// 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/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/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/optimistic-ethereum/OVM/chain/OVM_StateCommitmentChain.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;
pragma experimental ABIEncoderV2;

/* Library Imports */
import { Lib_OVMCodec } from "../../libraries/codec/Lib_OVMCodec.sol";
import { Lib_AddressResolver } from "../../libraries/resolver/Lib_AddressResolver.sol";
import { Lib_MerkleTree } from "../../libraries/utils/Lib_MerkleTree.sol";

/* Interface Imports */
import { iOVM_FraudVerifier } from "../../iOVM/verification/iOVM_FraudVerifier.sol";
import { iOVM_StateCommitmentChain } from "../../iOVM/chain/iOVM_StateCommitmentChain.sol";
import { iOVM_CanonicalTransactionChain } from "../../iOVM/chain/iOVM_CanonicalTransactionChain.sol";
import { iOVM_BondManager } from "../../iOVM/verification/iOVM_BondManager.sol";
import { iOVM_ChainStorageContainer } from "../../iOVM/chain/iOVM_ChainStorageContainer.sol";

/* External Imports */
import '@openzeppelin/contracts/math/SafeMath.sol';

/**
 * @title OVM_StateCommitmentChain
 * @dev The State Commitment Chain (SCC) contract contains a list of proposed state roots which
 * Proposers assert to be a result of each transaction in the Canonical Transaction Chain (CTC).
 * Elements here have a 1:1 correspondence with transactions in the CTC, and should be the unique
 * state root calculated off-chain by applying the canonical transactions one by one.
 *
 * Compiler used: solc
 * Runtime target: EVM
 */
contract OVM_StateCommitmentChain is iOVM_StateCommitmentChain, Lib_AddressResolver {

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

    uint256 public FRAUD_PROOF_WINDOW;
    uint256 public SEQUENCER_PUBLISH_WINDOW;


    /***************
     * Constructor *
     ***************/

    /**
     * @param _libAddressManager Address of the Address Manager.
     */
    constructor(
        address _libAddressManager,
        uint256 _fraudProofWindow,
        uint256 _sequencerPublishWindow
    )
        Lib_AddressResolver(_libAddressManager)
    {
        FRAUD_PROOF_WINDOW = _fraudProofWindow;
        SEQUENCER_PUBLISH_WINDOW = _sequencerPublishWindow;
    }


    /********************
     * Public Functions *
     ********************/

    /**
     * Accesses the batch storage container.
     * @return Reference to the batch storage container.
     */
    function batches()
        public
        view
        returns (
            iOVM_ChainStorageContainer
        )
    {
        return iOVM_ChainStorageContainer(
            resolve("OVM_ChainStorageContainer-SCC-batches")
        );
    }

    /**
     * @inheritdoc iOVM_StateCommitmentChain
     */
    function getTotalElements()
        override
        public
        view
        returns (
            uint256 _totalElements
        )
    {
        (uint40 totalElements, ) = _getBatchExtraData();
        return uint256(totalElements);
    }

    /**
     * @inheritdoc iOVM_StateCommitmentChain
     */
    function getTotalBatches()
        override
        public
        view
        returns (
            uint256 _totalBatches
        )
    {
        return batches().length();
    }

    /**
     * @inheritdoc iOVM_StateCommitmentChain
     */
    function getLastSequencerTimestamp()
        override
        public
        view
        returns (
            uint256 _lastSequencerTimestamp
        )
    {
        (, uint40 lastSequencerTimestamp) = _getBatchExtraData();
        return uint256(lastSequencerTimestamp);
    }

    /**
     * @inheritdoc iOVM_StateCommitmentChain
     */
    function appendStateBatch(
        bytes32[] memory _batch,
        uint256 _shouldStartAtElement
    )
        override
        public
    {
        // Fail fast in to make sure our batch roots aren't accidentally made fraudulent by the
        // publication of batches by some other user.
        require(
            _shouldStartAtElement == getTotalElements(),
            "Actual batch start index does not match expected start index."
        );

        // Proposers must have previously staked at the BondManager
        require(
            iOVM_BondManager(resolve("OVM_BondManager")).isCollateralized(msg.sender),
            "Proposer does not have enough collateral posted"
        );

        require(
            _batch.length > 0,
            "Cannot submit an empty state batch."
        );

        require(
            getTotalElements() + _batch.length <= iOVM_CanonicalTransactionChain(resolve("OVM_CanonicalTransactionChain")).getTotalElements(),
            "Number of state roots cannot exceed the number of canonical transactions."
        );

        // Pass the block's timestamp and the publisher of the data
        // to be used in the fraud proofs
        _appendBatch(
            _batch,
            abi.encode(block.timestamp, msg.sender)
        );
    }

    /**
     * @inheritdoc iOVM_StateCommitmentChain
     */
    function deleteStateBatch(
        Lib_OVMCodec.ChainBatchHeader memory _batchHeader
    )
        override
        public
    {
        require(
            msg.sender == resolve("OVM_FraudVerifier"),
            "State batches can only be deleted by the OVM_FraudVerifier."
        );

        require(
            _isValidBatchHeader(_batchHeader),
            "Invalid batch header."
        );

        require(
            insideFraudProofWindow(_batchHeader),
            "State batches can only be deleted within the fraud proof window."
        );

        _deleteBatch(_batchHeader);
    }

    /**
     * @inheritdoc iOVM_StateCommitmentChain
     */
    function verifyStateCommitment(
        bytes32 _element,
        Lib_OVMCodec.ChainBatchHeader memory _batchHeader,
        Lib_OVMCodec.ChainInclusionProof memory _proof
    )
        override
        public
        view
        returns (
            bool
        )
    {
        require(
            _isValidBatchHeader(_batchHeader),
            "Invalid batch header."
        );

        require(
            Lib_MerkleTree.verify(
                _batchHeader.batchRoot,
                _element,
                _proof.index,
                _proof.siblings,
                _batchHeader.batchSize
            ),
            "Invalid inclusion proof."
        );

        return true;
    }

    /**
     * @inheritdoc iOVM_StateCommitmentChain
     */
    function insideFraudProofWindow(
        Lib_OVMCodec.ChainBatchHeader memory _batchHeader
    )
        override
        public
        view
        returns (
            bool _inside
        )
    {
        (uint256 timestamp,) = abi.decode(
            _batchHeader.extraData,
            (uint256, address)
        );

        require(
            timestamp != 0,
            "Batch header timestamp cannot be zero"
        );
        return SafeMath.add(timestamp, FRAUD_PROOF_WINDOW) > block.timestamp;
    }


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

    /**
     * Parses the batch context from the extra data.
     * @return Total number of elements submitted.
     * @return Timestamp of the last batch submitted by the sequencer.
     */
    function _getBatchExtraData()
        internal
        view
        returns (
            uint40,
            uint40
        )
    {
        bytes27 extraData = batches().getGlobalMetadata();

        uint40 totalElements;
        uint40 lastSequencerTimestamp;
        assembly {
            extraData              := shr(40, extraData)
            totalElements          :=         and(extraData, 0x000000000000000000000000000000000000000000000000000000FFFFFFFFFF)
            lastSequencerTimestamp := shr(40, and(extraData, 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF0000000000))
        }

        return (
            totalElements,
            lastSequencerTimestamp
        );
    }

    /**
     * Encodes the batch context for the extra data.
     * @param _totalElements Total number of elements submitted.
     * @param _lastSequencerTimestamp Timestamp of the last batch submitted by the sequencer.
     * @return Encoded batch context.
     */
    function _makeBatchExtraData(
        uint40 _totalElements,
        uint40 _lastSequencerTimestamp
    )
        internal
        pure
        returns (
            bytes27
        )
    {
        bytes27 extraData;
        assembly {
            extraData := _totalElements
            extraData := or(extraData, shl(40, _lastSequencerTimestamp))
            extraData := shl(40, extraData)
        }

        return extraData;
    }

    /**
     * Appends a batch to the chain.
     * @param _batch Elements within the batch.
     * @param _extraData Any extra data to append to the batch.
     */
    function _appendBatch(
        bytes32[] memory _batch,
        bytes memory _extraData
    )
        internal
    {
        address sequencer = resolve("OVM_Proposer");
        (uint40 totalElements, uint40 lastSequencerTimestamp) = _getBatchExtraData();

        if (msg.sender == sequencer) {
            lastSequencerTimestamp = uint40(block.timestamp);
        } else {
            // We keep track of the last batch submitted by the sequencer so there's a window in
            // which only the sequencer can publish state roots. A window like this just reduces
            // the chance of "system breaking" state roots being published while we're still in
            // testing mode. This window should be removed or significantly reduced in the future.
            require(
                lastSequencerTimestamp + SEQUENCER_PUBLISH_WINDOW < block.timestamp,
                "Cannot publish state roots within the sequencer publication window."
            );
        }

