{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PyQNormaliz: First example\n",
    "\n",
    "For computations over real embedded number fields with QNormaliz, first load the PyQNormaliz_cpp module"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import PyQNormaliz_cpp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we enter a dodecahedron"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "dodec = PyQNormaliz_cpp.NmzCone( number_field = \"min_poly (a^2 - 5) embedding [2.236067977499789 +/- 8.06e-16]\",\n",
    "                                cone = [\n",
    "[ \"-a + 3\", \"-a + 3\", \"-a + 3\", 1 ],\n",
    "[ \"2*a - 4\", 0, \"a - 1\", 1 ],\n",
    "[ \"-a + 3\", \"a - 3\", \"-a + 3\", 1 ],\n",
    "[ \"a - 1\", \"2*a - 4\", 0, 1 ],\n",
    "[ \"a - 1\", \"-2*a + 4\", 0, 1 ],\n",
    "[ 0, \"a - 1\", \"2*a - 4\", 1 ],\n",
    "[ 0, \"-a + 1\", \"2*a - 4\", 1 ],\n",
    "[ \"-2*a + 4\", 0, \"a - 1\", 1 ],\n",
    "[ \"a - 3\", \"-a + 3\", \"-a + 3\", 1 ],\n",
    "[ \"a - 3\", \"a - 3\", \"-a + 3\", 1 ],\n",
    "[ \"-a + 3\", \"-a + 3\", \"a - 3\", 1 ],\n",
    "[ 0, \"a - 1\", \"-2*a + 4\", 1 ],\n",
    "[ \"-a + 3\", \"a - 3\", \"a - 3\", 1 ],\n",
    "[ 0, \"-a + 1\", \"-2*a + 4\", 1 ],\n",
    "[ \"2*a - 4\", 0, \"-a + 1\", 1 ],\n",
    "[ \"-a + 1\", \"2*a - 4\", 0, 1 ],\n",
    "[ \"-a + 1\", \"-2*a + 4\", 0, 1 ],\n",
    "[ \"a - 3\", \"-a + 3\", \"a - 3\", 1 ],\n",
    "[ \"a - 3\", \"a - 3\", \"a - 3\", 1 ],\n",
    "[ \"-2*a + 4\", 0, \"-a + 1\", 1 ] ] )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The input for number field elements can be given as strings, or as a list of pairs of integers. The polynomial\n",
    "```\n",
    "a^2+1/2a-3/5\n",
    "```\n",
    "would be the list\n",
    "```\n",
    "[[-3,5],[1,2],[1,1]]\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can compute the support hyperplanes of the dodecahedron"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[[[-1, 4], [-1, 4]], [[0, 1]], [[-1, 2]], [[1, 1]]],\n",
       " [[[-1, 4], [-1, 4]], [[0, 1]], [[1, 2]], [[1, 1]]],\n",
       " [[[-1, 2]], [[-1, 4], [-1, 4]], [[0, 1]], [[1, 1]]],\n",
       " [[[-1, 2]], [[1, 4], [1, 4]], [[0, 1]], [[1, 1]]],\n",
       " [[[0, 1]], [[-1, 2]], [[-1, 4], [-1, 4]], [[1, 1]]],\n",
       " [[[0, 1]], [[-1, 2]], [[1, 4], [1, 4]], [[1, 1]]],\n",
       " [[[0, 1]], [[1, 2]], [[-1, 4], [-1, 4]], [[1, 1]]],\n",
       " [[[0, 1]], [[1, 2]], [[1, 4], [1, 4]], [[1, 1]]],\n",
       " [[[1, 2]], [[-1, 4], [-1, 4]], [[0, 1]], [[1, 1]]],\n",
       " [[[1, 2]], [[1, 4], [1, 4]], [[0, 1]], [[1, 1]]],\n",
       " [[[1, 4], [1, 4]], [[0, 1]], [[-1, 2]], [[1, 1]]],\n",
       " [[[1, 4], [1, 4]], [[0, 1]], [[1, 2]], [[1, 1]]]]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PyQNormaliz_cpp.NmzResult(dodec,\"SupportHyperplanes\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The output might be a bit hard to read. Therefore one can pass functions to the result function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rational_handler(list):\n",
    "    return list[0]/list[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[[-0.25, -0.25], [0.0], [-0.5], [1.0]],\n",
       " [[-0.25, -0.25], [0.0], [0.5], [1.0]],\n",
       " [[-0.5], [-0.25, -0.25], [0.0], [1.0]],\n",
       " [[-0.5], [0.25, 0.25], [0.0], [1.0]],\n",
       " [[0.0], [-0.5], [-0.25, -0.25], [1.0]],\n",
       " [[0.0], [-0.5], [0.25, 0.25], [1.0]],\n",
       " [[0.0], [0.5], [-0.25, -0.25], [1.0]],\n",
       " [[0.0], [0.5], [0.25, 0.25], [1.0]],\n",
       " [[0.5], [-0.25, -0.25], [0.0], [1.0]],\n",
       " [[0.5], [0.25, 0.25], [0.0], [1.0]],\n",
       " [[0.25, 0.25], [0.0], [-0.5], [1.0]],\n",
       " [[0.25, 0.25], [0.0], [0.5], [1.0]]]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PyQNormaliz_cpp.NmzResult(dodec,\"SupportHyperplanes\",RationalHandler=rational_handler)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The resulting number field elements are displayed as lists. One might want to display them as polynomial strings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def polynomial_string(list):\n",
    "    string = str(list[0][0]) + \"/\" + str(list[0][1])\n",
    "    for i in range(1,len(list)):\n",
    "        string += \" + \" + str(list[i][0]) + \"/\" + str(list[i][1]) + \"*a^\" + str(i)\n",
    "    return string\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[['-1/4 + -1/4*a^1', '0/1', '-1/2', '1/1'],\n",
       " ['-1/4 + -1/4*a^1', '0/1', '1/2', '1/1'],\n",
       " ['-1/2', '-1/4 + -1/4*a^1', '0/1', '1/1'],\n",
       " ['-1/2', '1/4 + 1/4*a^1', '0/1', '1/1'],\n",
       " ['0/1', '-1/2', '-1/4 + -1/4*a^1', '1/1'],\n",
       " ['0/1', '-1/2', '1/4 + 1/4*a^1', '1/1'],\n",
       " ['0/1', '1/2', '-1/4 + -1/4*a^1', '1/1'],\n",
       " ['0/1', '1/2', '1/4 + 1/4*a^1', '1/1'],\n",
       " ['1/2', '-1/4 + -1/4*a^1', '0/1', '1/1'],\n",
       " ['1/2', '1/4 + 1/4*a^1', '0/1', '1/1'],\n",
       " ['1/4 + 1/4*a^1', '0/1', '-1/2', '1/1'],\n",
       " ['1/4 + 1/4*a^1', '0/1', '1/2', '1/1']]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PyQNormaliz_cpp.NmzResult(dodec,\"SupportHyperplanes\",NumberfieldElementHandler=polynomial_string)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Furthermore, `RationalHandler` and `MatrixHandler` can also be used."
   ]
  }
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