// File: @openzeppelin/contracts/utils/math/Math.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

/**
 * @dev Standard math utilities missing in the Solidity language.
 */
library Math {
    /**
     * @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, so we distribute
        return (a / 2) + (b / 2) + ((a % 2 + b % 2) / 2);
    }
}

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



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

// CAUTION
// This version of SafeMath should only be used with Solidity 0.8 or later,
// because it relies on the compiler's built in overflow checks.

/**
 * @dev Wrappers over Solidity's arithmetic operations.
 *
 * NOTE: `SafeMath` is no longer needed starting with Solidity 0.8. The compiler
 * now has built in overflow checking.
 */
library SafeMath {
    /**
     * @dev Returns the addition of two unsigned integers, with an overflow flag.
     *
     * _Available since v3.4._
     */
    function tryAdd(uint256 a, uint256 b) internal pure returns (bool, uint256) {
        unchecked {
            uint256 c = a + b;
            if (c < a) return (false, 0);
            return (true, c);
        }
    }

    /**
     * @dev Returns the substraction of two unsigned integers, with an overflow flag.
     *
     * _Available since v3.4._
     */
    function trySub(uint256 a, uint256 b) internal pure returns (bool, uint256) {
        unchecked {
            if (b > a) return (false, 0);
            return (true, a - b);
        }
    }

    /**
     * @dev Returns the multiplication of two unsigned integers, with an overflow flag.
     *
     * _Available since v3.4._
     */
    function tryMul(uint256 a, uint256 b) internal pure returns (bool, uint256) {
        unchecked {
            // Gas optimization: this is cheaper than requiring 'a' not being zero, but the
            // benefit is lost if 'b' is also tested.
            // See: https://github.com/OpenZeppelin/openzeppelin-contracts/pull/522
            if (a == 0) return (true, 0);
            uint256 c = a * b;
            if (c / a != b) return (false, 0);
            return (true, c);
        }
    }

    /**
     * @dev Returns the division of two unsigned integers, with a division by zero flag.
     *
     * _Available since v3.4._
     */
    function tryDiv(uint256 a, uint256 b) internal pure returns (bool, uint256) {
        unchecked {
            if (b == 0) return (false, 0);
            return (true, a / b);
        }
    }

    /**
     * @dev Returns the remainder of dividing two unsigned integers, with a division by zero flag.
     *
     * _Available since v3.4._
     */
    function tryMod(uint256 a, uint256 b) internal pure returns (bool, uint256) {
        unchecked {
            if (b == 0) return (false, 0);
            return (true, a % b);
        }
    }

    /**
     * @dev Returns the addition of two unsigned integers, reverting on
     * overflow.
     *
     * Counterpart to Solidity's `+` operator.
     *
     * Requirements:
     *
     * - Addition cannot overflow.
     */
    function add(uint256 a, uint256 b) internal pure returns (uint256) {
        return a + b;
    }

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

    /**
     * @dev Returns the multiplication of two unsigned integers, reverting on
     * overflow.
     *
     * Counterpart to Solidity's `*` operator.
     *
     * Requirements:
     *
     * - Multiplication cannot overflow.
     */
    function mul(uint256 a, uint256 b) internal pure returns (uint256) {
        return a * b;
    }

    /**
     * @dev Returns the integer division of two unsigned integers, reverting on
     * division by zero. The result is rounded towards zero.
     *
     * Counterpart to Solidity's `/` operator.
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function div(uint256 a, uint256 b) internal pure returns (uint256) {
        return a / b;
    }

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

    /**
     * @dev Returns the subtraction of two unsigned integers, reverting with custom message on
     * overflow (when the result is negative).
     *
     * CAUTION: This function is deprecated because it requires allocating memory for the error
     * message unnecessarily. For custom revert reasons use {trySub}.
     *
     * Counterpart to Solidity's `-` operator.
     *
     * Requirements:
     *
     * - Subtraction cannot overflow.
     */
    function sub(uint256 a, uint256 b, string memory errorMessage) internal pure returns (uint256) {
        unchecked {
            require(b <= a, errorMessage);
            return a - b;
        }
    }

    /**
     * @dev Returns the integer division of two unsigned integers, reverting with custom message on
     * division by zero. The result is rounded towards zero.
     *
     * Counterpart to Solidity's `%` operator. This function uses a `revert`
     * opcode (which leaves remaining gas untouched) while Solidity uses an
     * invalid opcode to revert (consuming all remaining gas).
     *
     * Counterpart to Solidity's `/` operator. Note: this function uses a
     * `revert` opcode (which leaves remaining gas untouched) while Solidity
     * uses an invalid opcode to revert (consuming all remaining gas).
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function div(uint256 a, uint256 b, string memory errorMessage) internal pure returns (uint256) {
        unchecked {
            require(b > 0, errorMessage);
            return a / b;
        }
    }

    /**
     * @dev Returns the remainder of dividing two unsigned integers. (unsigned integer modulo),
     * reverting with custom message when dividing by zero.
     *
     * CAUTION: This function is deprecated because it requires allocating memory for the error
     * message unnecessarily. For custom revert reasons use {tryMod}.
     *
     * Counterpart to Solidity's `%` operator. This function uses a `revert`
     * opcode (which leaves remaining gas untouched) while Solidity uses an
     * invalid opcode to revert (consuming all remaining gas).
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function mod(uint256 a, uint256 b, string memory errorMessage) internal pure returns (uint256) {
        unchecked {
            require(b > 0, errorMessage);
            return a % b;
        }
    }
}