        // For efficiency reasons getMerkleRoot modifies the `_batch` argument in place
        // while calculating the root hash therefore any arguments passed to it must not
        // be used again afterwards
        Lib_OVMCodec.ChainBatchHeader memory batchHeader = Lib_OVMCodec.ChainBatchHeader({
            batchIndex: getTotalBatches(),
            batchRoot: Lib_MerkleTree.getMerkleRoot(_batch),
            batchSize: _batch.length,
            prevTotalElements: totalElements,
            extraData: _extraData
        });

        emit StateBatchAppended(
            batchHeader.batchIndex,
            batchHeader.batchRoot,
            batchHeader.batchSize,
            batchHeader.prevTotalElements,
            batchHeader.extraData
        );

        batches().push(
            Lib_OVMCodec.hashBatchHeader(batchHeader),
            _makeBatchExtraData(
                uint40(batchHeader.prevTotalElements + batchHeader.batchSize),
                lastSequencerTimestamp
            )
        );
    }

    /**
     * Removes a batch and all subsequent batches from the chain.
     * @param _batchHeader Header of the batch to remove.
     */
    function _deleteBatch(
        Lib_OVMCodec.ChainBatchHeader memory _batchHeader
    )
        internal
    {
        require(
            _batchHeader.batchIndex < batches().length(),
            "Invalid batch index."
        );

        require(
            _isValidBatchHeader(_batchHeader),
            "Invalid batch header."
        );

        batches().deleteElementsAfterInclusive(
            _batchHeader.batchIndex,
            _makeBatchExtraData(
                uint40(_batchHeader.prevTotalElements),
                0
            )
        );

        emit StateBatchDeleted(
            _batchHeader.batchIndex,
            _batchHeader.batchRoot
        );
    }

    /**
     * Checks that a batch header matches the stored hash for the given index.
     * @param _batchHeader Batch header to validate.
     * @return Whether or not the header matches the stored one.
     */
    function _isValidBatchHeader(
        Lib_OVMCodec.ChainBatchHeader memory _batchHeader
    )
        internal
        view
        returns (
            bool
        )
    {
        return Lib_OVMCodec.hashBatchHeader(_batchHeader) == batches().get(_batchHeader.batchIndex);
    }
}

// File: contracts/optimistic-ethereum/iOVM/chain/iOVM_CanonicalTransactionChain.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;
pragma experimental ABIEncoderV2;

/* Library Imports */
import { Lib_OVMCodec } from "../../libraries/codec/Lib_OVMCodec.sol";

/* Interface Imports */
import { iOVM_ChainStorageContainer } from "./iOVM_ChainStorageContainer.sol";

/**
 * @title iOVM_CanonicalTransactionChain
 */
interface iOVM_CanonicalTransactionChain {

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

    event TransactionEnqueued(
        address _l1TxOrigin,
        address _target,
        uint256 _gasLimit,
        bytes _data,
        uint256 _queueIndex,
        uint256 _timestamp
    );

    event QueueBatchAppended(
        uint256 _startingQueueIndex,
        uint256 _numQueueElements,
        uint256 _totalElements
    );

    event SequencerBatchAppended(
        uint256 _startingQueueIndex,
        uint256 _numQueueElements,
        uint256 _totalElements
    );

    event TransactionBatchAppended(
        uint256 indexed _batchIndex,
        bytes32 _batchRoot,
        uint256 _batchSize,
        uint256 _prevTotalElements,
        bytes _extraData
    );


    /***********
     * Structs *
     ***********/

    struct BatchContext {
        uint256 numSequencedTransactions;
        uint256 numSubsequentQueueTransactions;
        uint256 timestamp;
        uint256 blockNumber;
    }


    /********************
     * Public Functions *
     ********************/


    /**
     * Accesses the batch storage container.
     * @return Reference to the batch storage container.
     */
    function batches()
        external
        view
        returns (
            iOVM_ChainStorageContainer
        );

    /**
     * Accesses the queue storage container.
     * @return Reference to the queue storage container.
     */
    function queue()
        external
        view
        returns (
            iOVM_ChainStorageContainer
        );

    /**
     * Retrieves the total number of elements submitted.
     * @return _totalElements Total submitted elements.
     */
    function getTotalElements()
        external
        view
        returns (
            uint256 _totalElements
        );

    /**
     * Retrieves the total number of batches submitted.
     * @return _totalBatches Total submitted batches.
     */
    function getTotalBatches()
        external
        view
        returns (
            uint256 _totalBatches
        );

    /**
     * Returns the index of the next element to be enqueued.
     * @return Index for the next queue element.
     */
    function getNextQueueIndex()
        external
        view
        returns (
            uint40
        );

    /**
     * Gets the queue element at a particular index.
     * @param _index Index of the queue element to access.
     * @return _element Queue element at the given index.
     */
    function getQueueElement(
        uint256 _index
    )
        external
        view
        returns (
            Lib_OVMCodec.QueueElement memory _element
        );

    /**
     * Returns the timestamp of the last transaction.
     * @return Timestamp for the last transaction.
     */
    function getLastTimestamp()
        external
        view
        returns (
            uint40
        );

    /**
     * Returns the blocknumber of the last transaction.
     * @return Blocknumber for the last transaction.
     */
    function getLastBlockNumber()
        external
        view
        returns (
            uint40
        );

    /**
     * Get the number of queue elements which have not yet been included.
     * @return Number of pending queue elements.
     */
    function getNumPendingQueueElements()
        external
        view
        returns (
            uint40
        );

    /**
     * Retrieves the length of the queue, including
     * both pending and canonical transactions.
     * @return Length of the queue.
     */
    function getQueueLength()
        external
        view
        returns (
            uint40
        );


    /**
     * Adds a transaction to the queue.
     * @param _target Target contract to send the transaction to.
     * @param _gasLimit Gas limit for the given transaction.
     * @param _data Transaction data.
     */
    function enqueue(
        address _target,
        uint256 _gasLimit,
        bytes memory _data
    )
        external;

    /**
     * Appends a given number of queued transactions as a single batch.
     * @param _numQueuedTransactions Number of transactions to append.
     */
    function appendQueueBatch(
        uint256 _numQueuedTransactions
    )
        external;

    /**
     * Allows the sequencer to append a batch of transactions.
     * @dev This function uses a custom encoding scheme for efficiency reasons.
     * .param _shouldStartAtElement Specific batch we expect to start appending to.
     * .param _totalElementsToAppend Total number of batch elements we expect to append.
     * .param _contexts Array of batch contexts.
     * .param _transactionDataFields Array of raw transaction data.
     */
    function appendSequencerBatch(
        // uint40 _shouldStartAtElement,
        // uint24 _totalElementsToAppend,
        // BatchContext[] _contexts,
        // bytes[] _transactionDataFields
    )
        external;

    /**
     * Verifies whether a transaction is included in the chain.
     * @param _transaction Transaction to verify.
     * @param _txChainElement Transaction chain element corresponding to the transaction.
     * @param _batchHeader Header of the batch the transaction was included in.
     * @param _inclusionProof Inclusion proof for the provided transaction chain element.
     * @return True if the transaction exists in the CTC, false if not.
     */
    function verifyTransaction(
        Lib_OVMCodec.Transaction memory _transaction,
        Lib_OVMCodec.TransactionChainElement memory _txChainElement,
        Lib_OVMCodec.ChainBatchHeader memory _batchHeader,
        Lib_OVMCodec.ChainInclusionProof memory _inclusionProof
    )
        external
        view
        returns (
            bool
        );
}

// File: contracts/optimistic-ethereum/iOVM/chain/iOVM_ChainStorageContainer.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;

/**
 * @title iOVM_ChainStorageContainer
 */
interface iOVM_ChainStorageContainer {

    /********************
     * Public Functions *
     ********************/

    /**
     * Sets the container's global metadata field. We're using `bytes27` here because we use five
     * bytes to maintain the length of the underlying data structure, meaning we have an extra
     * 27 bytes to store arbitrary data.
     * @param _globalMetadata New global metadata to set.
     */
    function setGlobalMetadata(
        bytes27 _globalMetadata
    )
        external;

    /**
     * Retrieves the container's global metadata field.
     * @return Container global metadata field.
     */
    function getGlobalMetadata()
        external
        view
        returns (
            bytes27
        );

    /**
     * Retrieves the number of objects stored in the container.
     * @return Number of objects in the container.
     */
    function length()
        external
        view
        returns (
            uint256
        );

    /**
     * Pushes an object into the container.
     * @param _object A 32 byte value to insert into the container.
     */
    function push(
        bytes32 _object
    )
        external;

