// SPDX-License-Identifier: MIT

pragma solidity 0.6.12;



// Part: ABDKMath64x64

/**
 * Smart contract library of mathematical functions operating with signed
 * 64.64-bit fixed point numbers.  Signed 64.64-bit fixed point number is
 * basically a simple fraction whose numerator is signed 128-bit integer and
 * denominator is 2^64.  As long as denominator is always the same, there is no
 * need to store it, thus in Solidity signed 64.64-bit fixed point numbers are
 * represented by int128 type holding only the numerator.
 */
library ABDKMath64x64 {
  /*
   * Minimum value signed 64.64-bit fixed point number may have. 
   */
  int128 private constant MIN_64x64 = -0x80000000000000000000000000000000;

  /*
   * Maximum value signed 64.64-bit fixed point number may have. 
   */
  int128 private constant MAX_64x64 = 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

  /**
   * Convert signed 256-bit integer number into signed 64.64-bit fixed point
   * number.  Revert on overflow.
   *
   * @param x signed 256-bit integer number
   * @return signed 64.64-bit fixed point number
   */
  function fromInt (int256 x) internal pure returns (int128) {
    require (x >= -0x8000000000000000 && x <= 0x7FFFFFFFFFFFFFFF);
    return int128 (x << 64);
  }

  /**
   * Convert signed 64.64 fixed point number into signed 64-bit integer number
   * rounding down.
   *
   * @param x signed 64.64-bit fixed point number
   * @return signed 64-bit integer number
   */
  function toInt (int128 x) internal pure returns (int64) {
    return int64 (x >> 64);
  }

  /**
   * Convert unsigned 256-bit integer number into signed 64.64-bit fixed point
   * number.  Revert on overflow.
   *
   * @param x unsigned 256-bit integer number
   * @return signed 64.64-bit fixed point number
   */
  function fromUInt (uint256 x) internal pure returns (int128) {
    require (x <= 0x7FFFFFFFFFFFFFFF);
    return int128 (x << 64);
  }

  /**
   * Convert signed 64.64 fixed point number into unsigned 64-bit integer
   * number rounding down.  Revert on underflow.
   *
   * @param x signed 64.64-bit fixed point number
   * @return unsigned 64-bit integer number
   */
  function toUInt (int128 x) internal pure returns (uint64) {
    require (x >= 0);
    return uint64 (x >> 64);
  }

  /**
   * Convert signed 128.128 fixed point number into signed 64.64-bit fixed point
   * number rounding down.  Revert on overflow.
   *
   * @param x signed 128.128-bin fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function from128x128 (int256 x) internal pure returns (int128) {
    int256 result = x >> 64;
    require (result >= MIN_64x64 && result <= MAX_64x64);
    return int128 (result);
  }

  /**
   * Convert signed 64.64 fixed point number into signed 128.128 fixed point
   * number.
   *
   * @param x signed 64.64-bit fixed point number
   * @return signed 128.128 fixed point number
   */
  function to128x128 (int128 x) internal pure returns (int256) {
    return int256 (x) << 64;
  }

  /**
   * Calculate x + y.  Revert on overflow.
   *
   * @param x signed 64.64-bit fixed point number
   * @param y signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function add (int128 x, int128 y) internal pure returns (int128) {
    int256 result = int256(x) + y;
    require (result >= MIN_64x64 && result <= MAX_64x64);
    return int128 (result);
  }

  /**
   * Calculate x - y.  Revert on overflow.
   *
   * @param x signed 64.64-bit fixed point number
   * @param y signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function sub (int128 x, int128 y) internal pure returns (int128) {
    int256 result = int256(x) - y;
    require (result >= MIN_64x64 && result <= MAX_64x64);
    return int128 (result);
  }

  /**
   * Calculate x * y rounding down.  Revert on overflow.
   *
   * @param x signed 64.64-bit fixed point number
   * @param y signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function mul (int128 x, int128 y) internal pure returns (int128) {
    int256 result = int256(x) * y >> 64;
    require (result >= MIN_64x64 && result <= MAX_64x64);
    return int128 (result);
  }

  /**
   * Calculate x * y rounding towards zero, where x is signed 64.64 fixed point
   * number and y is signed 256-bit integer number.  Revert on overflow.
   *
   * @param x signed 64.64 fixed point number
   * @param y signed 256-bit integer number
   * @return signed 256-bit integer number
   */
  function muli (int128 x, int256 y) internal pure returns (int256) {
    if (x == MIN_64x64) {
      require (y >= -0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF &&
        y <= 0x1000000000000000000000000000000000000000000000000);
      return -y << 63;
    } else {
      bool negativeResult = false;
      if (x < 0) {
        x = -x;
        negativeResult = true;
      }
      if (y < 0) {
        y = -y; // We rely on overflow behavior here
        negativeResult = !negativeResult;
      }
      uint256 absoluteResult = mulu (x, uint256 (y));
      if (negativeResult) {
        require (absoluteResult <=
          0x8000000000000000000000000000000000000000000000000000000000000000);
        return -int256 (absoluteResult); // We rely on overflow behavior here
      } else {
        require (absoluteResult <=
          0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
        return int256 (absoluteResult);
      }
    }
  }

  /**
   * Calculate x * y rounding down, where x is signed 64.64 fixed point number
   * and y is unsigned 256-bit integer number.  Revert on overflow.
   *
   * @param x signed 64.64 fixed point number
   * @param y unsigned 256-bit integer number
   * @return unsigned 256-bit integer number
   */
  function mulu (int128 x, uint256 y) internal pure returns (uint256) {
    if (y == 0) return 0;

    require (x >= 0);

    uint256 lo = (uint256 (x) * (y & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)) >> 64;
    uint256 hi = uint256 (x) * (y >> 128);

    require (hi <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
    hi <<= 64;

    require (hi <=
      0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF - lo);
    return hi + lo;
  }

  /**
   * Calculate x / y rounding towards zero.  Revert on overflow or when y is
   * zero.
   *
   * @param x signed 64.64-bit fixed point number
   * @param y signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function div (int128 x, int128 y) internal pure returns (int128) {
    require (y != 0);
    int256 result = (int256 (x) << 64) / y;
    require (result >= MIN_64x64 && result <= MAX_64x64);
    return int128 (result);
  }

  /**
   * Calculate x / y rounding towards zero, where x and y are signed 256-bit
   * integer numbers.  Revert on overflow or when y is zero.
   *
   * @param x signed 256-bit integer number
   * @param y signed 256-bit integer number
   * @return signed 64.64-bit fixed point number
   */
  function divi (int256 x, int256 y) internal pure returns (int128) {
    require (y != 0);

    bool negativeResult = false;
    if (x < 0) {
      x = -x; // We rely on overflow behavior here
      negativeResult = true;
    }
    if (y < 0) {
      y = -y; // We rely on overflow behavior here
      negativeResult = !negativeResult;
    }
    uint128 absoluteResult = divuu (uint256 (x), uint256 (y));
    if (negativeResult) {
      require (absoluteResult <= 0x80000000000000000000000000000000);
      return -int128 (absoluteResult); // We rely on overflow behavior here
    } else {
      require (absoluteResult <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
      return int128 (absoluteResult); // We rely on overflow behavior here
    }
  }

  /**
   * Calculate x / y rounding towards zero, where x and y are unsigned 256-bit
   * integer numbers.  Revert on overflow or when y is zero.
   *
   * @param x unsigned 256-bit integer number
   * @param y unsigned 256-bit integer number
   * @return signed 64.64-bit fixed point number
   */
  function divu (uint256 x, uint256 y) internal pure returns (int128) {
    require (y != 0);
    uint128 result = divuu (x, y);
    require (result <= uint128 (MAX_64x64));
    return int128 (result);
  }

  /**
   * Calculate -x.  Revert on overflow.
   *
   * @param x signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function neg (int128 x) internal pure returns (int128) {
    require (x != MIN_64x64);
    return -x;
  }

  /**
   * Calculate |x|.  Revert on overflow.
   *
   * @param x signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function abs (int128 x) internal pure returns (int128) {
    require (x != MIN_64x64);
    return x < 0 ? -x : x;
  }

  /**
   * Calculate 1 / x rounding towards zero.  Revert on overflow or when x is
   * zero.
   *
   * @param x signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function inv (int128 x) internal pure returns (int128) {
    require (x != 0);
    int256 result = int256 (0x100000000000000000000000000000000) / x;
    require (result >= MIN_64x64 && result <= MAX_64x64);
    return int128 (result);
  }

  /**
   * Calculate arithmetics average of x and y, i.e. (x + y) / 2 rounding down.
   *
   * @param x signed 64.64-bit fixed point number
   * @param y signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function avg (int128 x, int128 y) internal pure returns (int128) {
    return int128 ((int256 (x) + int256 (y)) >> 1);
  }