// File: @openzeppelin/contracts/utils/math/SignedSafeMath.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

/**
 * @dev Wrappers over Solidity's arithmetic operations.
 *
 * NOTE: `SignedSafeMath` is no longer needed starting with Solidity 0.8. The compiler
 * now has built in overflow checking.
 */
library SignedSafeMath {
    /**
     * @dev Returns the multiplication of two signed integers, reverting on
     * overflow.
     *
     * Counterpart to Solidity's `*` operator.
     *
     * Requirements:
     *
     * - Multiplication cannot overflow.
     */
    function mul(int256 a, int256 b) internal pure returns (int256) {
        return a * b;
    }

    /**
     * @dev Returns the integer division of two signed integers. Reverts on
     * division by zero. The result is rounded towards zero.
     *
     * Counterpart to Solidity's `/` operator.
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function div(int256 a, int256 b) internal pure returns (int256) {
        return a / b;
    }

    /**
     * @dev Returns the subtraction of two signed integers, reverting on
     * overflow.
     *
     * Counterpart to Solidity's `-` operator.
     *
     * Requirements:
     *
     * - Subtraction cannot overflow.
     */
    function sub(int256 a, int256 b) internal pure returns (int256) {
        return a - b;
    }

    /**
     * @dev Returns the addition of two signed integers, reverting on
     * overflow.
     *
     * Counterpart to Solidity's `+` operator.
     *
     * Requirements:
     *
     * - Addition cannot overflow.
     */
    function add(int256 a, int256 b) internal pure returns (int256) {
        return a + b;
    }
}

// File: contracts/common/implementation/FixedPoint.sol



// SPDX-License-Identifier: AGPL-3.0-only
pragma solidity ^0.8.0;

import "@openzeppelin/contracts/utils/math/SafeMath.sol";
import "@openzeppelin/contracts/utils/math/SignedSafeMath.sol";

/**
 * @title Library for fixed point arithmetic on uints
 */
library FixedPoint {
    using SafeMath for uint256;
    using SignedSafeMath for int256;

    // Supports 18 decimals. E.g., 1e18 represents "1", 5e17 represents "0.5".
    // For unsigned values:
    //   This can represent a value up to (2^256 - 1)/10^18 = ~10^59. 10^59 will be stored internally as uint256 10^77.
    uint256 private constant FP_SCALING_FACTOR = 10**18;

    // --------------------------------------- UNSIGNED -----------------------------------------------------------------------------
    struct Unsigned {
        uint256 rawValue;
    }

    /**
     * @notice Constructs an `Unsigned` from an unscaled uint, e.g., `b=5` gets stored internally as `5*(10**18)`.
     * @param a uint to convert into a FixedPoint.
     * @return the converted FixedPoint.
     */
    function fromUnscaledUint(uint256 a) internal pure returns (Unsigned memory) {
        return Unsigned(a.mul(FP_SCALING_FACTOR));
    }

    /**
     * @notice Whether `a` is equal to `b`.
     * @param a a FixedPoint.
     * @param b a uint256.
     * @return True if equal, or False.
     */
    function isEqual(Unsigned memory a, uint256 b) internal pure returns (bool) {
        return a.rawValue == fromUnscaledUint(b).rawValue;
    }

    /**
     * @notice Whether `a` is equal to `b`.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return True if equal, or False.
     */
    function isEqual(Unsigned memory a, Unsigned memory b) internal pure returns (bool) {
        return a.rawValue == b.rawValue;
    }

    /**
     * @notice Whether `a` is greater than `b`.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return True if `a > b`, or False.
     */
    function isGreaterThan(Unsigned memory a, Unsigned memory b) internal pure returns (bool) {
        return a.rawValue > b.rawValue;
    }

    /**
     * @notice Whether `a` is greater than `b`.
     * @param a a FixedPoint.
     * @param b a uint256.
     * @return True if `a > b`, or False.
     */
    function isGreaterThan(Unsigned memory a, uint256 b) internal pure returns (bool) {
        return a.rawValue > fromUnscaledUint(b).rawValue;
    }

    /**
     * @notice Whether `a` is greater than `b`.
     * @param a a uint256.
     * @param b a FixedPoint.
     * @return True if `a > b`, or False.
     */
    function isGreaterThan(uint256 a, Unsigned memory b) internal pure returns (bool) {
        return fromUnscaledUint(a).rawValue > b.rawValue;
    }

    /**
     * @notice Whether `a` is greater than or equal to `b`.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return True if `a >= b`, or False.
     */
    function isGreaterThanOrEqual(Unsigned memory a, Unsigned memory b) internal pure returns (bool) {
        return a.rawValue >= b.rawValue;
    }

    /**
     * @notice Whether `a` is greater than or equal to `b`.
     * @param a a FixedPoint.
     * @param b a uint256.
     * @return True if `a >= b`, or False.
     */
    function isGreaterThanOrEqual(Unsigned memory a, uint256 b) internal pure returns (bool) {
        return a.rawValue >= fromUnscaledUint(b).rawValue;
    }

    /**
     * @notice Whether `a` is greater than or equal to `b`.
     * @param a a uint256.
     * @param b a FixedPoint.
     * @return True if `a >= b`, or False.
     */
    function isGreaterThanOrEqual(uint256 a, Unsigned memory b) internal pure returns (bool) {
        return fromUnscaledUint(a).rawValue >= b.rawValue;
    }