    /**
     * Pushes an object into the container. Function allows setting the global metadata since
     * we'll need to touch the "length" storage slot anyway, which also contains the global
     * metadata (it's an optimization).
     * @param _object A 32 byte value to insert into the container.
     * @param _globalMetadata New global metadata for the container.
     */
    function push(
        bytes32 _object,
        bytes27 _globalMetadata
    )
        external;

    /**
     * Retrieves an object from the container.
     * @param _index Index of the particular object to access.
     * @return 32 byte object value.
     */
    function get(
        uint256 _index
    )
        external
        view
        returns (
            bytes32
        );

    /**
     * Removes all objects after and including a given index.
     * @param _index Object index to delete from.
     */
    function deleteElementsAfterInclusive(
        uint256 _index
    )
        external;

    /**
     * Removes all objects after and including a given index. Also allows setting the global
     * metadata field.
     * @param _index Object index to delete from.
     * @param _globalMetadata New global metadata for the container.
     */
    function deleteElementsAfterInclusive(
        uint256 _index,
        bytes27 _globalMetadata
    )
        external;
}

// File: contracts/optimistic-ethereum/iOVM/chain/iOVM_StateCommitmentChain.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;
pragma experimental ABIEncoderV2;

/* Library Imports */
import { Lib_OVMCodec } from "../../libraries/codec/Lib_OVMCodec.sol";

/**
 * @title iOVM_StateCommitmentChain
 */
interface iOVM_StateCommitmentChain {

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

    event StateBatchAppended(
        uint256 indexed _batchIndex,
        bytes32 _batchRoot,
        uint256 _batchSize,
        uint256 _prevTotalElements,
        bytes _extraData
    );

    event StateBatchDeleted(
        uint256 indexed _batchIndex,
        bytes32 _batchRoot
    );


    /********************
     * Public Functions *
     ********************/

    /**
     * Retrieves the total number of elements submitted.
     * @return _totalElements Total submitted elements.
     */
    function getTotalElements()
        external
        view
        returns (
            uint256 _totalElements
        );

    /**
     * Retrieves the total number of batches submitted.
     * @return _totalBatches Total submitted batches.
     */
    function getTotalBatches()
        external
        view
        returns (
            uint256 _totalBatches
        );

    /**
     * Retrieves the timestamp of the last batch submitted by the sequencer.
     * @return _lastSequencerTimestamp Last sequencer batch timestamp.
     */
    function getLastSequencerTimestamp()
        external
        view
        returns (
            uint256 _lastSequencerTimestamp
        );

    /**
     * Appends a batch of state roots to the chain.
     * @param _batch Batch of state roots.
     * @param _shouldStartAtElement Index of the element at which this batch should start.
     */
    function appendStateBatch(
        bytes32[] calldata _batch,
        uint256 _shouldStartAtElement
    )
        external;

    /**
     * Deletes all state roots after (and including) a given batch.
     * @param _batchHeader Header of the batch to start deleting from.
     */
    function deleteStateBatch(
        Lib_OVMCodec.ChainBatchHeader memory _batchHeader
    )
        external;

    /**
     * Verifies a batch inclusion proof.
     * @param _element Hash of the element to verify a proof for.
     * @param _batchHeader Header of the batch in which the element was included.
     * @param _proof Merkle inclusion proof for the element.
     */
    function verifyStateCommitment(
        bytes32 _element,
        Lib_OVMCodec.ChainBatchHeader memory _batchHeader,
        Lib_OVMCodec.ChainInclusionProof memory _proof
    )
        external
        view
        returns (
            bool _verified
        );

    /**
     * Checks whether a given batch is still inside its fraud proof window.
     * @param _batchHeader Header of the batch to check.
     * @return _inside Whether or not the batch is inside the fraud proof window.
     */
    function insideFraudProofWindow(
        Lib_OVMCodec.ChainBatchHeader memory _batchHeader
    )
        external
        view
        returns (
            bool _inside
        );
}

// File: contracts/optimistic-ethereum/iOVM/verification/iOVM_BondManager.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;

interface ERC20 {
    function transfer(address, uint256) external returns (bool);
    function transferFrom(address, address, uint256) external returns (bool);
}

/// All the errors which may be encountered on the bond manager
library Errors {
    string constant ERC20_ERR = "BondManager: Could not post bond";
    string constant ALREADY_FINALIZED = "BondManager: Fraud proof for this pre-state root has already been finalized";
    string constant SLASHED = "BondManager: Cannot finalize withdrawal, you probably got slashed";
    string constant WRONG_STATE = "BondManager: Wrong bond state for proposer";
    string constant CANNOT_CLAIM = "BondManager: Cannot claim yet. Dispute must be finalized first";

    string constant WITHDRAWAL_PENDING = "BondManager: Withdrawal already pending";
    string constant TOO_EARLY = "BondManager: Too early to finalize your withdrawal";

    string constant ONLY_TRANSITIONER = "BondManager: Only the transitioner for this pre-state root may call this function";
    string constant ONLY_FRAUD_VERIFIER = "BondManager: Only the fraud verifier may call this function";
    string constant ONLY_STATE_COMMITMENT_CHAIN = "BondManager: Only the state commitment chain may call this function";
    string constant WAIT_FOR_DISPUTES = "BondManager: Wait for other potential disputes";
}

/**
 * @title iOVM_BondManager
 */
interface iOVM_BondManager {

    /*******************
     * Data Structures *
     *******************/

    /// The lifecycle of a proposer's bond
    enum State {
        // Before depositing or after getting slashed, a user is uncollateralized
        NOT_COLLATERALIZED,
        // After depositing, a user is collateralized
        COLLATERALIZED,
        // After a user has initiated a withdrawal
        WITHDRAWING
    }

    /// A bond posted by a proposer
    struct Bond {
        // The user's state
        State state;
        // The timestamp at which a proposer issued their withdrawal request
        uint32 withdrawalTimestamp;
        // The time when the first disputed was initiated for this bond
        uint256 firstDisputeAt;
        // The earliest observed state root for this bond which has had fraud
        bytes32 earliestDisputedStateRoot;
        // The state root's timestamp
        uint256 earliestTimestamp;
    }

    // Per pre-state root, store the number of state provisions that were made
    // and how many of these calls were made by each user. Payouts will then be
    // claimed by users proportionally for that dispute.
    struct Rewards {
        // Flag to check if rewards for a fraud proof are claimable
        bool canClaim;
        // Total number of `recordGasSpent` calls made
        uint256 total;
        // The gas spent by each user to provide witness data. The sum of all
        // values inside this map MUST be equal to the value of `total`
        mapping(address => uint256) gasSpent;
    }


    /********************
     * Public Functions *
     ********************/

    function recordGasSpent(
        bytes32 _preStateRoot,
        bytes32 _txHash,
        address _who,
        uint256 _gasSpent
    ) external;

    function finalize(
        bytes32 _preStateRoot,
        address _publisher,
        uint256 _timestamp
    ) external;

    function deposit() external;

    function startWithdrawal() external;

    function finalizeWithdrawal() external;

    function claim(
        address _who
    ) external;

    function isCollateralized(
        address _who
    ) external view returns (bool);

    function getGasSpent(
        bytes32 _preStateRoot,
        address _who
    ) external view returns (uint256);
}

// File: contracts/optimistic-ethereum/iOVM/verification/iOVM_FraudVerifier.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;
pragma experimental ABIEncoderV2;

/* Library Imports */
import { Lib_OVMCodec } from "../../libraries/codec/Lib_OVMCodec.sol";

/* Interface Imports */
import { iOVM_StateTransitioner } from "./iOVM_StateTransitioner.sol";

/**
 * @title iOVM_FraudVerifier
 */
interface iOVM_FraudVerifier {

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

    event FraudProofInitialized(
        bytes32 _preStateRoot,
        uint256 _preStateRootIndex,
        bytes32 _transactionHash,
        address _who
    );

    event FraudProofFinalized(
        bytes32 _preStateRoot,
        uint256 _preStateRootIndex,
        bytes32 _transactionHash,
        address _who
    );


    /***************************************
     * Public Functions: Transition Status *
     ***************************************/

    function getStateTransitioner(bytes32 _preStateRoot, bytes32 _txHash) external view returns (iOVM_StateTransitioner _transitioner);


