{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook, we are deriving what the alpha value in the @dev formula is based off of 1hive Sol [code](https://github.com/1Hive/conviction-voting-app/blob/bc81f4af266bab6bc4096a31966bd9fe8a89f4a2/contracts/ConvictionVoting.sol)\n",
    "\n",
    "    /**\n",
    "     * @dev Formula: ρ * supply / (1 - a) / (β - requestedAmount / total)**2\n",
    "     * For the Solidity implementation we amplify ρ and β and simplify the formula:\n",
    "     * weight = ρ * D\n",
    "     * maxRatio = β * D\n",
    "     * decay = a * D\n",
    "     * threshold = weight * supply * D ** 2 * funds ** 2 / (D - decay) / (maxRatio * funds - requestedAmount * D) ** 2\n",
    "     * @param _requestedAmount Requested amount of tokens on certain proposal\n",
    "     * @return Threshold a proposal's conviction should surpass in order to be able to\n",
    "     * executed it.\n",
    "     */\n",
    "    function calculateThreshold(uint256 _requestedAmount) public view returns (uint256 _threshold) {\n",
    "        uint256 funds = vault.balance(requestToken);\n",
    "        require(maxRatio.mul(funds) > _requestedAmount.mul(D), ERROR_AMOUNT_OVER_MAX_RATIO);\n",
    "        uint256 supply = stakeToken.totalSupply();\n",
    "        // denom = maxRatio * 2 ** 64 / D  - requestedAmount * 2 ** 64 / funds\n",
    "        uint256 denom = (maxRatio << 64).div(D).sub((_requestedAmount << 64).div(funds));\n",
    "        // _threshold = (weight * 2 ** 128 / D) / (denom ** 2 / 2 ** 64) * supply * D / 2 ** 128\n",
    "        _threshold = ((weight << 128).div(D).div(denom.mul(denom) >> 64)).mul(D).div(D.sub(decay)).mul(supply) >> 64;\n",
    "    }"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.025"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "# Z original code without alpha\n",
    "def trigger_threshold_original(requested, funds, supply, beta, rho):\n",
    "    '''\n",
    "    Function that determines threshold for proposals being accepted. \n",
    "    '''\n",
    "    share = requested/funds\n",
    "    if share < beta:\n",
    "        return rho*supply/(beta-share)**2\n",
    "    else: \n",
    "        return np.inf\n",
    "    \n",
    "#Our new code code\n",
    "def trigger_threshold(requested, funds, supply, beta, rho, alpha):\n",
    "    '''\n",
    "    Function that determines threshold for proposals being accepted. \n",
    "    '''\n",
    "    share = requested/funds\n",
    "    if share < beta:\n",
    "        threshold = rho*supply/(beta-share)**2  * 1/(1-alpha)\n",
    "        return threshold #* (1-alpha)\n",
    "    else: \n",
    "        return np.inf\n",
    "    \n",
    "# Assumed values\n",
    "funds = 4000\n",
    "requested = 100\n",
    "supply = 21706\n",
    "share = requested/funds\n",
    "share"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Sol values\n",
    "D = 10000000\n",
    "decay = 9999599\n",
    "maxRatio = 2000000\n",
    "weight = 20000"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Based on the above equations, we can see that $\\rho$ is equal to:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.002"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho = weight / D\n",
    "rho"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can derive $\\beta$ is equal to be:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta = maxRatio/D\n",
    "beta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1417.5346938775506"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "originalOutput = trigger_threshold_original(requested,funds,supply,beta,rho)\n",
    "originalOutput"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "sol code threshold:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "35349992.36602372"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solThreshold = weight * supply * D ** 2 * funds ** 2 / (D - decay) / (maxRatio * funds - requested * D) ** 2\n",
    "solThreshold"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our new code threshold to match 1hive dev formulata "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0.0000401'"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha = 1- 0.9999599\n",
    "\"{:.7f}\".format(alpha)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1417.5915392982765"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newThreshold = trigger_threshold(requested, funds, supply, beta, rho, alpha)\n",
    "newThreshold"
   ]
  }
 ],
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