    /**
     * @notice Whether `a` is less than `b`.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return True if `a < b`, or False.
     */
    function isLessThan(Unsigned memory a, Unsigned memory b) internal pure returns (bool) {
        return a.rawValue < b.rawValue;
    }

    /**
     * @notice Whether `a` is less than `b`.
     * @param a a FixedPoint.
     * @param b a uint256.
     * @return True if `a < b`, or False.
     */
    function isLessThan(Unsigned memory a, uint256 b) internal pure returns (bool) {
        return a.rawValue < fromUnscaledUint(b).rawValue;
    }

    /**
     * @notice Whether `a` is less than `b`.
     * @param a a uint256.
     * @param b a FixedPoint.
     * @return True if `a < b`, or False.
     */
    function isLessThan(uint256 a, Unsigned memory b) internal pure returns (bool) {
        return fromUnscaledUint(a).rawValue < b.rawValue;
    }

    /**
     * @notice Whether `a` is less than or equal to `b`.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return True if `a <= b`, or False.
     */
    function isLessThanOrEqual(Unsigned memory a, Unsigned memory b) internal pure returns (bool) {
        return a.rawValue <= b.rawValue;
    }

    /**
     * @notice Whether `a` is less than or equal to `b`.
     * @param a a FixedPoint.
     * @param b a uint256.
     * @return True if `a <= b`, or False.
     */
    function isLessThanOrEqual(Unsigned memory a, uint256 b) internal pure returns (bool) {
        return a.rawValue <= fromUnscaledUint(b).rawValue;
    }

    /**
     * @notice Whether `a` is less than or equal to `b`.
     * @param a a uint256.
     * @param b a FixedPoint.
     * @return True if `a <= b`, or False.
     */
    function isLessThanOrEqual(uint256 a, Unsigned memory b) internal pure returns (bool) {
        return fromUnscaledUint(a).rawValue <= b.rawValue;
    }

    /**
     * @notice The minimum of `a` and `b`.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return the minimum of `a` and `b`.
     */
    function min(Unsigned memory a, Unsigned memory b) internal pure returns (Unsigned memory) {
        return a.rawValue < b.rawValue ? a : b;
    }

    /**
     * @notice The maximum of `a` and `b`.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return the maximum of `a` and `b`.
     */
    function max(Unsigned memory a, Unsigned memory b) internal pure returns (Unsigned memory) {
        return a.rawValue > b.rawValue ? a : b;
    }

    /**
     * @notice Adds two `Unsigned`s, reverting on overflow.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return the sum of `a` and `b`.
     */
    function add(Unsigned memory a, Unsigned memory b) internal pure returns (Unsigned memory) {
        return Unsigned(a.rawValue.add(b.rawValue));
    }

    /**
     * @notice Adds an `Unsigned` to an unscaled uint, reverting on overflow.
     * @param a a FixedPoint.
     * @param b a uint256.
     * @return the sum of `a` and `b`.
     */
    function add(Unsigned memory a, uint256 b) internal pure returns (Unsigned memory) {
        return add(a, fromUnscaledUint(b));
    }

    /**
     * @notice Subtracts two `Unsigned`s, reverting on overflow.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return the difference of `a` and `b`.
     */
    function sub(Unsigned memory a, Unsigned memory b) internal pure returns (Unsigned memory) {
        return Unsigned(a.rawValue.sub(b.rawValue));
    }

    /**
     * @notice Subtracts an unscaled uint256 from an `Unsigned`, reverting on overflow.
     * @param a a FixedPoint.
     * @param b a uint256.
     * @return the difference of `a` and `b`.
     */
    function sub(Unsigned memory a, uint256 b) internal pure returns (Unsigned memory) {
        return sub(a, fromUnscaledUint(b));
    }

    /**
     * @notice Subtracts an `Unsigned` from an unscaled uint256, reverting on overflow.
     * @param a a uint256.
     * @param b a FixedPoint.
     * @return the difference of `a` and `b`.
     */
    function sub(uint256 a, Unsigned memory b) internal pure returns (Unsigned memory) {
        return sub(fromUnscaledUint(a), b);
    }

    /**
     * @notice Multiplies two `Unsigned`s, reverting on overflow.
     * @dev This will "floor" the product.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return the product of `a` and `b`.
     */
    function mul(Unsigned memory a, Unsigned memory b) internal pure returns (Unsigned memory) {
        // There are two caveats with this computation:
        // 1. Max output for the represented number is ~10^41, otherwise an intermediate value overflows. 10^41 is
        // stored internally as a uint256 ~10^59.
        // 2. Results that can't be represented exactly are truncated not rounded. E.g., 1.4 * 2e-18 = 2.8e-18, which
        // would round to 3, but this computation produces the result 2.
        // No need to use SafeMath because FP_SCALING_FACTOR != 0.
        return Unsigned(a.rawValue.mul(b.rawValue) / FP_SCALING_FACTOR);
    }

    /**
     * @notice Multiplies an `Unsigned` and an unscaled uint256, reverting on overflow.
     * @dev This will "floor" the product.
     * @param a a FixedPoint.
     * @param b a uint256.
     * @return the product of `a` and `b`.
     */
    function mul(Unsigned memory a, uint256 b) internal pure returns (Unsigned memory) {
        return Unsigned(a.rawValue.mul(b));
    }