    /****************************************
     * Public Functions: Fraud Verification *
     ****************************************/

    function initializeFraudVerification(
        bytes32 _preStateRoot,
        Lib_OVMCodec.ChainBatchHeader calldata _preStateRootBatchHeader,
        Lib_OVMCodec.ChainInclusionProof calldata _preStateRootProof,
        Lib_OVMCodec.Transaction calldata _transaction,
        Lib_OVMCodec.TransactionChainElement calldata _txChainElement,
        Lib_OVMCodec.ChainBatchHeader calldata _transactionBatchHeader,
        Lib_OVMCodec.ChainInclusionProof calldata _transactionProof
    ) external;

    function finalizeFraudVerification(
        bytes32 _preStateRoot,
        Lib_OVMCodec.ChainBatchHeader calldata _preStateRootBatchHeader,
        Lib_OVMCodec.ChainInclusionProof calldata _preStateRootProof,
        bytes32 _txHash,
        bytes32 _postStateRoot,
        Lib_OVMCodec.ChainBatchHeader calldata _postStateRootBatchHeader,
        Lib_OVMCodec.ChainInclusionProof calldata _postStateRootProof
    ) external;
}

// File: contracts/optimistic-ethereum/iOVM/verification/iOVM_StateTransitioner.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;
pragma experimental ABIEncoderV2;

/* Library Imports */
import { Lib_OVMCodec } from "../../libraries/codec/Lib_OVMCodec.sol";

/**
 * @title iOVM_StateTransitioner
 */
interface iOVM_StateTransitioner {

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

    event AccountCommitted(
        address _address
    );

    event ContractStorageCommitted(
        address _address,
        bytes32 _key
    );


    /**********************************
     * Public Functions: State Access *
     **********************************/

    function getPreStateRoot() external view returns (bytes32 _preStateRoot);
    function getPostStateRoot() external view returns (bytes32 _postStateRoot);
    function isComplete() external view returns (bool _complete);


    /***********************************
     * Public Functions: Pre-Execution *
     ***********************************/

    function proveContractState(
        address _ovmContractAddress,
        address _ethContractAddress,
        bytes calldata _stateTrieWitness
    ) external;

    function proveStorageSlot(
        address _ovmContractAddress,
        bytes32 _key,
        bytes calldata _storageTrieWitness
    ) external;


    /*******************************
     * Public Functions: Execution *
     *******************************/

    function applyTransaction(
        Lib_OVMCodec.Transaction calldata _transaction
    ) external;


    /************************************
     * Public Functions: Post-Execution *
     ************************************/

    function commitContractState(
        address _ovmContractAddress,
        bytes calldata _stateTrieWitness
    ) external;

    function commitStorageSlot(
        address _ovmContractAddress,
        bytes32 _key,
        bytes calldata _storageTrieWitness
    ) external;


    /**********************************
     * Public Functions: Finalization *
     **********************************/

    function completeTransition() external;
}

// File: contracts/optimistic-ethereum/libraries/codec/Lib_OVMCodec.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;
pragma experimental ABIEncoderV2;

/* Library Imports */
import { Lib_RLPReader } from "../rlp/Lib_RLPReader.sol";
import { Lib_RLPWriter } from "../rlp/Lib_RLPWriter.sol";
import { Lib_BytesUtils } from "../utils/Lib_BytesUtils.sol";
import { Lib_Bytes32Utils } from "../utils/Lib_Bytes32Utils.sol";

/**
 * @title Lib_OVMCodec
 */
library Lib_OVMCodec {

    /*********
     * Enums *
     *********/

    enum QueueOrigin {
        SEQUENCER_QUEUE,
        L1TOL2_QUEUE
    }


    /***********
     * Structs *
     ***********/

    struct Account {
        uint256 nonce;
        uint256 balance;
        bytes32 storageRoot;
        bytes32 codeHash;
        address ethAddress;
        bool isFresh;
    }

    struct EVMAccount {
        uint256 nonce;
        uint256 balance;
        bytes32 storageRoot;
        bytes32 codeHash;
    }

    struct ChainBatchHeader {
        uint256 batchIndex;
        bytes32 batchRoot;
        uint256 batchSize;
        uint256 prevTotalElements;
        bytes extraData;
    }

    struct ChainInclusionProof {
        uint256 index;
        bytes32[] siblings;
    }

    struct Transaction {
        uint256 timestamp;
        uint256 blockNumber;
        QueueOrigin l1QueueOrigin;
        address l1TxOrigin;
        address entrypoint;
        uint256 gasLimit;
        bytes data;
    }

    struct TransactionChainElement {
        bool isSequenced;
        uint256 queueIndex;  // QUEUED TX ONLY
        uint256 timestamp;   // SEQUENCER TX ONLY
        uint256 blockNumber; // SEQUENCER TX ONLY
        bytes txData;        // SEQUENCER TX ONLY
    }

    struct QueueElement {
        bytes32 transactionHash;
        uint40 timestamp;
        uint40 blockNumber;
    }


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

    /**
     * Encodes a standard OVM transaction.
     * @param _transaction OVM transaction to encode.
     * @return Encoded transaction bytes.
     */
    function encodeTransaction(
        Transaction memory _transaction
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        return abi.encodePacked(
            _transaction.timestamp,
            _transaction.blockNumber,
            _transaction.l1QueueOrigin,
            _transaction.l1TxOrigin,
            _transaction.entrypoint,
            _transaction.gasLimit,
            _transaction.data
        );
    }

    /**
     * Hashes a standard OVM transaction.
     * @param _transaction OVM transaction to encode.
     * @return Hashed transaction
     */
    function hashTransaction(
        Transaction memory _transaction
    )
        internal
        pure
        returns (
            bytes32
        )
    {
        return keccak256(encodeTransaction(_transaction));
    }

    /**
     * Converts an OVM account to an EVM account.
     * @param _in OVM account to convert.
     * @return Converted EVM account.
     */
    function toEVMAccount(
        Account memory _in
    )
        internal
        pure
        returns (
            EVMAccount memory
        )
    {
        return EVMAccount({
            nonce: _in.nonce,
            balance: _in.balance,
            storageRoot: _in.storageRoot,
            codeHash: _in.codeHash
        });
    }

    /**
     * @notice RLP-encodes an account state struct.
     * @param _account Account state struct.
     * @return RLP-encoded account state.
     */
    function encodeEVMAccount(
        EVMAccount memory _account
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        bytes[] memory raw = new bytes[](4);

        // Unfortunately we can't create this array outright because
        // Lib_RLPWriter.writeList will reject fixed-size arrays. Assigning
        // index-by-index circumvents this issue.
        raw[0] = Lib_RLPWriter.writeBytes(
            Lib_Bytes32Utils.removeLeadingZeros(
                bytes32(_account.nonce)
            )
        );
        raw[1] = Lib_RLPWriter.writeBytes(
            Lib_Bytes32Utils.removeLeadingZeros(
                bytes32(_account.balance)
            )
        );
        raw[2] = Lib_RLPWriter.writeBytes(abi.encodePacked(_account.storageRoot));
        raw[3] = Lib_RLPWriter.writeBytes(abi.encodePacked(_account.codeHash));

        return Lib_RLPWriter.writeList(raw);
    }

    /**
     * @notice Decodes an RLP-encoded account state into a useful struct.
     * @param _encoded RLP-encoded account state.
     * @return Account state struct.
     */
    function decodeEVMAccount(
        bytes memory _encoded
    )
        internal
        pure
        returns (
            EVMAccount memory
        )
    {
        Lib_RLPReader.RLPItem[] memory accountState = Lib_RLPReader.readList(_encoded);

        return EVMAccount({
            nonce: Lib_RLPReader.readUint256(accountState[0]),
            balance: Lib_RLPReader.readUint256(accountState[1]),
            storageRoot: Lib_RLPReader.readBytes32(accountState[2]),
            codeHash: Lib_RLPReader.readBytes32(accountState[3])
        });
    }

    /**
     * Calculates a hash for a given batch header.
     * @param _batchHeader Header to hash.
     * @return Hash of the header.
     */
    function hashBatchHeader(
        Lib_OVMCodec.ChainBatchHeader memory _batchHeader
    )
        internal
        pure
        returns (
            bytes32
        )
    {
        return keccak256(
            abi.encode(
                _batchHeader.batchRoot,
                _batchHeader.batchSize,
                _batchHeader.prevTotalElements,
                _batchHeader.extraData
            )
        );
    }
}

// File: contracts/optimistic-ethereum/libraries/resolver/Lib_AddressManager.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;

/* External Imports */
import { Ownable } from "@openzeppelin/contracts/access/Ownable.sol";

/**
 * @title Lib_AddressManager
 */
contract Lib_AddressManager is Ownable {

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

    event AddressSet(
        string indexed _name,
        address _newAddress,
        address _oldAddress
    );


    /*************
     * Variables *
     *************/

    mapping (bytes32 => address) private addresses;


