moved quote tokens.
     *  @param  from           Price bucket from which quote tokens were moved.
     *  @param  to             Price bucket where quote tokens were moved.
     *  @param  amount         Amount of quote tokens moved (`WAD` precision).
     *  @param  lpRedeemedFrom Amount of `LP` removed from the `from` bucket (`WAD` precision).
     *  @param  lpAwardedTo    Amount of `LP` credited to the `to` bucket (`WAD` precision).
     *  @param  lup            `LUP` calculated after removal.
     */
    event MoveQuoteToken(
        address indexed lender,
        uint256 indexed from,
        uint256 indexed to,
        uint256 amount,
        uint256 lpRedeemedFrom,
        uint256 lpAwardedTo,
        uint256 lup
    );

    /**
     *  @notice Emitted when lender removes quote token from the pool.
     *  @param  lender     Recipient that removed quote tokens.
     *  @param  index      Index at which quote tokens were removed.
     *  @param  amount     Amount of quote tokens removed from the pool (`WAD` precision).
     *  @param  lpRedeemed Amount of `LP` exchanged for quote token (`WAD` precision).
     *  @param  lup        `LUP` calculated after removal.
     */
    event RemoveQuoteToken(
        address indexed lender,
        uint256 indexed index,
        uint256 amount,
        uint256 lpRedeemed,
        uint256 lup
    );

    /**
     *  @notice Emitted when lender claims collateral from a bucket.
     *  @param  claimer    Recipient that claimed collateral.
     *  @param  index      Index at which collateral was claimed.
     *  @param  amount     The amount of collateral (`WAD` precision for `ERC20` pools, number of `NFT` tokens for `ERC721` pools) transferred to the claimer.
     *  @param  lpRedeemed Amount of `LP` exchanged for quote token (`WAD` precision).
     */
    event RemoveCollateral(
        address indexed claimer,
        uint256 indexed index,
        uint256 amount,
        uint256 lpRedeemed
    );

    /***********************/
    /*** Borrower events ***/
    /***********************/

    /**
     *  @notice Emitted when borrower repays quote tokens to the pool and/or pulls collateral from the pool.
     *  @param  borrower         `msg.sender` or on behalf of sender.
     *  @param  quoteRepaid      Amount of quote tokens repaid to the pool (`WAD` precision).
     *  @param  collateralPulled The amount of collateral (`WAD` precision for `ERC20` pools, number of `NFT` tokens for `ERC721` pools) transferred to the claimer.
     *  @param  lup              `LUP` after repay.
     */
    event RepayDebt(
        address indexed borrower,
        uint256 quoteRepaid,
        uint256 collateralPulled,
        uint256 lup
    );

    /**********************/
    /*** Auction events ***/
    /**********************/

    /**
     *  @notice Emitted when a liquidation is initiated.
     *  @param  borrower   Identifies the loan being liquidated.
     *  @param  debt       Debt the liquidation will attempt to cover (`WAD` precision).
     *  @param  collateral Amount of collateral up for liquidation (`WAD` precision for `ERC20` pools, number of `NFT` tokens for `ERC721` pools).
     *  @param  bond       Bond amount locked by kicker (`WAD` precision).
     */
    event Kick(
        address indexed borrower,
        uint256 debt,
        uint256 collateral,
        uint256 bond
    );

    /**
     *  @notice Emitted when kickers are withdrawing funds posted as auction bonds.
     *  @param  kicker   The kicker withdrawing bonds.
     *  @param  reciever The address receiving withdrawn bond amount.
     *  @param  amount   The bond amount that was withdrawn (`WAD` precision).
     */
    event BondWithdrawn(
        address indexed kicker,
        address indexed reciever,
        uint256 amount
    );

    /**
     *  @notice Emitted when an actor uses quote token to arb higher-priced deposit off the book.
     *  @param  borrower    Identifies the loan being liquidated.
     *  @param  index       The index of the `Highest Price Bucket` used for this take.
     *  @param  amount      Amount of quote token used to purchase collateral (`WAD` precision).
     *  @param  collateral  Amount of collateral purchased with quote token (`WAD` precision).
     *  @param  bondChange  Impact of this take to the liquidation bond (`WAD` precision).
     *  @param  isReward    `True` if kicker was rewarded with `bondChange` amount, `false` if kicker was penalized.
     *  @dev    amount / collateral implies the auction price.
     */
    event BucketTake(
        address indexed borrower,
        uint256 index,
        uint256 amount,
        uint256 collateral,
        uint256 bondChange,
        bool    isReward
    );