    /**
     * @notice Multiplies two `Unsigned`s and "ceil's" the product, reverting on overflow.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return the product of `a` and `b`.
     */
    function mulCeil(Unsigned memory a, Unsigned memory b) internal pure returns (Unsigned memory) {
        uint256 mulRaw = a.rawValue.mul(b.rawValue);
        uint256 mulFloor = mulRaw / FP_SCALING_FACTOR;
        uint256 mod = mulRaw.mod(FP_SCALING_FACTOR);
        if (mod != 0) {
            return Unsigned(mulFloor.add(1));
        } else {
            return Unsigned(mulFloor);
        }
    }

    /**
     * @notice Multiplies an `Unsigned` and an unscaled uint256 and "ceil's" the product, reverting on overflow.
     * @param a a FixedPoint.
     * @param b a FixedPoint.
     * @return the product of `a` and `b`.
     */
    function mulCeil(Unsigned memory a, uint256 b) internal pure returns (Unsigned memory) {
        // Since b is an int, there is no risk of truncation and we can just mul it normally
        return Unsigned(a.rawValue.mul(b));
    }

    /**
     * @notice Divides one `Unsigned` by an `Unsigned`, reverting on overflow or division by 0.
     * @dev This will "floor" the quotient.
     * @param a a FixedPoint numerator.
     * @param b a FixedPoint denominator.
     * @return the quotient of `a` divided by `b`.
     */
    function div(Unsigned memory a, Unsigned memory b) internal pure returns (Unsigned memory) {
        // There are two caveats with this computation:
        // 1. Max value for the number dividend `a` represents is ~10^41, otherwise an intermediate value overflows.
        // 10^41 is stored internally as a uint256 10^59.
        // 2. Results that can't be represented exactly are truncated not rounded. E.g., 2 / 3 = 0.6 repeating, which
        // would round to 0.666666666666666667, but this computation produces the result 0.666666666666666666.
        return Unsigned(a.rawValue.mul(FP_SCALING_FACTOR).div(b.rawValue));
    }

    /**
     * @notice Divides one `Unsigned` by an unscaled uint256, reverting on overflow or division by 0.
     * @dev This will "floor" the quotient.
     * @param a a FixedPoint numerator.
     * @param b a uint256 denominator.
     * @return the quotient of `a` divided by `b`.
     */
    function div(Unsigned memory a, uint256 b) internal pure returns (Unsigned memory) {
        return Unsigned(a.rawValue.div(b));
    }

    /**
     * @notice Divides one unscaled uint256 by an `Unsigned`, reverting on overflow or division by 0.
     * @dev This will "floor" the quotient.
     * @param a a uint256 numerator.
     * @param b a FixedPoint denominator.
     * @return the quotient of `a` divided by `b`.
     */
    function div(uint256 a, Unsigned memory b) internal pure returns (Unsigned memory) {
        return div(fromUnscaledUint(a), b);
    }

    /**
     * @notice Divides one `Unsigned` by an `Unsigned` and "ceil's" the quotient, reverting on overflow or division by 0.
     * @param a a FixedPoint numerator.
     * @param b a FixedPoint denominator.
     * @return the quotient of `a` divided by `b`.
     */
    function divCeil(Unsigned memory a, Unsigned memory b) internal pure returns (Unsigned memory) {
        uint256 aScaled = a.rawValue.mul(FP_SCALING_FACTOR);
        uint256 divFloor = aScaled.div(b.rawValue);
        uint256 mod = aScaled.mod(b.rawValue);
        if (mod != 0) {
            return Unsigned(divFloor.add(1));
        } else {
            return Unsigned(divFloor);
        }
    }

    /**
     * @notice Divides one `Unsigned` by an unscaled uint256 and "ceil's" the quotient, reverting on overflow or division by 0.
     * @param a a FixedPoint numerator.
     * @param b a uint256 denominator.
     * @return the quotient of `a` divided by `b`.
     */
    function divCeil(Unsigned memory a, uint256 b) internal pure returns (Unsigned memory) {
        // Because it is possible that a quotient gets truncated, we can't just call "Unsigned(a.rawValue.div(b))"
        // similarly to mulCeil with a uint256 as the second parameter. Therefore we need to convert b into an Unsigned.
        // This creates the possibility of overflow if b is very large.
        return divCeil(a, fromUnscaledUint(b));
    }

    /**
     * @notice Raises an `Unsigned` to the power of an unscaled uint256, reverting on overflow. E.g., `b=2` squares `a`.
     * @dev This will "floor" the result.
     * @param a a FixedPoint numerator.
     * @param b a uint256 denominator.
     * @return output is `a` to the power of `b`.
     */
    function pow(Unsigned memory a, uint256 b) internal pure returns (Unsigned memory output) {
        output = fromUnscaledUint(1);
        for (uint256 i = 0; i < b; i = i.add(1)) {
            output = mul(output, a);
        }
    }

    // ------------------------------------------------- SIGNED -------------------------------------------------------------
    // Supports 18 decimals. E.g., 1e18 represents "1", 5e17 represents "0.5".
    // For signed values:
    //   This can represent a value up (or down) to +-(2^255 - 1)/10^18 = ~10^58. 10^58 will be stored internally as int256 10^76.
    int256 private constant SFP_SCALING_FACTOR = 10**18;

    struct Signed {
        int256 rawValue;
    }

    function fromSigned(Signed memory a) internal pure returns (Unsigned memory) {
        require(a.rawValue >= 0, "Negative value provided");
        return Unsigned(uint256(a.rawValue));
    }

    function fromUnsigned(Unsigned memory a) internal pure returns (Signed memory) {
        require(a.rawValue <= uint256(type(int256).max), "Unsigned too large");
        return Signed(int256(a.rawValue));
    }

