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



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.0 <0.8.0;

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

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

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

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

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

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

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

// File: @openzeppelin/contracts/introspection/ERC165.sol



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.0 <0.8.0;

import "./IERC165.sol";

/**
 * @dev Implementation of the {IERC165} interface.
 *
 * Contracts may inherit from this and call {_registerInterface} to declare
 * their support of an interface.
 */
abstract contract ERC165 is IERC165 {
    /*
     * bytes4(keccak256('supportsInterface(bytes4)')) == 0x01ffc9a7
     */
    bytes4 private constant _INTERFACE_ID_ERC165 = 0x01ffc9a7;

    /**
     * @dev Mapping of interface ids to whether or not it's supported.
     */
    mapping(bytes4 => bool) private _supportedInterfaces;

    constructor () internal {
        // Derived contracts need only register support for their own interfaces,
        // we register support for ERC165 itself here
        _registerInterface(_INTERFACE_ID_ERC165);
    }

    /**
     * @dev See {IERC165-supportsInterface}.
     *
     * Time complexity O(1), guaranteed to always use less than 30 000 gas.
     */
    function supportsInterface(bytes4 interfaceId) public view virtual override returns (bool) {
        return _supportedInterfaces[interfaceId];
    }

    /**
     * @dev Registers the contract as an implementer of the interface defined by
     * `interfaceId`. Support of the actual ERC165 interface is automatic and
     * registering its interface id is not required.
     *
     * See {IERC165-supportsInterface}.
     *
     * Requirements:
     *
     * - `interfaceId` cannot be the ERC165 invalid interface (`0xffffffff`).
     */
    function _registerInterface(bytes4 interfaceId) internal virtual {
        require(interfaceId != 0xffffffff, "ERC165: invalid interface id");
        _supportedInterfaces[interfaceId] = true;
    }
}

// File: @openzeppelin/contracts/introspection/IERC165.sol



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.0 <0.8.0;

/**
 * @dev Interface of the ERC165 standard, as defined in the
 * https://eips.ethereum.org/EIPS/eip-165[EIP].
 *
 * Implementers can declare support of contract interfaces, which can then be
 * queried by others ({ERC165Checker}).
 *
 * For an implementation, see {ERC165}.
 */
interface IERC165 {
    /**
     * @dev Returns true if this contract implements the interface defined by
     * `interfaceId`. See the corresponding
     * https://eips.ethereum.org/EIPS/eip-165#how-interfaces-are-identified[EIP section]
     * to learn more about how these ids are created.
     *
     * This function call must use less than 30 000 gas.
     */
    function supportsInterface(bytes4 interfaceId) external view returns (bool);
}

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



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.0 <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/math/SafeMath.sol



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.0 <0.8.0;

/**
 * @dev Wrappers over Solidity's arithmetic operations with added overflow
 * checks.
 *
 * Arithmetic operations in Solidity wrap on overflow. This can easily result
 * in bugs, because programmers usually assume that an overflow raises an
 * error, which is the standard behavior in high level programming languages.
 * `SafeMath` restores this intuition by reverting the transaction when an
 * operation overflows.
 *
 * Using this library instead of the unchecked operations eliminates an entire
 * class of bugs, so it's recommended to use it always.
 */
library SafeMath {
    /**
     * @dev Returns the addition of two unsigned integers, with an overflow flag.
     *
     * _Available since v3.4._
     */
    function tryAdd(uint256 a, uint256 b) internal pure returns (bool, uint256) {
        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) {
        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) {
        // 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) {
        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) {
        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) {
        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) {
        require(b <= a, "SafeMath: subtraction overflow");
        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) {
        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, reverting 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) {
        require(b > 0, "SafeMath: division by zero");
        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) {
        require(b > 0, "SafeMath: modulo by zero");
        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) {
        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.
     *
     * CAUTION: This function is deprecated because it requires allocating memory for the error
     * message unnecessarily. For custom revert reasons use {tryDiv}.
     *
     * 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);
        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) {
        require(b > 0, errorMessage);
        return a % b;
    }
}

// File: @openzeppelin/contracts/token/ERC1155/ERC1155Holder.sol



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.0 <0.8.0;

import "./ERC1155Receiver.sol";

/**
 * @dev _Available since v3.1._
 */
contract ERC1155Holder is ERC1155Receiver {
    function onERC1155Received(address, address, uint256, uint256, bytes memory) public virtual override returns (bytes4) {
        return this.onERC1155Received.selector;
    }

    function onERC1155BatchReceived(address, address, uint256[] memory, uint256[] memory, bytes memory) public virtual override returns (bytes4) {
        return this.onERC1155BatchReceived.selector;
    }
}

// File: @openzeppelin/contracts/token/ERC1155/ERC1155Receiver.sol



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.0 <0.8.0;

import "./IERC1155Receiver.sol";
import "../../introspection/ERC165.sol";

/**
 * @dev _Available since v3.1._
 */
abstract contract ERC1155Receiver is ERC165, IERC1155Receiver {
    constructor() internal {
        _registerInterface(
            ERC1155Receiver(address(0)).onERC1155Received.selector ^
            ERC1155Receiver(address(0)).onERC1155BatchReceived.selector
        );
    }
}

// File: @openzeppelin/contracts/token/ERC1155/IERC1155.sol



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.2 <0.8.0;

import "../../introspection/IERC165.sol";

/**
 * @dev Required interface of an ERC1155 compliant contract, as defined in the
 * https://eips.ethereum.org/EIPS/eip-1155[EIP].
 *
 * _Available since v3.1._
 */
interface IERC1155 is IERC165 {
    /**
     * @dev Emitted when `value` tokens of token type `id` are transferred from `from` to `to` by `operator`.
     */
    event TransferSingle(address indexed operator, address indexed from, address indexed to, uint256 id, uint256 value);

    /**
     * @dev Equivalent to multiple {TransferSingle} events, where `operator`, `from` and `to` are the same for all
     * transfers.
     */
    event TransferBatch(address indexed operator, address indexed from, address indexed to, uint256[] ids, uint256[] values);

    /**
     * @dev Emitted when `account` grants or revokes permission to `operator` to transfer their tokens, according to
     * `approved`.
     */
    event ApprovalForAll(address indexed account, address indexed operator, bool approved);

    /**
     * @dev Emitted when the URI for token type `id` changes to `value`, if it is a non-programmatic URI.
     *
     * If an {URI} event was emitted for `id`, the standard
     * https://eips.ethereum.org/EIPS/eip-1155#metadata-extensions[guarantees] that `value` will equal the value
     * returned by {IERC1155MetadataURI-uri}.
     */
    event URI(string value, uint256 indexed id);

    /**
     * @dev Returns the amount of tokens of token type `id` owned by `account`.
     *
     * Requirements:
     *
     * - `account` cannot be the zero address.
     */
    function balanceOf(address account, uint256 id) external view returns (uint256);