  /**
   * Calculate geometric average of x and y, i.e. sqrt (x * y) rounding down.
   * Revert on overflow or in case x * y is negative.
   *
   * @param x signed 64.64-bit fixed point number
   * @param y signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function gavg (int128 x, int128 y) internal pure returns (int128) {
    int256 m = int256 (x) * int256 (y);
    require (m >= 0);
    require (m <
        0x4000000000000000000000000000000000000000000000000000000000000000);
    return int128 (sqrtu (uint256 (m)));
  }

  /**
   * Calculate x^y assuming 0^0 is 1, where x is signed 64.64 fixed point number
   * and y is unsigned 256-bit integer number.  Revert on overflow.
   *
   * @param x signed 64.64-bit fixed point number
   * @param y uint256 value
   * @return signed 64.64-bit fixed point number
   */
  function pow (int128 x, uint256 y) internal pure returns (int128) {
    uint256 absoluteResult;
    bool negativeResult = false;
    if (x >= 0) {
      absoluteResult = powu (uint256 (x) << 63, y);
    } else {
      // We rely on overflow behavior here
      absoluteResult = powu (uint256 (uint128 (-x)) << 63, y);
      negativeResult = y & 1 > 0;
    }

    absoluteResult >>= 63;

    if (negativeResult) {
      require (absoluteResult <= 0x80000000000000000000000000000000);
      return -int128 (absoluteResult); // We rely on overflow behavior here
    } else {
      require (absoluteResult <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
      return int128 (absoluteResult); // We rely on overflow behavior here
    }
  }

  /**
   * Calculate sqrt (x) rounding down.  Revert if x < 0.
   *
   * @param x signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function sqrt (int128 x) internal pure returns (int128) {
    require (x >= 0);
    return int128 (sqrtu (uint256 (x) << 64));
  }

  /**
   * Calculate binary logarithm of x.  Revert if x <= 0.
   *
   * @param x signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function log_2 (int128 x) internal pure returns (int128) {
    require (x > 0);

    int256 msb = 0;
    int256 xc = x;
    if (xc >= 0x10000000000000000) { xc >>= 64; msb += 64; }
    if (xc >= 0x100000000) { xc >>= 32; msb += 32; }
    if (xc >= 0x10000) { xc >>= 16; msb += 16; }
    if (xc >= 0x100) { xc >>= 8; msb += 8; }
    if (xc >= 0x10) { xc >>= 4; msb += 4; }
    if (xc >= 0x4) { xc >>= 2; msb += 2; }
    if (xc >= 0x2) msb += 1;  // No need to shift xc anymore

    int256 result = msb - 64 << 64;
    uint256 ux = uint256 (x) << uint256 (127 - msb);
    for (int256 bit = 0x8000000000000000; bit > 0; bit >>= 1) {
      ux *= ux;
      uint256 b = ux >> 255;
      ux >>= 127 + b;
      result += bit * int256 (b);
    }

    return int128 (result);
  }

  /**
   * Calculate natural logarithm of x.  Revert if x <= 0.
   *
   * @param x signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function ln (int128 x) internal pure returns (int128) {
    require (x > 0);

    return int128 (
        uint256 (log_2 (x)) * 0xB17217F7D1CF79ABC9E3B39803F2F6AF >> 128);
  }

  /**
   * Calculate binary exponent of x.  Revert on overflow.
   *
   * @param x signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function exp_2 (int128 x) internal pure returns (int128) {
    require (x < 0x400000000000000000); // Overflow

    if (x < -0x400000000000000000) return 0; // Underflow

    uint256 result = 0x80000000000000000000000000000000;

    if (x & 0x8000000000000000 > 0)
      result = result * 0x16A09E667F3BCC908B2FB1366EA957D3E >> 128;
    if (x & 0x4000000000000000 > 0)
      result = result * 0x1306FE0A31B7152DE8D5A46305C85EDEC >> 128;
    if (x & 0x2000000000000000 > 0)
      result = result * 0x1172B83C7D517ADCDF7C8C50EB14A791F >> 128;
    if (x & 0x1000000000000000 > 0)
      result = result * 0x10B5586CF9890F6298B92B71842A98363 >> 128;
    if (x & 0x800000000000000 > 0)
      result = result * 0x1059B0D31585743AE7C548EB68CA417FD >> 128;
    if (x & 0x400000000000000 > 0)
      result = result * 0x102C9A3E778060EE6F7CACA4F7A29BDE8 >> 128;
    if (x & 0x200000000000000 > 0)
      result = result * 0x10163DA9FB33356D84A66AE336DCDFA3F >> 128;
    if (x & 0x100000000000000 > 0)
      result = result * 0x100B1AFA5ABCBED6129AB13EC11DC9543 >> 128;
    if (x & 0x80000000000000 > 0)
      result = result * 0x10058C86DA1C09EA1FF19D294CF2F679B >> 128;
    if (x & 0x40000000000000 > 0)
      result = result * 0x1002C605E2E8CEC506D21BFC89A23A00F >> 128;
    if (x & 0x20000000000000 > 0)
      result = result * 0x100162F3904051FA128BCA9C55C31E5DF >> 128;
    if (x & 0x10000000000000 > 0)
      result = result * 0x1000B175EFFDC76BA38E31671CA939725 >> 128;
    if (x & 0x8000000000000 > 0)
      result = result * 0x100058BA01FB9F96D6CACD4B180917C3D >> 128;
    if (x & 0x4000000000000 > 0)
      result = result * 0x10002C5CC37DA9491D0985C348C68E7B3 >> 128;
    if (x & 0x2000000000000 > 0)
      result = result * 0x1000162E525EE054754457D5995292026 >> 128;
    if (x & 0x1000000000000 > 0)
      result = result * 0x10000B17255775C040618BF4A4ADE83FC >> 128;
    if (x & 0x800000000000 > 0)
      result = result * 0x1000058B91B5BC9AE2EED81E9B7D4CFAB >> 128;
    if (x & 0x400000000000 > 0)
      result = result * 0x100002C5C89D5EC6CA4D7C8ACC017B7C9 >> 128;
    if (x & 0x200000000000 > 0)
      result = result * 0x10000162E43F4F831060E02D839A9D16D >> 128;
    if (x & 0x100000000000 > 0)
      result = result * 0x100000B1721BCFC99D9F890EA06911763 >> 128;
    if (x & 0x80000000000 > 0)
      result = result * 0x10000058B90CF1E6D97F9CA14DBCC1628 >> 128;
    if (x & 0x40000000000 > 0)
      result = result * 0x1000002C5C863B73F016468F6BAC5CA2B >> 128;
    if (x & 0x20000000000 > 0)
      result = result * 0x100000162E430E5A18F6119E3C02282A5 >> 128;
    if (x & 0x10000000000 > 0)
      result = result * 0x1000000B1721835514B86E6D96EFD1BFE >> 128;
    if (x & 0x8000000000 > 0)
      result = result * 0x100000058B90C0B48C6BE5DF846C5B2EF >> 128;
    if (x & 0x4000000000 > 0)
      result = result * 0x10000002C5C8601CC6B9E94213C72737A >> 128;
    if (x & 0x2000000000 > 0)
      result = result * 0x1000000162E42FFF037DF38AA2B219F06 >> 128;
    if (x & 0x1000000000 > 0)
      result = result * 0x10000000B17217FBA9C739AA5819F44F9 >> 128;
    if (x & 0x800000000 > 0)
      result = result * 0x1000000058B90BFCDEE5ACD3C1CEDC823 >> 128;
    if (x & 0x400000000 > 0)
      result = result * 0x100000002C5C85FE31F35A6A30DA1BE50 >> 128;
    if (x & 0x200000000 > 0)
      result = result * 0x10000000162E42FF0999CE3541B9FFFCF >> 128;
    if (x & 0x100000000 > 0)
      result = result * 0x100000000B17217F80F4EF5AADDA45554 >> 128;
    if (x & 0x80000000 > 0)
      result = result * 0x10000000058B90BFBF8479BD5A81B51AD >> 128;
    if (x & 0x40000000 > 0)
      result = result * 0x1000000002C5C85FDF84BD62AE30A74CC >> 128;
    if (x & 0x20000000 > 0)
      result = result * 0x100000000162E42FEFB2FED257559BDAA >> 128;
    if (x & 0x10000000 > 0)
      result = result * 0x1000000000B17217F7D5A7716BBA4A9AE >> 128;
    if (x & 0x8000000 > 0)
      result = result * 0x100000000058B90BFBE9DDBAC5E109CCE >> 128;
    if (x & 0x4000000 > 0)
      result = result * 0x10000000002C5C85FDF4B15DE6F17EB0D >> 128;
    if (x & 0x2000000 > 0)
      result = result * 0x1000000000162E42FEFA494F1478FDE05 >> 128;
    if (x & 0x1000000 > 0)
      result = result * 0x10000000000B17217F7D20CF927C8E94C >> 128;
    if (x & 0x800000 > 0)
      result = result * 0x1000000000058B90BFBE8F71CB4E4B33D >> 128;
    if (x & 0x400000 > 0)
      result = result * 0x100000000002C5C85FDF477B662B26945 >> 128;
    if (x & 0x200000 > 0)
      result = result * 0x10000000000162E42FEFA3AE53369388C >> 128;
    if (x & 0x100000 > 0)
      result = result * 0x100000000000B17217F7D1D351A389D40 >> 128;
    if (x & 0x80000 > 0)
      result = result * 0x10000000000058B90BFBE8E8B2D3D4EDE >> 128;
    if (x & 0x40000 > 0)
      result = result * 0x1000000000002C5C85FDF4741BEA6E77E >> 128;
    if (x & 0x20000 > 0)
      result = result * 0x100000000000162E42FEFA39FE95583C2 >> 128;
    if (x & 0x10000 > 0)
      result = result * 0x1000000000000B17217F7D1CFB72B45E1 >> 128;
    if (x & 0x8000 > 0)
      result = result * 0x100000000000058B90BFBE8E7CC35C3F0 >> 128;
    if (x & 0x4000 > 0)
      result = result * 0x10000000000002C5C85FDF473E242EA38 >> 128;
    if (x & 0x2000 > 0)
      result = result * 0x1000000000000162E42FEFA39F02B772C >> 128;
    if (x & 0x1000 > 0)
      result = result * 0x10000000000000B17217F7D1CF7D83C1A >> 128;
    if (x & 0x800 > 0)
      result = result * 0x1000000000000058B90BFBE8E7BDCBE2E >> 128;
    if (x & 0x400 > 0)
      result = result * 0x100000000000002C5C85FDF473DEA871F >> 128;
    if (x & 0x200 > 0)
      result = result * 0x10000000000000162E42FEFA39EF44D91 >> 128;
    if (x & 0x100 > 0)
      result = result * 0x100000000000000B17217F7D1CF79E949 >> 128;
    if (x & 0x80 > 0)
      result = result * 0x10000000000000058B90BFBE8E7BCE544 >> 128;
    if (x & 0x40 > 0)
      result = result * 0x1000000000000002C5C85FDF473DE6ECA >> 128;
    if (x & 0x20 > 0)
      result = result * 0x100000000000000162E42FEFA39EF366F >> 128;
    if (x & 0x10 > 0)
      result = result * 0x1000000000000000B17217F7D1CF79AFA >> 128;
    if (x & 0x8 > 0)
      result = result * 0x100000000000000058B90BFBE8E7BCD6D >> 128;
    if (x & 0x4 > 0)
      result = result * 0x10000000000000002C5C85FDF473DE6B2 >> 128;
    if (x & 0x2 > 0)
      result = result * 0x1000000000000000162E42FEFA39EF358 >> 128;
    if (x & 0x1 > 0)
      result = result * 0x10000000000000000B17217F7D1CF79AB >> 128;