    /**
     * @notice Constructs a `Signed` from an unscaled int, e.g., `b=5` gets stored internally as `5*(10**18)`.
     * @param a int to convert into a FixedPoint.Signed.
     * @return the converted FixedPoint.Signed.
     */
    function fromUnscaledInt(int256 a) internal pure returns (Signed memory) {
        return Signed(a.mul(SFP_SCALING_FACTOR));
    }

    /**
     * @notice Whether `a` is equal to `b`.
     * @param a a FixedPoint.Signed.
     * @param b a int256.
     * @return True if equal, or False.
     */
    function isEqual(Signed memory a, int256 b) internal pure returns (bool) {
        return a.rawValue == fromUnscaledInt(b).rawValue;
    }

    /**
     * @notice Whether `a` is equal to `b`.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return True if equal, or False.
     */
    function isEqual(Signed memory a, Signed memory b) internal pure returns (bool) {
        return a.rawValue == b.rawValue;
    }

    /**
     * @notice Whether `a` is greater than `b`.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return True if `a > b`, or False.
     */
    function isGreaterThan(Signed memory a, Signed memory b) internal pure returns (bool) {
        return a.rawValue > b.rawValue;
    }

    /**
     * @notice Whether `a` is greater than `b`.
     * @param a a FixedPoint.Signed.
     * @param b an int256.
     * @return True if `a > b`, or False.
     */
    function isGreaterThan(Signed memory a, int256 b) internal pure returns (bool) {
        return a.rawValue > fromUnscaledInt(b).rawValue;
    }

    /**
     * @notice Whether `a` is greater than `b`.
     * @param a an int256.
     * @param b a FixedPoint.Signed.
     * @return True if `a > b`, or False.
     */
    function isGreaterThan(int256 a, Signed memory b) internal pure returns (bool) {
        return fromUnscaledInt(a).rawValue > b.rawValue;
    }

    /**
     * @notice Whether `a` is greater than or equal to `b`.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return True if `a >= b`, or False.
     */
    function isGreaterThanOrEqual(Signed memory a, Signed memory b) internal pure returns (bool) {
        return a.rawValue >= b.rawValue;
    }

    /**
     * @notice Whether `a` is greater than or equal to `b`.
     * @param a a FixedPoint.Signed.
     * @param b an int256.
     * @return True if `a >= b`, or False.
     */
    function isGreaterThanOrEqual(Signed memory a, int256 b) internal pure returns (bool) {
        return a.rawValue >= fromUnscaledInt(b).rawValue;
    }

    /**
     * @notice Whether `a` is greater than or equal to `b`.
     * @param a an int256.
     * @param b a FixedPoint.Signed.
     * @return True if `a >= b`, or False.
     */
    function isGreaterThanOrEqual(int256 a, Signed memory b) internal pure returns (bool) {
        return fromUnscaledInt(a).rawValue >= b.rawValue;
    }

    /**
     * @notice Whether `a` is less than `b`.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return True if `a < b`, or False.
     */
    function isLessThan(Signed memory a, Signed memory b) internal pure returns (bool) {
        return a.rawValue < b.rawValue;
    }

    /**
     * @notice Whether `a` is less than `b`.
     * @param a a FixedPoint.Signed.
     * @param b an int256.
     * @return True if `a < b`, or False.
     */
    function isLessThan(Signed memory a, int256 b) internal pure returns (bool) {
        return a.rawValue < fromUnscaledInt(b).rawValue;
    }

    /**
     * @notice Whether `a` is less than `b`.
     * @param a an int256.
     * @param b a FixedPoint.Signed.
     * @return True if `a < b`, or False.
     */
    function isLessThan(int256 a, Signed memory b) internal pure returns (bool) {
        return fromUnscaledInt(a).rawValue < b.rawValue;
    }

    /**
     * @notice Whether `a` is less than or equal to `b`.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return True if `a <= b`, or False.
     */
    function isLessThanOrEqual(Signed memory a, Signed memory b) internal pure returns (bool) {
        return a.rawValue <= b.rawValue;
    }

    /**
     * @notice Whether `a` is less than or equal to `b`.
     * @param a a FixedPoint.Signed.
     * @param b an int256.
     * @return True if `a <= b`, or False.
     */
    function isLessThanOrEqual(Signed memory a, int256 b) internal pure returns (bool) {
        return a.rawValue <= fromUnscaledInt(b).rawValue;
    }

    /**
     * @notice Whether `a` is less than or equal to `b`.
     * @param a an int256.
     * @param b a FixedPoint.Signed.
     * @return True if `a <= b`, or False.
     */
    function isLessThanOrEqual(int256 a, Signed memory b) internal pure returns (bool) {
        return fromUnscaledInt(a).rawValue <= b.rawValue;
    }

    /**
     * @notice The minimum of `a` and `b`.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return the minimum of `a` and `b`.
     */
    function min(Signed memory a, Signed memory b) internal pure returns (Signed memory) {
        return a.rawValue < b.rawValue ? a : b;
    }