    /**
     * @dev xref:ROOT:erc1155.adoc#batch-operations[Batched] version of {balanceOf}.
     *
     * Requirements:
     *
     * - `accounts` and `ids` must have the same length.
     */
    function balanceOfBatch(address[] calldata accounts, uint256[] calldata ids) external view returns (uint256[] memory);

    /**
     * @dev Grants or revokes permission to `operator` to transfer the caller's tokens, according to `approved`,
     *
     * Emits an {ApprovalForAll} event.
     *
     * Requirements:
     *
     * - `operator` cannot be the caller.
     */
    function setApprovalForAll(address operator, bool approved) external;

    /**
     * @dev Returns true if `operator` is approved to transfer ``account``'s tokens.
     *
     * See {setApprovalForAll}.
     */
    function isApprovedForAll(address account, address operator) external view returns (bool);

    /**
     * @dev Transfers `amount` tokens of token type `id` from `from` to `to`.
     *
     * Emits a {TransferSingle} event.
     *
     * Requirements:
     *
     * - `to` cannot be the zero address.
     * - If the caller is not `from`, it must be have been approved to spend ``from``'s tokens via {setApprovalForAll}.
     * - `from` must have a balance of tokens of type `id` of at least `amount`.
     * - If `to` refers to a smart contract, it must implement {IERC1155Receiver-onERC1155Received} and return the
     * acceptance magic value.
     */
    function safeTransferFrom(address from, address to, uint256 id, uint256 amount, bytes calldata data) external;

    /**
     * @dev xref:ROOT:erc1155.adoc#batch-operations[Batched] version of {safeTransferFrom}.
     *
     * Emits a {TransferBatch} event.
     *
     * Requirements:
     *
     * - `ids` and `amounts` must have the same length.
     * - If `to` refers to a smart contract, it must implement {IERC1155Receiver-onERC1155BatchReceived} and return the
     * acceptance magic value.
     */
    function safeBatchTransferFrom(address from, address to, uint256[] calldata ids, uint256[] calldata amounts, bytes calldata data) external;
}

// File: @openzeppelin/contracts/token/ERC1155/IERC1155Receiver.sol



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.0 <0.8.0;

import "../../introspection/IERC165.sol";

/**
 * _Available since v3.1._
 */
interface IERC1155Receiver is IERC165 {

    /**
        @dev Handles the receipt of a single ERC1155 token type. This function is
        called at the end of a `safeTransferFrom` after the balance has been updated.
        To accept the transfer, this must return
        `bytes4(keccak256("onERC1155Received(address,address,uint256,uint256,bytes)"))`
        (i.e. 0xf23a6e61, or its own function selector).
        @param operator The address which initiated the transfer (i.e. msg.sender)
        @param from The address which previously owned the token
        @param id The ID of the token being transferred
        @param value The amount of tokens being transferred
        @param data Additional data with no specified format
        @return `bytes4(keccak256("onERC1155Received(address,address,uint256,uint256,bytes)"))` if transfer is allowed
    */
    function onERC1155Received(
        address operator,
        address from,
        uint256 id,
        uint256 value,
        bytes calldata data
    )
        external
        returns(bytes4);

    /**
        @dev Handles the receipt of a multiple ERC1155 token types. This function
        is called at the end of a `safeBatchTransferFrom` after the balances have
        been updated. To accept the transfer(s), this must return
        `bytes4(keccak256("onERC1155BatchReceived(address,address,uint256[],uint256[],bytes)"))`
        (i.e. 0xbc197c81, or its own function selector).
        @param operator The address which initiated the batch transfer (i.e. msg.sender)
        @param from The address which previously owned the token
        @param ids An array containing ids of each token being transferred (order and length must match values array)
        @param values An array containing amounts of each token being transferred (order and length must match ids array)
        @param data Additional data with no specified format
        @return `bytes4(keccak256("onERC1155BatchReceived(address,address,uint256[],uint256[],bytes)"))` if transfer is allowed
    */
    function onERC1155BatchReceived(
        address operator,
        address from,
        uint256[] calldata ids,
        uint256[] calldata values,
        bytes calldata data
    )
        external
        returns(bytes4);
}

// File: @openzeppelin/contracts/token/ERC20/ERC20.sol



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.0 <0.8.0;

import "../../utils/Context.sol";
import "./IERC20.sol";
import "../../math/SafeMath.sol";

/**
 * @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 {ERC20PresetMinterPauser}.
 *
 * 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 ERC20 is Context, IERC20 {
    using SafeMath for uint256;

    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.
     */
    constructor (string memory name_, string memory symbol_) public {
        _name = name_;
        _symbol = symbol_;
        _decimals = 18;
    }

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

    /**
     * @dev Returns the symbol of the token, usually a shorter version of the
     * name.
     */
    function symbol() public view virtual 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 virtual returns (uint8) {
        return _decimals;
    }

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

    /**
     * @dev See {IERC20-balanceOf}.
     */
    function balanceOf(address account) public view virtual 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 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 virtual {
        _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 { }
}

// File: @openzeppelin/contracts/token/ERC20/IERC20.sol



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.0 <0.8.0;

/**
 * @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);
}

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



// SPDX-License-Identifier: MIT

pragma solidity >=0.6.0 <0.8.0;

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

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

// File: contracts/BadgerScarcityPool.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.6.0;
pragma experimental ABIEncoderV2;

import { SafeMath } from "@openzeppelin/contracts/math/SafeMath.sol";
import { Math } from "@openzeppelin/contracts/math/Math.sol";
import { IMemeLtd } from "./interfaces/IMemeLtd.sol";
import { ERC1155Holder } from "@openzeppelin/contracts/token/ERC1155/ERC1155Holder.sol";
import { Ownable } from "@openzeppelin/contracts/access/Ownable.sol";
import { BadgerScarcityPoolLib } from "./BadgerScarcityPoolLib.sol";
import { ERC20 } from "@openzeppelin/contracts/token/ERC20/ERC20.sol";