    result >>= uint256 (63 - (x >> 64));
    require (result <= uint256 (MAX_64x64));

    return int128 (result);
  }

  /**
   * Calculate natural exponent of x.  Revert on overflow.
   *
   * @param x signed 64.64-bit fixed point number
   * @return signed 64.64-bit fixed point number
   */
  function exp (int128 x) internal pure returns (int128) {
    require (x < 0x400000000000000000); // Overflow

    if (x < -0x400000000000000000) return 0; // Underflow

    return exp_2 (
        int128 (int256 (x) * 0x171547652B82FE1777D0FFDA0D23A7D12 >> 128));
  }

  /**
   * Calculate x / y rounding towards zero, where x and y are unsigned 256-bit
   * integer numbers.  Revert on overflow or when y is zero.
   *
   * @param x unsigned 256-bit integer number
   * @param y unsigned 256-bit integer number
   * @return unsigned 64.64-bit fixed point number
   */
  function divuu (uint256 x, uint256 y) private pure returns (uint128) {
    require (y != 0);

    uint256 result;

    if (x <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)
      result = (x << 64) / y;
    else {
      uint256 msb = 192;
      uint256 xc = x >> 192;
      if (xc >= 0x100000000) { xc >>= 32; msb += 32; }
      if (xc >= 0x10000) { xc >>= 16; msb += 16; }
      if (xc >= 0x100) { xc >>= 8; msb += 8; }
      if (xc >= 0x10) { xc >>= 4; msb += 4; }
      if (xc >= 0x4) { xc >>= 2; msb += 2; }
      if (xc >= 0x2) msb += 1;  // No need to shift xc anymore

      result = (x << 255 - msb) / ((y - 1 >> msb - 191) + 1);
      require (result <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);

      uint256 hi = result * (y >> 128);
      uint256 lo = result * (y & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);

      uint256 xh = x >> 192;
      uint256 xl = x << 64;

      if (xl < lo) xh -= 1;
      xl -= lo; // We rely on overflow behavior here
      lo = hi << 128;
      if (xl < lo) xh -= 1;
      xl -= lo; // We rely on overflow behavior here

      assert (xh == hi >> 128);

      result += xl / y;
    }

    require (result <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
    return uint128 (result);
  }

  /**
   * Calculate x^y assuming 0^0 is 1, where x is unsigned 129.127 fixed point
   * number and y is unsigned 256-bit integer number.  Revert on overflow.
   *
   * @param x unsigned 129.127-bit fixed point number
   * @param y uint256 value
   * @return unsigned 129.127-bit fixed point number
   */
  function powu (uint256 x, uint256 y) private pure returns (uint256) {
    if (y == 0) return 0x80000000000000000000000000000000;
    else if (x == 0) return 0;
    else {
      int256 msb = 0;
      uint256 xc = x;
      if (xc >= 0x100000000000000000000000000000000) { xc >>= 128; msb += 128; }
      if (xc >= 0x10000000000000000) { xc >>= 64; msb += 64; }
      if (xc >= 0x100000000) { xc >>= 32; msb += 32; }
      if (xc >= 0x10000) { xc >>= 16; msb += 16; }
      if (xc >= 0x100) { xc >>= 8; msb += 8; }
      if (xc >= 0x10) { xc >>= 4; msb += 4; }
      if (xc >= 0x4) { xc >>= 2; msb += 2; }
      if (xc >= 0x2) msb += 1;  // No need to shift xc anymore

      int256 xe = msb - 127;
      if (xe > 0) x >>= uint256 (xe);
      else x <<= uint256 (-xe);

      uint256 result = 0x80000000000000000000000000000000;
      int256 re = 0;

      while (y > 0) {
        if (y & 1 > 0) {
          result = result * x;
          y -= 1;
          re += xe;
          if (result >=
            0x8000000000000000000000000000000000000000000000000000000000000000) {
            result >>= 128;
            re += 1;
          } else result >>= 127;
          if (re < -127) return 0; // Underflow
          require (re < 128); // Overflow
        } else {
          x = x * x;
          y >>= 1;
          xe <<= 1;
          if (x >=
            0x8000000000000000000000000000000000000000000000000000000000000000) {
            x >>= 128;
            xe += 1;
          } else x >>= 127;
          if (xe < -127) return 0; // Underflow
          require (xe < 128); // Overflow
        }
      }

      if (re > 0) result <<= uint256 (re);
      else if (re < 0) result >>= uint256 (-re);

      return result;
    }
  }

  /**
   * Calculate sqrt (x) rounding down, where x is unsigned 256-bit integer
   * number.
   *
   * @param x unsigned 256-bit integer number
   * @return unsigned 128-bit integer number
   */
  function sqrtu (uint256 x) private pure returns (uint128) {
    if (x == 0) return 0;
    else {
      uint256 xx = x;
      uint256 r = 1;
      if (xx >= 0x100000000000000000000000000000000) { xx >>= 128; r <<= 64; }
      if (xx >= 0x10000000000000000) { xx >>= 64; r <<= 32; }
      if (xx >= 0x100000000) { xx >>= 32; r <<= 16; }
      if (xx >= 0x10000) { xx >>= 16; r <<= 8; }
      if (xx >= 0x100) { xx >>= 8; r <<= 4; }
      if (xx >= 0x10) { xx >>= 4; r <<= 2; }
      if (xx >= 0x8) { r <<= 1; }
      r = (r + x / r) >> 1;
      r = (r + x / r) >> 1;
      r = (r + x / r) >> 1;
      r = (r + x / r) >> 1;
      r = (r + x / r) >> 1;
      r = (r + x / r) >> 1;
      r = (r + x / r) >> 1; // Seven iterations should be enough
      uint256 r1 = x / r;
      return uint128 (r < r1 ? r : r1);
    }
  }
}