    /**
     * @notice The maximum of `a` and `b`.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return the maximum of `a` and `b`.
     */
    function max(Signed memory a, Signed memory b) internal pure returns (Signed memory) {
        return a.rawValue > b.rawValue ? a : b;
    }

    /**
     * @notice Adds two `Signed`s, reverting on overflow.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return the sum of `a` and `b`.
     */
    function add(Signed memory a, Signed memory b) internal pure returns (Signed memory) {
        return Signed(a.rawValue.add(b.rawValue));
    }

    /**
     * @notice Adds an `Signed` to an unscaled int, reverting on overflow.
     * @param a a FixedPoint.Signed.
     * @param b an int256.
     * @return the sum of `a` and `b`.
     */
    function add(Signed memory a, int256 b) internal pure returns (Signed memory) {
        return add(a, fromUnscaledInt(b));
    }

    /**
     * @notice Subtracts two `Signed`s, reverting on overflow.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return the difference of `a` and `b`.
     */
    function sub(Signed memory a, Signed memory b) internal pure returns (Signed memory) {
        return Signed(a.rawValue.sub(b.rawValue));
    }

    /**
     * @notice Subtracts an unscaled int256 from an `Signed`, reverting on overflow.
     * @param a a FixedPoint.Signed.
     * @param b an int256.
     * @return the difference of `a` and `b`.
     */
    function sub(Signed memory a, int256 b) internal pure returns (Signed memory) {
        return sub(a, fromUnscaledInt(b));
    }

    /**
     * @notice Subtracts an `Signed` from an unscaled int256, reverting on overflow.
     * @param a an int256.
     * @param b a FixedPoint.Signed.
     * @return the difference of `a` and `b`.
     */
    function sub(int256 a, Signed memory b) internal pure returns (Signed memory) {
        return sub(fromUnscaledInt(a), b);
    }

    /**
     * @notice Multiplies two `Signed`s, reverting on overflow.
     * @dev This will "floor" the product.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return the product of `a` and `b`.
     */
    function mul(Signed memory a, Signed memory b) internal pure returns (Signed memory) {
        // There are two caveats with this computation:
        // 1. Max output for the represented number is ~10^41, otherwise an intermediate value overflows. 10^41 is
        // stored internally as an int256 ~10^59.
        // 2. Results that can't be represented exactly are truncated not rounded. E.g., 1.4 * 2e-18 = 2.8e-18, which
        // would round to 3, but this computation produces the result 2.
        // No need to use SafeMath because SFP_SCALING_FACTOR != 0.
        return Signed(a.rawValue.mul(b.rawValue) / SFP_SCALING_FACTOR);
    }

    /**
     * @notice Multiplies an `Signed` and an unscaled int256, reverting on overflow.
     * @dev This will "floor" the product.
     * @param a a FixedPoint.Signed.
     * @param b an int256.
     * @return the product of `a` and `b`.
     */
    function mul(Signed memory a, int256 b) internal pure returns (Signed memory) {
        return Signed(a.rawValue.mul(b));
    }

    /**
     * @notice Multiplies two `Signed`s and "ceil's" the product, reverting on overflow.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return the product of `a` and `b`.
     */
    function mulAwayFromZero(Signed memory a, Signed memory b) internal pure returns (Signed memory) {
        int256 mulRaw = a.rawValue.mul(b.rawValue);
        int256 mulTowardsZero = mulRaw / SFP_SCALING_FACTOR;
        // Manual mod because SignedSafeMath doesn't support it.
        int256 mod = mulRaw % SFP_SCALING_FACTOR;
        if (mod != 0) {
            bool isResultPositive = isLessThan(a, 0) == isLessThan(b, 0);
            int256 valueToAdd = isResultPositive ? int256(1) : int256(-1);
            return Signed(mulTowardsZero.add(valueToAdd));
        } else {
            return Signed(mulTowardsZero);
        }
    }

    /**
     * @notice Multiplies an `Signed` and an unscaled int256 and "ceil's" the product, reverting on overflow.
     * @param a a FixedPoint.Signed.
     * @param b a FixedPoint.Signed.
     * @return the product of `a` and `b`.
     */
    function mulAwayFromZero(Signed memory a, int256 b) internal pure returns (Signed memory) {
        // Since b is an int, there is no risk of truncation and we can just mul it normally
        return Signed(a.rawValue.mul(b));
    }

    /**
     * @notice Divides one `Signed` by an `Signed`, reverting on overflow or division by 0.
     * @dev This will "floor" the quotient.
     * @param a a FixedPoint numerator.
     * @param b a FixedPoint denominator.
     * @return the quotient of `a` divided by `b`.
     */
    function div(Signed memory a, Signed memory b) internal pure returns (Signed memory) {
        // There are two caveats with this computation:
        // 1. Max value for the number dividend `a` represents is ~10^41, otherwise an intermediate value overflows.
        // 10^41 is stored internally as an int256 10^59.
        // 2. Results that can't be represented exactly are truncated not rounded. E.g., 2 / 3 = 0.6 repeating, which
        // would round to 0.666666666666666667, but this computation produces the result 0.666666666666666666.
        return Signed(a.rawValue.mul(SFP_SCALING_FACTOR).div(b.rawValue));
    }

    /**
     * @notice Divides one `Signed` by an unscaled int256, reverting on overflow or division by 0.
     * @dev This will "floor" the quotient.
     * @param a a FixedPoint numerator.
     * @param b an int256 denominator.
     * @return the quotient of `a` divided by `b`.
     */
    function div(Signed memory a, int256 b) internal pure returns (Signed memory) {
        return Signed(a.rawValue.div(b));
    }