contract BadgerScarcityPool is ERC1155Holder, Ownable {
  using BadgerScarcityPoolLib for *;
  using SafeMath for *;
  BadgerScarcityPoolLib.Isolate public isolate;
  function bdigg() public view returns (ERC20) {
     return isolate.bdigg;
  }
  function memeLtd() public view returns (IMemeLtd) {
    return isolate.memeLtd;
  }
  function poolTokens(uint256 i) public view returns (BadgerScarcityPoolLib.PoolToken memory) {
    return isolate.poolTokens[i];
  }
  constructor(address _bdigg, address _memeLtd, uint256[] memory _tokenIds, uint256[] memory _roots) Ownable() public {
    require(_tokenIds.length == _roots.length, "tokenIds must be same size as roots");
    for (uint256 i = 0 ; i < _tokenIds.length; i++) {
      isolate.poolTokens.push(BadgerScarcityPoolLib.PoolToken({
        tokenId: _tokenIds[i],
        root: _roots[i]
      }));
    }
    isolate.memeLtd = IMemeLtd(_memeLtd);
    isolate.bdigg = ERC20(_bdigg);
  }
  function _assertMemeLtd() internal view {
    require(msg.sender == address(isolate.memeLtd), "can only send MemeLtd tokens");
  }
  function handleTransfer(address operator, uint256 id, uint256 value, uint256 alreadyTransferred, uint256 _reserve) internal returns (bool) {
    BadgerScarcityPoolLib.PoolToken storage poolToken = isolate.getPoolTokenRecord(id);
    require(isolate.bdigg.transfer(operator, isolate.computePayoutForToken(poolToken, value, alreadyTransferred, _reserve)), "failed to transfer BDigg");
    return true;
  }
  function withdraw(uint256[] memory ids, uint256[] memory values) public onlyOwner {
    isolate.memeLtd.safeBatchTransferFrom(address(this), msg.sender, ids, values, new bytes(0));
  }
  function onERC1155Received(address operator, address /* from */, uint256 id, uint256 value, bytes memory /* data */) public virtual override returns (bytes4) {
    _assertMemeLtd();
    require(value == 1, "can only transfer one token at a time");
    uint256 _reserve = reserve();
    require(handleTransfer(operator, id, value, value, _reserve), "handleTransfer: failure");
    return ERC1155Holder.onERC1155Received.selector;
  }
  function reserve() public view returns (uint256 result) {
    result = isolate.reserve();
  }
  function onERC1155BatchReceived(address /* operator */, address /* from */, uint256[] memory /* ids */, uint256[] memory /* values */, bytes memory /* data */) public virtual override returns (bytes4) {
    revert("batch transfers unsupported");
	  /*
    _assertMemeLtd();
    uint256 alreadyTransferred = values.sum();
    uint256 back = 0;
    for (uint256 i = 0; i < ids.length; i++) {
      back = back.add(values[i]);
      require(handleTransfer(operator, ids[i], values[i], alreadyTransferred.sub(back), reserve()), "handleTransfer: failure");
    }
    return ERC1155Holder.onERC1155BatchReceived.selector;
   */
  }
}

// File: contracts/BadgerScarcityPoolLib.sol



// SPDX-License-Identifier: MIT
pragma solidity ^0.6.0;

import { StringLib } from "./utils/StringLib.sol";
import { IMemeLtd } from "./interfaces/IMemeLtd.sol";
import { ERC20 } from "@openzeppelin/contracts/token/ERC20/ERC20.sol";
import { SafeMath } from "@openzeppelin/contracts/math/SafeMath.sol";
import { PowerLib } from "./math/PowerLib.sol";

library BadgerScarcityPoolLib {
  using StringLib for *;
  uint256 constant FIXED_1 = 0x080000000000000000000000000000000;
  using SafeMath for *;
  struct PoolToken {
    uint256 tokenId;
    uint256 root;
  }
  struct Isolate {
    ERC20 bdigg;
    IMemeLtd memeLtd;
    BadgerScarcityPoolLib.PoolToken[] poolTokens;
  }
  function reserve(Isolate storage isolate) internal view returns (uint256 balance) {
    balance = isolate.bdigg.balanceOf(address(this));
  }
  function getPoolTokenRecord(Isolate storage isolate, uint256 tokenId) internal view returns (PoolToken storage) {
    for (uint256 i = 0; i < isolate.poolTokens.length; i++) {
      if (isolate.poolTokens[i].tokenId == tokenId) return isolate.poolTokens[i];
    }
    revert(abi.encodePacked("tokenId not found: ", bytes32(tokenId).toString()).toString());
  }
  function computeTotalOutstandingForToken(Isolate storage isolate, PoolToken storage poolToken) internal view returns (uint256 result) {
    result = isolate.memeLtd.totalSupply(poolToken.tokenId).sub(isolate.memeLtd.balanceOf(address(this), poolToken.tokenId));
  }
  function computeTotalOutstanding(Isolate storage isolate, uint256 alreadyTransferred) internal view returns (uint256 result) {
    for (uint256 i = 0; i < isolate.poolTokens.length; i++) {
      result = result.add(computeTotalOutstandingForToken(isolate, isolate.poolTokens[i]));
    }
    result = result.add(alreadyTransferred); //alreadyTransferred);
  }
  function sum(uint256[] memory values) internal pure returns (uint256 total) {
    for (uint256 i = 0; i < values.length; i++) {
      total = total.add(values[i]);
    }
  }
  function power(uint256 b, uint256 exp) internal pure returns (uint256 result) {
    (uint256 mantissa, uint8 exponent) = PowerLib.power(b, 1, uint8(exp/uint256(100)), uint8(uint256(100)));
    return mantissa >> exponent;
  }
  function computePayoutForToken(Isolate storage isolate, PoolToken storage poolToken, uint256 amount, uint256 alreadyTransferred, uint256 _reserve) internal view returns (uint256) {
    {
      return _reserve.mul(power(amount.mul(uint256(1e6)).div(computeTotalOutstanding(isolate, alreadyTransferred)), poolToken.root)).div(power(uint256(1e6), poolToken.root));
    }
  }
}

// File: contracts/interfaces/IMemeLtd.sol



// SPDX-License-Identifier: MIT
pragma solidity ^0.6.0;

import { IERC1155 } from "@openzeppelin/contracts/token/ERC1155/IERC1155.sol";

abstract contract IMemeLtd is IERC1155 {
  function totalSupply(uint256 tokenId) virtual external view returns (uint256);
}

// File: contracts/math/PowerImplLib.sol



// SPDX-License-Identifier: MIT
pragma solidity >=0.6.0;

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

library PowerImplLib {
    using SafeMath for uint256;

    uint256 private constant ONE = 1;
    uint32 private constant MAX_WEIGHT = 1000000;
    uint8 private constant MIN_PRECISION = 32;
    uint8 private constant MAX_PRECISION = 127;

    // Auto-generated via 'PrintIntScalingFactors.py'
    uint256 private constant FIXED_1 = 0x080000000000000000000000000000000;
    uint256 private constant FIXED_2 = 0x100000000000000000000000000000000;
    uint256 private constant MAX_NUM = 0x200000000000000000000000000000000;

    // Auto-generated via 'PrintLn2ScalingFactors.py'
    uint256 private constant LN2_NUMERATOR = 0x3f80fe03f80fe03f80fe03f80fe03f8;
    uint256 private constant LN2_DENOMINATOR = 0x5b9de1d10bf4103d647b0955897ba80;