// Part: IOracle

interface IOracle {
    function getPrice() external view returns (uint256);
}

// Part: Initializable

/**
 * @title Initializable
 *
 * @dev Helper contract to support initializer functions. To use it, replace
 * the constructor with a function that has the `initializer` modifier.
 * WARNING: Unlike constructors, initializer functions must be manually
 * invoked. This applies both to deploying an Initializable contract, as well
 * as extending an Initializable contract via inheritance.
 * WARNING: When used with inheritance, manual care must be taken to not invoke
 * a parent initializer twice, or ensure that all initializers are idempotent,
 * because this is not dealt with automatically as with constructors.
 */
contract Initializable {

  /**
   * @dev Indicates that the contract has been initialized.
   */
  bool private initialized;

  /**
   * @dev Indicates that the contract is in the process of being initialized.
   */
  bool private initializing;

  /**
   * @dev Modifier to use in the initializer function of a contract.
   */
  modifier initializer() {
    require(initializing || isConstructor() || !initialized, "Contract instance has already been initialized");

    bool isTopLevelCall = !initializing;
    if (isTopLevelCall) {
      initializing = true;
      initialized = true;
    }

    _;

    if (isTopLevelCall) {
      initializing = false;
    }
  }

  /// @dev Returns true if and only if the function is running in the constructor
  function isConstructor() private view returns (bool) {
    // extcodesize checks the size of the code stored in an address, and
    // address returns the current address. Since the code is still not
    // deployed when running a constructor, any checks on its code size will
    // yield zero, making it an effective way to detect if a contract is
    // under construction or not.
    address self = address(this);
    uint256 cs;
    assembly { cs := extcodesize(self) }
    return cs == 0;
  }

  // Reserved storage space to allow for layout changes in the future.
  uint256[50] private ______gap;
}

// Part: OpenZeppelin/openzeppelin-contracts@3.2.0/Address

/**
 * @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
     * ====
     */
    function isContract(address account) internal view returns (bool) {
        // This method relies in extcodesize, which returns 0 for contracts in
        // construction, since the code is only stored at the end of the
        // constructor execution.

        uint256 size;
        // solhint-disable-next-line no-inline-assembly
        assembly { size := extcodesize(account) }
        return size > 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");

        // solhint-disable-next-line avoid-low-level-calls, avoid-call-value
        (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 functionCall(target, data, "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");
        return _functionCallWithValue(target, data, value, errorMessage);
    }

    function _functionCallWithValue(address target, bytes memory data, uint256 weiValue, string memory errorMessage) private returns (bytes memory) {
        require(isContract(target), "Address: call to non-contract");

        // solhint-disable-next-line avoid-low-level-calls
        (bool success, bytes memory returndata) = target.call{ value: weiValue }(data);
        if (success) {
            return returndata;
        } else {
            // 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

                // solhint-disable-next-line no-inline-assembly
                assembly {
                    let returndata_size := mload(returndata)
                    revert(add(32, returndata), returndata_size)
                }
            } else {
                revert(errorMessage);
            }
        }
    }
}

// Part: OpenZeppelin/openzeppelin-contracts@3.2.0/IERC20

/**
 * @dev Interface of the ERC20 standard as defined in the EIP.
 */
interface IERC20 {
    /**
     * @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 `recipient`.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * Emits a {Transfer} event.
     */
    function transfer(address recipient, 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 `sender` to `recipient` 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 sender, address recipient, uint256 amount) external returns (bool);

    /**
     * @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);
}

// Part: OpenZeppelin/openzeppelin-contracts@3.2.0/Math

/**
 * @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);
    }
}

// Part: OpenZeppelin/openzeppelin-contracts@3.2.0/SafeMath

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

        return c;
    }

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

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

        return c;
    }

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

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

        return c;
    }

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

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

        return c;
    }

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

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

// Part: ContextUpgradeSafe

/*
 * @dev Provides information about the current execution context, including the
 * sender of the transaction and its data. While these are generally available
 * via msg.sender and msg.data, they should not be accessed in such a direct
 * manner, since when dealing with GSN meta-transactions the account sending and
 * paying for execution may not be the actual sender (as far as an application
 * is concerned).
 *
 * This contract is only required for intermediate, library-like contracts.
 */
contract ContextUpgradeSafe is Initializable {
    // Empty internal constructor, to prevent people from mistakenly deploying
    // an instance of this contract, which should be used via inheritance.

    function __Context_init() internal initializer {
        __Context_init_unchained();
    }

    function __Context_init_unchained() internal initializer {


    }


    function _msgSender() internal view virtual returns (address payable) {
        return msg.sender;
    }

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

    uint256[50] private __gap;
}

// Part: OpenZeppelin/openzeppelin-contracts@3.2.0/SafeERC20

/**
 * @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 SafeMath for uint256;
    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'
        // solhint-disable-next-line max-line-length
        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).add(value);
        _callOptionalReturn(token, abi.encodeWithSelector(token.approve.selector, spender, newAllowance));
    }

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

    /**
     * @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
            // solhint-disable-next-line max-line-length
            require(abi.decode(returndata, (bool)), "SafeERC20: ERC20 operation did not succeed");
        }
    }
}

// Part: OptionMath

library OptionMath {
    using SafeMath for uint256;

    uint256 public constant SCALE = 1e18;

    /**
     * Converts total supplies of options into the tokenized payoff quantities used
     * by the LMSR
     *
     * For puts, multiply by strike price since option quantity is in terms of the
     * underlying, but lmsr quantities should be in terms of the strike currency
     */
    function calcQuantities(
        uint256[] memory strikePrices,
        bool isPut,
        uint256[] memory longSupplies,
        uint256[] memory shortSupplies
    ) internal pure returns (uint256[] memory quantities) {
        require(longSupplies.length == strikePrices.length, "Lengths do not match");
        require(shortSupplies.length == strikePrices.length, "Lengths do not match");
        uint256 n = strikePrices.length;

        // this mutates the method arguments, but costs less gas
        if (isPut) {
            for (uint256 i = 0; i < n; i++) {
                uint256 strikePrice = strikePrices[i];
                longSupplies[i] = longSupplies[i].mul(strikePrice).div(SCALE);
                shortSupplies[i] = shortSupplies[i].mul(strikePrice).div(SCALE);
            }
        }

        uint256[] memory leftSupplies = isPut ? shortSupplies : longSupplies;
        uint256[] memory rightSupplies = isPut ? longSupplies : shortSupplies;

        // initially set runningSum = sum(rightSupplies)
        uint256 runningSum;
        for (uint256 i = 0; i < n; i++) {
            runningSum = runningSum.add(rightSupplies[i]);
        }

        quantities = new uint256[](n + 1);
        quantities[0] = runningSum;

        // set quantities[i] = leftSupplies[:i] + rightSupplies[i:]
        for (uint256 i = 0; i < n; i++) {
            runningSum = runningSum.add(leftSupplies[i]).sub(rightSupplies[i]);
            quantities[i + 1] = runningSum;
        }
        return quantities;
    }

    /**
     * Calculates the LMSR cost function
     *
     *   C(q_1, ..., q_n) = b * log(exp(q_1 / b) + ... + exp(q_n / b))
     *
     * where
     *
     *   q_i = total supply of ith tokenized payoff
     *   b = liquidity parameter
     *
     * An equivalent expression for C is used to avoid overflow when calculating exponentials
     *
     *   C(q_1, ..., q_n) = m + b * log(exp((q_1 - m) / b) + ... + exp((q_n - m) / b))
     *
     * where
     *
     *   m = max(q_1, ..., q_n)
     */
    function calcLmsrCost(uint256[] memory quantities, uint256 b) internal pure returns (uint256) {
        if (b == 0) {
            return 0;
        }

        uint256 maxQuantity = quantities[0];
        for (uint256 i = 1; i < quantities.length; i++) {
            maxQuantity = Math.max(maxQuantity, quantities[i]);
        }

        int128 sumExp;
        for (uint256 i = 0; i < quantities.length; i++) {
            // max(q) - q_i
            uint256 diff = maxQuantity.sub(quantities[i]);

            // (max(q) - q_i) / b
            int128 div = ABDKMath64x64.divu(diff, b);

            // exp((q_i - max(q)) / b)
            int128 exp = ABDKMath64x64.exp(ABDKMath64x64.neg(div));
            sumExp = ABDKMath64x64.add(sumExp, exp);
        }

        // log(sumExp)
        int128 log = ABDKMath64x64.ln(sumExp);