    /**
     * @notice Divides one unscaled int256 by an `Signed`, reverting on overflow or division by 0.
     * @dev This will "floor" the quotient.
     * @param a an int256 numerator.
     * @param b a FixedPoint denominator.
     * @return the quotient of `a` divided by `b`.
     */
    function div(int256 a, Signed memory b) internal pure returns (Signed memory) {
        return div(fromUnscaledInt(a), b);
    }

    /**
     * @notice Divides one `Signed` by an `Signed` and "ceil's" the quotient, reverting on overflow or division by 0.
     * @param a a FixedPoint numerator.
     * @param b a FixedPoint denominator.
     * @return the quotient of `a` divided by `b`.
     */
    function divAwayFromZero(Signed memory a, Signed memory b) internal pure returns (Signed memory) {
        int256 aScaled = a.rawValue.mul(SFP_SCALING_FACTOR);
        int256 divTowardsZero = aScaled.div(b.rawValue);
        // Manual mod because SignedSafeMath doesn't support it.
        int256 mod = aScaled % b.rawValue;
        if (mod != 0) {
            bool isResultPositive = isLessThan(a, 0) == isLessThan(b, 0);
            int256 valueToAdd = isResultPositive ? int256(1) : int256(-1);
            return Signed(divTowardsZero.add(valueToAdd));
        } else {
            return Signed(divTowardsZero);
        }
    }

    /**
     * @notice Divides one `Signed` by an unscaled int256 and "ceil's" the quotient, reverting on overflow or division by 0.
     * @param a a FixedPoint numerator.
     * @param b an int256 denominator.
     * @return the quotient of `a` divided by `b`.
     */
    function divAwayFromZero(Signed memory a, int256 b) internal pure returns (Signed memory) {
        // Because it is possible that a quotient gets truncated, we can't just call "Signed(a.rawValue.div(b))"
        // similarly to mulCeil with an int256 as the second parameter. Therefore we need to convert b into an Signed.
        // This creates the possibility of overflow if b is very large.
        return divAwayFromZero(a, fromUnscaledInt(b));
    }

    /**
     * @notice Raises an `Signed` to the power of an unscaled uint256, reverting on overflow. E.g., `b=2` squares `a`.
     * @dev This will "floor" the result.
     * @param a a FixedPoint.Signed.
     * @param b a uint256 (negative exponents are not allowed).
     * @return output is `a` to the power of `b`.
     */
    function pow(Signed memory a, uint256 b) internal pure returns (Signed memory output) {
        output = fromUnscaledInt(1);
        for (uint256 i = 0; i < b; i = i.add(1)) {
            output = mul(output, a);
        }
    }
}

// File: contracts/common/implementation/Lockable.sol



// SPDX-License-Identifier: AGPL-3.0-only
pragma solidity ^0.8.0;

/**
 * @title A contract that provides modifiers to prevent reentrancy to state-changing and view-only methods. This contract
 * is inspired by https://github.com/OpenZeppelin/openzeppelin-contracts/blob/master/contracts/utils/ReentrancyGuard.sol
 * and https://github.com/balancer-labs/balancer-core/blob/master/contracts/BPool.sol.
 */
contract Lockable {
    bool private _notEntered;

    constructor() {
        // Storing an initial non-zero value makes deployment a bit more expensive, but in exchange the refund on every
        // call to nonReentrant will be lower in amount. Since refunds are capped to a percentage of the total
        // transaction's gas, it is best to keep them low in cases like this one, to increase the likelihood of the full
        // refund coming into effect.
        _notEntered = true;
    }

    /**
     * @dev Prevents a contract from calling itself, directly or indirectly.
     * Calling a `nonReentrant` function from another `nonReentrant` function is not supported. It is possible to
     * prevent this from happening by making the `nonReentrant` function external, and making it call a `private`
     * function that does the actual state modification.
     */
    modifier nonReentrant() {
        _preEntranceCheck();
        _preEntranceSet();
        _;
        _postEntranceReset();
    }

    /**
     * @dev Designed to prevent a view-only method from being re-entered during a call to a `nonReentrant()` state-changing method.
     */
    modifier nonReentrantView() {
        _preEntranceCheck();
        _;
    }

    // Internal methods are used to avoid copying the require statement's bytecode to every `nonReentrant()` method.
    // On entry into a function, `_preEntranceCheck()` should always be called to check if the function is being
    // re-entered. Then, if the function modifies state, it should call `_postEntranceSet()`, perform its logic, and
    // then call `_postEntranceReset()`.
    // View-only methods can simply call `_preEntranceCheck()` to make sure that it is not being re-entered.
    function _preEntranceCheck() internal view {
        // On the first call to nonReentrant, _notEntered will be true
        require(_notEntered, "ReentrancyGuard: reentrant call");
    }

    function _preEntranceSet() internal {
        // Any calls to nonReentrant after this point will fail
        _notEntered = false;
    }

    function _postEntranceReset() internal {
        // By storing the original value once again, a refund is triggered (see
        // https://eips.ethereum.org/EIPS/eip-2200)
        _notEntered = true;
    }
}

// File: contracts/financial-templates/common/financial-product-libraries/long-short-pair-libraries/CappedYieldDollarLongShortPairFinancialProductLibrary.sol