    /********************
     * Public Functions *
     ********************/

    /**
     * Changes the address associated with a particular name.
     * @param _name String name to associate an address with.
     * @param _address Address to associate with the name.
     */
    function setAddress(
        string memory _name,
        address _address
    )
        external
        onlyOwner
    {
        bytes32 nameHash = _getNameHash(_name);
        address oldAddress = addresses[nameHash];
        addresses[nameHash] = _address;

        emit AddressSet(
            _name,
            _address,
            oldAddress
        );
    }

    /**
     * Retrieves the address associated with a given name.
     * @param _name Name to retrieve an address for.
     * @return Address associated with the given name.
     */
    function getAddress(
        string memory _name
    )
        external
        view
        returns (
            address
        )
    {
        return addresses[_getNameHash(_name)];
    }


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

    /**
     * Computes the hash of a name.
     * @param _name Name to compute a hash for.
     * @return Hash of the given name.
     */
    function _getNameHash(
        string memory _name
    )
        internal
        pure
        returns (
            bytes32
        )
    {
        return keccak256(abi.encodePacked(_name));
    }
}

// File: contracts/optimistic-ethereum/libraries/resolver/Lib_AddressResolver.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;

/* Library Imports */
import { Lib_AddressManager } from "./Lib_AddressManager.sol";

/**
 * @title Lib_AddressResolver
 */
abstract contract Lib_AddressResolver {

    /*************
     * Variables *
     *************/

    Lib_AddressManager public libAddressManager;


    /***************
     * Constructor *
     ***************/

    /**
     * @param _libAddressManager Address of the Lib_AddressManager.
     */
    constructor(
        address _libAddressManager
    ) {
        libAddressManager = Lib_AddressManager(_libAddressManager);
    }


    /********************
     * Public Functions *
     ********************/

    /**
     * Resolves the address associated with a given name.
     * @param _name Name to resolve an address for.
     * @return Address associated with the given name.
     */
    function resolve(
        string memory _name
    )
        public
        view
        returns (
            address
        )
    {
        return libAddressManager.getAddress(_name);
    }
}

// File: contracts/optimistic-ethereum/libraries/rlp/Lib_RLPReader.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;

/**
 * @title Lib_RLPReader
 * @dev Adapted from "RLPReader" by Hamdi Allam (hamdi.allam97@gmail.com).
 */
library Lib_RLPReader {

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

    uint256 constant internal MAX_LIST_LENGTH = 32;


    /*********
     * Enums *
     *********/

    enum RLPItemType {
        DATA_ITEM,
        LIST_ITEM
    }


    /***********
     * Structs *
     ***********/

    struct RLPItem {
        uint256 length;
        uint256 ptr;
    }


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

    /**
     * Converts bytes to a reference to memory position and length.
     * @param _in Input bytes to convert.
     * @return Output memory reference.
     */
    function toRLPItem(
        bytes memory _in
    )
        internal
        pure
        returns (
            RLPItem memory
        )
    {
        uint256 ptr;
        assembly {
            ptr := add(_in, 32)
        }

        return RLPItem({
            length: _in.length,
            ptr: ptr
        });
    }

    /**
     * Reads an RLP list value into a list of RLP items.
     * @param _in RLP list value.
     * @return Decoded RLP list items.
     */
    function readList(
        RLPItem memory _in
    )
        internal
        pure
        returns (
            RLPItem[] memory
        )
    {
        (
            uint256 listOffset,
            ,
            RLPItemType itemType
        ) = _decodeLength(_in);

        require(
            itemType == RLPItemType.LIST_ITEM,
            "Invalid RLP list value."
        );

        // Solidity in-memory arrays can't be increased in size, but *can* be decreased in size by
        // writing to the length. Since we can't know the number of RLP items without looping over
        // the entire input, we'd have to loop twice to accurately size this array. It's easier to
        // simply set a reasonable maximum list length and decrease the size before we finish.
        RLPItem[] memory out = new RLPItem[](MAX_LIST_LENGTH);

        uint256 itemCount = 0;
        uint256 offset = listOffset;
        while (offset < _in.length) {
            require(
                itemCount < MAX_LIST_LENGTH,
                "Provided RLP list exceeds max list length."
            );

            (
                uint256 itemOffset,
                uint256 itemLength,
            ) = _decodeLength(RLPItem({
                length: _in.length - offset,
                ptr: _in.ptr + offset
            }));

            out[itemCount] = RLPItem({
                length: itemLength + itemOffset,
                ptr: _in.ptr + offset
            });

            itemCount += 1;
            offset += itemOffset + itemLength;
        }

        // Decrease the array size to match the actual item count.
        assembly {
            mstore(out, itemCount)
        }

        return out;
    }

    /**
     * Reads an RLP list value into a list of RLP items.
     * @param _in RLP list value.
     * @return Decoded RLP list items.
     */
    function readList(
        bytes memory _in
    )
        internal
        pure
        returns (
            RLPItem[] memory
        )
    {
        return readList(
            toRLPItem(_in)
        );
    }

    /**
     * Reads an RLP bytes value into bytes.
     * @param _in RLP bytes value.
     * @return Decoded bytes.
     */
    function readBytes(
        RLPItem memory _in
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        (
            uint256 itemOffset,
            uint256 itemLength,
            RLPItemType itemType
        ) = _decodeLength(_in);

        require(
            itemType == RLPItemType.DATA_ITEM,
            "Invalid RLP bytes value."
        );

        return _copy(_in.ptr, itemOffset, itemLength);
    }

    /**
     * Reads an RLP bytes value into bytes.
     * @param _in RLP bytes value.
     * @return Decoded bytes.
     */
    function readBytes(
        bytes memory _in
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        return readBytes(
            toRLPItem(_in)
        );
    }

    /**
     * Reads an RLP string value into a string.
     * @param _in RLP string value.
     * @return Decoded string.
     */
    function readString(
        RLPItem memory _in
    )
        internal
        pure
        returns (
            string memory
        )
    {
        return string(readBytes(_in));
    }

    /**
     * Reads an RLP string value into a string.
     * @param _in RLP string value.
     * @return Decoded string.
     */
    function readString(
        bytes memory _in
    )
        internal
        pure
        returns (
            string memory
        )
    {
        return readString(
            toRLPItem(_in)
        );
    }

    /**
     * Reads an RLP bytes32 value into a bytes32.
     * @param _in RLP bytes32 value.
     * @return Decoded bytes32.
     */
    function readBytes32(
        RLPItem memory _in
    )
        internal
        pure
        returns (
            bytes32
        )
    {
        require(
            _in.length <= 33,
            "Invalid RLP bytes32 value."
        );

        (
            uint256 itemOffset,
            uint256 itemLength,
            RLPItemType itemType
        ) = _decodeLength(_in);

        require(
            itemType == RLPItemType.DATA_ITEM,
            "Invalid RLP bytes32 value."
        );

        uint256 ptr = _in.ptr + itemOffset;
        bytes32 out;
        assembly {
            out := mload(ptr)

            // Shift the bytes over to match the item size.
            if lt(itemLength, 32) {
                out := div(out, exp(256, sub(32, itemLength)))
            }
        }

        return out;
    }

    /**
     * Reads an RLP bytes32 value into a bytes32.
     * @param _in RLP bytes32 value.
     * @return Decoded bytes32.
     */
    function readBytes32(
        bytes memory _in
    )
        internal
        pure
        returns (
            bytes32
        )
    {
        return readBytes32(
            toRLPItem(_in)
        );
    }

    /**
     * Reads an RLP uint256 value into a uint256.
     * @param _in RLP uint256 value.
     * @return Decoded uint256.
     */
    function readUint256(
        RLPItem memory _in
    )
        internal
        pure
        returns (
            uint256
        )
    {
        return uint256(readBytes32(_in));
    }

    /**
     * Reads an RLP uint256 value into a uint256.
     * @param _in RLP uint256 value.
     * @return Decoded uint256.
     */
    function readUint256(
        bytes memory _in
    )
        internal
        pure
        returns (
            uint256
        )
    {
        return readUint256(
            toRLPItem(_in)
        );
    }

    /**
     * Reads an RLP bool value into a bool.
     * @param _in RLP bool value.
     * @return Decoded bool.
     */
    function readBool(
        RLPItem memory _in
    )
        internal
        pure
        returns (
            bool
        )
    {
        require(
            _in.length == 1,
            "Invalid RLP boolean value."
        );

        uint256 ptr = _in.ptr;
        uint256 out;
        assembly {
            out := byte(0, mload(ptr))
        }

        require(
            out == 0 || out == 1,
            "Lib_RLPReader: Invalid RLP boolean value, must be 0 or 1"
        );

        return out != 0;
    }

    /**
     * Reads an RLP bool value into a bool.
     * @param _in RLP bool value.
     * @return Decoded bool.
     */
    function readBool(
        bytes memory _in
    )
        internal
        pure
        returns (
            bool
        )
    {
        return readBool(
            toRLPItem(_in)
        );
    }