    /**
     *  @notice Emitted when `LP` are awarded to a taker or kicker in a bucket take.
     *  @param  taker           Actor who invoked the bucket take.
     *  @param  kicker          Actor who started the auction.
     *  @param  lpAwardedTaker  Amount of `LP` awarded to the taker (`WAD` precision).
     *  @param  lpAwardedKicker Amount of `LP` awarded to the actor who started the auction (`WAD` precision).
     */
    event BucketTakeLPAwarded(
        address indexed taker,
        address indexed kicker,
        uint256 lpAwardedTaker,
        uint256 lpAwardedKicker
    );

    /**
     *  @notice Emitted when an actor uses quote token outside of the book to purchase collateral under liquidation.
     *  @param  borrower   Identifies the loan being liquidated.
     *  @param  amount     Amount of quote token used to purchase collateral (`WAD` precision).
     *  @param  collateral Amount of collateral purchased with quote token (for `ERC20` pool, `WAD` precision) or number of `NFT`s purchased (for `ERC721` pool).
     *  @param  bondChange Impact of this take to the liquidation bond (`WAD` precision).
     *  @param  isReward   `True` if kicker was rewarded with `bondChange` amount, `false` if kicker was penalized.
     *  @dev    amount / collateral implies the auction price.
     */
    event Take(
        address indexed borrower,
        uint256 amount,
        uint256 collateral,
        uint256 bondChange,
        bool    isReward
    );

    /**
     *  @notice Emitted when an actor settles debt in a completed liquidation
     *  @param  borrower    Identifies the loan under liquidation.
     *  @param  settledDebt Amount of pool debt settled in this transaction (`WAD` precision).
     *  @dev    When `amountRemaining_ == 0`, the auction has been completed cleared and removed from the queue.
     */
    event Settle(
        address indexed borrower,
        uint256 settledDebt
    );

    /**
     *  @notice Emitted when auction is completed.
     *  @param  borrower   Address of borrower that exits auction.
     *  @param  collateral Borrower's remaining collateral when auction completed (`WAD` precision).
     */
    event AuctionSettle(
        address indexed borrower,
        uint256 collateral
    );

    /**
     *  @notice Emitted when `NFT` auction is completed.
     *  @param  borrower   Address of borrower that exits auction.
     *  @param  collateral Borrower's remaining collateral when auction completed.
     *  @param  lp         Amount of `LP` given to the borrower to compensate fractional collateral (if any, `WAD` precision).
     *  @param  index      Index of the bucket with `LP` to compensate fractional collateral.
     */
    event AuctionNFTSettle(
        address indexed borrower,
        uint256 collateral,
        uint256 lp,
        uint256 index
    );

    /**
     *  @notice Emitted when a `Claimaible Reserve Auction` is started.
     *  @param  claimableReservesRemaining Amount of claimable reserves which has not yet been taken (`WAD` precision).
     *  @param  auctionPrice               Current price at which `1` quote token may be purchased, denominated in `Ajna`.
     *  @param  currentBurnEpoch           Current burn epoch.
     */
    event KickReserveAuction(
        uint256 claimableReservesRemaining,
        uint256 auctionPrice,
        uint256 currentBurnEpoch
    );

    /**
     *  @notice Emitted when a `Claimaible Reserve Auction` is taken.
     *  @param  claimableReservesRemaining Amount of claimable reserves which has not yet been taken (`WAD` precision).
     *  @param  auctionPrice               Current price at which `1` quote token may be purchased, denominated in `Ajna`.
     *  @param  currentBurnEpoch           Current burn epoch.
     */
    event ReserveAuction(
        uint256 claimableReservesRemaining,
        uint256 auctionPrice,
        uint256 currentBurnEpoch
    );

    /**************************/
    /*** LP transfer events ***/
    /**************************/

    /**
     *  @notice Emitted when owner increase the `LP` allowance of a spender at specified indexes with specified amounts.
     *  @param  owner     `LP` owner.
     *  @param  spender   Address approved to transfer `LP`.
     *  @param  indexes   Bucket indexes of `LP` approved.
     *  @param  amounts   `LP` amounts added (ordered by indexes, `WAD` precision).
     */
    event IncreaseLPAllowance(
        address indexed owner,
        address indexed spender,
        uint256[] indexes,
        uint256[] amounts
    );