    // Auto-generated via 'PrintFunctionOptimalLog.py' and 'PrintFunctionOptimalExp.py'
    uint256 private constant OPT_LOG_MAX_VAL = 0x15bf0a8b1457695355fb8ac404e7a79e3;
    uint256 private constant OPT_EXP_MAX_VAL = 0x800000000000000000000000000000000;

    // Auto-generated via 'PrintLambertFactors.py'
    uint256 private constant LAMBERT_CONV_RADIUS = 0x002f16ac6c59de6f8d5d6f63c1482a7c86;
    uint256 private constant LAMBERT_POS2_SAMPLE = 0x0003060c183060c183060c183060c18306;
    uint256 private constant LAMBERT_POS2_MAXVAL = 0x01af16ac6c59de6f8d5d6f63c1482a7c80;
    uint256 private constant LAMBERT_POS3_MAXVAL = 0x6b22d43e72c326539cceeef8bb48f255ff;

    // Auto-generated via 'PrintWeightFactors.py'
    uint256 private constant MAX_UNF_WEIGHT = 0x10c6f7a0b5ed8d36b4c7f34938583621fafc8b0079a2834d26fa3fcc9ea9;

    // Auto-generated via 'PrintMaxExpArray.py'

    /**
     * @dev General Description:
     *     Determine a value of precision.
     *     Calculate an integer approximation of (_baseN / _baseD) ^ (_expN / _expD) * 2 ^ precision.
     *     Return the result along with the precision used.
     *
     * Detailed Description:
     *     Instead of calculating "base ^ exp", we calculate "e ^ (log(base) * exp)".
     *     The value of "log(base)" is represented with an integer slightly smaller than "log(base) * 2 ^ precision".
     *     The larger "precision" is, the more accurately this value represents the real value.
     *     However, the larger "precision" is, the more bits are required in order to store this value.
     *     And the exponentiation function, which takes "x" and calculates "e ^ x", is limited to a maximum exponent (maximum value of "x").
     *     This maximum exponent depends on the "precision" used, and it is given by "maxExpArray[precision] >> (MAX_PRECISION - precision)".
     *     Hence we need to determine the highest precision which can be used for the given input, before calling the exponentiation function.
     *     This allows us to compute "base ^ exp" with maximum accuracy and without exceeding 256 bits in any of the intermediate computations.
     *     This functions assumes that "_expN < 2 ^ 256 / log(MAX_NUM - 1)", otherwise the multiplication should be replaced with a "safeMul".
     *     Since we rely on unsigned-integer arithmetic and "base < 1" ==> "log(base) < 0", this function does not support "_baseN < _baseD".
     */
    function _power(
        uint256 _baseN,
        uint256 _baseD,
        uint32 _expN,
        uint32 _expD
    ) internal pure returns (uint256, uint8) {
        require(_baseN < MAX_NUM);

        uint256 baseLog;
        uint256 base = (_baseN * FIXED_1) / _baseD;
        if (base < OPT_LOG_MAX_VAL) {
            baseLog = optimalLog(base);
        } else {
            baseLog = _generalLog(base);
        }

        uint256 baseLogTimesExp = (baseLog * _expN) / _expD;
        if (baseLogTimesExp < OPT_EXP_MAX_VAL) {
            return (optimalExp(baseLogTimesExp), MAX_PRECISION);
        } else {
            uint8 precision = findPositionInMaxExpArray(baseLogTimesExp);
            return (_generalExp(baseLogTimesExp >> (MAX_PRECISION - precision), precision), precision);
        }
    }

    /**
     * @dev computes log(x / FIXED_1) * FIXED_1.
     * This functions assumes that "x >= FIXED_1", because the output would be negative otherwise.
     */
    function _generalLog(uint256 x) internal pure returns (uint256) {
        uint256 res = 0;

        // If x >= 2, then we compute the integer part of log2(x), which is larger than 0.
        if (x >= FIXED_2) {
            uint8 count = _floorLog2(x / FIXED_1);
            x >>= count; // now x < 2
            res = count * FIXED_1;
        }

        // If x > 1, then we compute the fraction part of log2(x), which is larger than 0.
        if (x > FIXED_1) {
            for (uint8 i = MAX_PRECISION; i > 0; --i) {
                x = (x * x) / FIXED_1; // now 1 < x < 4
                if (x >= FIXED_2) {
                    x >>= 1; // now 1 < x < 2
                    res += ONE << (i - 1);
                }
            }
        }