    /**
     * Reads an RLP address value into a address.
     * @param _in RLP address value.
     * @return Decoded address.
     */
    function readAddress(
        RLPItem memory _in
    )
        internal
        pure
        returns (
            address
        )
    {
        if (_in.length == 1) {
            return address(0);
        }

        require(
            _in.length == 21,
            "Invalid RLP address value."
        );

        return address(readUint256(_in));
    }

    /**
     * Reads an RLP address value into a address.
     * @param _in RLP address value.
     * @return Decoded address.
     */
    function readAddress(
        bytes memory _in
    )
        internal
        pure
        returns (
            address
        )
    {
        return readAddress(
            toRLPItem(_in)
        );
    }

    /**
     * Reads the raw bytes of an RLP item.
     * @param _in RLP item to read.
     * @return Raw RLP bytes.
     */
    function readRawBytes(
        RLPItem memory _in
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        return _copy(_in);
    }


    /*********************
     * Private Functions *
     *********************/

    /**
     * Decodes the length of an RLP item.
     * @param _in RLP item to decode.
     * @return Offset of the encoded data.
     * @return Length of the encoded data.
     * @return RLP item type (LIST_ITEM or DATA_ITEM).
     */
    function _decodeLength(
        RLPItem memory _in
    )
        private
        pure
        returns (
            uint256,
            uint256,
            RLPItemType
        )
    {
        require(
            _in.length > 0,
            "RLP item cannot be null."
        );

        uint256 ptr = _in.ptr;
        uint256 prefix;
        assembly {
            prefix := byte(0, mload(ptr))
        }

        if (prefix <= 0x7f) {
            // Single byte.

            return (0, 1, RLPItemType.DATA_ITEM);
        } else if (prefix <= 0xb7) {
            // Short string.

            uint256 strLen = prefix - 0x80;

            require(
                _in.length > strLen,
                "Invalid RLP short string."
            );

            return (1, strLen, RLPItemType.DATA_ITEM);
        } else if (prefix <= 0xbf) {
            // Long string.
            uint256 lenOfStrLen = prefix - 0xb7;

            require(
                _in.length > lenOfStrLen,
                "Invalid RLP long string length."
            );

            uint256 strLen;
            assembly {
                // Pick out the string length.
                strLen := div(
                    mload(add(ptr, 1)),
                    exp(256, sub(32, lenOfStrLen))
                )
            }

            require(
                _in.length > lenOfStrLen + strLen,
                "Invalid RLP long string."
            );

            return (1 + lenOfStrLen, strLen, RLPItemType.DATA_ITEM);
        } else if (prefix <= 0xf7) {
            // Short list.
            uint256 listLen = prefix - 0xc0;

            require(
                _in.length > listLen,
                "Invalid RLP short list."
            );

            return (1, listLen, RLPItemType.LIST_ITEM);
        } else {
            // Long list.
            uint256 lenOfListLen = prefix - 0xf7;

            require(
                _in.length > lenOfListLen,
                "Invalid RLP long list length."
            );

            uint256 listLen;
            assembly {
                // Pick out the list length.
                listLen := div(
                    mload(add(ptr, 1)),
                    exp(256, sub(32, lenOfListLen))
                )
            }

            require(
                _in.length > lenOfListLen + listLen,
                "Invalid RLP long list."
            );

            return (1 + lenOfListLen, listLen, RLPItemType.LIST_ITEM);
        }
    }

    /**
     * Copies the bytes from a memory location.
     * @param _src Pointer to the location to read from.
     * @param _offset Offset to start reading from.
     * @param _length Number of bytes to read.
     * @return Copied bytes.
     */
    function _copy(
        uint256 _src,
        uint256 _offset,
        uint256 _length
    )
        private
        pure
        returns (
            bytes memory
        )
    {
        bytes memory out = new bytes(_length);
        if (out.length == 0) {
            return out;
        }

        uint256 src = _src + _offset;
        uint256 dest;
        assembly {
            dest := add(out, 32)
        }

        // Copy over as many complete words as we can.
        for (uint256 i = 0; i < _length / 32; i++) {
            assembly {
                mstore(dest, mload(src))
            }

            src += 32;
            dest += 32;
        }

        // Pick out the remaining bytes.
        uint256 mask = 256 ** (32 - (_length % 32)) - 1;
        assembly {
            mstore(
                dest,
                or(
                    and(mload(src), not(mask)),
                    and(mload(dest), mask)
                )
            )
        }

        return out;
    }

    /**
     * Copies an RLP item into bytes.
     * @param _in RLP item to copy.
     * @return Copied bytes.
     */
    function _copy(
        RLPItem memory _in
    )
        private
        pure
        returns (
            bytes memory
        )
    {
        return _copy(_in.ptr, 0, _in.length);
    }
}

// File: contracts/optimistic-ethereum/libraries/rlp/Lib_RLPWriter.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;
pragma experimental ABIEncoderV2;

/**
 * @title Lib_RLPWriter
 * @author Bakaoh (with modifications)
 */
library Lib_RLPWriter {

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

    /**
     * RLP encodes a byte string.
     * @param _in The byte string to encode.
     * @return The RLP encoded string in bytes.
     */
    function writeBytes(
        bytes memory _in
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        bytes memory encoded;

        if (_in.length == 1 && uint8(_in[0]) < 128) {
            encoded = _in;
        } else {
            encoded = abi.encodePacked(_writeLength(_in.length, 128), _in);
        }

        return encoded;
    }

    /**
     * RLP encodes a list of RLP encoded byte byte strings.
     * @param _in The list of RLP encoded byte strings.
     * @return The RLP encoded list of items in bytes.
     */
    function writeList(
        bytes[] memory _in
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        bytes memory list = _flatten(_in);
        return abi.encodePacked(_writeLength(list.length, 192), list);
    }

    /**
     * RLP encodes a string.
     * @param _in The string to encode.
     * @return The RLP encoded string in bytes.
     */
    function writeString(
        string memory _in
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        return writeBytes(bytes(_in));
    }

    /**
     * RLP encodes an address.
     * @param _in The address to encode.
     * @return The RLP encoded address in bytes.
     */
    function writeAddress(
        address _in
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        return writeBytes(abi.encodePacked(_in));
    }

    /**
     * RLP encodes a bytes32 value.
     * @param _in The bytes32 to encode.
     * @return _out The RLP encoded bytes32 in bytes.
     */
    function writeBytes32(
        bytes32 _in
    )
        internal
        pure
        returns (
            bytes memory _out
        )
    {
        return writeBytes(abi.encodePacked(_in));
    }

    /**
     * RLP encodes a uint.
     * @param _in The uint256 to encode.
     * @return The RLP encoded uint256 in bytes.
     */
    function writeUint(
        uint256 _in
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        return writeBytes(_toBinary(_in));
    }

    /**
     * RLP encodes a bool.
     * @param _in The bool to encode.
     * @return The RLP encoded bool in bytes.
     */
    function writeBool(
        bool _in
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        bytes memory encoded = new bytes(1);
        encoded[0] = (_in ? bytes1(0x01) : bytes1(0x80));
        return encoded;
    }


    /*********************
     * Private Functions *
     *********************/

    /**
     * Encode the first byte, followed by the `len` in binary form if `length` is more than 55.
     * @param _len The length of the string or the payload.
     * @param _offset 128 if item is string, 192 if item is list.
     * @return RLP encoded bytes.
     */
    function _writeLength(
        uint256 _len,
        uint256 _offset
    )
        private
        pure
        returns (
            bytes memory
        )
    {
        bytes memory encoded;

        if (_len < 56) {
            encoded = new bytes(1);
            encoded[0] = byte(uint8(_len) + uint8(_offset));
        } else {
            uint256 lenLen;
            uint256 i = 1;
            while (_len / i != 0) {
                lenLen++;
                i *= 256;
            }

            encoded = new bytes(lenLen + 1);
            encoded[0] = byte(uint8(lenLen) + uint8(_offset) + 55);
            for(i = 1; i <= lenLen; i++) {
                encoded[i] = byte(uint8((_len / (256**(lenLen-i))) % 256));
            }
        }

        return encoded;
    }

    /**
     * Encode integer in big endian binary form with no leading zeroes.
     * @notice TODO: This should be optimized with assembly to save gas costs.
     * @param _x The integer to encode.
     * @return RLP encoded bytes.
     */
    function _toBinary(
        uint256 _x
    )
        private
        pure
        returns (
            bytes memory
        )
    {
        bytes memory b = abi.encodePacked(_x);

        uint256 i = 0;
        for (; i < 32; i++) {
            if (b[i] != 0) {
                break;
            }
        }

        bytes memory res = new bytes(32 - i);
        for (uint256 j = 0; j < res.length; j++) {
            res[j] = b[i++];
        }

        return res;
    }

    /**
     * Copies a piece of memory to another location.
     * @notice From: https://github.com/Arachnid/solidity-stringutils/blob/master/src/strings.sol.
     * @param _dest Destination location.
     * @param _src Source location.
     * @param _len Length of memory to copy.
     */
    function _memcpy(
        uint256 _dest,
        uint256 _src,
        uint256 _len
    )
        private
        pure
    {
        uint256 dest = _dest;
        uint256 src = _src;
        uint256 len = _len;

        for(; len >= 32; len -= 32) {
            assembly {
                mstore(dest, mload(src))
            }
            dest += 32;
            src += 32;
        }

        uint256 mask = 256 ** (32 - len) - 1;
        assembly {
            let srcpart := and(mload(src), not(mask))
            let destpart := and(mload(dest), mask)
            mstore(dest, or(destpart, srcpart))
        }
    }