        // b * log(sumExp) + max(q)
        return ABDKMath64x64.mulu(log, b).add(maxQuantity);
    }

    /**
     * Calculate total payoff of all outstanding options
     *
     * This value will decrease as options are redeemed
     *
     * For calls, divide by expiry price since payoff should be in terms of the
     * `baseToken`
     */
    function calcPayoff(
        uint256[] memory strikePrices,
        uint256 expiryPrice,
        bool isPut,
        uint256[] memory longSupplies,
        uint256[] memory shortSupplies
    ) internal pure returns (uint256 payoff) {
        require(longSupplies.length == strikePrices.length, "Lengths do not match");
        require(shortSupplies.length == strikePrices.length, "Lengths do not match");

        if (expiryPrice == 0) {
            return 0;
        }

        for (uint256 i = 0; i < strikePrices.length; i++) {
            uint256 strikePrice = strikePrices[i];

            if (isPut && expiryPrice < strikePrice) {
                // put payoff = max(K - S, 0)
                payoff = payoff.add(longSupplies[i].mul(strikePrice.sub(expiryPrice)));
            } else if (!isPut && expiryPrice > strikePrice) {
                // call payoff = max(S - K, 0)
                payoff = payoff.add(longSupplies[i].mul(expiryPrice.sub(strikePrice)));
            }

            // short payoff = min(S, K)
            payoff = payoff.add(shortSupplies[i].mul(Math.min(expiryPrice, strikePrice)));
        }

        payoff = payoff.div(isPut ? SCALE : expiryPrice);
    }
}

// Part: ReentrancyGuardUpgradeSafe

/**
 * @dev Contract module that helps prevent reentrant calls to a function.
 *
 * Inheriting from `ReentrancyGuard` will make the {nonReentrant} modifier
 * available, which can be applied to functions to make sure there are no nested
 * (reentrant) calls to them.
 *
 * Note that because there is a single `nonReentrant` guard, functions marked as
 * `nonReentrant` may not call one another. This can be worked around by making
 * those functions `private`, and then adding `external` `nonReentrant` entry
 * points to them.
 *
 * TIP: If you would like to learn more about reentrancy and alternative ways
 * to protect against it, check out our blog post
 * https://blog.openzeppelin.com/reentrancy-after-istanbul/[Reentrancy After Istanbul].
 */
contract ReentrancyGuardUpgradeSafe is Initializable {
    bool private _notEntered;


    function __ReentrancyGuard_init() internal initializer {
        __ReentrancyGuard_init_unchained();
    }

    function __ReentrancyGuard_init_unchained() internal initializer {


        // 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 percetange 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 make it call a
     * `private` function that does the actual work.
     */
    modifier nonReentrant() {
        // On the first call to nonReentrant, _notEntered will be true
        require(_notEntered, "ReentrancyGuard: reentrant call");

        // Any calls to nonReentrant after this point will fail
        _notEntered = false;

        _;

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

    uint256[49] private __gap;
}

// Part: ERC20UpgradeSafe

/**
 * @dev Implementation of the {IERC20} interface.
 *
 * This implementation is agnostic to the way tokens are created. This means
 * that a supply mechanism has to be added in a derived contract using {_mint}.
 * For a generic mechanism see {ERC20MinterPauser}.
 *
 * TIP: For a detailed writeup see our guide
 * https://forum.zeppelin.solutions/t/how-to-implement-erc20-supply-mechanisms/226[How
 * to implement supply mechanisms].
 *
 * We have followed general OpenZeppelin guidelines: functions revert instead
 * of returning `false` on failure. This behavior is nonetheless conventional
 * and does not conflict with the expectations of ERC20 applications.
 *
 * Additionally, an {Approval} event is emitted on calls to {transferFrom}.
 * This allows applications to reconstruct the allowance for all accounts just
 * by listening to said events. Other implementations of the EIP may not emit
 * these events, as it isn't required by the specification.
 *
 * Finally, the non-standard {decreaseAllowance} and {increaseAllowance}
 * functions have been added to mitigate the well-known issues around setting
 * allowances. See {IERC20-approve}.
 */
contract ERC20UpgradeSafe is Initializable, ContextUpgradeSafe, IERC20 {
    using SafeMath for uint256;
    using Address for address;

    mapping (address => uint256) private _balances;

    mapping (address => mapping (address => uint256)) private _allowances;

    uint256 private _totalSupply;

    string private _name;
    string private _symbol;
    uint8 private _decimals;

    /**
     * @dev Sets the values for {name} and {symbol}, initializes {decimals} with
     * a default value of 18.
     *
     * To select a different value for {decimals}, use {_setupDecimals}.
     *
     * All three of these values are immutable: they can only be set once during
     * construction.
     */

    function __ERC20_init(string memory name, string memory symbol) internal initializer {
        __Context_init_unchained();
        __ERC20_init_unchained(name, symbol);
    }

    function __ERC20_init_unchained(string memory name, string memory symbol) internal initializer {


        _name = name;
        _symbol = symbol;
        _decimals = 18;

    }


    /**
     * @dev Returns the name of the token.
     */
    function name() public view returns (string memory) {
        return _name;
    }

    /**
     * @dev Returns the symbol of the token, usually a shorter version of the
     * name.
     */
    function symbol() public view returns (string memory) {
        return _symbol;
    }

    /**
     * @dev Returns the number of decimals used to get its user representation.
     * For example, if `decimals` equals `2`, a balance of `505` tokens should
     * be displayed to a user as `5,05` (`505 / 10 ** 2`).
     *
     * Tokens usually opt for a value of 18, imitating the relationship between
     * Ether and Wei. This is the value {ERC20} uses, unless {_setupDecimals} is
     * called.
     *
     * NOTE: This information is only used for _display_ purposes: it in
     * no way affects any of the arithmetic of the contract, including
     * {IERC20-balanceOf} and {IERC20-transfer}.
     */
    function decimals() public view returns (uint8) {
        return _decimals;
    }

    /**
     * @dev See {IERC20-totalSupply}.
     */
    function totalSupply() public view override returns (uint256) {
        return _totalSupply;
    }

    /**
     * @dev See {IERC20-balanceOf}.
     */
    function balanceOf(address account) public view override returns (uint256) {
        return _balances[account];
    }

    /**
     * @dev See {IERC20-transfer}.
     *
     * Requirements:
     *
     * - `recipient` cannot be the zero address.
     * - the caller must have a balance of at least `amount`.
     */
    function transfer(address recipient, uint256 amount) public virtual override returns (bool) {
        _transfer(_msgSender(), recipient, amount);
        return true;
    }

    /**
     * @dev See {IERC20-allowance}.
     */
    function allowance(address owner, address spender) public view virtual override returns (uint256) {
        return _allowances[owner][spender];
    }

    /**
     * @dev See {IERC20-approve}.
     *
     * Requirements:
     *
     * - `spender` cannot be the zero address.
     */
    function approve(address spender, uint256 amount) public virtual override returns (bool) {
        _approve(_msgSender(), spender, amount);
        return true;
    }

    /**
     * @dev See {IERC20-transferFrom}.
     *
     * Emits an {Approval} event indicating the updated allowance. This is not
     * required by the EIP. See the note at the beginning of {ERC20};
     *
     * Requirements:
     * - `sender` and `recipient` cannot be the zero address.
     * - `sender` must have a balance of at least `amount`.
     * - the caller must have allowance for ``sender``'s tokens of at least
     * `amount`.
     */
    function transferFrom(address sender, address recipient, uint256 amount) public virtual override returns (bool) {
        _transfer(sender, recipient, amount);
        _approve(sender, _msgSender(), _allowances[sender][_msgSender()].sub(amount, "ERC20: transfer amount exceeds allowance"));
        return true;
    }

    /**
     * @dev Atomically increases the allowance granted to `spender` by the caller.
     *
     * This is an alternative to {approve} that can be used as a mitigation for
     * problems described in {IERC20-approve}.
     *
     * Emits an {Approval} event indicating the updated allowance.
     *
     * Requirements:
     *
     * - `spender` cannot be the zero address.
     */
    function increaseAllowance(address spender, uint256 addedValue) public virtual returns (bool) {
        _approve(_msgSender(), spender, _allowances[_msgSender()][spender].add(addedValue));
        return true;
    }

    /**
     * @dev Atomically decreases the allowance granted to `spender` by the caller.
     *
     * This is an alternative to {approve} that can be used as a mitigation for
     * problems described in {IERC20-approve}.
     *
     * Emits an {Approval} event indicating the updated allowance.
     *
     * Requirements:
     *
     * - `spender` cannot be the zero address.
     * - `spender` must have allowance for the caller of at least
     * `subtractedValue`.
     */
    function decreaseAllowance(address spender, uint256 subtractedValue) public virtual returns (bool) {
        _approve(_msgSender(), spender, _allowances[_msgSender()][spender].sub(subtractedValue, "ERC20: decreased allowance below zero"));
        return true;
    }