        return (res * LN2_NUMERATOR) / LN2_DENOMINATOR;
    }
    function getMaxExpArray() internal pure returns (uint256[128] memory maxExpArray) {
        //  maxExpArray[  0] = 0x6bffffffffffffffffffffffffffffffff;
        //  maxExpArray[  1] = 0x67ffffffffffffffffffffffffffffffff;
        //  maxExpArray[  2] = 0x637fffffffffffffffffffffffffffffff;
        //  maxExpArray[  3] = 0x5f6fffffffffffffffffffffffffffffff;
        //  maxExpArray[  4] = 0x5b77ffffffffffffffffffffffffffffff;
        //  maxExpArray[  5] = 0x57b3ffffffffffffffffffffffffffffff;
        //  maxExpArray[  6] = 0x5419ffffffffffffffffffffffffffffff;
        //  maxExpArray[  7] = 0x50a2ffffffffffffffffffffffffffffff;
        //  maxExpArray[  8] = 0x4d517fffffffffffffffffffffffffffff;
        //  maxExpArray[  9] = 0x4a233fffffffffffffffffffffffffffff;
        //  maxExpArray[ 10] = 0x47165fffffffffffffffffffffffffffff;
        //  maxExpArray[ 11] = 0x4429afffffffffffffffffffffffffffff;
        //  maxExpArray[ 12] = 0x415bc7ffffffffffffffffffffffffffff;
        //  maxExpArray[ 13] = 0x3eab73ffffffffffffffffffffffffffff;
        //  maxExpArray[ 14] = 0x3c1771ffffffffffffffffffffffffffff;
        //  maxExpArray[ 15] = 0x399e96ffffffffffffffffffffffffffff;
        //  maxExpArray[ 16] = 0x373fc47fffffffffffffffffffffffffff;
        //  maxExpArray[ 17] = 0x34f9e8ffffffffffffffffffffffffffff;
        //  maxExpArray[ 18] = 0x32cbfd5fffffffffffffffffffffffffff;
        //  maxExpArray[ 19] = 0x30b5057fffffffffffffffffffffffffff;
        //  maxExpArray[ 20] = 0x2eb40f9fffffffffffffffffffffffffff;
        //  maxExpArray[ 21] = 0x2cc8340fffffffffffffffffffffffffff;
        //  maxExpArray[ 22] = 0x2af09481ffffffffffffffffffffffffff;
        //  maxExpArray[ 23] = 0x292c5bddffffffffffffffffffffffffff;
        //  maxExpArray[ 24] = 0x277abdcdffffffffffffffffffffffffff;
        //  maxExpArray[ 25] = 0x25daf6657fffffffffffffffffffffffff;
        //  maxExpArray[ 26] = 0x244c49c65fffffffffffffffffffffffff;
        //  maxExpArray[ 27] = 0x22ce03cd5fffffffffffffffffffffffff;
        //  maxExpArray[ 28] = 0x215f77c047ffffffffffffffffffffffff;
        //  maxExpArray[ 29] = 0x1fffffffffffffffffffffffffffffffff;
        //  maxExpArray[ 30] = 0x1eaefdbdabffffffffffffffffffffffff;
        //  maxExpArray[ 31] = 0x1d6bd8b2ebffffffffffffffffffffffff;
        maxExpArray[32] = 0x1c35fedd14ffffffffffffffffffffffff;
        maxExpArray[33] = 0x1b0ce43b323fffffffffffffffffffffff;
        maxExpArray[34] = 0x19f0028ec1ffffffffffffffffffffffff;
        maxExpArray[35] = 0x18ded91f0e7fffffffffffffffffffffff;
        maxExpArray[36] = 0x17d8ec7f0417ffffffffffffffffffffff;
        maxExpArray[37] = 0x16ddc6556cdbffffffffffffffffffffff;
        maxExpArray[38] = 0x15ecf52776a1ffffffffffffffffffffff;
        maxExpArray[39] = 0x15060c256cb2ffffffffffffffffffffff;
        maxExpArray[40] = 0x1428a2f98d72ffffffffffffffffffffff;
        maxExpArray[41] = 0x13545598e5c23fffffffffffffffffffff;
        maxExpArray[42] = 0x1288c4161ce1dfffffffffffffffffffff;
        maxExpArray[43] = 0x11c592761c666fffffffffffffffffffff;
        maxExpArray[44] = 0x110a688680a757ffffffffffffffffffff;
        maxExpArray[45] = 0x1056f1b5bedf77ffffffffffffffffffff;
        maxExpArray[46] = 0x0faadceceeff8bffffffffffffffffffff;
        maxExpArray[47] = 0x0f05dc6b27edadffffffffffffffffffff;
        maxExpArray[48] = 0x0e67a5a25da4107fffffffffffffffffff;
        maxExpArray[49] = 0x0dcff115b14eedffffffffffffffffffff;
        maxExpArray[50] = 0x0d3e7a392431239fffffffffffffffffff;
        maxExpArray[51] = 0x0cb2ff529eb71e4fffffffffffffffffff;
        maxExpArray[52] = 0x0c2d415c3db974afffffffffffffffffff;
        maxExpArray[53] = 0x0bad03e7d883f69bffffffffffffffffff;
        maxExpArray[54] = 0x0b320d03b2c343d5ffffffffffffffffff;
        maxExpArray[55] = 0x0abc25204e02828dffffffffffffffffff;
        maxExpArray[56] = 0x0a4b16f74ee4bb207fffffffffffffffff;
        maxExpArray[57] = 0x09deaf736ac1f569ffffffffffffffffff;
        maxExpArray[58] = 0x0976bd9952c7aa957fffffffffffffffff;
        maxExpArray[59] = 0x09131271922eaa606fffffffffffffffff;
        maxExpArray[60] = 0x08b380f3558668c46fffffffffffffffff;
        maxExpArray[61] = 0x0857ddf0117efa215bffffffffffffffff;
        maxExpArray[62] = 0x07ffffffffffffffffffffffffffffffff;
        maxExpArray[63] = 0x07abbf6f6abb9d087fffffffffffffffff;
        maxExpArray[64] = 0x075af62cbac95f7dfa7fffffffffffffff;
        maxExpArray[65] = 0x070d7fb7452e187ac13fffffffffffffff;
        maxExpArray[66] = 0x06c3390ecc8af379295fffffffffffffff;
        maxExpArray[67] = 0x067c00a3b07ffc01fd6fffffffffffffff;
        maxExpArray[68] = 0x0637b647c39cbb9d3d27ffffffffffffff;
        maxExpArray[69] = 0x05f63b1fc104dbd39587ffffffffffffff;
        maxExpArray[70] = 0x05b771955b36e12f7235ffffffffffffff;
        maxExpArray[71] = 0x057b3d49dda84556d6f6ffffffffffffff;
        maxExpArray[72] = 0x054183095b2c8ececf30ffffffffffffff;
        maxExpArray[73] = 0x050a28be635ca2b888f77fffffffffffff;
        maxExpArray[74] = 0x04d5156639708c9db33c3fffffffffffff;
        maxExpArray[75] = 0x04a23105873875bd52dfdfffffffffffff;
        maxExpArray[76] = 0x0471649d87199aa990756fffffffffffff;
        maxExpArray[77] = 0x04429a21a029d4c1457cfbffffffffffff;
        maxExpArray[78] = 0x0415bc6d6fb7dd71af2cb3ffffffffffff;
        maxExpArray[79] = 0x03eab73b3bbfe282243ce1ffffffffffff;
        maxExpArray[80] = 0x03c1771ac9fb6b4c18e229ffffffffffff;
        maxExpArray[81] = 0x0399e96897690418f785257fffffffffff;
        maxExpArray[82] = 0x0373fc456c53bb779bf0ea9fffffffffff;
        maxExpArray[83] = 0x034f9e8e490c48e67e6ab8bfffffffffff;
        maxExpArray[84] = 0x032cbfd4a7adc790560b3337ffffffffff;
        maxExpArray[85] = 0x030b50570f6e5d2acca94613ffffffffff;
        maxExpArray[86] = 0x02eb40f9f620fda6b56c2861ffffffffff;
        maxExpArray[87] = 0x02cc8340ecb0d0f520a6af58ffffffffff;
        maxExpArray[88] = 0x02af09481380a0a35cf1ba02ffffffffff;
        maxExpArray[89] = 0x0292c5bdd3b92ec810287b1b3fffffffff;
        maxExpArray[90] = 0x0277abdcdab07d5a77ac6d6b9fffffffff;
        maxExpArray[91] = 0x025daf6654b1eaa55fd64df5efffffffff;
        maxExpArray[92] = 0x0244c49c648baa98192dce88b7ffffffff;
        maxExpArray[93] = 0x022ce03cd5619a311b2471268bffffffff;
        maxExpArray[94] = 0x0215f77c045fbe885654a44a0fffffffff;
        maxExpArray[95] = 0x01ffffffffffffffffffffffffffffffff;
        maxExpArray[96] = 0x01eaefdbdaaee7421fc4d3ede5ffffffff;
        maxExpArray[97] = 0x01d6bd8b2eb257df7e8ca57b09bfffffff;
        maxExpArray[98] = 0x01c35fedd14b861eb0443f7f133fffffff;
        maxExpArray[99] = 0x01b0ce43b322bcde4a56e8ada5afffffff;
        maxExpArray[100] = 0x019f0028ec1fff007f5a195a39dfffffff;
        maxExpArray[101] = 0x018ded91f0e72ee74f49b15ba527ffffff;
        maxExpArray[102] = 0x017d8ec7f04136f4e5615fd41a63ffffff;
        maxExpArray[103] = 0x016ddc6556cdb84bdc8d12d22e6fffffff;
        maxExpArray[104] = 0x015ecf52776a1155b5bd8395814f7fffff;
        maxExpArray[105] = 0x015060c256cb23b3b3cc3754cf40ffffff;
        maxExpArray[106] = 0x01428a2f98d728ae223ddab715be3fffff;
        maxExpArray[107] = 0x013545598e5c23276ccf0ede68034fffff;
        maxExpArray[108] = 0x01288c4161ce1d6f54b7f61081194fffff;
        maxExpArray[109] = 0x011c592761c666aa641d5a01a40f17ffff;
        maxExpArray[110] = 0x0110a688680a7530515f3e6e6cfdcdffff;
        maxExpArray[111] = 0x01056f1b5bedf75c6bcb2ce8aed428ffff;
        maxExpArray[112] = 0x00faadceceeff8a0890f3875f008277fff;
        maxExpArray[113] = 0x00f05dc6b27edad306388a600f6ba0bfff;
        maxExpArray[114] = 0x00e67a5a25da41063de1495d5b18cdbfff;
        maxExpArray[115] = 0x00dcff115b14eedde6fc3aa5353f2e4fff;
        maxExpArray[116] = 0x00d3e7a3924312399f9aae2e0f868f8fff;
        maxExpArray[117] = 0x00cb2ff529eb71e41582cccd5a1ee26fff;
        maxExpArray[118] = 0x00c2d415c3db974ab32a51840c0b67edff;
        maxExpArray[119] = 0x00bad03e7d883f69ad5b0a186184e06bff;
        maxExpArray[120] = 0x00b320d03b2c343d4829abd6075f0cc5ff;
        maxExpArray[121] = 0x00abc25204e02828d73c6e80bcdb1a95bf;
        maxExpArray[122] = 0x00a4b16f74ee4bb2040a1ec6c15fbbf2df;
        maxExpArray[123] = 0x009deaf736ac1f569deb1b5ae3f36c130f;
        maxExpArray[124] = 0x00976bd9952c7aa957f5937d790ef65037;
        maxExpArray[125] = 0x009131271922eaa6064b73a22d0bd4f2bf;
        maxExpArray[126] = 0x008b380f3558668c46c91c49a2f8e967b9;
        maxExpArray[127] = 0x00857ddf0117efa215952912839f6473e6;
    }