    /**
     *  @notice Emitted when owner decrease the `LP` allowance of a spender at specified indexes with specified amounts.
     *  @param  owner     `LP` owner.
     *  @param  spender   Address approved to transfer `LP`.
     *  @param  indexes   Bucket indexes of `LP` approved.
     *  @param  amounts   `LP` amounts removed (ordered by indexes, `WAD` precision).
     */
    event DecreaseLPAllowance(
        address indexed owner,
        address indexed spender,
        uint256[] indexes,
        uint256[] amounts
    );

    /**
     *  @notice Emitted when lender removes the allowance of a spender for their `LP`.
     *  @param  owner   `LP` owner.
     *  @param  spender Address that is having it's allowance revoked.
     *  @param  indexes List of bucket index to remove the allowance from.
     */
    event RevokeLPAllowance(
        address indexed owner,
        address indexed spender,
        uint256[] indexes
    );

    /**
     *  @notice Emitted when lender whitelists addresses to accept `LP` from.
     *  @param  lender      Recipient that approves new owner for `LP`.
     *  @param  transferors List of addresses that can transfer `LP` to lender.
     */
    event ApproveLPTransferors(
        address indexed lender,
        address[] transferors
    );

    /**
     *  @notice Emitted when lender removes addresses from the `LP` transferors whitelist.
     *  @param  lender      Recipient that approves new owner for `LP`.
     *  @param  transferors List of addresses that won't be able to transfer `LP` to lender anymore.
     */
    event RevokeLPTransferors(
        address indexed lender,
        address[] transferors
    );

    /**
     *  @notice Emitted when a lender transfers their `LP` to a different address.
     *  @dev    Used by `PositionManager.memorializePositions()`.
     *  @param  owner    The original owner address of the position.
     *  @param  newOwner The new owner address of the position.
     *  @param  indexes  Array of price bucket indexes at which `LP` were transferred.
     *  @param  lp       Amount of `LP` transferred (`WAD` precision).
     */
    event TransferLP(
        address owner,
        address newOwner,
        uint256[] indexes,
        uint256 lp
    );

    /**************************/
    /*** Pool common events ***/
    /**************************/

    /**
     *  @notice Emitted when `LP` are forfeited as a result of the bucket losing all assets.
     *  @param  index       The index of the bucket.
     *  @param  lpForfeited Amount of `LP` forfeited by lenders (`WAD` precision).
     */
    event BucketBankruptcy(
        uint256 indexed index,
        uint256 lpForfeited
    );

    /**
     *  @notice Emitted when a flashloan is taken from pool.
     *  @param  receiver The address receiving the flashloan.
     *  @param  token    The address of token flashloaned from pool.
     *  @param  amount   The amount of tokens flashloaned from pool (token precision).
     */
    event Flashloan(
        address indexed receiver,
        address indexed token,
        uint256 amount
    );

    /**
     *  @notice Emitted when a loan `Np to Tp ratio` is restamped.
     *  @param  borrower Identifies the loan to update the `Np to Tp ratio`.
     */
    event LoanStamped(
        address indexed borrower
    );

    /**
     *  @notice Emitted when pool interest rate is reset. This happens when `interest rate > 10%` and `debtEma < 5%` of `depositEma`
     *  @param  oldRate Old pool interest rate.
     *  @param  newRate New pool interest rate.
     */
    event ResetInterestRate(
        uint256 oldRate,
        uint256 newRate
    );

    /**
     *  @notice Emitted when pool interest rate is updated.
     *  @param  oldRate Old pool interest rate.
     *  @param  newRate New pool interest rate.
     */
    event UpdateInterestRate(
        uint256 oldRate,
        uint256 newRate
    );

    /**
     *  @notice Emitted when interest accural or update interest overflows.
     */
    event InterestUpdateFailure();

}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;

/**
 * @title Pool Errors.
 */
interface IPoolErrors {
    /**************************/
    /*** Common Pool Errors ***/
    /**************************/

    /**
     *  @notice Adding liquidity above current auction price.
     */
    error AddAboveAuctionPrice();

    /**
     *  @notice The action cannot be executed on an active auction.
     */
    error AuctionActive();

    /**
     *  @notice Attempted auction to clear doesn't meet conditions.
     */
    error AuctionNotClearable();

    /**
     *  @notice Auction does not meet requirements to take liquidity.
     */
    error AuctionNotTakeable();