    /**
     * @dev Moves tokens `amount` from `sender` to `recipient`.
     *
     * This is internal function is equivalent to {transfer}, and can be used to
     * e.g. implement automatic token fees, slashing mechanisms, etc.
     *
     * Emits a {Transfer} event.
     *
     * Requirements:
     *
     * - `sender` cannot be the zero address.
     * - `recipient` cannot be the zero address.
     * - `sender` must have a balance of at least `amount`.
     */
    function _transfer(address sender, address recipient, uint256 amount) internal virtual {
        require(sender != address(0), "ERC20: transfer from the zero address");
        require(recipient != address(0), "ERC20: transfer to the zero address");

        _beforeTokenTransfer(sender, recipient, amount);

        _balances[sender] = _balances[sender].sub(amount, "ERC20: transfer amount exceeds balance");
        _balances[recipient] = _balances[recipient].add(amount);
        emit Transfer(sender, recipient, amount);
    }

    /** @dev Creates `amount` tokens and assigns them to `account`, increasing
     * the total supply.
     *
     * Emits a {Transfer} event with `from` set to the zero address.
     *
     * Requirements
     *
     * - `to` cannot be the zero address.
     */
    function _mint(address account, uint256 amount) internal virtual {
        require(account != address(0), "ERC20: mint to the zero address");

        _beforeTokenTransfer(address(0), account, amount);

        _totalSupply = _totalSupply.add(amount);
        _balances[account] = _balances[account].add(amount);
        emit Transfer(address(0), account, amount);
    }

    /**
     * @dev Destroys `amount` tokens from `account`, reducing the
     * total supply.
     *
     * Emits a {Transfer} event with `to` set to the zero address.
     *
     * Requirements
     *
     * - `account` cannot be the zero address.
     * - `account` must have at least `amount` tokens.
     */
    function _burn(address account, uint256 amount) internal virtual {
        require(account != address(0), "ERC20: burn from the zero address");

        _beforeTokenTransfer(account, address(0), amount);

        _balances[account] = _balances[account].sub(amount, "ERC20: burn amount exceeds balance");
        _totalSupply = _totalSupply.sub(amount);
        emit Transfer(account, address(0), amount);
    }

    /**
     * @dev Sets `amount` as the allowance of `spender` over the `owner`s tokens.
     *
     * This is internal function is equivalent to `approve`, and can be used to
     * e.g. set automatic allowances for certain subsystems, etc.
     *
     * Emits an {Approval} event.
     *
     * Requirements:
     *
     * - `owner` cannot be the zero address.
     * - `spender` cannot be the zero address.
     */
    function _approve(address owner, address spender, uint256 amount) internal virtual {
        require(owner != address(0), "ERC20: approve from the zero address");
        require(spender != address(0), "ERC20: approve to the zero address");

        _allowances[owner][spender] = amount;
        emit Approval(owner, spender, amount);
    }

    /**
     * @dev Sets {decimals} to a value other than the default one of 18.
     *
     * WARNING: This function should only be called from the constructor. Most
     * applications that interact with token contracts will not expect
     * {decimals} to ever change, and may work incorrectly if it does.
     */
    function _setupDecimals(uint8 decimals_) internal {
        _decimals = decimals_;
    }

    /**
     * @dev Hook that is called before any transfer of tokens. This includes
     * minting and burning.
     *
     * Calling conditions:
     *
     * - when `from` and `to` are both non-zero, `amount` of ``from``'s tokens
     * will be to transferred to `to`.
     * - when `from` is zero, `amount` tokens will be minted for `to`.
     * - when `to` is zero, `amount` of ``from``'s tokens will be burned.
     * - `from` and `to` are never both zero.
     *
     * To learn more about hooks, head to xref:ROOT:extending-contracts.adoc#using-hooks[Using Hooks].
     */
    function _beforeTokenTransfer(address from, address to, uint256 amount) internal virtual { }

    uint256[44] private __gap;
}

// Part: OwnableUpgradeSafe

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

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

    /**
     * @dev Initializes the contract setting the deployer as the initial owner.
     */

    function __Ownable_init() internal initializer {
        __Context_init_unchained();
        __Ownable_init_unchained();
    }

    function __Ownable_init_unchained() internal initializer {


        address msgSender = _msgSender();
        _owner = msgSender;
        emit OwnershipTransferred(address(0), msgSender);

    }


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

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

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

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

    uint256[49] private __gap;
}

// Part: UniERC20

library UniERC20 {
    using SafeMath for uint256;
    using SafeERC20 for IERC20;

    function isETH(IERC20 token) internal pure returns (bool) {
        return (address(token) == address(0));
    }

    function uniBalanceOf(IERC20 token, address account) internal view returns (uint256) {
        if (isETH(token)) {
            return account.balance;
        } else {
            return token.balanceOf(account);
        }
    }

    function uniTransfer(
        IERC20 token,
        address payable to,
        uint256 amount
    ) internal {
        if (amount > 0) {
            if (isETH(token)) {
                (bool success, ) = to.call{value: amount}("");
                require(success, "Transfer failed");
            } else {
                token.safeTransfer(to, amount);
            }
        }
    }

    function uniTransferFromSenderToThis(IERC20 token, uint256 amount) internal {
        if (amount > 0) {
            if (isETH(token)) {
                require(msg.value >= amount, "UniERC20: not enough value");
                if (msg.value > amount) {
                    // Return remainder if exist
                    uint256 refundAmount = msg.value.sub(amount);
                    (bool success, ) = msg.sender.call{value: refundAmount}("");
                    require(success, "Transfer failed");
                }
            } else {
                token.safeTransferFrom(msg.sender, address(this), amount);
            }
        }
    }

    function uniSymbol(IERC20 token) internal view returns (string memory) {
        if (isETH(token)) {
            return "ETH";
        }

        (bool success, bytes memory data) = address(token).staticcall{gas: 20000}(abi.encodeWithSignature("symbol()"));
        if (!success) {
            (success, data) = address(token).staticcall{gas: 20000}(abi.encodeWithSignature("SYMBOL()"));
        }

        if (success && data.length >= 96) {
            (uint256 offset, uint256 len) = abi.decode(data, (uint256, uint256));
            if (offset == 0x20 && len > 0 && len <= 256) {
                return string(abi.decode(data, (bytes)));
            }
        }

        if (success && data.length == 32) {
            uint256 len = 0;
            while (len < data.length && data[len] >= 0x20 && data[len] <= 0x7E) {
                len++;
            }

            if (len > 0) {
                bytes memory result = new bytes(len);
                for (uint256 i = 0; i < len; i++) {
                    result[i] = data[i];
                }
                return string(result);
            }
        }

        return _toHex(address(token));
    }

    function _toHex(address account) private pure returns (string memory) {
        return _toHex(abi.encodePacked(account));
    }

    function _toHex(bytes memory data) private pure returns (string memory) {
        bytes memory str = new bytes(2 + data.length * 2);
        str[0] = "0";
        str[1] = "x";
        uint256 j = 2;
        for (uint256 i = 0; i < data.length; i++) {
            uint256 a = uint8(data[i]) >> 4;
            uint256 b = uint8(data[i]) & 0x0f;
            str[j++] = bytes1(uint8(a + 48 + (a / 10) * 39));
            str[j++] = bytes1(uint8(b + 48 + (b / 10) * 39));
        }

        return string(str);
    }
}

// Part: OptionToken

/**
 * ERC20 token representing a long or short option position. It is intended to be
 * used by `OptionMarket`, which mints/burns these tokens when users buy/sell options
 *
 * Note that `decimals` should match the decimals of the `baseToken` in `OptionMarket`
 */
contract OptionToken is ERC20UpgradeSafe {
    using Address for address;
    using SafeERC20 for IERC20;
    using SafeMath for uint256;

    address public market;

    function initialize(
        address _market,
        string memory name,
        string memory symbol,
        uint8 decimals
    ) public initializer {
        __ERC20_init(name, symbol);
        _setupDecimals(decimals);
        market = _market;
    }

    function mint(address account, uint256 amount) public {
        require(msg.sender == market, "!market");
        _mint(account, amount);
    }

    function burn(address account, uint256 amount) public {
        require(msg.sender == market, "!market");
        _burn(account, amount);
    }
}