    /**
     * @dev computes the largest integer smaller than or equal to the binary logarithm of the input.
     */
    function _floorLog2(uint256 _n) internal pure returns (uint8) {
        uint8 res = 0;

        if (_n < 256) {
            // At most 8 iterations
            while (_n > 1) {
                _n >>= 1;
                res += 1;
            }
        } else {
            // Exactly 8 iterations
            for (uint8 s = 128; s > 0; s >>= 1) {
                if (_n >= (ONE << s)) {
                    _n >>= s;
                    res |= s;
                }
            }
        }

        return res;
    }

    /**
     * @dev the global "maxExpArray" is sorted in descending order, and therefore the following statements are equivalent:
     * - This function finds the position of [the smallest value in "maxExpArray" larger than or equal to "x"]
     * - This function finds the highest position of [a value in "maxExpArray" larger than or equal to "x"]
     */
    function findPositionInMaxExpArray(uint256 _x) internal pure returns (uint8) {
        uint256[128] memory maxExpArray = getMaxExpArray();
        uint8 lo = MIN_PRECISION;
        uint8 hi = MAX_PRECISION;

        while (lo + 1 < hi) {
            uint8 mid = (lo + hi) / 2;
            if (maxExpArray[mid] >= _x) lo = mid;
            else hi = mid;
        }

        if (maxExpArray[hi] >= _x) return hi;
        if (maxExpArray[lo] >= _x) return lo;
        revert("PowerImplLib.sol: out of bounds");
    }

    /**
     * @dev this function can be auto-generated by the script 'PrintFunctionGeneralExp.py'.
     * it approximates "e ^ x" via maclaurin summation: "(x^0)/0! + (x^1)/1! + ... + (x^n)/n!".
     * it returns "e ^ (x / 2 ^ precision) * 2 ^ precision", that is, the result is upshifted for accuracy.
     * the global "maxExpArray" maps each "precision" to "((maximumExponent + 1) << (MAX_PRECISION - precision)) - 1".
     * the maximum permitted value for "x" is therefore given by "maxExpArray[precision] >> (MAX_PRECISION - precision)".
     */
    function _generalExp(uint256 _x, uint8 _precision) internal pure returns (uint256) {
        uint256 xi = _x;
        uint256 res = 0;

        xi = (xi * _x) >> _precision;
        res += xi * 0x3442c4e6074a82f1797f72ac0000000; // add x^02 * (33! / 02!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x116b96f757c380fb287fd0e40000000; // add x^03 * (33! / 03!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x045ae5bdd5f0e03eca1ff4390000000; // add x^04 * (33! / 04!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x00defabf91302cd95b9ffda50000000; // add x^05 * (33! / 05!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x002529ca9832b22439efff9b8000000; // add x^06 * (33! / 06!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x00054f1cf12bd04e516b6da88000000; // add x^07 * (33! / 07!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x0000a9e39e257a09ca2d6db51000000; // add x^08 * (33! / 08!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x000012e066e7b839fa050c309000000; // add x^09 * (33! / 09!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x000001e33d7d926c329a1ad1a800000; // add x^10 * (33! / 10!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x0000002bee513bdb4a6b19b5f800000; // add x^11 * (33! / 11!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x00000003a9316fa79b88eccf2a00000; // add x^12 * (33! / 12!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x0000000048177ebe1fa812375200000; // add x^13 * (33! / 13!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x0000000005263fe90242dcbacf00000; // add x^14 * (33! / 14!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x000000000057e22099c030d94100000; // add x^15 * (33! / 15!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x0000000000057e22099c030d9410000; // add x^16 * (33! / 16!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x00000000000052b6b54569976310000; // add x^17 * (33! / 17!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x00000000000004985f67696bf748000; // add x^18 * (33! / 18!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x000000000000003dea12ea99e498000; // add x^19 * (33! / 19!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x00000000000000031880f2214b6e000; // add x^20 * (33! / 20!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x000000000000000025bcff56eb36000; // add x^21 * (33! / 21!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x000000000000000001b722e10ab1000; // add x^22 * (33! / 22!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x0000000000000000001317c70077000; // add x^23 * (33! / 23!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x00000000000000000000cba84aafa00; // add x^24 * (33! / 24!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x00000000000000000000082573a0a00; // add x^25 * (33! / 25!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x00000000000000000000005035ad900; // add x^26 * (33! / 26!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x000000000000000000000002f881b00; // add x^27 * (33! / 27!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x0000000000000000000000001b29340; // add x^28 * (33! / 28!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x00000000000000000000000000efc40; // add x^29 * (33! / 29!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x0000000000000000000000000007fe0; // add x^30 * (33! / 30!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x0000000000000000000000000000420; // add x^31 * (33! / 31!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x0000000000000000000000000000021; // add x^32 * (33! / 32!)
        xi = (xi * _x) >> _precision;
        res += xi * 0x0000000000000000000000000000001; // add x^33 * (33! / 33!)