    /**
     * Flattens a list of byte strings into one byte string.
     * @notice From: https://github.com/sammayo/solidity-rlp-encoder/blob/master/RLPEncode.sol.
     * @param _list List of byte strings to flatten.
     * @return The flattened byte string.
     */
    function _flatten(
        bytes[] memory _list
    )
        private
        pure
        returns (
            bytes memory
        )
    {
        if (_list.length == 0) {
            return new bytes(0);
        }

        uint256 len;
        uint256 i = 0;
        for (; i < _list.length; i++) {
            len += _list[i].length;
        }

        bytes memory flattened = new bytes(len);
        uint256 flattenedPtr;
        assembly { flattenedPtr := add(flattened, 0x20) }

        for(i = 0; i < _list.length; i++) {
            bytes memory item = _list[i];

            uint256 listPtr;
            assembly { listPtr := add(item, 0x20)}

            _memcpy(flattenedPtr, listPtr, item.length);
            flattenedPtr += _list[i].length;
        }

        return flattened;
    }
}

// File: contracts/optimistic-ethereum/libraries/utils/Lib_Bytes32Utils.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;

/**
 * @title Lib_Byte32Utils
 */
library Lib_Bytes32Utils {

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

    /**
     * Converts a bytes32 value to a boolean. Anything non-zero will be converted to "true."
     * @param _in Input bytes32 value.
     * @return Bytes32 as a boolean.
     */
    function toBool(
        bytes32 _in
    )
        internal
        pure
        returns (
            bool
        )
    {
        return _in != 0;
    }

    /**
     * Converts a boolean to a bytes32 value.
     * @param _in Input boolean value.
     * @return Boolean as a bytes32.
     */
    function fromBool(
        bool _in
    )
        internal
        pure
        returns (
            bytes32
        )
    {
        return bytes32(uint256(_in ? 1 : 0));
    }

    /**
     * Converts a bytes32 value to an address. Takes the *last* 20 bytes.
     * @param _in Input bytes32 value.
     * @return Bytes32 as an address.
     */
    function toAddress(
        bytes32 _in
    )
        internal
        pure
        returns (
            address
        )
    {
        return address(uint160(uint256(_in)));
    }

    /**
     * Converts an address to a bytes32.
     * @param _in Input address value.
     * @return Address as a bytes32.
     */
    function fromAddress(
        address _in
    )
        internal
        pure
        returns (
            bytes32
        )
    {
        return bytes32(uint256(_in));
    }

    /**
     * Removes the leading zeros from a bytes32 value and returns a new (smaller) bytes value.
     * @param _in Input bytes32 value.
     * @return Bytes32 without any leading zeros.
     */
    function removeLeadingZeros(
        bytes32 _in
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        bytes memory out;

        assembly {
            // Figure out how many leading zero bytes to remove.
            let shift := 0
            for { let i := 0 } and(lt(i, 32), eq(byte(i, _in), 0)) { i := add(i, 1) } {
                shift := add(shift, 1)
            }

            // Reserve some space for our output and fix the free memory pointer.
            out := mload(0x40)
            mstore(0x40, add(out, 0x40))

            // Shift the value and store it into the output bytes.
            mstore(add(out, 0x20), shl(mul(shift, 8), _in))

            // Store the new size (with leading zero bytes removed) in the output byte size.
            mstore(out, sub(32, shift))
        }

        return out;
    }
}

// File: contracts/optimistic-ethereum/libraries/utils/Lib_BytesUtils.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;

/**
 * @title Lib_BytesUtils
 */
library Lib_BytesUtils {

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

    function slice(
        bytes memory _bytes,
        uint256 _start,
        uint256 _length
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        require(_length + 31 >= _length, "slice_overflow");
        require(_start + _length >= _start, "slice_overflow");
        require(_bytes.length >= _start + _length, "slice_outOfBounds");

        bytes memory tempBytes;

        assembly {
            switch iszero(_length)
            case 0 {
                // Get a location of some free memory and store it in tempBytes as
                // Solidity does for memory variables.
                tempBytes := mload(0x40)

                // The first word of the slice result is potentially a partial
                // word read from the original array. To read it, we calculate
                // the length of that partial word and start copying that many
                // bytes into the array. The first word we copy will start with
                // data we don't care about, but the last `lengthmod` bytes will
                // land at the beginning of the contents of the new array. When
                // we're done copying, we overwrite the full first word with
                // the actual length of the slice.
                let lengthmod := and(_length, 31)

                // The multiplication in the next line is necessary
                // because when slicing multiples of 32 bytes (lengthmod == 0)
                // the following copy loop was copying the origin's length
                // and then ending prematurely not copying everything it should.
                let mc := add(add(tempBytes, lengthmod), mul(0x20, iszero(lengthmod)))
                let end := add(mc, _length)

                for {
                    // The multiplication in the next line has the same exact purpose
                    // as the one above.
                    let cc := add(add(add(_bytes, lengthmod), mul(0x20, iszero(lengthmod))), _start)
                } lt(mc, end) {
                    mc := add(mc, 0x20)
                    cc := add(cc, 0x20)
                } {
                    mstore(mc, mload(cc))
                }

                mstore(tempBytes, _length)

                //update free-memory pointer
                //allocating the array padded to 32 bytes like the compiler does now
                mstore(0x40, and(add(mc, 31), not(31)))
            }
            //if we want a zero-length slice let's just return a zero-length array
            default {
                tempBytes := mload(0x40)

                //zero out the 32 bytes slice we are about to return
                //we need to do it because Solidity does not garbage collect
                mstore(tempBytes, 0)

                mstore(0x40, add(tempBytes, 0x20))
            }
        }

        return tempBytes;
    }

    function slice(
        bytes memory _bytes,
        uint256 _start
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        if (_start >= _bytes.length) {
            return bytes('');
        }

        return slice(_bytes, _start, _bytes.length - _start);
    }

    function toBytes32PadLeft(
        bytes memory _bytes
    )
        internal
        pure
        returns (
            bytes32
        )
    {
        bytes32 ret;
        uint256 len = _bytes.length <= 32 ? _bytes.length : 32;
        assembly {
            ret := shr(mul(sub(32, len), 8), mload(add(_bytes, 32)))
        }
        return ret;
    }

    function toBytes32(
        bytes memory _bytes
    )
        internal
        pure
        returns (
            bytes32
        )
    {
        if (_bytes.length < 32) {
            bytes32 ret;
            assembly {
                ret := mload(add(_bytes, 32))
            }
            return ret;
        }

        return abi.decode(_bytes,(bytes32)); // will truncate if input length > 32 bytes
    }

    function toUint256(
        bytes memory _bytes
    )
        internal
        pure
        returns (
            uint256
        )
    {
        return uint256(toBytes32(_bytes));
    }

    function toUint24(
        bytes memory _bytes,
        uint256 _start
    )
        internal
        pure
        returns (
            uint24
        )
    {
        require(_start + 3 >= _start, "toUint24_overflow");
        require(_bytes.length >= _start + 3 , "toUint24_outOfBounds");
        uint24 tempUint;

        assembly {
            tempUint := mload(add(add(_bytes, 0x3), _start))
        }

        return tempUint;
    }

    function toUint8(
        bytes memory _bytes,
        uint256 _start
    )
        internal
        pure
        returns (
            uint8
        )
    {
        require(_start + 1 >= _start, "toUint8_overflow");
        require(_bytes.length >= _start + 1 , "toUint8_outOfBounds");
        uint8 tempUint;

        assembly {
            tempUint := mload(add(add(_bytes, 0x1), _start))
        }