    /**
     *  @notice Head auction should be cleared prior of executing this action.
     */
    error AuctionNotCleared();

    /**
     *  @notice The auction price is greater than the arbed bucket price.
     */
    error AuctionPriceGtBucketPrice();

    /**
     *  @notice Pool already initialized.
     */
    error AlreadyInitialized();

    /**
     *  @notice Borrower is attempting to create or modify a loan such that their loan's quote token would be less than the pool's minimum debt amount.
     */
    error AmountLTMinDebt();

    /**
     *  @notice Recipient of borrowed quote tokens doesn't match the caller of the `drawDebt` function.
     */
    error BorrowerNotSender();

    /**
     *  @notice Borrower has a healthy over-collateralized position.
     */
    error BorrowerOk();

    /**
     *  @notice Borrower is attempting to borrow more quote token than they have collateral for.
     */
    error BorrowerUnderCollateralized();

    /**
     *  @notice Operation cannot be executed in the same block when bucket becomes insolvent.
     */
    error BucketBankruptcyBlock();

    /**
     *  @notice User attempted to merge collateral from a lower price bucket into a higher price bucket.
     */
    error CannotMergeToHigherPrice();

    /**
     *  @notice User attempted an operation which does not exceed the dust amount, or leaves behind less than the dust amount.
     */
    error DustAmountNotExceeded();

    /**
     *  @notice Callback invoked by `flashLoan` function did not return the expected hash (see `ERC-3156` spec).
     */
    error FlashloanCallbackFailed();

    /**
     *  @notice Balance of pool contract before flashloan is different than the balance after flashloan.
     */
    error FlashloanIncorrectBalance();

    /**
     *  @notice Pool cannot facilitate a flashloan for the specified token address.
     */
    error FlashloanUnavailableForToken();

    /**
     *  @notice User is attempting to move or pull more collateral than is available.
     */
    error InsufficientCollateral();

    /**
     *  @notice Lender is attempting to move or remove more collateral they have claim to in the bucket.
     *  @notice Lender is attempting to remove more collateral they have claim to in the bucket.
     *  @notice Lender must have enough `LP` to claim the desired amount of quote from the bucket.
     */
    error InsufficientLP();

    /**
     *  @notice Bucket must have more quote available in the bucket than the lender is attempting to claim.
     */
    error InsufficientLiquidity();

    /**
     *  @notice When increasing / decreasing `LP` allowances indexes and amounts arrays parameters should have same length.
     */
    error InvalidAllowancesInput();

    /**
     *  @notice When transferring `LP` between indices, the new index must be a valid index.
     */
    error InvalidIndex();

    /**
     *  @notice The amount used for performed action should be greater than `0`.
     */
    error InvalidAmount();

    /**
     *  @notice Borrower is attempting to borrow more quote token than is available before the supplied `limitIndex`.
     */
    error LimitIndexExceeded();

    /**
     *  @notice When moving quote token `HTP` must stay below `LUP`.
     *  @notice When removing quote token `HTP` must stay below `LUP`.
     */
    error LUPBelowHTP();

    /**
     *  @notice From index and to index arguments to move are the same.
     */
    error MoveToSameIndex();

    /**
     *  @notice Owner of the `LP` must have approved the new owner prior to transfer.
     */
    error NoAllowance();

    /**
     *  @notice Actor is attempting to take or clear an inactive auction.
     */
    error NoAuction();

    /**
     *  @notice No pool reserves are claimable.
     */
    error NoReserves();

    /**
     *  @notice Actor is attempting to take or clear an inactive reserves auction.
     */
    error NoReservesAuction();

    /**
     *  @notice Lender must have non-zero `LP` when attemptign to remove quote token from the pool.
     */
    error NoClaim();

    /**
     *  @notice Borrower has no debt to liquidate.
     *  @notice Borrower is attempting to repay when they have no outstanding debt.
     */
    error NoDebt();

    /**
     *  @notice Actor is attempting to kick with bucket price below the `LUP`.
     */
    error PriceBelowLUP();

    /**
     *  @notice Lender is attempting to remove quote tokens from a bucket that exists above active auction debt from top-of-book downward.
     */
    error RemoveDepositLockedByAuctionDebt();

    /**
     * @notice User attempted to kick off a new auction less than `2` weeks since the last auction completed.
     */
    error ReserveAuctionTooSoon();