        return res / 0x688589cc0e9505e2f2fee5580000000 + _x + (ONE << _precision); // divide by 33! and then add x^1 / 1! + x^0 / 0!
    }

    /**
     * @dev computes log(x / FIXED_1) * FIXED_1
     * Input range: FIXED_1 <= x <= OPT_LOG_MAX_VAL - 1
     * Auto-generated via 'PrintFunctionOptimalLog.py'
     * Detailed description:
     * - Rewrite the input as a product of natural exponents and a single residual r, such that 1 < r < 2
     * - The natural logarithm of each (pre-calculated) exponent is the degree of the exponent
     * - The natural logarithm of r is calculated via Taylor series for log(1 + x), where x = r - 1
     * - The natural logarithm of the input is calculated by summing up the intermediate results above
     * - For example: log(250) = log(e^4 * e^1 * e^0.5 * 1.021692859) = 4 + 1 + 0.5 + log(1 + 0.021692859)
     */
    function optimalLog(uint256 x) internal pure returns (uint256) {
        uint256 res = 0;

        uint256 y;
        uint256 z;
        uint256 w;

        if (x >= 0xd3094c70f034de4b96ff7d5b6f99fcd8) {
            res += 0x40000000000000000000000000000000;
            x = (x * FIXED_1) / 0xd3094c70f034de4b96ff7d5b6f99fcd8;
        } // add 1 / 2^1
        if (x >= 0xa45af1e1f40c333b3de1db4dd55f29a7) {
            res += 0x20000000000000000000000000000000;
            x = (x * FIXED_1) / 0xa45af1e1f40c333b3de1db4dd55f29a7;
        } // add 1 / 2^2
        if (x >= 0x910b022db7ae67ce76b441c27035c6a1) {
            res += 0x10000000000000000000000000000000;
            x = (x * FIXED_1) / 0x910b022db7ae67ce76b441c27035c6a1;
        } // add 1 / 2^3
        if (x >= 0x88415abbe9a76bead8d00cf112e4d4a8) {
            res += 0x08000000000000000000000000000000;
            x = (x * FIXED_1) / 0x88415abbe9a76bead8d00cf112e4d4a8;
        } // add 1 / 2^4
        if (x >= 0x84102b00893f64c705e841d5d4064bd3) {
            res += 0x04000000000000000000000000000000;
            x = (x * FIXED_1) / 0x84102b00893f64c705e841d5d4064bd3;
        } // add 1 / 2^5
        if (x >= 0x8204055aaef1c8bd5c3259f4822735a2) {
            res += 0x02000000000000000000000000000000;
            x = (x * FIXED_1) / 0x8204055aaef1c8bd5c3259f4822735a2;
        } // add 1 / 2^6
        if (x >= 0x810100ab00222d861931c15e39b44e99) {
            res += 0x01000000000000000000000000000000;
            x = (x * FIXED_1) / 0x810100ab00222d861931c15e39b44e99;
        } // add 1 / 2^7
        if (x >= 0x808040155aabbbe9451521693554f733) {
            res += 0x00800000000000000000000000000000;
            x = (x * FIXED_1) / 0x808040155aabbbe9451521693554f733;
        } // add 1 / 2^8

        z = y = x - FIXED_1;
        w = (y * y) / FIXED_1;
        res += (z * (0x100000000000000000000000000000000 - y)) / 0x100000000000000000000000000000000;
        z = (z * w) / FIXED_1; // add y^01 / 01 - y^02 / 02
        res += (z * (0x0aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa - y)) / 0x200000000000000000000000000000000;
        z = (z * w) / FIXED_1; // add y^03 / 03 - y^04 / 04
        res += (z * (0x099999999999999999999999999999999 - y)) / 0x300000000000000000000000000000000;
        z = (z * w) / FIXED_1; // add y^05 / 05 - y^06 / 06
        res += (z * (0x092492492492492492492492492492492 - y)) / 0x400000000000000000000000000000000;
        z = (z * w) / FIXED_1; // add y^07 / 07 - y^08 / 08
        res += (z * (0x08e38e38e38e38e38e38e38e38e38e38e - y)) / 0x500000000000000000000000000000000;
        z = (z * w) / FIXED_1; // add y^09 / 09 - y^10 / 10
        res += (z * (0x08ba2e8ba2e8ba2e8ba2e8ba2e8ba2e8b - y)) / 0x600000000000000000000000000000000;
        z = (z * w) / FIXED_1; // add y^11 / 11 - y^12 / 12
        res += (z * (0x089d89d89d89d89d89d89d89d89d89d89 - y)) / 0x700000000000000000000000000000000;
        z = (z * w) / FIXED_1; // add y^13 / 13 - y^14 / 14
        res += (z * (0x088888888888888888888888888888888 - y)) / 0x800000000000000000000000000000000; // add y^15 / 15 - y^16 / 16

        return res;
    }

    /**
     * @dev computes e ^ (x / FIXED_1) * FIXED_1
     * input range: 0 <= x <= OPT_EXP_MAX_VAL - 1
     * auto-generated via 'PrintFunctionOptimalExp.py'
     * Detailed description:
     * - Rewrite the input as a sum of binary exponents and a single residual r, as small as possible
     * - The exponentiation of each binary exponent is given (pre-calculated)
     * - The exponentiation of r is calculated via Taylor series for e^x, where x = r
     * - The exponentiation of the input is calculated by multiplying the intermediate results above
     * - For example: e^5.521692859 = e^(4 + 1 + 0.5 + 0.021692859) = e^4 * e^1 * e^0.5 * e^0.021692859
     */
    function optimalExp(uint256 x) internal pure returns (uint256) {
        uint256 res = 0;

        uint256 y;
        uint256 z;