// File: OptionMarket.sol

/**
 * Automated market-maker for options
 *
 * This contract allows an asset to be split up into tokenized payoffs such that
 * different combinations of payoffs sum up to different call/put option payoffs.
 * An LMSR (Hanson's market-maker) is used to provide liquidity for the tokenized
 * payoffs.
 *
 * The parameter `b` in the LMSR represents the market depth. `b` is increased when
 * users provide liquidity by depositing funds and it is decreased when they withdraw
 * liquidity. Trading fees are distributed proportionally to liquidity providers
 * at the time of the trade.
 *
 * Call and put option with any of the supported strikes are provided. Short options
 * (equivalent to owning 1 underlying + sell 1 option) are provided, which let users
 * take on short option exposure
 *
 * `buy`, `sell`, `deposit` and `withdraw` are the main methods used to interact with
 * this contract.
 *
 * After expiration, `settle` can be called to fetch the expiry price from a
 * price oracle. `buy` and `deposit` cannot be called after expiration, but `sell`
 * can be called to redeem options for their corresponding payouts and `withdraw`
 * can be called to redeem LP tokens for a stake of the remaining funds left
 * in the contract.
 *
 * Methods to calculate the LMSR cost and option payoffs can be found in `OptionMath`.
 * `OptionToken` is an ERC20 token representing a long or short option position
 * that's minted or burned when users buy or sell options.
 *
 * This contract is also an ERC20 token itself representing shares in the liquidity
 * pool.
 *
 * The intended way to deploy this contract is to call `createMarket` in `OptionFactory`
 * Then liquidity has to be provided using `deposit` before trades can occur.
 *
 * Please note that the deployer of this contract is highly privileged and has
 * permissions such as being able to pause trading, modify the market parameters
 * and override the settlement price. These permissions will be removed in future
 * versions.
 */
contract OptionMarket is ERC20UpgradeSafe, ReentrancyGuardUpgradeSafe, OwnableUpgradeSafe {
    using Address for address;
    using SafeERC20 for IERC20;
    using UniERC20 for IERC20;
    using SafeMath for uint256;

    event Buy(
        address indexed account,
        bool isLongToken,
        uint256 strikeIndex,
        uint256 optionsOut,
        uint256 amountIn,
        uint256 newSupply
    );

    event Sell(
        address indexed account,
        bool isLongToken,
        uint256 strikeIndex,
        uint256 optionsIn,
        uint256 amountOut,
        uint256 newSupply,
        bool isSettled
    );

    event Deposit(address indexed account, uint256 sharesOut, uint256 amountIn, uint256 newB);
    event Withdraw(address indexed account, uint256 sharesIn, uint256 amountOut, uint256 newB, bool isSettled);
    event Settle(uint256 expiryPrice);

    uint256 public constant SCALE = 1e18;
    uint256 public constant SCALE_SCALE = 1e36;

    IERC20 public baseToken;
    IOracle public oracle;
    OptionToken[] public longTokens;
    OptionToken[] public shortTokens;
    uint256[] public strikePrices;
    uint256 public expiryTime;
    bool public isPut;
    uint256 public tradingFee;
    uint256 public balanceCap;
    uint256 public disputePeriod;

    bool public isPaused;
    bool public isSettled;
    uint256 public expiryPrice;

    // cache getCurrentCost and getCurrentPayoff between trades to save gas
    uint256 public lastCost;
    uint256 public lastPayoff;

    // total value of fees owed to LPs
    uint256 public poolValue;

    /**
     * @param _baseToken        Underlying asset if call. Strike currency if put
     *                          Represents ETH if equal to 0x0
     * @param _oracle           Oracle from which settlement price is obtained
     * @param _longTokens       Tokens representing long calls/puts
     * @param _shortTokens      Tokens representing short calls/puts
     * @param _strikePrices     Strike prices expressed in wei. Must be in increasing order
     * @param _expiryTime       Expiration time as a unix timestamp
     * @param _isPut            Whether this market provides calls or puts
     * @param _tradingFee       Trading fee as fraction of underlying expressed in wei
     * @param _balanceCap       Cap on TVL for guarded launch. No limit if equal to 0
     * @param _disputePeriod    How long after expiry the oracle price can be disputed by deployer
     * @param _symbol           Name and symbol of LP tokens
     */
    function initialize(
        address _baseToken,
        address _oracle,
        address[] memory _longTokens,
        address[] memory _shortTokens,
        uint256[] memory _strikePrices,
        uint256 _expiryTime,
        bool _isPut,
        uint256 _tradingFee,
        uint256 _balanceCap,
        uint256 _disputePeriod,
        string memory _symbol
    ) public payable initializer {
        // this contract is also an ERC20 token, representing shares in the liquidity pool
        __ERC20_init(_symbol, _symbol);
        __ReentrancyGuard_init();
        __Ownable_init();

        // LP tokens use same decimals as `baseToken`
        uint8 decimals = IERC20(_baseToken).isETH() ? 18 : ERC20UpgradeSafe(_baseToken).decimals();
        _setupDecimals(decimals);

        require(_longTokens.length == _strikePrices.length, "Lengths do not match");
        require(_shortTokens.length == _strikePrices.length, "Lengths do not match");

        require(_strikePrices.length > 0, "Strike prices must not be empty");
        require(_strikePrices[0] > 0, "Strike prices must be > 0");

        // check strike prices are increasing
        for (uint256 i = 0; i < _strikePrices.length - 1; i++) {
            require(_strikePrices[i] < _strikePrices[i + 1], "Strike prices must be increasing");
        }

        // trading fee can be 0
        require(_tradingFee < SCALE, "Trading fee must be < 1");

        baseToken = IERC20(_baseToken);
        oracle = IOracle(_oracle);
        strikePrices = _strikePrices;
        expiryTime = _expiryTime;
        isPut = _isPut;
        tradingFee = _tradingFee;
        balanceCap = _balanceCap;
        disputePeriod = _disputePeriod;

        for (uint256 i = 0; i < _strikePrices.length; i++) {
            longTokens.push(OptionToken(_longTokens[i]));
            shortTokens.push(OptionToken(_shortTokens[i]));
        }

        require(!isExpired(), "Already expired");
    }

    /**
     * Buy options
     *
     * The option bought is specified by `isLongToken` and `strikeIndex` and the
     * amount by `optionsOut`
     *
     * This method reverts if the resulting cost is greater than `maxAmountIn`
     */
    function buy(
        bool isLongToken,
        uint256 strikeIndex,
        uint256 optionsOut,
        uint256 maxAmountIn
    ) external payable nonReentrant returns (uint256 amountIn) {
        require(totalSupply() > 0, "No liquidity");
        require(!isExpired(), "Already expired");
        require(msg.sender == owner() || !isPaused, "Paused");
        require(strikeIndex < strikePrices.length, "Index too large");
        require(optionsOut > 0, "options out must be > 0");

        // mint options to user
        OptionToken option = isLongToken ? longTokens[strikeIndex] : shortTokens[strikeIndex];
        option.mint(msg.sender, optionsOut);

        // calculate trading fee and allocate it to the LP pool
        // like LMSR cost, fees have to be multiplied by strike price
        uint256 fee = optionsOut.mul(tradingFee);
        fee = isPut ? fee.mul(strikePrices[strikeIndex]).div(SCALE_SCALE) : fee.div(SCALE);
        poolValue = poolValue.add(fee);

        // calculate amount that needs to be paid by user to buy these options
        // it's equal to the increase in LMSR cost after minting the options
        uint256 costAfter = getCurrentCost();
        amountIn = costAfter.sub(lastCost).add(fee); // do sub first as a check since should not fail
        lastCost = costAfter;
        require(amountIn > 0, "Amount in must be > 0");
        require(amountIn <= maxAmountIn, "Max slippage exceeded");

        // transfer in amount from user
        _transferIn(amountIn);
        emit Buy(msg.sender, isLongToken, strikeIndex, optionsOut, amountIn, option.totalSupply());
    }

    /**
     * Sell options
     *
     * The option sold is specified by `isLongToken` and `strikeIndex` and the
     * amount by `optionsIn`
     *
     * This method reverts if the resulting amount returned is less than `minAmountOut`
     */
    function sell(
        bool isLongToken,
        uint256 strikeIndex,
        uint256 optionsIn,
        uint256 minAmountOut
    ) external nonReentrant returns (uint256 amountOut) {
        require(totalSupply() > 0, "No liquidity");
        require(!isExpired() || isSettled, "Must be called before expiry or after settlement");
        require(!isDisputePeriod(), "Dispute period");
        require(msg.sender == owner() || !isPaused, "Paused");
        require(strikeIndex < strikePrices.length, "Index too large");
        require(optionsIn > 0, "options in must be > 0");

        // burn user's options
        OptionToken option = isLongToken ? longTokens[strikeIndex] : shortTokens[strikeIndex];
        option.burn(msg.sender, optionsIn);