    /**
     *  @notice Current block timestamp has reached or exceeded a user-provided expiration.
     */
    error TransactionExpired();

    /**
     *  @notice The address that transfer `LP` is not approved by the `LP` receiving address.
     */
    error TransferorNotApproved();

    /**
     *  @notice Owner of the `LP` attemps to transfer `LP` to same address.
     */
    error TransferToSameOwner();

    /**
     *  @notice The DebtToCollateral of the loan to be inserted in loans heap is zero.
     */
    error ZeroDebtToCollateral();

}

// SPDX-License-Identifier: MIT

pragma solidity 0.8.18;
import { IERC3156FlashBorrower } from "./IERC3156FlashBorrower.sol";


interface IERC3156FlashLender {

    /**
     * @dev    The amount of currency available to be lent.
     * @param  token_ The loan currency.
     * @return The amount of `token` that can be borrowed (token precision).
     */
    function maxFlashLoan(
        address token_
    ) external view returns (uint256);

    /**
     * @dev    The fee to be charged for a given loan.
     * @param  token_    The loan currency.
     * @param  amount_   The amount of tokens lent (token precision).
     * @return The amount of `token` to be charged for the loan (token precision), on top of the returned principal .
     */
    function flashFee(
        address token_,
        uint256 amount_
    ) external view returns (uint256);

    /**
     * @dev    Initiate a flash loan.
     * @param  receiver_ The receiver of the tokens in the loan, and the receiver of the callback.
     * @param  token_    The loan currency.
     * @param  amount_   The amount of tokens lent (token precision).
     * @param  data_     Arbitrary data structure, intended to contain user-defined parameters.
     * @return `True` when successful flashloan, `false` otherwise.
     */
    function flashLoan(
        IERC3156FlashBorrower receiver_,
        address token_,
        uint256 amount_,
        bytes   calldata data_
    ) external returns (bool);
}

{
  "remappings": [
    "@solmate/=lib/solmate/src/",
    "@std/=lib/forge-std/src/",
    "@clones/=lib/clones-with-immutable-args/src/",
    "@openzeppelin/contracts/=lib/openzeppelin-contracts/contracts/",
    "@prb-math/=lib/prb-math/",
    "@base64-sol/=lib/base64/",
    "src/=src/",
    "base64/=lib/base64/",
    "clones-with-immutable-args/=lib/clones-with-immutable-args/src/",
    "ds-test/=lib/clones-with-immutable-args/lib/ds-test/src/",
    "forge-std/=lib/forge-std/src/",
    "openzeppelin-contracts/=lib/openzeppelin-contracts/",
    "prb-math/=lib/prb-math/contracts/"
  ],
  "optimizer": {
    "enabled": true,
    "runs": 0
  },
  "metadata": {
    "useLiteralContent": false,
    "bytecodeHash": "ipfs",
    "appendCBOR": true
  },
  "outputSelection": {
    "*": {
      "*": [
        "evm.bytecode",
        "evm.deployedBytecode",
        "devdoc",
        "userdoc",
        "metadata",
        "abi"
      ]
    }
  },
  "evmVersion": "paris",
  "libraries": {
    "src/libraries/external/BorrowerActions.sol": {
      "BorrowerActions": "0x37ed1d5d903adddda4fcc6d003b840c883d05402"
    },
    "src/libraries/external/KickerActions.sol": {
      "KickerActions": "0xaf983b52aec0f6e127ddd51ce3367552d7916387"
    },
    "src/libraries/external/LPActions.sol": {
      "LPActions": "0xac8892dd81ee0fec9c11ecf6ef3bd1a773d003fb"
    },
    "src/libraries/external/LenderActions.sol": {
      "LenderActions": "0xdce7fd455e1a65b40186292657e6231f87d81c49"
    },
    "src/libraries/external/PoolCommons.sol": {
      "PoolCommons": "0xe88aaf46c9124b7b08c2dcc2505429ce72979648"
    },
    "src/libraries/external/PositionNFTSVG.sol": {
      "PositionNFTSVG": "0x83fcb77b91288173175a4ac70f848d57ff95a9dd"
    },
    "src/libraries/external/SettlerActions.sol": {
      "SettlerActions": "0x4418b6a45d785b85e87c022d99b0ff9e267268fe"
    },
    "src/libraries/external/TakerActions.sol": {
      "TakerActions": "0x9c09a67a622650037fe70f21a5f6770a363009e1"
    }
  }
}

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