        z = y = x % 0x10000000000000000000000000000000; // get the input modulo 2^(-3)
        z = (z * y) / FIXED_1;
        res += z * 0x10e1b3be415a0000; // add y^02 * (20! / 02!)
        z = (z * y) / FIXED_1;
        res += z * 0x05a0913f6b1e0000; // add y^03 * (20! / 03!)
        z = (z * y) / FIXED_1;
        res += z * 0x0168244fdac78000; // add y^04 * (20! / 04!)
        z = (z * y) / FIXED_1;
        res += z * 0x004807432bc18000; // add y^05 * (20! / 05!)
        z = (z * y) / FIXED_1;
        res += z * 0x000c0135dca04000; // add y^06 * (20! / 06!)
        z = (z * y) / FIXED_1;
        res += z * 0x0001b707b1cdc000; // add y^07 * (20! / 07!)
        z = (z * y) / FIXED_1;
        res += z * 0x000036e0f639b800; // add y^08 * (20! / 08!)
        z = (z * y) / FIXED_1;
        res += z * 0x00000618fee9f800; // add y^09 * (20! / 09!)
        z = (z * y) / FIXED_1;
        res += z * 0x0000009c197dcc00; // add y^10 * (20! / 10!)
        z = (z * y) / FIXED_1;
        res += z * 0x0000000e30dce400; // add y^11 * (20! / 11!)
        z = (z * y) / FIXED_1;
        res += z * 0x000000012ebd1300; // add y^12 * (20! / 12!)
        z = (z * y) / FIXED_1;
        res += z * 0x0000000017499f00; // add y^13 * (20! / 13!)
        z = (z * y) / FIXED_1;
        res += z * 0x0000000001a9d480; // add y^14 * (20! / 14!)
        z = (z * y) / FIXED_1;
        res += z * 0x00000000001c6380; // add y^15 * (20! / 15!)
        z = (z * y) / FIXED_1;
        res += z * 0x000000000001c638; // add y^16 * (20! / 16!)
        z = (z * y) / FIXED_1;
        res += z * 0x0000000000001ab8; // add y^17 * (20! / 17!)
        z = (z * y) / FIXED_1;
        res += z * 0x000000000000017c; // add y^18 * (20! / 18!)
        z = (z * y) / FIXED_1;
        res += z * 0x0000000000000014; // add y^19 * (20! / 19!)
        z = (z * y) / FIXED_1;
        res += z * 0x0000000000000001; // add y^20 * (20! / 20!)
        res = res / 0x21c3677c82b40000 + y + FIXED_1; // divide by 20! and then add y^1 / 1! + y^0 / 0!

        if ((x & 0x010000000000000000000000000000000) != 0)
            res = (res * 0x1c3d6a24ed82218787d624d3e5eba95f9) / 0x18ebef9eac820ae8682b9793ac6d1e776; // multiply by e^2^(-3)
        if ((x & 0x020000000000000000000000000000000) != 0)
            res = (res * 0x18ebef9eac820ae8682b9793ac6d1e778) / 0x1368b2fc6f9609fe7aceb46aa619baed4; // multiply by e^2^(-2)
        if ((x & 0x040000000000000000000000000000000) != 0)
            res = (res * 0x1368b2fc6f9609fe7aceb46aa619baed5) / 0x0bc5ab1b16779be3575bd8f0520a9f21f; // multiply by e^2^(-1)
        if ((x & 0x080000000000000000000000000000000) != 0)
            res = (res * 0x0bc5ab1b16779be3575bd8f0520a9f21e) / 0x0454aaa8efe072e7f6ddbab84b40a55c9; // multiply by e^2^(+0)
        if ((x & 0x100000000000000000000000000000000) != 0)
            res = (res * 0x0454aaa8efe072e7f6ddbab84b40a55c5) / 0x00960aadc109e7a3bf4578099615711ea; // multiply by e^2^(+1)
        if ((x & 0x200000000000000000000000000000000) != 0)
            res = (res * 0x00960aadc109e7a3bf4578099615711d7) / 0x0002bf84208204f5977f9a8cf01fdce3d; // multiply by e^2^(+2)
        if ((x & 0x400000000000000000000000000000000) != 0)
            res = (res * 0x0002bf84208204f5977f9a8cf01fdc307) / 0x0000003c6ab775dd0b95b4cbee7e65d11; // multiply by e^2^(+3)

        return res;
    }
}

// File: contracts/math/PowerLib.sol



// SPDX-License-Identifier: MIT
pragma solidity >=0.6.0;

import { PowerImplLib } from "./PowerImplLib.sol";

library PowerLib {
  function power(uint256 baseN, uint256 baseD, uint8 expN, uint8 expD) external pure returns (uint256, uint8) {
    return PowerImplLib._power(baseN, baseD, expN, expD);
  }
}

// File: contracts/utils/StringLib.sol



pragma experimental ABIEncoderV2;
// SPDX-License-Identifier: MIT
pragma solidity ^0.6.0;

library StringLib {
    /// @notice Convert a uint value to its decimal string representation
    // solium-disable-next-line security/no-assign-params
    function toString(uint256 _i) internal pure returns (string memory) {
        if (_i == 0) {
            return "0";
        }
        uint256 j = _i;
        uint256 len;
        while (j != 0) {
            len++;
            j /= 10;
        }
        bytes memory bstr = new bytes(len);
        uint256 k = len - 1;
        while (_i != 0) {
            bstr[k--] = bytes1(uint8(48 + (_i % 10)));
            _i /= 10;
        }
        return string(bstr);
    }

    /// @notice Convert a bytes32 value to its hex string representation
    function toString(bytes32 _value) internal pure returns (string memory) {
        bytes memory alphabet = "0123456789abcdef";

        bytes memory str = new bytes(32 * 2 + 2);
        str[0] = "0";
        str[1] = "x";
        for (uint256 i = 0; i < 32; i++) {
            str[2 + i * 2] = alphabet[uint256(uint8(_value[i] >> 4))];
            str[3 + i * 2] = alphabet[uint256(uint8(_value[i] & 0x0f))];
        }
        return string(str);
    }

    /// @notice Convert an address to its hex string representation
    function toString(address _addr) internal pure returns (string memory) {
        bytes32 value = bytes32(uint256(uint160(_addr)));
        bytes memory alphabet = "0123456789abcdef";

        bytes memory str = new bytes(20 * 2 + 2);
        str[0] = "0";
        str[1] = "x";
        for (uint256 i = 0; i < 20; i++) {
            str[2 + i * 2] = alphabet[uint256(uint8(value[i + 12] >> 4))];
            str[3 + i * 2] = alphabet[uint256(uint8(value[i + 12] & 0x0f))];
        }
        return string(str);
    }
    function toString(bytes memory input) internal pure returns (string memory) {
      return string(input);
    }
}