        // calculate amount that needs to be returned to user
        if (isSettled) {
            // after settlement, amount is the option payoff
            uint256 payoffAfter = getCurrentPayoff();
            amountOut = lastPayoff.sub(payoffAfter);
            lastPayoff = payoffAfter;
        } else {
            // before expiry, amount is the decrease in LMSR cost after burning the options
            uint256 costAfter = getCurrentCost();
            amountOut = lastCost.sub(costAfter);
            lastCost = costAfter;
        }
        require(amountOut > 0, "Amount out must be > 0");
        require(amountOut >= minAmountOut, "Max slippage exceeded");

        // transfer amount to user
        baseToken.uniTransfer(msg.sender, amountOut);
        emit Sell(msg.sender, isLongToken, strikeIndex, optionsIn, amountOut, option.totalSupply(), isSettled);
    }

    /**
     * Deposit liquidity
     *
     * `sharesOut` is the intended increase in the parameter `b`
     *
     * This method reverts if the resulting cost is greater than `maxAmountIn`
     */
    function deposit(uint256 sharesOut, uint256 maxAmountIn) external payable nonReentrant returns (uint256 amountIn) {
        require(!isExpired(), "Already expired");
        require(msg.sender == owner() || !isPaused, "Paused");
        require(sharesOut > 0, "sharesOut must be > 0");

        // user needs to contribute proportional amount of fees to pool, which
        // ensures they are only earning fees generated after they have deposited
        uint256 poolAmountIn;
        if (totalSupply() > 0) {
            // add 1 to round up
            poolAmountIn = poolValue.mul(sharesOut).div(totalSupply()).add(1);
            poolValue = poolValue.add(poolAmountIn);
        }
        _mint(msg.sender, sharesOut);

        // calculate amount that needs to be deposited by user
        // it's equal to the increase in LMSR cost after increasing b
        uint256 costAfter = getCurrentCost();
        amountIn = costAfter.sub(lastCost).add(poolAmountIn); // do sub first as a check since should not fail
        lastCost = costAfter;
        require(amountIn > 0, "Amount in must be > 0");
        require(amountIn <= maxAmountIn, "Max slippage exceeded");

        // transfer in amount from user
        _transferIn(amountIn);
        emit Deposit(msg.sender, sharesOut, amountIn, totalSupply());
    }

    /**
     * Withdraw liquidity
     *
     * `sharesIn` is the intended decrease in the parameter `b`
     *
     * This method reverts if the resulting amount returned is less than `minAmountOut`
     */
    function withdraw(uint256 sharesIn, uint256 minAmountOut) external nonReentrant returns (uint256 amountOut) {
        require(!isExpired() || isSettled, "Must be called before expiry or after settlement");
        require(!isDisputePeriod(), "Dispute period");
        require(msg.sender == owner() || !isPaused, "Paused");
        require(sharesIn > 0, "sharesIn must be > 0");

        // calculate the cut of fees earned by user
        uint256 poolAmountOut = poolValue.mul(sharesIn).div(totalSupply());
        poolValue = poolValue.sub(poolAmountOut);
        _burn(msg.sender, sharesIn);

        // calculate amount that needs to be returned to user
        if (isSettled) {
            // after settlement, amount is the user's share in pool
            amountOut = poolAmountOut;
        } else {
            // before expiry, amount is the decrease in LMSR cost after decreasing b
            uint256 costAfter = getCurrentCost();
            amountOut = lastCost.sub(costAfter).add(poolAmountOut); // do sub first as a check since should not fail
            lastCost = costAfter;
        }
        require(amountOut > 0, "Amount out must be > 0");
        require(amountOut >= minAmountOut, "Max slippage exceeded");

        // return amount to user
        baseToken.uniTransfer(msg.sender, amountOut);
        emit Withdraw(msg.sender, sharesIn, amountOut, totalSupply(), isSettled);
    }

    /**
     * Retrieve and store the underlying price from the oracle
     *
     * This method can be called by anyone after expiration but cannot be called
     * more than once. In practice it should be called as soon as possible after the
     * expiration time.
     */
    function settle() external nonReentrant {
        require(isExpired(), "Cannot be called before expiry");
        require(!isSettled, "Already settled");

        // fetch expiry price from oracle
        isSettled = true;
        expiryPrice = oracle.getPrice();
        require(expiryPrice > 0, "Price from oracle must be > 0");

        // update cached payoff and pool value
        lastPayoff = getCurrentPayoff();
        poolValue = baseToken.uniBalanceOf(address(this)).sub(lastPayoff);
        emit Settle(expiryPrice);
    }

    /**
     * Calculate LMSR cost
     *
     * Represents total amount locked in the LMSR
     *
     * This value will increase as options are bought and decrease as options
     * are sold. The change in value corresponds to the total cost of a purchase
     * or the amount returned from a sale.
     *
     * This method is only used before expiry. Before expiry, the `baseToken`
     * balance of this contract is always at least current cost + pool value.
     * Current cost is maximum possible amount that needs to be paid out to
     * option holders. Pool value is the fees earned by LPs.
     */
    function getCurrentCost() public view returns (uint256) {
        uint256[] memory longSupplies = getTotalSupplies(longTokens);
        uint256[] memory shortSupplies = getTotalSupplies(shortTokens);
        uint256[] memory quantities = OptionMath.calcQuantities(strikePrices, isPut, longSupplies, shortSupplies);
        return OptionMath.calcLmsrCost(quantities, totalSupply());
    }

    /**
     * Calculate option payoff
     *
     * Represents total payoff to option holders
     *
     * This value will decrease as options are redeemed. The change in value
     * corresponds to the payoff returned from a redemption.
     *
     * This method is only used after expiry. After expiry, the `baseToken` balance
     * of this contract is always at least current payoff + pool value. Current
     * payoff is the amount owed to option holders and pool value is the amount
     * owed to LPs.
     */
    function getCurrentPayoff() public view returns (uint256) {
        uint256[] memory longSupplies = getTotalSupplies(longTokens);
        uint256[] memory shortSupplies = getTotalSupplies(shortTokens);
        return OptionMath.calcPayoff(strikePrices, expiryPrice, isPut, longSupplies, shortSupplies);
    }

    function getTotalSupplies(OptionToken[] memory optionTokens) public view returns (uint256[] memory totalSupplies) {
        totalSupplies = new uint256[](optionTokens.length);
        for (uint256 i = 0; i < optionTokens.length; i++) {
            totalSupplies[i] = optionTokens[i].totalSupply();
        }
    }

    function isExpired() public view returns (bool) {
        return block.timestamp >= expiryTime;
    }

    function isDisputePeriod() public view returns (bool) {
        return block.timestamp >= expiryTime && block.timestamp < expiryTime.add(disputePeriod);
    }

    function numStrikes() external view returns (uint256) {
        return strikePrices.length;
    }

    /**
     * Transfer amount from sender and do additional checks
     */
    function _transferIn(uint256 amountIn) private {
        uint256 balanceBefore = baseToken.uniBalanceOf(address(this));
        baseToken.uniTransferFromSenderToThis(amountIn);
        uint256 balanceAfter = baseToken.uniBalanceOf(address(this));
        require(baseToken.isETH() || balanceAfter.sub(balanceBefore) == amountIn, "Deflationary tokens not supported");
        require(balanceCap == 0 || baseToken.uniBalanceOf(address(this)) <= balanceCap, "Balance cap exceeded");
    }

    // used for guarded launch. to be removed in future versions
    function setBalanceCap(uint256 _balanceCap) external onlyOwner {
        balanceCap = _balanceCap;
    }

    // emergency use only. to be removed in future versions
    function pause() external onlyOwner {
        isPaused = true;
    }

    // emergency use only. to be removed in future versions
    function unpause() external onlyOwner {
        isPaused = false;
    }

    // emergency use only. to be removed in future versions
    function setOracle(IOracle _oracle) external onlyOwner {
        oracle = _oracle;
    }

    // emergency use only. to be removed in future versions
    function setExpiryTime(uint256 _expiryTime) external onlyOwner {
        expiryTime = _expiryTime;
    }

    // emergency use only. to be removed in future versions
    function setDisputePeriod(uint256 _disputePeriod) external onlyOwner {
        disputePeriod = _disputePeriod;
    }

    // emergency use only. to be removed in future versions
    function disputeExpiryPrice(uint256 _expiryPrice) external onlyOwner {
        require(isDisputePeriod(), "Not dispute period");
        require(isSettled, "Cannot be called before settlement");
        expiryPrice = _expiryPrice;

        // update cached payoff and pool value
        lastPayoff = getCurrentPayoff();
        poolValue = baseToken.uniBalanceOf(address(this)).sub(lastPayoff);
        emit Settle(expiryPrice);
    }
}