        return tempUint;
    }

    function toAddress(
        bytes memory _bytes,
        uint256 _start
    )
        internal
        pure
        returns (
            address
        )
    {
        require(_start + 20 >= _start, "toAddress_overflow");
        require(_bytes.length >= _start + 20, "toAddress_outOfBounds");
        address tempAddress;

        assembly {
            tempAddress := div(mload(add(add(_bytes, 0x20), _start)), 0x1000000000000000000000000)
        }

        return tempAddress;
    }

    function toNibbles(
        bytes memory _bytes
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        bytes memory nibbles = new bytes(_bytes.length * 2);

        for (uint256 i = 0; i < _bytes.length; i++) {
            nibbles[i * 2] = _bytes[i] >> 4;
            nibbles[i * 2 + 1] = bytes1(uint8(_bytes[i]) % 16);
        }

        return nibbles;
    }

    function fromNibbles(
        bytes memory _bytes
    )
        internal
        pure
        returns (
            bytes memory
        )
    {
        bytes memory ret = new bytes(_bytes.length / 2);

        for (uint256 i = 0; i < ret.length; i++) {
            ret[i] = (_bytes[i * 2] << 4) | (_bytes[i * 2 + 1]);
        }

        return ret;
    }

    function equal(
        bytes memory _bytes,
        bytes memory _other
    )
        internal
        pure
        returns (
            bool
        )
    {
        return keccak256(_bytes) == keccak256(_other);
    }
}

// File: contracts/optimistic-ethereum/libraries/utils/Lib_MerkleTree.sol



// SPDX-License-Identifier: MIT
pragma solidity >0.5.0 <0.8.0;

/**
 * @title Lib_MerkleTree
 * @author River Keefer
 */
library Lib_MerkleTree {

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

    /**
     * Calculates a merkle root for a list of 32-byte leaf hashes.  WARNING: If the number
     * of leaves passed in is not a power of two, it pads out the tree with zero hashes.
     * If you do not know the original length of elements for the tree you are verifying,
     * then this may allow empty leaves past _elements.length to pass a verification check down the line.
     * Note that the _elements argument is modified, therefore it must not be used again afterwards
     * @param _elements Array of hashes from which to generate a merkle root.
     * @return Merkle root of the leaves, with zero hashes for non-powers-of-two (see above).
     */
    function getMerkleRoot(
        bytes32[] memory _elements
    )
        internal
        pure
        returns (
            bytes32
        )
    {
        require(
            _elements.length > 0,
            "Lib_MerkleTree: Must provide at least one leaf hash."
        );

        if (_elements.length == 1) {
            return _elements[0];
        }

        uint256[16] memory defaults = [
            0x290decd9548b62a8d60345a988386fc84ba6bc95484008f6362f93160ef3e563,
            0x633dc4d7da7256660a892f8f1604a44b5432649cc8ec5cb3ced4c4e6ac94dd1d,
            0x890740a8eb06ce9be422cb8da5cdafc2b58c0a5e24036c578de2a433c828ff7d,
            0x3b8ec09e026fdc305365dfc94e189a81b38c7597b3d941c279f042e8206e0bd8,
            0xecd50eee38e386bd62be9bedb990706951b65fe053bd9d8a521af753d139e2da,
            0xdefff6d330bb5403f63b14f33b578274160de3a50df4efecf0e0db73bcdd3da5,
            0x617bdd11f7c0a11f49db22f629387a12da7596f9d1704d7465177c63d88ec7d7,
            0x292c23a9aa1d8bea7e2435e555a4a60e379a5a35f3f452bae60121073fb6eead,
            0xe1cea92ed99acdcb045a6726b2f87107e8a61620a232cf4d7d5b5766b3952e10,
            0x7ad66c0a68c72cb89e4fb4303841966e4062a76ab97451e3b9fb526a5ceb7f82,
            0xe026cc5a4aed3c22a58cbd3d2ac754c9352c5436f638042dca99034e83636516,
            0x3d04cffd8b46a874edf5cfae63077de85f849a660426697b06a829c70dd1409c,
            0xad676aa337a485e4728a0b240d92b3ef7b3c372d06d189322bfd5f61f1e7203e,
            0xa2fca4a49658f9fab7aa63289c91b7c7b6c832a6d0e69334ff5b0a3483d09dab,
            0x4ebfd9cd7bca2505f7bef59cc1c12ecc708fff26ae4af19abe852afe9e20c862,
            0x2def10d13dd169f550f578bda343d9717a138562e0093b380a1120789d53cf10
        ];

        // Reserve memory space for our hashes.
        bytes memory buf = new bytes(64);

        // We'll need to keep track of left and right siblings.
        bytes32 leftSibling;
        bytes32 rightSibling;

        // Number of non-empty nodes at the current depth.
        uint256 rowSize = _elements.length;

        // Current depth, counting from 0 at the leaves
        uint256 depth = 0;

        // Common sub-expressions
        uint256 halfRowSize;         // rowSize / 2
        bool rowSizeIsOdd;           // rowSize % 2 == 1

        while (rowSize > 1) {
            halfRowSize = rowSize / 2;
            rowSizeIsOdd = rowSize % 2 == 1;

            for (uint256 i = 0; i < halfRowSize; i++) {
                leftSibling  = _elements[(2 * i)    ];
                rightSibling = _elements[(2 * i) + 1];
                assembly {
                    mstore(add(buf, 32), leftSibling )
                    mstore(add(buf, 64), rightSibling)
                }

                _elements[i] = keccak256(buf);
            }

            if (rowSizeIsOdd) {
                leftSibling  = _elements[rowSize - 1];
                rightSibling = bytes32(defaults[depth]);
                assembly {
                    mstore(add(buf, 32), leftSibling)
                    mstore(add(buf, 64), rightSibling)
                }

                _elements[halfRowSize] = keccak256(buf);
            }

            rowSize = halfRowSize + (rowSizeIsOdd ? 1 : 0);
            depth++;
        }

        return _elements[0];
    }

    /**
     * Verifies a merkle branch for the given leaf hash.  Assumes the original length
     * of leaves generated is a known, correct input, and does not return true for indices
     * extending past that index (even if _siblings would be otherwise valid.)
     * @param _root The Merkle root to verify against.
     * @param _leaf The leaf hash to verify inclusion of.
     * @param _index The index in the tree of this leaf.
     * @param _siblings Array of sibline nodes in the inclusion proof, starting from depth 0 (bottom of the tree).
     * @param _totalLeaves The total number of leaves originally passed into.
     * @return Whether or not the merkle branch and leaf passes verification.
     */
    function verify(
        bytes32 _root,
        bytes32 _leaf,
        uint256 _index,
        bytes32[] memory _siblings,
        uint256 _totalLeaves
    )
        internal
        pure
        returns (
            bool
        )
    {
        require(
            _totalLeaves > 0,
            "Lib_MerkleTree: Total leaves must be greater than zero."
        );

        require(
            _index < _totalLeaves,
            "Lib_MerkleTree: Index out of bounds."
        );

        require(
            _siblings.length == _ceilLog2(_totalLeaves),
            "Lib_MerkleTree: Total siblings does not correctly correspond to total leaves."
        );

        bytes32 computedRoot = _leaf;

        for (uint256 i = 0; i < _siblings.length; i++) {
            if ((_index & 1) == 1) {
                computedRoot = keccak256(
                    abi.encodePacked(
                        _siblings[i],
                        computedRoot
                    )
                );
            } else {
                computedRoot = keccak256(
                    abi.encodePacked(
                        computedRoot,
                        _siblings[i]
                    )
                );
            }

            _index >>= 1;
        }

        return _root == computedRoot;
    }


    /*********************
     * Private Functions *
     *********************/

    /**
     * Calculates the integer ceiling of the log base 2 of an input.
     * @param _in Unsigned input to calculate the log.
     * @return ceil(log_base_2(_in))
     */
    function _ceilLog2(
        uint256 _in
    )
        private
        pure
        returns (
            uint256
        )
    {
        require(
            _in > 0,
            "Lib_MerkleTree: Cannot compute ceil(log_2) of 0."
        );

        if (_in == 1) {
            return 0;
        }

        // Find the highest set bit (will be floor(log_2)).
        // Borrowed with <3 from https://github.com/ethereum/solidity-examples
        uint256 val = _in;
        uint256 highest = 0;
        for (uint256 i = 128; i >= 1; i >>= 1) {
            if (val & (uint(1) << i) - 1 << i != 0) {
                highest += i;
                val >>= i;
            }
        }

        // Increment by one if this is not a perfect logarithm.
        if ((uint(1) << highest) != _in) {
            highest += 1;
        }

        return highest;
    }
}