// SPDX-License-Identifier: AGPL-3.0-only
pragma solidity ^0.8.0;

import "@openzeppelin/contracts/utils/math/SignedSafeMath.sol";
import "@openzeppelin/contracts/utils/math/Math.sol";

import "./LongShortPairFinancialProductLibrary.sol";
import "../../../../common/implementation/Lockable.sol";

/**
 * @title Capped Yield Dollar Long Short Pair Financial Product Library
 * @notice Adds settlement logic to create Yield Dollar LSPs. A range bond is the combination of a Yield dollar and short
 * put option enabling the token sponsor to issue structured products to unlock DeFi treasuries. This library is like
 * a Range Bond, but with no embedded call option.
 * A Capped Yield Dollar is defined as = Yield Dollar - Put Option. In order for the Capped Yield Dollar to be fully
 * collateralized and non-liquidatable, there is a low price for the collateral token below which the Capped Yield Dollar
 * will be worth < $1.
 * Numerically this is found using:
 * N = Notional of bond
 * P = price of token
 * T = number of tokens
 * R1 = low price range
 * C = collateral per pair, should be N/R1
 * T = min(1,(R1/P)*C)
 * If you want a yield dollar denominated as N = $1, you should set C to 1/R1. In that case, T = min(1,1/P).
 * - At any price below the low price range (R1) the long side effectively holds a fixed number of collateral equal to
 * collateralPerPair from the LSP with the value of expiryPercentLong = 1. This is the max payout in collateral.
 * - Any price equal to or above R1 gives a payout equivalent to a yield dollar (bond) of notional N. In this range the
 * expiryPercentLong shifts to keep the payout in dollar terms equal to the bond notional.
 * With this equation, the contract deployer does not need to specify the bond notional N. The notional can be calculated
 * by taking R1*collateralPerPair from the LSP.
 */
contract CappedYieldDollarLongShortPairFinancialProductLibrary is LongShortPairFinancialProductLibrary, Lockable {
    using FixedPoint for FixedPoint.Unsigned;
    using SignedSafeMath for int256;

    mapping(address => uint256) public lowPriceRanges;

    /**
     * @notice Enables any address to set the low price range for an associated financial product.
     * @param longShortPair address of the LSP contract.
     * @param lowPriceRange low price range below which the payout transforms from a yield dollar to a short put option.
     * @dev above the lowPriceRange the contract will payout a fixed amount of
     * lowPriceRange*collateralPerPair (i.e the "notional" of the yield dollar).
     * @dev Note: a) Any address can set these parameters b) existing LSP parameters for address not set.
     * c) low price range can only be set once to prevent the deployer from changing the parameters after the fact.
     * d) For safety, a low price range should be set before depositing any synthetic tokens in a liquidity pool.
     * e) longShortPair must expose an expirationTimestamp method to validate it is correctly deployed.
     */
    function setLongShortPairParameters(address longShortPair, uint256 lowPriceRange) public nonReentrant() {
        require(ExpiringContractInterface(longShortPair).expirationTimestamp() != 0, "Invalid LSP address");
        require(lowPriceRanges[longShortPair] == 0, "Parameters already set");

        lowPriceRanges[longShortPair] = lowPriceRange;
    }

    /**
     * @notice Returns a number between 0 and 1e18 to indicate how much collateral each long and short token are
     * entitled to per collateralPerPair.
     * @param expiryPrice price from the optimistic oracle for the LSP price identifier.
     * @return expiryPercentLong to indicate how much collateral should be sent between long and short tokens.
     */
    function percentageLongCollateralAtExpiry(int256 expiryPrice)
        public
        view
        override
        nonReentrantView()
        returns (uint256)
    {
        uint256 contractLowPriceRange = lowPriceRanges[msg.sender];
        require(contractLowPriceRange != 0, "Params not set for calling LSP");

        // This function returns a value between 0 and 1e18 to be used in conjunction with the LSP collateralPerPair
        // that allocates collateral between the short and long tokens on expiry. This can be simplified by considering
        // the price in two discrete ranges: 1) below the low price range, 2) above the low price range.
        uint256 positiveExpiryPrice = expiryPrice > 0 ? uint256(expiryPrice) : 0;

        // For expiry prices below lower bound, return expiryPercentLong = 1 (full position)
        if (positiveExpiryPrice <= contractLowPriceRange) return FixedPoint.fromUnscaledUint(1).rawValue;
        /**
         * For expiry prices above lower bound. For example, if the lower bound of Sushi is $4, collateral per pair
         * is 0.25, and the expiry price is $12, the payout will be (4/12)*0.25, or .08333 Sushi. With Sushi at $12,
         * .08333 Sushi is equal to $1.
         */
        return FixedPoint.Unsigned(contractLowPriceRange).div(FixedPoint.Unsigned(positiveExpiryPrice)).rawValue;
    }
}

// File: contracts/financial-templates/common/financial-product-libraries/long-short-pair-libraries/LongShortPairFinancialProductLibrary.sol



// SPDX-License-Identifier: AGPL-3.0-only
pragma solidity ^0.8.0;
import "../../../../common/implementation/FixedPoint.sol";
import "@openzeppelin/contracts/utils/math/SafeMath.sol";

interface ExpiringContractInterface {
    function expirationTimestamp() external view returns (uint256);
}

abstract contract LongShortPairFinancialProductLibrary {
    function percentageLongCollateralAtExpiry(int256 expiryPrice) public view virtual returns (uint256);
}